Proof. To compute the coefficient of \(g\in C_X(Q)\) in a product of two \(Q\)-invariant elements, let \(Q\) conjugate the pairs \((a,b)\) with \(ab=g\). Coefficient products are constant on the orbits. Every nontrivial orbit has size divisible by \(p\), and the fixed pairs are exactly those with \(a,b\in C_X(Q)\). This proves multiplicativity; the identity is preserved.

Each coset \(hQ\) is stable under conjugation by \(Q\), because \(h\in N_X(Q)\). Applying the same orbit cancellation to the sum of coefficients over that coset proves the first identity in (3). Also \(\operatorname{Br}_Q(v)\in Z(kH)\), since its support lies in \(C_X(Q)\leq H\) and it is invariant under \(H\).

Every nonzero module for a finite \(p\)-group over \(k\) has nonzero invariants. For completeness, choose a central element \(z\) of order \(p\); the nonzero kernel of \(z-1\), whose \(p\)th power is zero, is a module for the smaller group \(Q/\langle z\rangle\), and induction applies. If the module is simple for \(H\), its \(Q\)-invariants are \(H\)-stable, and hence the whole module. Thus every simple \(kH\)-module factors through \(H/Q\), proving the second identity. The composition with \(\operatorname{Br}_Q\) is a unital algebra character. Characters of the finite commutative algebra \(Z(kX)\) are its local residue characters, one for each block, giving uniqueness. ◻

Proof. We identify the terms of \(T^r l\). After \(r\) steps they correspond to normal chains of length \(r\), with group \(H_{\mathcal C}/Q_r\) and idempotent \[\pi_{Q_r}(s_{H_{\mathcal C}}(E)) =\pi_{Q_r}(E_{\mathcal C}).\] For the induction, the preimage \(Q_{r+1}\) of a nontrivial \(p\)-subgroup of \(H_{\mathcal C}/Q_r\) is a \(p\)-group strictly containing \(Q_r\). It lies in \(H_{\mathcal C}\) and therefore normalizes every preceding \(Q_i\). Conversely every normal one-step extension arises in this way. Its normalizer in the quotient has preimage \(H_{\mathcal C}\cap N_X(Q_{r+1})=H_{\mathcal C'}\). After fixing \(\mathcal C\), conjugation is by its stabilizer \(H_{\mathcal C}\), which is exactly quotient conjugation at the next step. This proves the orbit assertion without an extra multiplicity.

To check the idempotent at the next step, write \(E=\sum_{x\in X}e_xx\) and put \(H'=H_{\mathcal C'}\). The new normalizer in \(H_{\mathcal C}/Q_r\) is \(H'/Q_r\). Restriction to this normalizer retains exactly the coefficient sums over the \(Q_r\)-cosets contained in \(H'\). Choose a set \(A\) of representatives for the cosets of \(Q_r\) in \(Q_{r+1}\). For \(h\in H'\), the subsequent quotient has coefficient \[\sum_{a\in A}\ \sum_{q\in Q_r}e_{haq} =\sum_{t\in Q_{r+1}}e_{ht}\] at \(hQ_{r+1}\): the sets \(aQ_r\), \(a\in A\), partition \(Q_{r+1}\). These are precisely the coefficients of \(\pi_{Q_{r+1}}(s_{H'}(E))\). By 4, this idempotent is \(\pi_{Q_{r+1}}(E_{\mathcal C'})\), as required. Every simple \(kH_{\mathcal C}\)-module factors through \(H_{\mathcal C}/Q_r\) and has the indicated idempotent action. Hence the contribution to \(T^rl\) is the corresponding term in (5). The finite inverse of \(1+T\) now gives the equivalence. ◻

Proof. In the expansion of \((a+b)^p\), cyclic rotations of any mixed word give the same cocenter class. Their orbits have size \(p\), leaving only \(a^p\) and \(b^p\). Thus the power operation into the cocenter is additive and is Frobenius semilinear. Moreover \(\overline{(ab)^p}=\overline{(ba)^p}\), by one cyclic rotation; additivity shows that this operation kills the linear commutator space. It therefore descends to \(F_A\).

Let \(R\) be the Jacobson radical of \(A\). The kernel of \(V(A)\longrightarrow V(A/R)\) is represented by elements of \(R\): lift any commutator expression downstairs and subtract it upstairs. Since \(R\) is nilpotent, a sufficiently high iterate of \(F_A\) kills this kernel. On the split semisimple algebra \(A/R\), the cocenter has one matrix-trace coordinate per simple factor, and \(F_A\) raises each coordinate to the \(p\)th power. It is bijective because \(k\) is perfect. For example, the trace identity here follows by triangularizing a matrix over \(k\) and taking the \(p\)th powers of its diagonal entries.

The descending images of \(F_A\) stabilize, since they are \(k\)-subspaces of a finite-dimensional space. Denote the stable image by \(S\) and the quotient map by \(q:V(A)\to V(A/R)\). Choose \(m\) large enough that \(S=\operatorname{im}F_A^m\) and \(F_A^m\) kills \(\ker q\). On \(S\), the map \(F_A\) is surjective and hence bijective, because it is semilinear for the automorphism \(a\mapsto a^p\) of the perfect field \(k\). If \(x\in S\cap\ker q\), then \(F_A^m(x)=0\); injectivity on \(S\) gives \(x=0\). To prove surjectivity of \(q|_S\), take \(y\in V(A/R)\). The Frobenius map \(F_{A/R}\) is bijective, so choose \(y'\) with \(F_{A/R}^m(y')=y\) and lift \(y'\) to \(x\in V(A)\). Then \(F_A^m(x)\in S\) and \(q(F_A^m(x))=y\). Thus \(q|_S\) is an isomorphism. Finally \((Ea)^p=Ea^p\) for a central idempotent \(E\), so all these statements respect its summand.

For a group algebra the cocenter is free on conjugacy classes, represented by individual group elements: the commutator relations identify exactly conjugate elements. Sufficiently high \(p\)-powers land on \(p'\)-elements, and taking \(p\)th powers permutes the \(p'\)-classes. This identifies the stable image. The integral cocenter \(V(\mathcal O X)\) is free on the same class basis. The lift of \(E\) projects it onto a free direct summand, whose rank is unchanged over \(k\) and \(K\). Split semisimplicity over \(K\) identifies this rank with \(k_{\mathrm{ord}}(X,E)\). ◻

Proof. Fix a \(p\)-element \(u\) and set \(C=C_X(u)\). The map \(\overline t\mapsto\overline{ut}\) identifies the \(p'\)-class span of \(V(kC)\) with the section span belonging to \(u\) in \(V(kX)\). It is a bijection of class bases: conjugacy of \(ut\) and \(ut'\) forces a conjugating element to centralize their common \(p\)-part \(u\), and the converse is immediate.

For \(t\in C\) a \(p'\)-element, conjugation by \(\langle u\rangle\) in the expansion of \(Eut\) gives \[\overline{Eut} =\overline{\operatorname{Br}_{\langle u\rangle}(E)ut}.\] Indeed, the coefficients are constant on these orbits and the cocenter classes of their products with \(ut\) agree; only fixed orbits survive. The restricted idempotent is central in \(kC\) and preserves its \(p'\)-class span by 7. Multiplication by \(u\) is central in \(kC\), so the displayed identification intertwines the two idempotent actions. Their ranks are thus the terms in (6). Summing the ranks over the sections proves that identity.

For the center, use the pairing \[Z(kX)\times V(kX)\longrightarrow k, \qquad (a,\overline b)\longmapsto\tau(ab),\] where \(\tau\) takes the coefficient of \(1\). It is nondegenerate: a class sum pairs with an individual representative of its inverse class with coefficient one. Multiplication by \(E\) is self-adjoint for this pairing. Sections on the center pair with the inverse sections of the cocenter, so preservation there proves preservation here. In particular no class-size division is involved. ◻

Proof. For a nonnormal chain, let \(i\) be the largest index for which \(A=Q_i\) is not normal in \(P=Q_r\), and let \(R\) be the normal closure of \(A\) in \(P\). Then \(A<R\leq Q_{i+1}\): every later member is normal in \(P\). Also \(R<P\). To see the latter, put the proper subgroup \(A\) in a maximal subgroup of \(P\), which is normal because \(P\) is a finite \(p\)-group; it contains \(R\).

Insert \(R\) immediately after \(A\) if it is absent, and remove it if it is present. The last group \(P\) and the last nonnormal member \(A\) are unchanged, so this operation is an involution. Every element normalizing \(A\) and \(P\) normalizes their normal closure \(R\); hence insertion or removal leaves \(H_{\mathcal C}\) unchanged. The operation commutes with \(X\)-conjugation and changes the length by one. It induces a fixed-point-free sign reversal on the orbits, which proves cancellation. ◻

Proof. By 9, both sums may be taken over all chains. Expand the ordinary count by 8. The \(u=1\) terms give the simple-module sum. The other terms are indexed by \(X\)-orbits of pairs \((\mathcal C,u)\), with \(u\ne1\) a \(p\)-element in \(H_{\mathcal C}\). This orbit description follows because the stabilizer of a fixed chain is \(H_{\mathcal C}\). The contribution of such a pair is \[ l\bigl(H_{\mathcal C}\cap C_X(u), \operatorname{Br}_{Q_r\langle u\rangle}(E)\bigr). \tag{8}\] Here \(u\) normalizes \(Q_r\), so \(Q_r\langle u\rangle\) is a \(p\)-group; successive coefficient restrictions have support \(C_X(Q_r)\cap C_X(u)=C_X(Q_r\langle u\rangle)\), proving the idempotent in this expression.

Let \(i\) be the largest index, allowing \(0\) and \(r\), for which \(u\notin Q_i\). Such an index exists since \(Q_0=1\). Toggle \(R=Q_i\langle u\rangle\) immediately after \(Q_i\): insert it if absent, and remove it if present. It is a \(p\)-group, since \(u\) normalizes \(Q_i\), and it is contained in every later member, all of which contain \(u\). The last member not containing \(u\) therefore remains \(Q_i\); this proves involutivity, including an insertion after the terminal member and its inverse deletion.

Every element centralizing \(u\) and normalizing \(Q_i\) normalizes \(R\). Thus \(H_{\mathcal C}\cap C_X(u)\) is unchanged. The product of the terminal member with \(\langle u\rangle\) is also unchanged, even when the terminal member is inserted or deleted. Hence (8) is preserved. The operation is \(X\)-equivariant and reverses the sign, so all \(u\ne1\) terms cancel. ◻

Proof. Radical annihilation. The form \((a,b)\mapsto\tau(ab)\) on \(kX\) is symmetric and nondegenerate. For dual bases \(v_i,v_i^*\), the tensor \(\Omega=\sum_i v_i\otimes v_i^*\) is independent of the chosen bases. Apply the linear map \(u\otimes v\mapsto uav\) to this tensor. In the group basis, whose dual basis consists of the inverses, the result is \(I_X(a)\). Similarly, apply \(u\otimes v\otimes w\otimes z\mapsto uwvz\) to \(\Omega\otimes\Omega\). In the group basis the result is \(\sum_{x,y\in X}xyx^{-1}y^{-1}=d_X\). Thus in any dual bases \[ I_X(a)=\sum_i v_iav_i^*,\qquad d_X=\sum_{i,j}v_iv_jv_i^*v_j^* =\sum_j I_X(v_j)v_j^*. \tag{9}\] For \(b\) in the radical \(R\), the trace identity \[\tau(I_X(a)b)=\operatorname{Tr}(L_bR_a)\] follows by writing the trace in these dual bases; \(L\) and \(R\) denote left and right multiplication. Since \(L_b\) is nilpotent and commutes with \(R_a\), this trace is zero. Replace \(b\) by \(bc\) for arbitrary \(c\in kX\), and use nondegeneracy to obtain \(I_X(a)b=0\). The final expression in (9) then shows that \(d_XR=0\) as well, because \(v_j^*R\subseteq R\).

Semisimple blocks and ordinary defect zero. Distinct block ideals are orthogonal for the form. On a semisimple block \(M_d(k)\) its restriction is \(t\operatorname{Tr}\) for some \(t\in k^\times\). Indeed a symmetric functional kills commutators, whose quotient in a matrix algebra is the trace line, and nondegeneracy forces \(t\ne0\). Matrix units \(e_{ij}\) have duals \(t^{-1}e_{ji}\). Since \(\sum_{i,j}e_{ij}ae_{ji}=\operatorname{Tr}(a)1\), the first contraction in (9) gives \(I_X(a)=t^{-1}\operatorname{Tr}(a)1\). In particular \(I_X(e_{ij})=t^{-1}\delta_{ij}1\), so its last expression gives \[d_X=\sum_{i,j}I_X(e_{ij})t^{-1}e_{ji} =t^{-2}\sum_i e_{ii}=t^{-2}1.\] We have therefore proved \[ I_X(a)=t^{-1}\operatorname{Tr}(a)1, \qquad d_X=t^{-2}1 \quad\text{on this block}. \tag{10}\] In particular its idempotent lies in the image of \(I_X\); one can take \(a=te_{11}\). These formulas do not require \(d\) to be nonzero in \(k\). On a nonsemisimple block, a central element with nonzero residue scalar is a unit in its local center. Such a unit cannot annihilate the nonzero radical. This proves the assertion about the central character of \(d_X\).

Now compute the same commutator sum in characteristic zero. If \(\chi\) has degree \(d\), Schur’s lemma gives \[\sum_{x\in X}\rho(xyx^{-1}) =\frac{|X|\chi(y)}d\,1.\] Multiply by \(\rho(y^{-1})\), sum over \(y\), and take traces. Ordinary character orthogonality gives the scalar \[ \lambda_\chi(d_X)=\frac{|X|^2}{\chi(1)^2}. \tag{11}\] It is integral, by the integrality of ordinary central scalars. Its reduction is nonzero exactly when \(v_p(\chi(1))=v_p(|X|)\), namely in defect zero. The modular calculation therefore puts exactly these characters in semisimple blocks. Such a block has cocenter dimension one, so 7 shows that its lifted block selects exactly one ordinary irreducible. This also proves existence of that character.

