Proof. On an elliptic curve, a degree-zero divisor is principal precisely when its sum in the elliptic-curve group is \(O\). The divisor in (6) has sum \[S_t-(-S_t)-nR_t+n(-R_t)=2S_t-2nR_t=O.\] It therefore determines a rational function, uniquely after imposing \(f_t(O)=1\). Applying this argument over \(k(U)\) and spreading out gives a rational function on the family. Its divisor changes sign under \(Z\mapsto-Z\), and its normalization at \(O\) consequently gives \[ f_t(-Z)=f_t(Z)^{-1}. \tag{8}\] This also shows that the function has no vertical divisor: every fiber component of the smooth elliptic family is preserved by inversion, so its valuation is its own negative. Thus the prescribed disjoint sections are the entire divisor on \(U\).

The valuation at \(S_t\) is \(1\), so the Kummer extension has degree \(m\) even geometrically. It is totally ramified at \(S_t\) and \(-S_t\) and unramified elsewhere, since the remaining nonzero valuations are \(\pm n\), divisible by \(m\). Riemann–Hurwitz gives \(2g(C_{m,t})-2=2(m-1)\). The same local description proves that the normalization is smooth over \(U\). Indeed, a zero or pole of order divisible by \(m\) can be removed from the Kummer equation; at a section of order \(1\) or \(-1\), after taking a root of a unit étale locally, the equation is the \(m\)th-root cover of a parameter transverse to that section. Normalization is finite here, so the resulting smooth family is proper. Equation (8) gives the involution, and the remaining assertions follow from the Kummer equations. ◻

Proof. At \(R=O\), choose parameters \(r\) on the parameter curve and \(z\) on the constant elliptic curve, both at \(O\), with the same tangent coordinate. The sections \(R,-R,[n]R,-[n]R\) have first-order equations \(z=r,-r,nr,-nr\). Their strict transforms meet the exceptional line at four distinct points with coordinates \(1,-1,n,-n\); the node joining this line to the main component has coordinate \(\infty\). At the second boundary parameter, only the sections \([n]R\) and \(-[n]R\) collide. Their tangent slopes are distinct, so one blowup again separates them from each other and from the node.

The involution \(Z\mapsto-Z\) preserves both components of each blown-up fiber. Equation (8) therefore forces the valuation of \(f_R\) on each component to be zero. None of its horizontal divisor sections passes through the node. Since the surface is regular, \(f_R\) is consequently a unit at that node. Taking its \(m\)th root is étale there. The node downstairs has local smoothing equation \(xy=b\), with \(b\) a parameter on the base. Over an algebraic closure, its \(m\) lifts have the same local equation. In particular, the smoothing parameter is \(b\), rather than a higher power of \(b\).

At the separated sections the normalization has the smooth local description used in Lemma 2. This proves regularity of the total space and semistability of the family. On the exceptional line, only the two sections of multiplicities \(1\) and \(-1\) cause ramification. The cyclic cover is connected, is totally ramified at these two points, and has genus zero by Riemann–Hurwitz. ◻

Proof. Let \(v=z/r\) be the coordinate on the exceptional line. Its normalized function is \[ f_{\mathrm{exc}}(v)= \frac{v-n}{v+n}\left(\frac{v+1}{v-1}\right)^n. \tag{10}\] Indeed, the divisor gives this expression up to a scalar. The section \(Z=O\) meets the exceptional line at \(v=0\), where the function is a unit with value \(1\). Since \(n\) is odd, the displayed expression also has value \(1\) at \(0\), so the scalar is \(1\).

The restriction of \(f_R\) to the main elliptic component has no divisor, and is therefore constant. At its node with the exceptional component this constant is \(f_{\mathrm{exc}}(\infty)=1\). The cover of the main component is consequently the disjoint union of the \(m\) copies of \(E\) given by \(Y\in\mu_m\). On the exceptional line the substitution \[Y'=Y\left(\frac{v-1}{v+1}\right)^{n/m}\] gives \(Y'^m=(v-n)/(v+n)\). Its \(m\) points over \(v=\infty\) are defined over \(k\) and attach to the \(m\) elliptic components, one each.

The dual graph is a tree, as illustrated in Figure 1. Thus the fiber is of compact type: its identity-component Picard variety is the product of the Jacobians of its components, namely \(E^m\). The marked point \((O,1)\) extends to a section through the smooth locus. The relative Picard theorem for a family of nodal curves of compact type then makes the Jacobian family an abelian scheme across this fiber. More precisely, the relative identity-component Picard scheme is semi-abelian, and its torus vanishes when the dual graph is a tree; see (Bosch et al. 1990, sec. 9.4, Theorem 1, and Section 9.2, Proposition 10). Shrinking the parameter neighborhood removes the other singular fibers and gives the asserted extension. ◻

Proof. Write \(Q=a/2=[n]R_*\) and use additive analytic coordinates \(t=R-R_*\) and \(z=Z-Q\). The colliding sections have equations \(z=nt\) and \(z=-nt\), while the other divisor sections avoid this neighborhood. Thus \[f_R=g(z,t)\frac{z-nt}{z+nt}\] with \(g\) a holomorphic unit. Choose a local \(m\)th root of \(g\) and put \(V=Y/g^{1/m}\). The Kummer equation becomes \[ (z+nt)(V^m-1)=-2nt. \tag{11}\] For each \(V=\zeta\), \(\zeta\in\mu_m\), the function \(U=V^m-1\) is a local coordinate. Taking \(X=z+nt\) gives the asserted node equation. It is the local chart of the normalization of the blowup from Lemma 3; the exceptional component has \(X=0\), and the main component has \(U=0\). In particular, the nonzero constant \(2n\) changes no vanishing order.

On the main component, the divisor of the limiting function is \(n(-R_*)-n(R_*)\). The associated unramified Kummer cover is determined by the \(m\)-torsion divisor class \[\frac{n}{m}\