Support and the \(p\)-element sum. For a group element \(g\) one has \[I_X(g)=|C_X(g)|\sum_{a\in[g]}a.\] This proves assertion (i). The support of a semisimple block idempotent, already in this image, therefore contains no element centralized by a nontrivial \(p\)-subgroup, proving (ii).

Finally fix a nontrivial \(p\)-subgroup \(P\). Choose \(z\in Z(P)\) of order \(p\). It is central also in \(C_X(P)\), and multiplication by \(z\) preserves the set of \(p\)-elements of that centralizer. With \(q=\sum_{j=0}^{p-1}z^j\), the sum of those \(p\)-elements factors as \(q\) times a sum of orbit representatives. As \(q\) is central and \(q^2=pq=0\), this proves \[\operatorname{Br}_P(c_X)^2=0, \qquad \operatorname{Br}_P(c_X^2)=0.\] If \(p\) divides \(|C_X(g)|\), a nontrivial \(p\)-subgroup \(P\leq C_X(g)\) makes the coefficient of \(g\) in \(c_X^2\) vanish by this identity. Assertion (i) gives \(c_X^2\in\operatorname{im}I_X\). On a nonsemisimple block its central character must therefore vanish, by the same radical-annihilator argument. Since \(\lambda_b(c_X)^2=\lambda_b(c_X^2)=0\), assertion (iii) follows. ◻

Proof. The element \(1-c_X\) is the negative sum of the nonidentity \(p\)-elements, hence lies entirely in \(p\)-singular sections. Section preservation from 8 proves the same support assertion for \(v\). Let \(B_0\) be a semisimple block idempotent. Since \(v\) is central and \(B_0kX\) is a full matrix algebra, \(B_0v\) lies in its one-dimensional center \(kB_0\). On the other hand, section preservation makes the support of \(B_0v\) \(p\)-singular. By 11, the support of \(B_0\) contains only elements whose centralizers have order prime to \(p\), and therefore no \(p\)-singular elements. The two support conditions force the scalar multiple \(B_0v\) to be zero. On a nonsemisimple block, 11 gives \(\lambda_b(v)=\lambda_b(E)\), proving the length-zero assertion.

For positive length, put \(Q=Q_r\ne1\) and \(H=H_{\mathcal C}\). By 4, \(\lambda_b\circ s_H\) is a central character of \(X\) and factors through \(\operatorname{Br}_Q\). It cannot be the character of a semisimple block \(B_0\): that would give value one on \(B_0\), whereas \(\operatorname{Br}_Q(B_0)=0\) by 11. On its nonsemisimple block, \(\lambda(v)=\lambda(E)\). Applying 4 to \(E\) gives the asserted value \(\lambda_b(E_{\mathcal C})\). ◻

Proof. The nonnormal chains cancel by 9, since their trace contribution depends only on the stabilizer. For a remaining chain, the cocenter of \(KH_{\mathcal C}\) has one coordinate per ordinary irreducible. The unpowered operator has on these coordinates integral central eigenvalues reducing to the scalars in 12.

If an eigenvalue reduces to zero, its \(p^s\)th powers tend to zero. If it is \(1+a\) with \(\operatorname{val}(a)>0\), the binomial formula gives \[\operatorname{val}((1+a)^p-1) \geq \min\{\operatorname{val}(p)+\operatorname{val}(a), p\operatorname{val}(a)\}.\] After normalizing the discrete valuation to integer values, each iteration increases the positive valuation by at least one, unless the difference has already become zero. These powers tend to one. This argument applies also to \(p=2\) and requires no finiteness of the residue field.

For positive-length chains the limiting trace therefore counts all ordinary irreducibles selected by \(E_{\mathcal C}\). At length zero it omits the semisimple blocks, each of which contributes exactly one defect-zero character by 11. There are finitely many chains and irreducibles, so limits may be summed, proving (13). ◻

Proof. The sum on the right before division is \(\tau(d_Hw^n)\). The regular trace of any element of \(KH\) is \(|H|\) times its identity coefficient. On the other hand, (11) and the regular decomposition give \[\begin{align*} |H|\tau(d_Hw^n) &=\sum_{\chi\in\operatorname{Irr}(H)} \chi(1)^2\frac{|H|^2}{\chi(1)^2}\lambda_\chi(w)^n\\ &=|H|^2\operatorname{Tr}(w^n\mid V(KH)). \end{align*}\] Dividing proves exactly the single denominator in (14). ◻

Proof. Put \(P=O_p(J)\). In any chain normalized by \(J\), each \(Q_i\) is normalized by \(P\leq J\), so \(Q_iP\) is a \(p\)-subgroup. Both factors are normalized by \(J\), making their product \(J\)-normalized. Let \(i\) be the last index such that \(P\nleq Q_i\); it exists since \(Q_0=1\) and \(P\ne1\). Toggle \(Q_iP\) immediately after \(Q_i\). It strictly contains \(Q_i\) and is contained in all later members. Insertion and deletion preserve the last member not containing \(P\), so this is an involution. It is defined also for the length-zero chain and for terminal insertion or deletion. It preserves the property of being normalized by \(J\) and reverses the sign. The sum of actual chains in (15) therefore vanishes. ◻

Proof. The orbit of a chain \(\mathcal C\) has size \(|X|/|H_{\mathcal C}|\). Consequently 14 changes the orbit sum (12) into the following sum over actual tuples and actual chains: \[ \Phi_n=\frac1{|X|} \sum_{\substack{x,y,u_1,\ldots,u_n\in X\\ [x,y]u_1\cdots u_n=1}} a_X\bigl(\langle x,y,u_1,\ldots,u_n\rangle\bigr) \prod_{i=1}^n\alpha_{u_i}. \tag{16}\] To verify the factor, each orbit contribution has prefactor \(1/|H_{\mathcal C}|\); replacing its representative by all \(|X|/|H_{\mathcal C}|\) conjugates gives the common prefactor \(1/|X|\). For a fixed tuple the admissible chains are exactly those normalized by its generated subgroup. This is precisely \(a_X(J)\).

By 15, only generated subgroups \(J\) with \(O_p(J)=1\) contribute. By the support of \(\widehat v\), only tuples with every \(u_i\) \(p\)-singular contribute. Fix such a subgroup \(J\) and group its generating tuples according to the multiset \(\mathcal M\) of \(J\)-conjugacy classes of the \(u_i\). The product \(\prod_i\alpha_{u_i}\) is constant on this multiset: the coefficients are constant on \(X\)-classes, hence on \(J\)-classes, and their product is unchanged by reordering. Denote this integral product by \(\alpha(\mathcal M)\). A generating tuple has simultaneous conjugation stabilizer \(Z(J)\), since an element centralizing all its entries centralizes the group they generate. Thus its number of actual tuples is \(|J|/|Z(J)|\) times its orbit count. For \(n=p^s\), the numerator in (16) is therefore \[ \sum_{\substack{J\leq X\\O_p(J)=1}} a_X(J)\frac{|J|}{|Z(J)|} \sum_{\mathcal M} \alpha(\mathcal M)N_{J,s}(\mathcal M). \tag{17}\]

Fix any \(h\geq1\). By 3, each summand of the inner sum belongs to \(p^h\mathcal O\) for all sufficiently large \(s\), with a threshold independent of \(\mathcal M\). Although the number of multisets grows with \(s\), the sum is finite for each \(s\) and remains in that same ideal. There are only finitely many subgroups \(J\leq X\), so take the largest of their thresholds. The integer factors in (17) preserve divisibility. We conclude that its numerator tends to zero in the valuation topology. The fixed denominator \(|X|\) does not change this conclusion, so \(\Phi_{p^s}\longrightarrow0\).

Combine this with 13 and 10. The resulting equality is (5); its integer terms agree in the characteristic-zero field \(K\), and hence agree as integers. The same reasoning applies to every pair and to the splitting subquotients arising in the transform. 6 now gives \(l=(1+T)z\). The weight interpretation of (4) proves the asserted block equality. ◻

Proof. For \(p=2\), take the Weierstrass cubic \[y^2+xy=x^3+1.\] The affine partial derivatives could vanish simultaneously only at \((x,y)=(0,0)\), which is not on the curve. In its projective equation \(Y^2Z+XYZ=X^3+Z^3\), the origin \([0:1:0]\) is smooth because the partial derivative with respect to \(Z\) is nonzero there. Its points over \(\mathbb F_2\) are \[[0:1:0],\quad (0,1),\quad (1,0),\quad (1,1).\] Thus its finite group of rational points contains an element of order \(2\), by Sylow theory.

Suppose that \(p\) is odd and put \(r=(p-1)/2\). Consider \[f_\lambda(x)=x(x-1)(x-\lambda),\qquad H(\lambda)=[x^{p-1}]f_\lambda(x)^r.\] Here \([x^a]\) means the coefficient of \(x^a\). The coefficient of \(\lambda^r\) in \(H\) is \((-1)^r\): in \(x^r(x-1)^r(x-\lambda)^r\), take \((-\lambda)^r\) from the last factor and \(x^r\) from the middle factor. In particular \(H\) is not the zero polynomial. Choose \(\lambda\in\kappa\setminus\{0,1\}\) with \(H(\lambda)\ne0\). The cubic \[A_0:\quad y^2=f_\lambda(x)\] is smooth, since \(f_\lambda\) has three distinct roots and the point at infinity is smooth.

Choose \(q=p^\ell\) with \(\lambda\in\mathbb F_q\). For each \(x\in\mathbb F_q\), the number of solutions for \(y\) is \(1+\chi_q(f_\lambda(x))\), where \(\chi_q\) is the quadratic character with values in \(\{0,1,-1\}\). Its reduction in \(\mathbb F_q\) is \(f_\lambda(x)^{(q-1)/2}\). Reducing the point count modulo \(p\) therefore gives \[ \#A_0(\mathbb F_q) \equiv 1+\sum_{x\in\mathbb F_q}f_\lambda(x)^{(q-1)/2} =1-H(\lambda)^{1+p+\cdots+p^{\ell-1}}. \tag{18}\] We verify the last equality, including its coefficient calculation. The exponents occurring in \(f_\lambda(x)^{(q-1)/2}\) range from \((q-1)/2\) to \(3(q-1)/2\). Among them, only \(q-1\) is a positive multiple of \(q-1\). Since \(\mathbb F_q^\times\) is cyclic, the sum of \(x^a\) over \(\mathbb F_q\) is zero for the other positive exponents and is \(-1\) for \(a=q-1\). Furthermore, \[f_\lambda(x)^{(q-1)/2} =\prod_{j=0}^{\ell-1}\bigl(f_\lambda(x)^r\bigr)^{p^j}.\] A contribution to the coefficient of \(x^{q-1}\) would choose exponents \(a_j\in[r,3r]\) satisfying \[\sum_{j=0}^{\ell-1}p^j a_j=p^\ell-1.\] Reduction modulo \(p\) forces \(a_0\equiv p-1\pmod p\). The interval \([r,3r]\) has length \(p-1\) and contains exactly one such integer, namely \(p-1\). Subtract this integer and divide the displayed equality by \(p\). Repetition forces \(a_j=p-1\) for every \(j\). Thus the desired coefficient is exactly \(\prod_jH(\lambda)^{p^j}\), proving (18).

The nonzero element \(c=H(\lambda)^{1+p+\cdots+p^{\ell-1}}\) lies in \(\mathbb F_p^\times\). Replace \(\mathbb F_q\) by \(\mathbb F_{q^d}\), where \(d\) is a multiple of the multiplicative order of \(c\). The corresponding expression is \(c^d=1\), so (18) shows that \(p\mid\#A_0(\mathbb F_{q^d})\). Sylow theory again supplies a point of order \(p\).

Finally, multiplication by \(p\) is a finite locally free map of degree \(p^2\) on an elliptic curve, by 11. It is surjective, hence surjective on points over the algebraically closed field \(\kappa\). If \(Q\) has order \(p^a\) and \([p]Q'=Q\), then \([p^{a+1}]Q'=0\) whereas \([p^a]Q'\ne0\). Starting from a point of order \(p\) and repeatedly choosing such a preimage proves the assertion. ◻

Proof. For a point with inseparable residue field, its multiplicity in the geometric cycle must be included before applying Galois descent to the sum. We first show that multiplication by a power of \(p\) moves the point into the corresponding field of \(p\)th powers. Write \(K_A=\kappa(A)\). A nonzero invariant differential \(\eta\) generates \(\Omega_{K_A/\kappa}\), and \([p]^*\eta=0\) by 11. Thus every rational function pulled back by \([p]\) has differential zero. The kernel of \(d:K_A\longrightarrow\Omega_{K_A/\kappa}\) is exactly \(K_A^p\). Indeed, it is a subfield containing \(K_A^p\), it is a proper subfield because \(\Omega_{K_A/\kappa}\) has dimension one, and \([K_A:K_A^p]=p\) by 4. Consequently \[ [p]^*K_A\subseteq K_A^p. \tag{19}\]

For every extension \(V/\kappa\), this entails \[ [p]\bigl(A(V)\bigr)\subseteq A(V^p). \tag{20}\] Here \(\kappa\subseteq V^p\) because \(\kappa\) is perfect. To justify evaluation in (20), take \(R\in A(V)\) and a \(\kappa\)-affine open \(W\) containing \([p]R\). For \(f\in\mathcal O_A(W)\), write \([p]^*f=g^p\) with \(g\in K_A\), using (19). On \([p]^{-1}W\) the rational function \(g\) is regular: its \(p\)th power is regular, and each local ring of this smooth curve is integrally closed. Hence \(f([p]R)=g(R)^p\in V^p\) for every such \(f\). The point \([p]R\) therefore factors through \(W(V^p)\), as claimed. Iteration gives \([p^u](A(V))\subseteq A(V^{p^u})\).

It suffices to treat a closed point \(z\) with residue field \(T/U\). Let \(T_{\mathrm{sep}}\) be the maximal separable subextension and write \([T:T_{\mathrm{sep}}]=p^u\). Then \(T^{p^u}\subseteq T_{\mathrm{sep}}\). Over an algebraic closure \(\overline U\), the cycle of \(z\) consists of the points \(R_\tau\) indexed by the \(U\)-embeddings \(\tau:T_{\mathrm{sep}}\hookrightarrow\overline U\), each with multiplicity \(p^u\). To see the multiplicity, first split the separable extension; each remaining purely inseparable tensor factor is local of dimension \(p^u\) over \(\overline U\), with residue field \(\overline U\).

Each embedding \(\tau\) extends uniquely to \(T\) inside \(\overline U\). By (20), \[[p^u]R_\tau\in A\bigl(\tau(T_{\mathrm{sep}})\bigr).\] The geometric group-law sum is therefore \(\sum_\tau[p^u]R_\tau\in A(U^{\mathrm{sep}})\), and Galois permutes its summands. Galois descent for points gives a point of \(A(U)\). Additivity proves the assertion for arbitrary signed cycles. Finally, in the geometric sum each copy of each point gains \(Q\) under translation, so the change is exactly \([b]Q\). ◻

Proof. We first prove the homomorphism assertion in (i). It can be checked after extending \(V\) to an algebraic closure. If \(f(0)=0\), form \[g(x,y)=f(x+y)-f(x)-f(y).\] Choose a point \(a\ne0\) of the target elliptic curve. The projection to the second factor of \(g^{-1}(a)\) is closed, by properness, and does not contain \(0\), because \(g(x,0)=0\). Its complement is a nonempty open set \(W\). For \(y\in W\), the morphism \(x\mapsto g(x,y)\) misses \(a\). A nonconstant morphism of projective integral curves is surjective, so this morphism is constant; its value at \(x=0\) is zero. Thus \(g\) is zero on \(A\times W\), and hence everywhere. This proves that \(f\) is a homomorphism.

We next descend the homomorphism to \(\kappa\), first by controlling its values on \(A(\kappa)\) and then by recovering its rational functions from those values. The zero homomorphism already descends, so suppose that \(f\) is nonconstant. Because \(\kappa=\overline{\mathbb F}_p\), the curve, its origin, and any chosen point of \(A(\kappa)\) are defined over a common finite field. The curve has a finite group of points over that field, so every point of \(A(\kappa)\) is torsion. Since \(f\) is a homomorphism, each such point has torsion image. For each positive integer \(N\), the scheme \(A[N]\) is finite over the algebraically closed field \(\kappa\); all its points with values in an extension field are therefore \(\kappa\)-points. Consequently \[ f(A(\kappa))\subseteq A(\kappa). \tag{22}\]

For a rational function \(h\in\kappa(A)\), write its pullback as \[f^*h=\frac{a}{b},\qquad a=\sum_{j=1}^t c_j a_j,\quad b=\sum_{j=1}^t c_j b_j, \qquad a_j,b_j\in\kappa(A),\] where the finitely many \(c_j\in V\) are linearly independent over \(\kappa\). Such an expression follows by putting a rational function in \(V(A)\) over a common denominator and choosing a basis for its finitely many scalar coefficients. Fix \(j_0\) with \(b_{j_0}\ne0\). For all but finitely many \(R\in A(\kappa)\), all functions in this expression can be evaluated, \(b(R)b_{j_0}(R)\ne0\), and \(f^*h(R)\in\kappa\), by (22). Writing \(\lambda_R=f^*h(R)\), scalar independence gives \[a_j(R)=\lambda_R b_j(R)\quad\text{for every }j, \qquad f^*h(R)=\frac{a_{j_0}(R)}{b_{j_0}(R)}.\] There are infinitely many such \(\kappa\)-points. A nonzero rational function on a projective integral curve has only finitely many zeros, so \(f^*h=a_{j_0}/b_{j_0}\in\kappa(A)\). Apply this to coordinate functions on a \(\kappa\)-affine open of the target. The morphism \(f\) is thus a rational map defined over \(\kappa\). It extends over every point of the smooth source curve by the valuative criterion for properness, and the extension agrees with \(f\) after scalar extension. This proves (i).

For (ii), view a rational map in the first variable over \(V=\kappa(A^{n-1})\), the function field of the remaining factors. It extends to a morphism \(A_V\to A_V\): the local rings at closed points of the smooth source curve are DVRs, and properness extends the map uniquely over each of them. These extensions are morphisms on neighborhoods and glue by separatedness. Subtracting its value at the origin leaves, by (i), an endomorphism \(\beta_1\in\operatorname{End}_\kappa(A)\). The subtracted value is an element of \(A(V)\), hence a rational map from the remaining \(n-1\) factors. Repeat to obtain (21). The right-hand side is a morphism everywhere. Its value at the origin determines \(c\), and its restriction to each factor with the other coordinates zero determines the corresponding \(\beta_i\). This also proves uniqueness. ◻

Proof. The cyclic permutation followed by translation on the smooth product \(A_0^n\) defines the automorphism of its function field in (23). Its powers satisfy \[\sigma^a(t_i)=t_{i+a}-[a]P.\] Thus \(\sigma^n=1\). If \(0<a<n\), the maps given by the independent projections \(t_i\) and \(t_{i+a}\) cannot differ by a constant: vary one factor while holding the other fixed. Therefore \(\sigma^a\ne1\). Set \(F_0=(F')^{\langle\sigma\rangle}\). Finite fixed-field theory gives \([F':F_0]=n\) with the indicated cyclic Galois group. Since \(F'/\kappa\) is finitely generated of transcendence degree \(n\), the same is true of \(F_0/\kappa\). More explicitly, choose a transcendence basis of \(F_0/\kappa\); it is also a transcendence basis of \(F'/\kappa\), and \(F'\) is finite over the rational function field on this basis, as is its intermediate field \(F_0\).

On \(A_{0,F'}\) use the semilinear action \[\delta(x)=\sigma(x)+P.\] Since \(\sigma\) fixes \(P\) and \([n]P=0\), this action satisfies \(\delta^n=1\). The translation by \(-P\) in the field action cancels the translation by \(P\) here, so \(\delta\) sends \(t_i\) to \(t_{i+1}\). It also sends \(jP\) to \((j+1)P\). Thus the generic projections give the invariant marked divisor \(\mathcal B_{F'}\), and the orbit of the origin gives the separate invariant divisor \(D_{F'}\) of degree \(n\). All their points are distinct, and the two supports are disjoint: a generic projection is nonconstant, so cannot equal a constant point \(jP\).

We use \(D_{F'}\) to descend the curve projectively; once descended, its degree will also give the upper bound \(n\) for the index. The invertible sheaf \(\mathcal O(D_{F'})\) has a canonical linearization, obtained by applying the action to rational functions with prescribed poles in the invariant divisor. It is ample: its third power is the tensor product of the translates of \(\mathcal O(3[0])\) by the points \(jP\), and \(\mathcal O(3[0])\) is the ample Weierstrass hyperplane bundle. Thus projective Galois descent, in the form of 6, constructs \(E\) and descends both divisors. Smoothness, geometric integrality, finiteness, and the étale property follow after the splitting extension; ampleness descends as well. Cohomology and base change give \(\dim_{F_0}H^1(E,\mathcal O_E)=1\), so \(E\) has genus one.

It remains to prove that every divisor degree is divisible by \(n\). The two preparatory lemmas reduce this to comparing endomorphisms in the cyclic descent relation. Let \(Z\) be an arbitrary divisor on \(E\) of degree \(b\), and pull it back to \(A_{0,F'}\). Pullback preserves its degree. By 18, its geometric group-law sum, including inseparable multiplicities and signed coefficients, is a point \(v\in A_0(F')\). The pulled-back divisor is invariant under \(\delta\). Applying the translation formula in that lemma yields \[ \sigma(v)+[b]P=v. \tag{25}\] By 19, the point \(v\), viewed as a rational map from \(A_0^n\) to \(A_0\), has the form \[v=c+\sum_{i=1}^n\beta_i(t_i),\qquad c\in A_0(\kappa),\quad \beta_i\in\operatorname{End}_\kappa(A_0).\] Substitution in (25) gives \[ \sum_{i=1}^n(\beta_{i-1}-\beta_i)(t_i) +[b]P-\sum_{i=1}^n\beta_i(P)=0, \qquad \beta_0=\beta_n. \tag{26}\] This equality at the generic point is an equality of morphisms on the product. Evaluate it first at the origin and then on one factor at a time. Each \(\beta_{i-1}-\beta_i\) must be zero, so all \(\beta_i\) equal one endomorphism \(\beta\). Their translation terms sum to \[\sum_i\beta_i(P)=\beta([n]P)=0.\] Equation (26) now gives \([b]P=0\). Since \(P\) has exact order \(n\), this proves \(n\mid b\). The divisor \(D\) has degree \(n\), so the index is exactly \(n\). ◻

Proof. Put \(N=[F:F_0]\). For a closed point \(z\) of \(E_F\) mapping to \(x\) of \(E\), finite pushforward is \(z\mapsto[\kappa(z):\kappa(x)]x\). Its degree over \(F_0\) is \[[\kappa(z):\kappa(x)]\,[\kappa(x):F_0] =[\kappa(z):F_0]=N[\kappa(z):F].\] Thus a divisor of degree \(b\) over \(F\) pushes forward to a divisor of degree \(Nb\) over \(F_0\), including when \(F/F_0\) or a point’s residue extension is inseparable. By 20, \(p^s\mid Nb\). Since \(N_p\leq p^h\), it follows that \(p^{s-h}\mid b\).

For \(C\to E_F\), the same residue-field tower calculation shows that pushforward preserves the degree over \(F\). Apply the first assertion to this pushforward. Finally every invertible sheaf on a regular integral curve has a nonzero rational section and is the invertible sheaf of its divisor, so the line-bundle assertion follows. ◻

Proof. Since \(F'/F_0\) is Galois, its compositum with any finite extension \(F/F_0\), including an inseparable one, is Galois over \(F\), with Galois group a subgroup of \(\operatorname{Gal}(F'/F_0)\). Consequently \([S:F]\) divides \(n=p^s\). The tower identity \[[S:F']\,[F':F_0]=[S:F]\,[F:F_0]\] then gives (28).

The forms \(t_i^*\eta\) are the basis in (27). Their loss of independence over \(S\) is measured by the kernel of the first map in the exact sequence \[S\otimes_{F'}\Omega_{F'/\kappa} \longrightarrow\Omega_{S/\kappa} \longrightarrow\Omega_{S/F'}\longrightarrow0.\] The first two spaces both have dimension \(n\) over \(S\), since the fields are finitely generated of transcendence degree \(n\) over the perfect field \(\kappa\). The dimension of the kernel of the first map is therefore \(\dim_S\Omega_{S/F'}\). Let \(S_{\mathrm{sep}}/F'\) be the maximal separable subextension and write \([S:S_{\mathrm{sep}}]=p^u\). By (28), \(u\leq h\). The purely inseparable extension \(S/S_{\mathrm{sep}}\) can be generated by at most \(u\) elements: every nontrivial successive adjunction has degree at least \(p\). Their differentials generate \(\Omega_{S/S_{\mathrm{sep}}}\), while \(\Omega_{S_{\mathrm{sep}}/F'}=0\). The differential sequence thus gives \(\dim_S\Omega_{S/F'}\leq u\leq h\).

Any subspace spanned by \(|I|\) of the original basis vectors loses at most \(h\) dimensions under a map with kernel dimension at most \(h\). This proves the assertion for the forms. Projection of (29) onto the second summand gives their negatives, so the rank of the covectors is at least the rank of the forms. ◻

Proof. For a closed point \(x\) put \(\deg_S(x)=[\kappa(x):S]\). For any divisor \(D\) the quotient in \[0\longrightarrow\mathcal O_C(D) \longrightarrow\mathcal O_C(D+x) \longrightarrow\mathcal Q_x\longrightarrow0\] has length one over the DVR \(\mathcal O_{C,x}\), so its dimension over \(S\) is \([\kappa(x):S]\). Finite-support sheaves have no higher cohomology. Adding and subtracting closed points proves \(\chi(\mathcal O_C(D))=\chi(C)+\deg_S D\). A rational trivialization of \(\mathcal L\) identifies it with \(\mathcal O_C(D)\) for a divisor \(D\), giving (32). Applied to a principal divisor, the same formula gives degree zero, so this degree depends only on \(\mathcal L\).

Locally on \(C'\), the finite module \(f_*\mathcal O_C\) is torsion-free over a DVR and therefore free, of generic rank \(\delta\). For a closed point \(x\in C'\) and a uniformizer \(t\) there, reduction of this free module modulo \(t\) gives \[ \sum_{y\mapsto x}e_y[\kappa(y):\kappa(x)]=\delta, \tag{33}\] where \(e_y\) is the valuation of \(t\) in \(\mathcal O_{C,y}\). Indeed the summand at \(y\) has a filtration of length \(e_y\) with successive quotients \(\kappa(y)\). Multiplication by \([\kappa(x):S]\) proves the pullback degree formula. The pushforward formula follows from \(f_*y=[\kappa(y):\kappa(x)]x\) and the tower formula for residue degrees.

Finite separable scalar extension preserves regularity, and a regular curve after such extension is a disjoint union of its integral components. Flat base change for coherent cohomology in 9 gives \[H^i(C_{S_1},\mathcal G_{S_1}) =H^i(C,\mathcal G)\otimes_S S_1.\] Thus the dimensions over the respective fields, and hence Euler characteristics, agree. Applying (32) proves the assertion for line bundles and their Cartier divisors. In particular it also applies to a reduced finite divisor: separable scalar extension preserves its reducedness. ◻

Proof. An affine coordinate ring of \(C\) is a localization of a finitely generated integral \(\kappa\)-algebra: choose a finitely generated domain with fraction field \(S\), clear the finitely many coefficients in a presentation over \(S\), and take the contraction of the defining prime ideal before localizing. Its local rings are consequently essentially of finite type over \(\kappa\). The regularity criterion over a perfect field in 8 says that these regular local rings are local rings at smooth points of the corresponding \(\kappa\)-variety. Their absolute differentials are free of rank \[\operatorname{trdeg}_\kappa\kappa(C)=\rho+1.\] The sheaf is coherent: the relative differentials over \(S\) are coherent, and \(\Omega_{S/\kappa}\) is finite-dimensional, so the cotangent sequence gives a finite presentation locally. This proves the asserted local freeness without making a smoothness assumption over \(S\).

Choose a projective embedding \(i:C\hookrightarrow\mathbb P^b_S\) and let \(\mathcal I\) be its ideal sheaf. This is a regular immersion of codimension \(b-1\). Here is a local verification. Let \(R\) be an ambient regular local ring, \(I\) the local ideal, and \(R/I\) the regular local ring of \(C\). Their dimensions differ by \(b-1\), by the dimension formula at the common residue field. Since both are regular, the kernel of \[\mathfrak m_R/\mathfrak m_R^2 \longrightarrow \mathfrak m_{R/I}/\mathfrak m_{R/I}^2\] has dimension \(b-1\). Choose \(b-1\) elements of \(I\) lifting a basis of this kernel. They form part of a regular system of parameters of \(R\) and cut out a regular local quotient of dimension \(\dim(R/I)\). The remaining ideal defining \(R/I\) in that quotient is prime and must be zero: a nonzero prime in a finite-dimensional local domain strictly lowers the quotient dimension. Thus the chosen elements generate \(I\). It follows that \(\mathcal I/\mathcal I^2\) is locally free of rank \(b-1\).

The absolute conormal sequence is in fact short exact: \[ 0\longrightarrow\mathcal I/\mathcal I^2 \longrightarrow i^*\Omega_{\mathbb P^b_S/\kappa} \longrightarrow\Omega_{C/\kappa}\longrightarrow0. \tag{35}\] To check the left end, the kernel of the last arrow is locally free, since the last module is locally free. Its rank is \((b+\rho)-(1+\rho)=b-1\). The conormal sequence surjects \(\mathcal I/\mathcal I^2\) onto this kernel. A surjection between locally free modules of the same rank is an isomorphism, proving (35).

The projective space is obtained from \(\mathbb P^b_\kappa\) by extending constants. Its absolute cotangents therefore split into relative cotangents over \(S\) and the constant bundle of \(\Omega_{S/\kappa}\). Taking determinants in (35) gives \[\det\Omega_{C/\kappa} \simeq\mathcal L_c\otimes_S\det\Omega_{S/\kappa}, \qquad \mathcal L_c= \mathcal O_C(-b-1)\otimes \det(\mathcal I/\mathcal I^2)^\vee.\] The last tensor factor is a constant line and has degree zero.

We compute the degree of \(\mathcal L_c\) using the projective-space duality in 10. A Koszul resolution for the local regular sequence defining \(C\) shows that \[\mathcal E xt^v_{\mathbb P^b_S} (i_*\mathcal O_C,\mathcal O(-b-1)) =\begin{cases} i_*\mathcal L_c,&v=b-1,\\ 0,&v\ne b-1. \end{cases}\] After dualizing the Koszul resolution, its top cohomology is the dual of the top exterior power of the free module of equations, restricted to \(C\). A change of those equations acts on this exterior power by the determinant of their change on \(\mathcal I/\mathcal I^2\). Thus the local top cohomology modules glue to \(\det(\mathcal I/\mathcal I^2)^\vee\otimes\mathcal O_C(-b-1)\).

The local-to-global Ext spectral sequence has only this one nonzero row. Projective-space duality consequently gives, for \(r=0,1\), \[H^r(C,\mathcal O_C)^\vee \simeq \operatorname{Ext}^{b-r}_{\mathbb P^b_S} (i_*\mathcal O_C,\mathcal O(-b-1)) \simeq H^{1-r}(C,\mathcal L_c).\] Hence \(\chi(C,\mathcal L_c)=-\chi(C)\). Applying (32) yields \(\deg_S\mathcal L_c=-2\chi(C)\), which proves (34). ◻

Proof. The morphism \(f^*\Omega_{C'/\kappa}\to\Omega_{C/\kappa}\) is generically an isomorphism: at the function fields a finite separable extension has no relative differentials, and both absolute differential spaces have dimension \(\rho+1\). Its determinant is therefore a nonzero regular morphism of line bundles. The divisor of this morphism is effective; call it \(R_f\). Its degree is the difference of the two determinant degrees, which is \(\nu(C)-\delta\nu(C')\) by [lem:curve-degrees,prop:absolute-genus].

Write \(x=f(y)\) and choose a uniformizer \(t\) of the DVR \(B=\mathcal O_{C',x}\). Its absolute differential is primitive in \(\Omega_{B/\kappa}\), meaning that it is part of a \(B\)-basis. Indeed the conormal sequence for the residue field is \[(t)/(t^2)\longrightarrow \Omega_{B/\kappa}\otimes_B\kappa(x) \longrightarrow\Omega_{\kappa(x)/\kappa}\longrightarrow0.\] The middle term has dimension \(\rho+1\) and the right term dimension \(\rho\), since \(\kappa(x)\) is finite over \(S\) and 4 computes absolute differential dimension over the perfect field \(\kappa\). Thus the first map has nonzero one-dimensional image, generated by \(dt\). Consequently \(dt\) is nonzero modulo the maximal ideal, and it extends to a basis of the free module \(\Omega_{B/\kappa}\).

In \(A=\mathcal O_{C,y}\) write \(t=u\tau^{a_y}\), where \(u\) is a unit and \(\tau\) a uniformizer. Absolute differentiation gives \[dt=\tau^{a_y}\,du+ a_yu\tau^{a_y-1}\,d\tau.\] All coordinates of this column of the differential matrix are divisible by \(\tau^{a_y-1}\); when \(p\mid a_y\), the second term is zero, so they are divisible by \(\tau^{a_y}\). Expanding the determinant in this column proves (36). Every step uses absolute differentials over \(\kappa\), so an inseparable extension of residue fields causes no change to the argument. ◻

Proof. The integral presentation. On an affine open of \(C'\), every element of the finite upstairs coordinate ring has its \(p\)th power in the downstairs ring: the power lies in the downstairs fraction field and is integral, so normality puts it in that ring. There is therefore a unique prime upstairs over each downstairs prime, since membership of an element is determined by membership of its \(p\)th power. At a closed point we may consequently write \(B\subset A\) for the downstairs and upstairs DVRs, with \(A\) still finite over \(B\). It is finite free of rank \(q\) and satisfies \(A^p\subset B\).

Let \(\tau\) be an upstairs uniformizer, let \(\epsilon\) be the ramification index, and put \(f=[\kappa(A):\kappa(B)]\). The element \(\tau^p\in B\) has upstairs valuation \(p\); all nonzero elements of \(B\) have upstairs valuation in \(\epsilon\mathbb Z\). Thus \(\epsilon\mid p\), so \(\epsilon\) is \(1\) or \(p\). Reduction of the rank-\(q\) free module modulo a downstairs uniformizer gives \(q=\epsilon f\) by (33). The residue extension has exponent at most one. Successively adjoining a generator outside the preceding residue field gives a degree-\(p\) extension each time. Choose residue generators \(\bar b_1,\ldots,\bar b_r\) in this manner, so their monomials with exponents in \(\{0,\ldots,p-1\}\) form a \(\kappa(B)\)-basis of \(\kappa(A)\) and \(f=p^r\). Lift them to \(b_1,\ldots,b_r\in A\). If \(\epsilon=p\), append \(b_{r+1}=\tau\); if \(\epsilon=1\), append nothing. In both cases there are exactly \(a\) generators, and \(c_\ell=b_\ell^p\) belongs to \(B\).

For a downstairs uniformizer \(t\), the ideal \(tA\) is \(\tau^\epsilon A\). Filter \(A/tA\) by the powers of \(\tau\) from \(0\) to \(\epsilon\). Each successive quotient is \(\kappa(A)\), and the lifted residue monomials give its basis after multiplication by the corresponding power of \(\tau\). It follows that the \(q\) monomials \[b_1^{i_1}\cdots b_a^{i_a},\qquad 0\le i_\ell<p,\] are a \(\kappa(B)\)-basis of \(A/tA\). By Nakayama they generate \(A\) over \(B\); since \(A\) is free of rank \(q\), they are a \(B\)-basis. The homomorphism from the monic-relation algebra therefore gives the actual presentation \[ A\simeq B[X_1,\ldots,X_a]/(X_1^p-c_1,\ldots,X_a^p-c_a), \qquad X_\ell\longmapsto b_\ell. \tag{39}\] Both sides have the same free monomial basis over \(B\), which verifies that there are no further relations. This includes the cases of a nontrivial inseparable residue extension and simultaneous residue extension and ramification.

These presentations are available on open neighborhoods. Shrink an open downstairs around the chosen point so that all the \(b_\ell\) are sections of \(\pi_*\mathcal O_C\), all the \(c_\ell\) are regular downstairs, and the determinant of the displayed monomial basis is invertible. Then (39) holds there. Such neighborhoods give local presentations throughout the curves.

Free kernel and free cokernel. Set \(N=A\otimes_B\Omega_{B/\kappa}\), and let \(W\subset N\) be the span of \(dc_1,\ldots,dc_a\). Differentiating the displayed presentation, using characteristic \(p\), gives \[ \Omega_{A/\kappa}\simeq (N/W)\oplus\bigoplus_{\ell=1}^a A\,db_\ell. \tag{40}\] Both \(N\) and \(\Omega_{A/\kappa}\) are free of rank \(\rho+1\) by 25. The summand \(N/W\) in (40) is therefore free, of rank \(\rho+1-a\). The sequence \(0\to W\to N\to N/W\to0\) splits, so \(W\) is free of rank \(a\). Its \(a\) generators \(dc_\ell\) are consequently a basis.

Dualizing now shows that the tangent kernel consists of the \(\kappa\)-derivations \(A\to A\) killing \(B\), and has free coordinates \[D\longmapsto(D(b_1),\ldots,D(b_a)).\] The tangent cokernel is \(W^\vee\), with coordinates \[[\delta]\longmapsto (\delta(c_1),\ldots,\delta(c_a)), \qquad \delta\in\operatorname{Der}_\kappa(B,A).\] Indeed a derivation \(B\to A\) extends to \(A\) exactly when it kills the \(c_\ell\); when it does, the values assigned to the \(b_\ell\) are arbitrary. Thus both kernel and cokernel are locally free of rank \(a\).

Gluing the determinant lines. On an overlap, write another system of generators as \(b'_k=P_k(b_1,\ldots,b_a)\) with coefficients in \(B\), and put \[\mathsf J_{k\ell}= \frac{\partial P_k}{\partial X_\ell}(b_1,\ldots,b_a).\] A derivation in the kernel kills the coefficients, so its new coordinate vector is \(\mathsf J\) times its old coordinate vector. The matrix \(\mathsf J\) is invertible since both coordinate systems are bases. Taking \(p\)th powers of the generator identities gives downstairs \[c'_k=(b'_k)^p=P_k^{[p]}(c_1,\ldots,c_a),\] where \(P_k^{[p]}\) means that the coefficients of \(P_k\) are raised to the \(p\)th power. Their absolute differentials vanish. Differentiating this identity by a representative \(\delta:B\to A\) of a cokernel class therefore gives \[\delta(c'_k)=\sum_\ell \mathsf J_{k\ell}^{p}\,\delta(c_\ell).\] Thus the cokernel coordinate transition is the entrywise Frobenius power \(\mathsf J^{[p]}\). Coordinate transitions for determinant lines are \(\det\mathsf J\) and \(\det(\mathsf J^{[p]})=(\det\mathsf J)^p\); if one uses local frames instead, both transitions are inverted. In either convention they give exactly (37), with the exponent \(+p\).

Finally the exact sequence \[0\longrightarrow\mathcal F\longrightarrow T_C \longrightarrow\pi^*T_{C'}\longrightarrow\mathcal Q \longrightarrow0\] and (37) give \[\deg T_C=q\deg T_{C'}-(p-1)\deg\mathcal F.\] Since \(\deg T_C=-\nu(C)\), this is (38). ◻

Proof of 23. Normalization and the separable splitting field. Normalizations are finite by 7, and normal curves over fields are regular. We first record why the finite purely inseparable maps in this argument are radicial, that is, universally injective. If \(A\) is the integral closure of a normal affine domain \(B\) in a finite purely inseparable extension, there is an exponent \(v\) such that \(a^{p^v}\in B\) for every \(a\in A\). The power belongs to \(\operatorname{Frac}(B)\) and is integral over \(B\), so it belongs to \(B\) by normality. After tensoring with any \(B\)-algebra, the \(p^v\)th power of every element still lies in the image of that algebra. Consequently over a prime of the base there is at most one prime upstairs: membership of an element in a prime is determined by membership of its \(p^v\)th power downstairs. The residue extension is purely inseparable for the same reason. This proves the assertion after every base change.

The extension \(S/F\) is finite separable, since \(F'/F_0\) is finite Galois. Thus \(Z_S\) is regular and reduced by 8. Its map to the integral curve \(E_S\) is still finite, surjective, and radicial. This map is a homeomorphism on underlying spaces, so \(Z_S\) is irreducible and hence integral. It is the normalization of \(E_S\) in its function field: it is finite and normal with that field, and an element integral over the base is also integral over its finite normal coordinate ring. Its degree remains \(d\) by finite flatness. By 24, \[\nu(Z_S)=\nu(Z),\] where the two Euler characteristics are computed over their stated fields. We may therefore do the differential calculation over \(S\). If \(d=1\), this curve is \(E_S\), every \(r_i=1\), and the theorem holds. Assume henceforth that \(d>1\).

The Frobenius tower and its recurrence. Let \(M_e=S(E)\) and \(M_z=S(Z)\), and set \[M_j=M_eM_z^{p^j}\quad(0\le j\le b),\qquad M_0=M_z,\qquad M_b=M_e,\] where \(b\) is the first index for which the last equality holds. Let \(Y_j\) be the normalization of \(E_S\) in \(M_j\), and write \[\gamma_j:Y_j\longrightarrow E_S,\qquad d_j=[M_j:M_e],\qquad \pi_j:Y_j\longrightarrow Y_{j+1},\qquad q_j=d_j/d_{j+1}=p^{m_j}.\] In particular \(Y_0=Z_S\), \(d_0=d\), \(Y_b=E_S\), and \(d_b=1\). The defining fields satisfy \[ M_{j+1}=M_eM_j^p. \tag{41}\] Thus each \(\pi_j\) has exponent at most one and 27 applies.

Put \(T_j=(\Omega_{Y_j/\kappa})^\vee\) and \[\mathcal F_j=\ker(T_j\longrightarrow\pi_j^*T_{j+1}).\] This is also the kernel of the composite tangent map \(T_j\to\gamma_j^*T_{E_S}\). To see this, at the function field a \(\kappa\)-derivation kills \(M_e\) if and only if it kills \(M_eM_j^p=M_{j+1}\), since every derivation kills \(p\)th powers. The two kernels are therefore equal generically. They are equal as sheaves: a section whose image vanishes generically has zero image everywhere in either of the locally free, hence torsion-free, targets.

The genus identity involves \(\deg_S\mathcal F_j\). We will estimate this degree through the image of the tangent map to \(E_S\). Choose a nonzero invariant differential \(\eta\) on \(A_0\). The constant-curve identification of \(E_S\) gives the splitting \[\Omega_{E_S/\kappa} \simeq\mathcal O_{E_S}\eta \oplus(\mathcal O_{E_S}\otimes_S\Omega_{S/\kappa}).\] Thus its dual is a trivial bundle of rank \(n+1\). In this trivial target we can use the marked points to impose linear vanishing conditions on the tangent-map image.

Let \(\mathcal H_j\) be the image in \(\gamma_j^*T_{E_S}\), and put \(b_j^\circ=-\deg_S\mathcal H_j\). It is a torsion-free coherent sheaf on a regular curve, hence locally free. Its rank is \(n+1-m_j\), since \(\operatorname{trdeg}_\kappa S=n\). The exact sequence with kernel \(\mathcal F_j\) and image \(\mathcal H_j\), together with (38), gives \[\nu(Y_j)=q_j\nu(Y_{j+1})+(p-1)\deg_S\mathcal F_j, \qquad b_j^\circ=\nu(Y_j)+\deg_S\mathcal F_j.\] Eliminating \(\deg_S\mathcal F_j\) and using \(d_j=q_jd_{j+1}\) yields \[ \frac{\nu(Y_j)}{d_j} =\frac1p\frac{\nu(Y_{j+1})}{d_{j+1}} +\left(1-\frac1p\right)\frac{b_j^\circ}{d_j}. \tag{42}\] The terminal term \(\nu(Y_b)\) is zero because \(E_S\) has genus one. It remains to give a lower bound for \(b_j^\circ/d_j=-\deg_S\mathcal H_j/d_j\).

Vanishing at a ramified label. For each \(j\) let \(S_j\) be the set of labels at which \(\gamma_j\) has ramification index greater than one. For \(i\in S_j\) write \(D_{j,i}=\gamma_j^*(t_i)\) and \[v_i=(1,-t_i^*\eta)\in S\oplus\Omega_{S/\kappa}.\] Regard \(v_i\) as a constant linear functional on \(\gamma_j^*T_{E_S}\). We claim that its restriction to \(\mathcal H_j\) vanishes along the entire divisor \(D_{j,i}\); equivalently, its values lie in \(\mathcal O_{Y_j}(-D_{j,i})\).

Let \(a\) be a local parameter defining the rational point \(t_i\) on \(E_S\). Restriction of absolute differentials along \(t_i\) sends \((\lambda,\beta)\) in the displayed splitting to \(\lambda t_i^*\eta+\beta\). Its kernel is the line spanned by \(v_i\). Since \(a\) restricts to zero, \(da\) belongs to this kernel modulo \((a)\). Moreover its relative component is nonzero: \(t_i\) is an \(S\)-rational point on a smooth curve over \(S\), and \(a\) generates its relative cotangent space. Hence \[da\equiv\lambda_i v_i\pmod{a\Omega_{E_S/\kappa}} \quad\text{for some }\lambda_i\in S^\times.\] At the unique point upstairs write \(a=u\tau^r\), with \(u\) a unit. The nontrivial purely inseparable index \(r\) is divisible by \(p\); therefore \[d(\gamma_j^*a)=\tau^r\,du=(\gamma_j^*a)\,u^{-1}du.\] It lies in the ideal of the full divisor \(D_{j,i}\) times \(\Omega_{Y_j/\kappa}\). Pulling back the preceding congruence and dividing by the unit \(\lambda_i\) shows that the pullback of \(v_i\) lies in that same ideal times \(\Omega_{Y_j/\kappa}\). To relate this differential to the claimed functional, denote the tangent map by \(d\gamma_j:T_j\to\gamma_j^*T_{E_S}\) and write \(\gamma_j^*v_i\) for the image of the pulled-back covector in \(\Omega_{Y_j/\kappa}\). For a local section \(\theta\) of \(T_j\), duality gives \[v_i(d\gamma_j(\theta))=(\gamma_j^*v_i)(\theta).\] The divisibility just proved makes this value a section of \(\mathcal O_{Y_j}(-D_{j,i})\). Since \(\mathcal H_j\) is the image of \(d\gamma_j\), the claim follows. The vanishing is along \(\gamma_j^*(t_i)\) with its full multiplicity, not just its reduced support.

From independent vanishing conditions to a degree bound. By 22, among the \(v_i\) for \(i\in S_j\) there are \(c\ge |S_j|-h\) independent covectors over \(S\).

Complete the \(c\) chosen covectors to a constant basis of the dual of the target trivial tangent bundle. The full-divisor constraint gives an injection \[ \mathcal H_j\hookrightarrow \mathcal O_{Y_j}^{n+1-c}\oplus \bigoplus_{i\ \mathrm{chosen}}\mathcal O_{Y_j}(-D_{j,i}). \tag{43}\] Let \(R=n+1-m_j\) be the rank of \(\mathcal H_j\). The \(R\)th exterior power of the bundle on the right is a direct sum of line bundles, each the tensor product of a choice of \(R\) summands. The induced map from \(\det\mathcal H_j\) is nonzero, so it has a nonzero component in at least one of these lines. Only \(n+1-c\) of the available summands are trivial. The chosen line must therefore contain at least \(R-(n+1-c)=c-m_j\) negative divisor summands when this number is positive, and at least zero otherwise. Each \(D_{j,i}\) has degree \(d_j\) by finite flatness and rationality of \(t_i\). If the nonzero component uses \(k\) negative summands, its target line has degree \(-kd_j\). A nonzero regular map from one line bundle to another has an effective zero divisor, and consequently \[\deg_S\mathcal H_j\le-kd_j, \qquad \frac{b_j^\circ}{d_j}\ge c-m_j\ge |S_j|-h-m_j.\] The argument also covers rank zero, with \(\det(0)=\mathcal O\).

The weighted estimate. Write \(r_i=p^{u_i}\). Every height-one step has local index at most \(p\), as proved in 27. The composite \(Y_0\to Y_j\) consequently has ramification index at most \(p^j\) at this label. In particular, \(u_i\leq b\). Since ramification indices multiply, the remaining map \(Y_j\to E_S\) has index at least \(p^{u_i-j}\) whenever \(j<u_i\). Hence \(i\in S_j\) for every \(0\le j<u_i\). Iterating (42) with terminal value zero gives \[\frac{\nu(Z)}d =\sum_{j=0}^{b-1} p^{-j}\left(1-\frac1p\right)\frac{b_j^\circ}{d_j}.\] Write \(w_j=p^{-j}(1-1/p)\) for the weight of stage \(j\). For each label, the weights from those first \(u_i\) stages sum to \[\sum_{j=0}^{u_i-1}p^{-j}\left(1-\frac1p\right) =1-p^{-u_i}=1-\frac1{r_i}.\] Summing these contributions by label gives \[\sum_{j=0}^{b-1}w_j|S_j| =\sum_{i=1}^n\sum_{\substack{0\leq j<b\\i\in S_j}}w_j \geq\sum_{i=1}^n\left(1-\frac1{r_i}\right).\] The error terms in the degree bound contribute separately. Since \(m_j\le\log_p d\) and the sum of all the weights is \(1-p^{-b}<1\), \[\sum_{j=0}^{b-1}w_j(h+m_j)\leq h+\log_p d.\] Substitute \(b_j^\circ/d_j\geq |S_j|-h-m_j\) in the recurrence sum. Subtracting the error bound from the label contribution yields \[ \frac{\nu(Z)}d \ge\sum_{i=1}^n\left(1-\frac1{r_i}\right) -\bigl(h+\log_p d\bigr). \tag{44}\]

Divisors over \(F\) and exact rounding. Multiplying (44) by \(d\) bounds the possible deficit by \(d(h+\log_p d)\). Set \(N=p^{s-h}\) and consider the difference \[A=\nu(Z)-d\sum_i\left(1-\frac1{r_i}\right).\] We will realize both terms as degrees of divisors on \(Z\), each retaining its degree under pushforward to \(E_F\). The index property 21 then makes both degrees divisible by \(N\). These divisors must be defined over \(F\), before the separable extension to \(S\). By 25, the integer \(\nu(Z)\) is the degree over \(F\) of \(\det\Omega_{Z/\kappa}\). A rational trivialization represents this line bundle by a Cartier divisor on the regular integral curve \(Z\). Its pushforward to \(E_F\) preserves degree over \(F\), by 24. Thus \[\nu(Z)\in N\mathbb Z.\] This uses the absolute determinant on \(Z/F\) and its Euler characteristic over \(F\), including any inseparable constant field.

Let \(T\) be the reduced inverse image in \(Z\) of \(\mathcal B_F\). It is an effective divisor on the regular curve \(Z\). Finite separable scalar extension preserves reducedness, so \(T_S\) is the reduced inverse image of the split labels. If \(z_i\) is the unique point above the rational point \(t_i\), the local degree formula is \[r_i[\kappa(z_i):S]=d.\] It follows that \[\deg_F T=\deg_S T_S=\sum_i\frac d{r_i}.\] In particular this sum counts the full residue degrees of the reduced points; those residue fields may be inseparable over \(S\). The divisor \(\gamma^*\mathcal B_F-T\) is defined on \(Z\) over \(F\), where \(\gamma:Z\to E_F\), and its degree is \[dn-\sum_i\frac d{r_i} =d\sum_i\left(1-\frac1{r_i}\right).\] Push it to \(E_F\) to conclude that this integer also belongs to \(N\mathbb Z\).

Consequently \(A\) is a multiple of \(N\). By (44) and \(d\le m\), \[A\ge-d(h+\log_p d) \ge-m(h+\log_p m)>-N.\] A multiple of \(N\) strictly greater than \(-N\) is nonnegative. This proves (31). The sufficient threshold (30) contains no datum from \(F\), \(Z\), or a tuple-class multiset. If a threshold independent of the prime is desired, the stronger condition \[2^{s-h}>m(h+\log_2m)\] suffices for every \(p\ge2\). This proves the stated uniformity. ◻

Proof. We first construct the constant complete DVR. Starting with \(\mathbb Z_p\), successively lift monic separable irreducible polynomials over the residue fields so that the resulting finite residue fields exhaust \(\kappa=\overline{\mathbb F}_p\). To justify this construction, if \(R\) is the current complete DVR and \(f\in R[T]\) is such a monic lift, then \(R[T]/(f)\) is finite free, its reduction is a field, and its maximal ideal is generated by the original uniformizer. The polynomial \(f\) is irreducible over the fraction field: a monic factorization there would have coefficients integral over \(R\), hence in \(R\), and would give a factorization of its irreducible reduction. Consequently the quotient is a local domain with principal maximal ideal, hence a DVR; its residue extension is the prescribed one. Its derivative is a unit, so the extension is unramified. The union of this tower is a valuation ring with value group \(\mathbb Z\) and residue \(\kappa\). Complete it. Completion preserves its uniformizer and residue field, giving a complete DVR of mixed characteristic with residue \(\kappa\). Denote it for the moment by \(R_c^{(0)}\), with fraction field \(K_c^{(0)}\).

Lift the coefficients of the smooth Weierstrass equation for \(A_0\) to this DVR. Its discriminant remains a unit, so it defines a smooth elliptic scheme \(\mathcal A\) by 11. We next lift \(P\) without assuming that the torsion scheme is étale. The kernel \(\mathcal A[n]\) is finite locally free of degree \(n^2\). Its generic fiber is dense: in its finite flat coordinate algebra the uniformizer is a nonzerodivisor, so no irreducible component is supported in the special fiber. Since this dense generic fiber is a finite set, one of its points \(Q\) has closure containing \(P\). Its field of definition \(K_Q\) is finite over \(K_c^{(0)}\); this is a characteristic-zero field, not a valuation residue field. The reduced closure of \(Q\) is finite and integral over \(R_c^{(0)}\). Pass to the integral closure of \(R_c^{(0)}\) in \(K_Q\). By 5 it is a complete DVR, and lying over shows that its closed point maps to \(P\). We obtain an \(n\)-torsion section \(\widetilde P\) specializing to \(P\). Its order is exactly \(n\), since its order divides \(n\) and reduction has order \(n\). Enlarge the constant field once more to contain all \(m\)th roots of unity. These are finite extensions of complete discretely valued fields, and their residue fields are still \(\kappa\), which is algebraically closed. Denote the resulting constant valuation ring by \(R_c\) and choose its uniformizer \(\varpi\).

Consider the smooth product \(\mathcal A^n/R_c\). Its generic fiber is integral, and flatness makes the total space integral. At the generic point of the integral special fiber, its local ring is a noetherian local domain whose maximal ideal is generated by \(\varpi\) and whose residue field is \(F'\). It is therefore a DVR. Complete its fraction field for this valuation and call the result \(K_b'\), with valuation ring \(R_b'\). The product projections give points \(\xi_i\in\mathcal A(K_b')\). Their sections over \(R_b'\) reduce to the generic projections \(t_i\).

The automorphism of the product specified by \[\sigma(\xi_i)=\xi_{i+1}-\widetilde P\] fixes the constants, has order \(n\), and preserves the special fiber. It preserves the valuation and extends to \(K_b'\). Its residue action is the specified action on \(F'\), of order \(n\). Put \(K_b=(K_b')^{\langle\sigma\rangle}\). The fixed field is closed in the complete valued field \(K_b'\), so it is complete. The fixed uniformizer \(\varpi\) shows that its value group is the same discrete group. Finite fixed-field theory gives \([K_b':K_b]=n\) with Galois group \(\langle\sigma\rangle\).

Let \(k_b\) be the residue field of \(K_b\). Then \(k_b\subseteq F_0\). Lifting a linearly independent system in a residue extension to units gives a linearly independent system over the valued base field: normalize a putative relation by a coefficient of minimum value and reduce. Applying this observation here gives \[n=[F':F_0]\leq [F':k_b]\leq[K_b':K_b]=n.\] Thus \(k_b=F_0\).

To verify the stronger assertion about valuation rings, choose a primitive element \(\bar a\) of the separable extension \(F'/F_0\) and a lift \(a\in R_b'\). The polynomial \[f(T)=\prod_{j=0}^{n-1}(T-\sigma^j a)\in R_b[T]\] reduces to the separable irreducible polynomial of \(\bar a\). Consequently \(R_b[a]\) is finite free of rank \(n\), local with maximal ideal \((\varpi)\), and a domain with fraction field \(K_b'\). It is a DVR, and the derivative \(f'(a)\) is a unit. Uniqueness of the extension of the complete-DVR valuation identifies it with \(R_b'\). This proves that \(R_b'/R_b\) is finite étale. The Galois torsor map \[R_b'\otimes_{R_b}R_b'\longrightarrow\prod_{j=0}^{n-1}R_b', \qquad a\otimes b\longmapsto (a\sigma^j(b))_j\] is an isomorphism: after reduction it is the usual isomorphism for \(F'/F_0\), and both sides are free \(R_b'\)-modules of rank \(n\).

On \(\mathcal A_{R_b'}\) define the semilinear descent action \(\delta(x)=\sigma(x)+\widetilde P\). Here \(\sigma\) acts on the coefficients, whereas \(\delta\) also translates the elliptic-curve point. Its \(n\)th power is the identity. The divisor \[D'=\sum_{j=0}^{n-1}[j\widetilde P]\] is invariant. Its sections are disjoint because their reductions are distinct. The line bundle \(\mathcal O(3D')\) is a tensor product of translates of the relatively ample Weierstrass line bundle \(\mathcal O(3[0])\). It is relatively ample, and therefore so is \(\mathcal O(D')\). The invariant divisor supplies its canonical compatible linearization. Finite étale Galois descent (6) gives \(\mathcal X\) and \(D\), with a splitting identification \(\mathcal X_{R_b'}\simeq\mathcal A_{R_b'}\). Their reduction is precisely the twist \(E\) and the corresponding origin-orbit divisor. Smoothness and relative ampleness descend.

Under this identification, continue to write \(\xi_i\) for the corresponding sections of \(\mathcal X_{R_b'}\). They have distinct reductions, all different from the constant points \(jP\). They are therefore pairwise disjoint and avoid \(D'\). Under the descent action they satisfy \(\delta(\xi_i)=\xi_{i+1}\). Their union is an invariant disjoint union of sections, hence a finite étale divisor of degree \(n\), and descends with these properties, although the individual sections need not be defined over \(R_b\). ◻

Construction of \(K_s\). Normalize the valuation of 28 by \(v(\varpi)=1\). In an algebraic closure with an extended valuation, choose compatible factorial roots: take \(\theta_1=\varpi\) and \(\theta_{j+1}^{j+1}=\theta_j\) for \(j\geq1\). Thus \(\theta_j^{j!}=\varpi\) and the fields \(K_b(\theta_j)\) are nested. Put \(K_\infty=\bigcup_{j\geq1}K_b(\theta_j)\). For \(r=j!\) and \(\theta=\theta_j\), one has \(\theta^r=\varpi\). The values of \(1,\theta,\ldots,\theta^{r-1}\) lie in distinct cosets of \(\mathbb Z\). These elements are linearly independent over \(K_b\), and the equation for \(\theta\) shows that they are a basis of \(K_b(\theta)\). For \(x=\sum_{j=0}^{r-1}a_j\theta^j\ne0\), the distinct value cosets give \[v(x)=\min_j\bigl(v(a_j)+j/r\bigr).\] The value group is \(r^{-1}\mathbb Z\). If \(v(x)=0\), the minimum can only occur at \(j=0\), and all other terms have positive value; hence the residue of \(x\) is the residue of \(a_0\). This proves that \(K_\infty\) has value group \(\mathbb Q\) and residue field \(F_0\).

Here is the set-theoretic detail needed to choose a maximal immediate extension. Any valued field with countable value group and countable residue field has cardinality at most \(\mathfrak c=2^{\aleph_0}\). Indeed, for every \(q\) in the value group and every closed \(q\)-ball \(B\), the open \(q\)-balls contained in \(B\) are in bijection with the residue field, after choosing a center and an element of value \(q\). Label these open subballs injectively by natural numbers. For a point \(x\), record, for each \(q\), the label of its open \(q\)-ball inside its closed \(q\)-ball. If \(x\ne y\), at \(q=v(x-y)\) they belong to the same closed ball but different open subballs, so these records differ. This gives an injection into a countable product of countable sets. In our situation \(F_0\) is countable, being a subfield of the finitely generated field \(F'/\kappa\).

Choose a fixed set of cardinality greater than \(\mathfrak c\), containing the underlying set of \(K_\infty\). Order the immediate valued extensions of \(K_\infty\) carried by subsets of this set by literal extension of all their structures. The union of a chain is an immediate valued field: each of its elements and every finite field computation occur in one member of the chain. The cardinality bound applies to the union as well. Zorn’s lemma gives a maximal member \(K_s\). Any further immediate extension has cardinality at most \(\mathfrak c\) and could be relabeled, fixing \(K_s\), into the unused part of the fixed set. Thus \(K_s\) has no proper immediate extension in the unrestricted sense.

We now prove spherical completeness directly. Suppose a nest of nonempty closed balls has empty intersection. Distinct balls in a nest with the same radius coincide, so the countability of the value group gives a countable cofinal subnest. Successively go deeper in that subnest, also choosing each next ball to omit the preceding center. We obtain cofinal balls \(B(a_i,\delta_i)\) such that \(a_i\notin B(a_{i+1},\delta_{i+1})\) and \[ v(a_j-a_i)=\gamma_i\quad(j>i),\qquad \delta_i\leq\gamma_i<\delta_{i+1},\qquad \gamma_1<\gamma_2<\cdots . \tag{46}\] The balls \(B(a_i,\gamma_i)\) are nested, lie inside \(B(a_i,\delta_i)\), and still have empty intersection.

Extend the valuation to an algebraic closure of \(K_s\). Call \(b\) a pseudo-limit of this sequence if \(v(b-a_i)=\gamma_i\) for all sufficiently large \(i\). No element of \(K_s\) is a pseudo-limit, since a pseudo-limit belongs to every ball \(B(a_i,\gamma_i)\). For any point \(b\) which is not a pseudo-limit, the values \(v(b-a_i)\) are eventually constant and smaller than \(\gamma_i\). To see this, if \(v(b-a_i)<\gamma_i\) at one stage, the strong triangle law and (46) give the same value at every later stage. If \(v(b-a_i)>\gamma_i\), the next value is \(\gamma_i\), which is smaller than \(\gamma_{i+1}\), reducing to the preceding case. Otherwise equality holds eventually, as required for a pseudo-limit.

Suppose first that an algebraic pseudo-limit \(\alpha\) exists, and choose one of smallest degree \(d\) over \(K_s\). Every root \(\beta\) of a nonzero polynomial \(f\in K_s[T]\) with \(\deg f<d\) is not a pseudo-limit. For large \(i\) we therefore have \[v(a_i-\beta)=v(\alpha-\beta)<v(\alpha-a_i),\qquad \frac{\alpha-\beta}{a_i-\beta}\equiv1 \pmod{\mathfrak m_{\overline{K_s}}}.\] Factoring \(f\) over the algebraic closure yields \[ f(\alpha)/f(a_i)\equiv1 \pmod{\mathfrak m_{\overline{K_s}}} \qquad(i\gg0). \tag{47}\] Every nonzero element of \(K_s(\alpha)\) is \(f(\alpha)\) for a polynomial of degree less than \(d\). Equation (47) therefore gives, for each such element \(x\), a scalar \(c=f(a_i)\in K_s\) with \(v(x-c)>v(x)\). It follows that \(v(c)=v(x)\), so no new values occur. If \(v(x)=0\), the same inequality says that \(x\) and \(c\) have the same residue, so no new residue-field elements occur either. Thus \(K_s(\alpha)/K_s\) is a proper immediate extension, a contradiction.

Suppose instead that no algebraic pseudo-limit exists. Factoring any nonzero polynomial over the fixed algebraic closure shows that its values at \(a_i\) are eventually constant and finite; the same holds for nonzero rational functions. Define, on \(K_s(T)\), \[w(f)=\text{the eventual value of }v(f(a_i)), \qquad w(0)=+\infty.\] Evaluation proves multiplicativity and the strong triangle inequality, so \(w\) is a valuation extending that of \(K_s\). For fixed \(f\ne0\) and \(j>i\gg0\), every linear factor in its numerator and denominator satisfies \((a_j-\beta)/(a_i-\beta)\equiv1\pmod{\mathfrak m_{\overline{K_s}}}\), where \(\beta\) is its root. Indeed \(v(a_i-\beta)<\gamma_i=v(a_j-a_i)\). Consequently \[v(f(a_j)-f(a_i))>v(f(a_i))=w(f).\] For a fixed sufficiently large \(i\), the left side has an eventual value as \(j\) increases, since it evaluates the rational function \(f(T)-f(a_i)\). Thus \(w(f-f(a_i))>w(f)\) unless the difference is zero, when the assertion is immediate. Thus every nonzero element \(f\) of \(K_s(T)\) admits a scalar \(f(a_i)\) with this strict approximation inequality. As in the algebraic case, this shows that neither the value group nor the residue field changes. This is again a proper immediate extension, contradicting maximality. Both cases being impossible, \(K_s\) is spherically complete. ◻

Proof. A subspace with an orthogonal basis is spherically complete: its closed balls are products of closed balls in the finitely many coordinates, with the radii shifted by the norms of the basis vectors. Each coordinate nest has a common point.

Suppose an orthogonal basis has already been constructed for a subspace \(U\), and choose \(x\notin U\). For every value \(\gamma\) achieved by \(\lambda(x-u)\), \(u\in U\), consider \[B_\gamma=\{u\in U:\lambda(x-u)\geq\gamma\}.\] This is a nonempty closed ball in \(U\): if \(u_\gamma\) belongs to it, then it equals \(\{u:\lambda(u-u_\gamma)\geq\gamma\}\). These balls form a nest, so they have a common point \(u_0\). The finite value \(\ell=\lambda(x-u_0)\) is at least every achieved value and itself is achieved. Thus \(u_0\) is a best approximation to \(x\). Put \(y=x-u_0\). For \(u\in U\), maximality of \(\ell\) gives \(\lambda(y+u)\leq\ell\). If \(\lambda(u)<\ell\), the strong triangle law gives \(\lambda(y+u)=\lambda(u)\); if \(\lambda(u)\geq\ell\), it gives \(\lambda(y+u)\geq\ell\), hence equality. Therefore \[\lambda(y+u)=\min\{\ell,\lambda(u)\}.\] Homogeneity shows that adjoining \(y\) preserves orthogonality. Induction gives an orthogonal basis of \(V\). Since every basis norm is in the base value group, scale its vectors to norm zero. Formulae (48) and (49) follow, and the same coordinate argument proves spherical completeness of \(V\). ◻

Finite-extension assertions of 29. For any algebraic extension of a field with value group \(\mathbb Q\), every nonzero algebraic element has value in \(\mathbb Q\). Indeed, in a polynomial relation \(\sum_i a_i x^i=0\), at least two nonzero terms must have the same minimum value, so \[(i-j)v(x)=v(a_j)-v(a_i)\in\mathbb Q \quad(i\ne j).\] Divisibility of \(\mathbb Q\) and torsion-freeness of ordered value groups imply \(v(x)\in\mathbb Q\). Apply 30 to a finite extension \(L/K_s\) with any extended valuation, viewed as a normed vector space over \(K_s\). Its integral and positive balls are exactly its valuation ring and maximal ideal. Formula (49) proves (45), including the full inseparable degree of the residue extension. Orthogonal coordinates also show that \(L\) is spherically complete. The same argument applies to every finite extension of \(L\). ◻

Proof. The preceding polynomial-relation argument puts all the values in \(\mathbb Q\), with their normalization fixed by the valuation on \(T\). For \(i\ne j\), distinctness gives an element whose two values differ. Inverting it if necessary and multiplying by an element of \(T\) of a suitable rational value produces \(z\in U\) with \(w_i(z)>0>w_j(z)\). Choose a unit \(c\in T\) such that, at every \(w_\ell\) for which \(w_\ell(z)=0\), the residue of \(1+cz\) is nonzero. Each such condition excludes at most one value for the residue of \(c\) in \(k_T^\times\), so the infinitude of \(k_T\) permits this choice. Then \[q_{ij}=\frac{1}{1+cz}\] is integral at every \(w_\ell\), has residue \(1\) at \(i\), and has residue \(0\) at \(j\). Indeed, a positive value of \(z\) makes the denominator a unit with residue \(1\); a negative value of \(z\) gives the denominator that negative value; and value zero is covered by the choice of \(c\).

Put \(q_i=\prod_{j\ne i}q_{ij}\), with \(q_i=1\) if \(r=1\). It has residue \(1\) at \(i\) and positive value at all other places. For a prescribed residue \(b_i\in k_i\), choose any lift \(\widetilde b_i\) integral at \(i\). If the lift is nonzero, choose \(N_i\) so that \[N_iw_\ell(q_i)+w_\ell(\widetilde b_i)>0 \qquad(\ell\ne i).\] There are only finitely many conditions, and every \(w_\ell(q_i)\) in them is positive. At \(i\), the factor \(q_i^{N_i}\) has residue one. Thus \(q_i^{N_i}\widetilde b_i\) is integral everywhere, has residue \(b_i\) at \(i\), and has residue zero at every other place. A zero lift contributes the zero term. Summing these terms realizes any tuple \((b_i)_i\).

For completeness, each \([k_i:k_T]\) is finite and at most \([U:T]\) by the single-place lifting argument already used in the proof of 28. Choose a \(k_T\)-basis of each \(k_i\), and use the surjectivity just proved to lift its elements with zero residues in the other coordinates. These lifts are linearly independent over \(T\). For if they satisfied a nonzero relation, divide it by a coefficient of minimum value. Every coefficient then is integral and at least one is a unit. Reducing in a coordinate which contains such a unit contradicts the chosen residue basis there. Their total number is therefore at most \([U:T]\), proving (50). ◻

Completion of the proof of 29. The residue of \(K_s\), and that of every finite extension \(L/K_s\), is infinite. If two distinct valuations extended the valuation of \(L\) to a finite extension \(U/L\), the finite-extension argument above would give residue degree \([U:L]\) for each. This contradicts (50). Existence of prolongations is part of 5; uniqueness on finite extensions implies uniqueness on arbitrary algebraic extensions by restriction to finite subextensions.

Finally \(|K_s|\leq\mathfrak c\) by the cardinality argument, while \(\mathbb Q_p\subset K_s\) gives \(|K_s|\geq\mathfrak c\). Its algebraic closure has the same cardinality. An uncountable algebraically closed field of characteristic zero has cardinality equal to its transcendence degree over \(\mathbb Q\): the field generated by a transcendence basis, and then its algebraic closure, both have cardinality the maximum of that basis cardinality and \(\aleph_0\). Both \(\overline{K_s}\) and \(\mathbb C\) therefore have transcendence degree \(\mathfrak c\) over \(\mathbb Q\). The algebraically closed field classification in 4 gives the asserted abstract isomorphism. No compatibility with either field’s topology is asserted or used. ◻

Proof. Under the chosen complex identification the base is a smooth Weierstrass cubic. In characteristic zero, completing the square gives an equation \(y^2=f_3(x)\) with three distinct roots. Its double cover of the projective line branches simply at those roots and at infinity. Cut the sphere along two disjoint arcs joining pairs of these four points and glue the two sheets crosswise. The resulting compact surface is a torus. This identifies the topological type of our particular base without a general analytic–algebraic genus comparison.

Remove the \(n\) marked points. Its complex fundamental group has generators and relation \([x,y]u_1\cdots u_n=1\) as in 12. Choose a point in a fiber of a marked cover and identify the fiber with the regular left \(J\)-set. Path lifting commutes with the left deck action, so monodromy acts by right translations. With path multiplication chosen in the order of successive traversal, these translations specify a homomorphism from the fundamental group to \(J\). The cover is connected exactly when the monodromy action is transitive, which for this regular action means that the homomorphism is surjective. Changing the chosen fiber point conjugates this homomorphism by one element of \(J\). Conversely, a surjective homomorphism defines such a fiber action and therefore a connected unramified cover. An equivariant fiber bijection is a right translation, and compatibility with monodromy says exactly that the two homomorphisms are simultaneously conjugate. Thus the quotient is by inner conjugation; keeping the specified \(J\)-action does not identify covers through outer automorphisms of \(J\).

Riemann existence in 12 algebraizes the unramified cover of this punctured algebraic curve, with its morphisms and specified deck action. Normalize the original projective base in the resulting function field. Finiteness of normalization gives a projective algebraic cover; normality makes its source smooth in characteristic zero. The deck action and isomorphisms extend uniquely across normalization. In a local power-map chart \(z\mapsto z^e\), the ramification index is \(e\), the order of the associated monodromy element. These extensions give precisely the branched covers and isomorphisms in the statement.

We describe the local conjugacy class algebraically in order to check Galois invariance. For each divisor \(e\) of \(m\), use the root \(\zeta_e\in K_s\) which maps to \(\exp(2\pi i/e)\) under the chosen complex identification. All these roots belong to \(K_s\) by 28. Represent the based puncture loop as an approach path, a positive circle around the puncture, and the reverse approach path. In the chart \(z\mapsto z^e\), lifting the positive circle sends \(z\) to \(\zeta_e z\). Let \(Q\) be the point above the puncture selected by the lifted approach path, and let \(g\in J\) be the inertia element acting this way in its local chart. Apply \(g\) to the lifted approach path. The reverse of the resulting path is the lift of the return path starting at the endpoint of the circle lift. Thus the complete based loop sends the chosen fiber point to its image under \(g\). Under the preceding fiber identification, its monodromy element is therefore \(g\). The inertia group is cyclic, and \(g\) is its unique generator acting on the tangent line at \(Q\) by \(\zeta_e\). Choosing another point above the puncture conjugates the element in \(J\). This recovers its conjugacy class from the algebraic tangent action. (Reversing all local-loop conventions replaces all these generators by their inverses consistently.)

A Galois automorphism over \(K_s\) sends the tangent action at \(Q\) to the tangent action at its conjugate point, and fixes \(\zeta_e\). Consequently it preserves the local \(J\)-conjugacy class at the transported branch point. It permutes the branch points, since their union is defined over \(K_s\), and hence preserves their prescribed class multiset and the condition that all inertia orders are divisible by \(p\). This proves invariance of \(\mathcal T_{\mathcal C}\). Its finiteness follows from the finite tuple description. Although Galois can permute the marked points, an isomorphism of the marked covers is over the fixed geometric base curve. The cover classes are therefore still parametrized by simultaneous inner conjugacy, without quotienting by those permutations. ◻

Proof. Choose a representative for each of the finitely many marked-cover classes. Their projective equations, maps, and specified deck actions are finite data, and hence are defined over one finite extension of \(K_s\) by 6. Take a finite Galois extension \(N/K_s\) containing these fields of definition, and enlarge it if necessary to contain \(K_sK_b'\). For every \(\tau\in\operatorname{Gal}(N/K_s)\) and every chosen representative, choose a marked isomorphism between its \(\tau\)-conjugate and the representative of the resulting class. Such an isomorphism exists over \(\overline{K_s}\) by 32; there are only finitely many choices to make. Their finite coefficient data lie in a further finite extension, which we enclose in a finite Galois extension \(M/K_s\). This two-stage choice ensures that all required isomorphisms are defined over \(M\). Conjugates of the original models under \(\operatorname{Gal}(M/K_s)\) depend only on restriction to \(N\) and are scalar extensions of the already considered conjugates. The action on classes therefore factors through \(\Gamma\).

Let \(S_p\leq P_\Gamma\) stabilize a class in a small orbit and let \(Y_M\) be its chosen model. Since \(S_p\) fixes this isomorphism class, for each \(\tau\in S_p\) the chosen isomorphisms supply a \(\tau\)-semilinear isomorphism of \(Y_M\) over the corresponding action on \(\mathcal X_M\), commuting with the specified \(J\)-action. Here semilinear means that the map changes scalars by \(\tau\) rather than fixing \(M\). These maps have not been chosen to respect multiplication in \(S_p\); that compatibility is the remaining descent problem.

Form the group \(\mathcal E\) of all such marked semilinear isomorphisms, for all \(\tau\in S_p\), with composition as its group operation. Projection to the scalar action is a surjection \(\mathcal E\to S_p\). An automorphism of \(Y_M\) over \(\mathcal X_M\) is a deck transformation: after extension to \(\overline{K_s}\) the Galois cover has exactly its \(m\) specified deck transformations, all already defined over \(M\). Those commuting with the specified \(J\)-action are exactly \(Z(J)\). Every marked semilinear map also commutes with these specified deck transformations. Thus this kernel is central, and \(\mathcal E\) is finite and fits into an exact sequence \[ 1\longrightarrow Z(J)\longrightarrow\mathcal E \longrightarrow S_p\longrightarrow1. \tag{52}\] The Sylow \(p\)-subgroup of the abelian group \(Z(J)\) is characteristic there, hence normal in \(J\). Since \(O_p(J)=1\), it is trivial; thus \(p\nmid|Z(J)|\). A Sylow \(p\)-subgroup of \(\mathcal E\) has order \(|S_p|\), intersects \(Z(J)\) trivially, and therefore maps isomorphically onto \(S_p\). Its inverse supplies a section of (52), assigning maps \(\phi_\tau\) with \[\phi_\tau\phi_\rho=\phi_{\tau\rho},\qquad \phi_1=\mathrm{id}.\] These are exactly the compatibility identities for a descent datum on \(Y_M\) and its cover map. Each \(\phi_\tau\) commutes with every specified element of \(J\), so the datum descends the marked action as well.

Take the pullback of an ample line bundle on \(X_s\), for example \(\mathcal O(D)\) on its generic fiber. It is ample on \(Y_M\), and the semilinear cover maps give its compatible linearization. Galois descent in 6 therefore gives the marked cover over \(L=M^{S_p}\). Its claimed geometric properties can be checked after extension to \(\overline{K_s}\), where they are those of the chosen cover. In particular its function-field extension is Galois of degree \(m\) with the specified group \(J\). Finally \[[L:K_s]_p=[\Gamma:S_p]_p =\frac{|P_\Gamma|}{|S_p|} =|P_\Gamma:S_p|<p^h.\] Equation (45) gives the residue assertion in (51). ◻

Proof. The uniqueness proved in 29 gives one valuation on \(L'\). Its restriction to \(K_b'\) is the valuation of 28, by uniqueness over the complete discretely valued field \(K_b\). Put \(T=\operatorname{res}(L')\). It contains both \(F\) and \(F'\), so it contains their compositum \(S\). Base change of the finite Galois extension \(K_b'/K_b\) gives the Galois extension \(L'/L\), and restriction embeds \(H=\operatorname{Gal}(L'/L)\) in \(\operatorname{Gal}(K_b'/K_b)\). Every element of \(H\) preserves the unique valuation and hence acts on \(T\), fixing \(F\) and preserving \(F'\). Its induced action on \(S\) is faithful: if it is trivial on \(S\), it is trivial on \(F'\); the faithful residue action of \(\operatorname{Gal}(K_b'/K_b)\) then makes its restriction to \(K_b'\) trivial; and \(L\) and \(K_b'\) generate \(L'\). Thus reduction injects \(H\) into \(\operatorname{Aut}_F(S)\). Also \(S/F\) is Galois, being a compositum with the finite Galois extension \(F'/F_0\). The residue-degree bound now gives the complete degree comparison \[ |H|\leq[S:F]\leq[T:F]\leq[L':L]=|H|. \tag{53}\] All these inequalities are equalities. In particular \(T=S\) and the injection of groups is an isomorphism.

The sections \(\xi_i\) extend over the valuation ring of \(L'\) by base change from the split good-reduction model, and reduce to \(t_i\). Their reductions are distinct. Applying a member of \(H\) to a section and then reducing gives the same section as first reducing and then applying its image in \(\operatorname{Gal}(S/F)\). This proves equality of the two permutations of the index set. An invariant subset of a split finite étale branch divisor descends along the respective finite Galois extension. The final assertion follows by taking the same invariant subset of indices under these identified permutation actions: it selects the \(\xi_i\) on the generic side and the \(t_i\) on the residue side. ◻

Proof. Work initially over the complete DVR \(R_b\), and write \(D_E=D|_E\). For \(j\) sufficiently large, 9 and the special-fiber sequence give \[H^0(\mathcal X,\mathcal O(jD))\otimes_{R_b}F_0 \simeq H^0(E,\mathcal O_E(jD_E)).\] Indeed Serre vanishing kills \(H^1(\mathcal X,\mathcal O(jD))\), so the sequence obtained by multiplication by a uniformizer gives surjectivity onto the special sections. The module on the left before reduction is finite and torsion-free, hence free over \(R_b\); torsion-freeness follows from flatness of \(\mathcal X\) and invertibility of \(\mathcal O(jD)\). Choose a basis \(b_{j,1},\ldots,b_{j,t_j}\) whose reductions are a basis of the special section space. Flat base change supplies bases over \(L\) on the generic side and over \(F\) on the special side.

As \(D_X\) is effective, regard these sections as rational functions. For \(b=\sum_\alpha c_\alpha b_{j,\alpha}\) put \[v_X(b)=\min_\alpha v(c_\alpha).\] Integral coefficients give a special rational function with poles bounded by \(jD_F\), and this function is nonzero if the displayed minimum is zero. The inclusions for \(jD\leq j'D\) are integral maps on the original model and reduce to injective inclusions of special section spaces. Consequently the assigned value is independent of the sufficiently large space containing \(b\): scale the coefficients so that their minimum is zero, and use the nonzero special image. The same argument proves multiplicativity. After scaling two nonzero sections to value zero, their reductions are nonzero, and their product is nonzero because \(E_F\) is integral. The strong triangle inequality follows directly from the coefficient rule.

The union of these section spaces is a ring whose fraction field is \(L(X)\): sufficiently high ample multiples give a projective embedding, and their section ratios generate the function field. Thus the rule extends to a valuation on \(L(X)\). Its values lie in \(\mathbb Q\), and all values in \(\mathbb Q\) occur already on \(L\). A fraction of value zero can be written with numerator and denominator both of value zero, by a common scalar rescaling. Its residue is the ratio of their nonzero special reductions, hence lies in \(F(E_F)\). Conversely the ratios of high-degree special sections generate \(F(E_F)\), and each such section has an integral lift. This identifies the residue field.

Finally the basis functions are regular along the splitting sections over \(R_b'\), since those sections avoid \(D\). Their values are integral and reduce to their evaluations at \(t_i\). The same holds after extension of coefficients from \(R_b\) to \(R\) and evaluation over the valuation ring of \(L'\). Since its residue is \(S\), this proves (56). ◻

Proof. Let \(a\in A_j\). Form the polynomial using every element of \(J\), including any repeated conjugates: \[ P_a(T)=\prod_{g\in J}(T-g(a)) =T^m+c_1T^{m-1}+\cdots+c_m. \tag{59}\] The coefficients are \(J\)-invariant rational functions, hence belong to \(L(Y)^J=L(X)\). At a fixed \(w\in\mathcal V\) every \(g(a)\) is integral, because \(g\) permutes \(\mathcal V\). Thus each \(c_k\) has nonnegative Gauss value. Moreover \[c_k\in H^0(X,\mathcal O_X(kjD_X)).\] To see the pole bound, each conjugate has poles bounded by \(jD_Y\), so the elementary symmetric function of degree \(k\) has poles bounded by \(kjD_Y\). Since the coefficient is a rational function on \(X\), its pole bound on \(X\) follows from that on its finite surjective pullback.

By 36, \(\bar c_k\) has poles bounded by \(kjD_F\). The residue \(\bar a_w\in K(w)\) satisfies the reduction of (59). Let \(t\) be a local equation for \(jD_F\) at any point of \(E_F\). Then \(t\bar a_w\) satisfies the monic equation whose coefficient of degree \(m-k\) is \(t^k\bar c_k\). These coefficients are regular there. At every point of \(C_w\) above this neighborhood, normality therefore makes \(t\bar a_w\) regular. This proves the required pole bound and hence (58); injectivity was established in (57). Residue maps themselves respect multiplication.

For the last assertion, work at a selected marked point after splitting. Every \(g(a)\) vanishes at every point of its reduced inverse fiber, because this fiber is \(J\)-invariant. The non-leading coefficients \(c_k\) consequently vanish at the marked point downstairs. There are no poles there, since \(D_X\) avoids all the marked points. By (56), every \(\bar c_k\) vanishes at \(t_i\). The reduced monic equation at any point above \(t_i\) is then \(\bar a_w^m=0\) in its residue field, so \(\bar a_w\) vanishes there. ◻

Proof. Take \(x\in K(w)^{\Delta_w}\). Simultaneous approximation from 31 gives an element \(a\in L(Y)\) integral at every member of \(\mathcal V\), with residue \(x\) at \(w\) and zero at the other members. The coefficients of the conjugate polynomial \(P_a(T)\) in (59) have nonnegative Gauss value. At \(w\), its reduction is \[ \overline{P_a}(T) =T^{m-|\Delta_w|}(T-x)^{|\Delta_w|} \in F(E_F)[T]. \tag{64}\] Indeed, elements of \(\Delta_w\) act on the residue \(x\) trivially, and each other conjugate draws its residue from a different member of the orbit, where the prescribed residue is zero.

Write \(|\Delta_w|=p^v q\) with \(p\nmid q\). In characteristic \(p\), \[(T-x)^{|\Delta_w|}=(T^{p^v}-x^{p^v})^q.\] The coefficient of \(T^{m-p^v}\) in (64) is therefore \(-q x^{p^v}\). The scalar \(q\) is nonzero in \(F(E_F)\), so \(x^{p^v}\in F(E_F)\). This also covers \(x=0\), for which the coefficient is zero. Thus \(M_w/F(E_F)\) is purely inseparable.

The effective automorphism group on \(K(w)\) is \(\Delta_w/I_w\). Finite fixed-field theory gives \[[K(w):M_w]=|\Delta_w/I_w|.\] Combining this with (62) and the tower formula proves (63). The assertions about the normalizations and their separable function fields follow. Conjugation by \(J\) supplies the asserted isomorphisms between the towers, so \(d\) is independent of \(w\). ◻

Proof. Form the graded \(R\)-algebra \[A=\bigoplus_{j\geq0}A_j.\] Multiplication is defined because each \(w\) is a valuation: \(A_iA_j\subseteq A_{i+j}\). Each piece is finite free by (57). Moreover \(A_0=R\), since \(Y\) is projective and geometrically integral and hence \(H^0(Y,\mathcal O_Y)=L\). Put \[C=\coprod_{w\in\mathcal V}C_w, \qquad B_j=\bigoplus_{w\in\mathcal V} H^0(C_w,\mathcal O_{C_w}(jD_w)), \qquad B=\bigoplus_{j\geq0}B_j.\] The two equalities in the proposition make the source and target dimensions in (58) equal for every sufficiently large \(j\). Thus there is an integer \(j_0\) such that \[ A_j/\mathfrak m_R A_j\simeq B_j \qquad(j\geq j_0), \tag{71}\] compatibly with products. Equality in each high degree, rather than only equality of leading terms, is where the Euler-characteristic equality is used.

We first choose a Veronese regrading for which the reduced algebra is generated in degree one. The divisor on \(C\) given by the \(D_w\) is ample, so \(B\) is finitely generated by 9. Choose positive-degree homogeneous generators \(b_1,\ldots,b_t\) over \(B_0\), of degrees \(d_1,\ldots,d_t\). Let \(W\) be a common multiple of these positive degrees. In a monomial in the \(b_i\), remove pure powers \(b_i^{W/d_i}\), each of weight \(W\), until the remaining exponent of \(b_i\) is less than \(W/d_i\). The weight of the remainder is less than \(tW\). Choose \(N=qW\geq j_0\) with \(q\geq t\).

If the original monomial has degree \(kN\), its remainder has weight \(cW\) for an integer \(0\leq c<t\leq q\), since the removed blocks and the original total weight are multiples of \(W\). There are \(kq-c\) removed blocks. Combine the remainder with \(q-c\) of these blocks to make one factor of weight \(N\), and group the remaining \((k-1)q\) blocks in groups of \(q\). This factors the monomial into \(k\) factors of weight \(N\). Any coefficient from \(B_0\) is absorbed into the first factor. It follows that \[ F\ \oplus\!\bigoplus_{k\geq1} B_{kN} \quad\text{is generated over $F$ by }B_N. \tag{72}\] This argument permits \(B_0=H^0(C,\mathcal O_C)\) to be larger than \(F\), as can happen for a disconnected curve or a curve with extra constants. All the positive-degree factors used in the argument belong to the full spaces in (71).

Let \(A^{(N)}=\bigoplus_{k\geq0}A_{kN}\), with \(A_{kN}\) assigned degree \(k\). Choose an \(R\)-basis of \(A_N\), and map the variables of a degree-one polynomial ring \[P=R[x_1,\ldots,x_u]\longrightarrow A^{(N)}\] to that basis. In each degree the target is a finite free \(R\)-module. By (71) and (72), the map is surjective after reduction modulo \(\mathfrak m_R\). Its degreewise cokernel is finite, so Nakayama’s lemma makes the map surjective in every degree. This proves finite generation over \(R\).

For finite presentation, let \(K_k\) be the kernel of \(P_k\to A_{kN}\). This surjection splits as an \(R\)-module map because \(A_{kN}\) is free. Hence \(K_k\) is a finite direct summand of the finite free module \(P_k\), and reduction gives the exact kernel of the corresponding polynomial presentation over \(F\). The special relation ideal is a homogeneous ideal in the Noetherian ring \(F[x_1,\ldots,x_u]\), so it has finitely many homogeneous generators. Lift them, in their respective degrees, to elements \(g_1,\ldots,g_v\) of \(\ker(P\to A^{(N)})\), and let \(I\) be the ideal they generate. For every \(k\), the quotient \(K_k/I_k\) is a finite \(R\)-module. Its reduction is zero, since the reductions of the \(g_i\) generate the special relation ideal. Degreewise Nakayama gives \(K_k=I_k\) for every \(k\). Thus \[A^{(N)}\simeq R[x_1,\ldots,x_u]/(g_1,\ldots,g_v)\] is a finite homogeneous presentation. In particular, this argument does not require the nondiscrete valuation ring \(R\) to be Noetherian.

Define \(\mathcal Y=\operatorname{Proj}A^{(N)}\). The degree-one presentation makes it a projective \(R\)-scheme of finite presentation. It is flat over \(R\): the graded algebra is a direct sum of free \(R\)-modules, homogeneous localization is a filtered colimit of such flat modules, and the degree-zero part of the resulting graded module is a direct summand. Thus every standard affine chart of \(\mathcal Y\) has a flat coordinate ring over \(R\).

Since \(A_j\) spans \(V_j\), the generic fiber is the Proj of the ample section ring of \(Y\) after Veronese regrading, and hence is \(Y\). By (71), the special fiber is the Proj of the algebra in (72). Replacing its degree-zero term \(F\) by \(B_0\) has no effect on Proj. Indeed, on a localization at a positive-degree element \(f\), a missing constant \(b\in B_0\) is present as \((bf)/f\), since \(bf\) has positive degree. The degree-zero coordinate rings of these localizations therefore agree. The full ample section ring recovers \(C\), proving the special-fiber assertion. ◻

Proof. Choose a finite set of homogeneous projective equations for \(\mathcal U\). Their coefficients generate a finite-type \(\mathbb Z\)-subdomain \(R_0\subset R\). Normalize \(R_0\) in its fraction field, obtaining \(R_1\). By 7, this normalization is finite, so \(R_1\) is a normal Noetherian domain. It remains a subring of \(R\): its elements lie in \(\operatorname{Frac}(R_0)\) and are integral over \(R\), which is integrally closed.

The equations define a projective scheme \(\mathcal U_1\) over \(R_1\). Its generic fiber is geometrically integral, because this can be checked after extension of its fraction field to \(L\), where the fiber is \(\mathcal U_L\). Let \(\mathcal T\subseteq\mathcal U_1\) be the scheme-theoretic closure of that generic fiber. Then \(\mathcal T\) is integral and projective over \(R_1\): locally its coordinate ring is the image in the coordinate ring of the integral generic fiber, and the generic fiber is dense.

The pullback \(\mathcal T_R\) is a closed subscheme of \(\mathcal U\) and agrees with it over \(L\). Its defining ideal in \(\mathcal O_{\mathcal U}\) therefore vanishes upon localization at \(R\setminus\{0\}\). On each affine chart, every element of this ideal is annihilated by a nonzero element of \(R\). Flatness of \(\mathcal U\) makes its affine coordinate rings torsion-free over \(R\), so the ideal is already zero. Thus \[ \mathcal T_R=\mathcal U. \tag{73}\]

Write \(K_1=\operatorname{Frac}(R_1)\) and \(\pi:\mathcal T\to\operatorname{Spec}R_1\). Proper coherent finiteness over this Noetherian base gives a finite \(R_1\)-algebra \(H^0(\mathcal T,\mathcal O_{\mathcal T})\). Restriction to the generic fiber is injective because \(\mathcal T\) is integral and that fiber is dense. Since the generic fiber is projective and geometrically integral, its global functions are \(K_1\). Hence \[R_1\subseteq H^0(\mathcal T,\mathcal O_{\mathcal T})\subseteq K_1.\] The middle ring is finite over \(R_1\), so all its elements are integral over \(R_1\). Normality yields \(H^0(\mathcal T,\mathcal O_{\mathcal T})=R_1\). Equivalently, \[\pi_*\mathcal O_{\mathcal T}=\mathcal O_{\operatorname{Spec}R_1};\] one can check this equality on principal affine opens by flat base change for coherent cohomology, as in 9.

Stein connectedness over the Noetherian base, in the form stated in 13, now gives geometrically connected fibers of \(\pi\). The special fiber of \(\mathcal U\) is, by (73), an extension of one of these fibers to the residue field of \(R\). It is therefore geometrically connected. All uses of proper coherent cohomology and Stein factorization in this proof occur over \(R_1\). ◻

Completion of the proof of 35. Apply 40 to the flat projective model of 39. Its generic fiber \(Y\) is geometrically integral, so its special fiber \(\coprod_{w\in\mathcal V}C_w\) is connected. Every \(C_w\) is nonempty; hence \(\mathcal V\) has one member. The stabilizer of that member is all of \(J\), so \(I_w\) is a normal subgroup of \(J\). By (63) and (70), it is a nonidentity \(p\)-group. This contradicts \(O_p(J)=1\). ◻

Proof of 3. Fix \(h\geq1\) and choose \(s_0\) satisfying (54) with \(s=s_0\). The same inequalities hold for every \(s\geq s_0\); fix any such \(s\). For any prescribed multiset of conjugacy classes of the \(p\)-singular puncture entries, [lem:marked-cover-classification,prop:small-orbit-descent] identify the finite tuple-class set with the corresponding marked-cover classes and supply a Sylow \(p\)-group action on that set. If an orbit had size less than \(p^h\), the same proposition would descend one of its covers to a finite extension \(L/K_s\) with \([L:K_s]_p<p^h\). This is excluded by 35.

Every orbit of a finite \(p\)-group has \(p\)-power size. Thus all the orbits have sizes divisible by \(p^h\), and so does their total. The threshold depends only on \(p,h,m\), not on the multiset. An empty tuple set has count zero and satisfies the same assertion; in particular this covers groups with no \(p\)-singular elements. ◻

Proof of 1. The tuple divisibility just proved supplies the hypothesis of 16. Therefore \(l=(1+T)z\) for every pair \((X,E)\) of a finite group and a central block sum considered there. Taking \((X,E)=(G,B)\) gives \[l(B)= \sum_{\substack{Q\leq G\text{ a }p\text{-subgroup}\\ \text{up to }G\text{-conjugacy}}} z\bigl(N_G(Q)/Q,\bar B_Q\bigr).\] The normalizer block-assignment calculation following 4 identifies this sum, including \(Q=1\), with \(|W_p(B)|\) under the central-character convention (1). Hence \(l(B)=|W_p(B)|\) for every prime, finite group, and block, as required. ◻

Proof of 2. The local inequality and its abelian-defect equality are Alperin’s Consequences 1–2 (Alperin 1987), as restated in (Hung et al. 2024, sec. 3, p. 334); their numerical weight-conjecture hypotheses follow from 1. Put \(N=N_G(P)\). Since \(P\) is a normal Sylow \(p\)-subgroup of \(N\), it is a defect group of every block of \(N\). Brauer’s first main theorem therefore matches all \(p\)-blocks of \(N\) with the maximal-defect \(p\)-blocks of \(G\); this indexing is also used in (Feng, Martínez, et al. 2026, proof of Lemma 2.14). Writing these two block sets as \(\operatorname{Bl}_p(N)\) and \(\operatorname{Bl}_{p,\max}(G)\), respectively, the local inequality gives \[|\mathop{\mathrm{IBr}}_p(G)| \geq\sum_{B'\in\operatorname{Bl}_{p,\max}(G)}l(B') \geq\sum_{b'\in\operatorname{Bl}_p(N)}l(b') =|\mathop{\mathrm{IBr}}_p(N)|.\] Finally, 1 supplies the only conjectural input in Hung–Sambale–Tiep’s Propositions 3.1–3.2 (Hung et al. 2024), which give the two principal-block bounds under their respective hypotheses. ◻

Proof. Hung, Sambale, and Tiep proved this numerical classification assuming the numerical weight conjecture for \(B\) (Hung et al. 2024, Theorem 3.3). 1 supplies that hypothesis. Their examples establishing that every listed value occurs were already unconditional. ◻

Proof. Boltje–Bouc–Yılmaz first prove that, for every group–block pair, the alternating sum above is an integer multiple of \([S_{1,1,F}]\): every other simple-functor multiplicity cancels (Boltje et al. 2026, Theorem 3.5(1)). The universal numerical blockwise Alperin weight conjecture determines the remaining coefficient through its chain formulation. Their Theorem 3.5(2) therefore gives the stated identity from 1. ◻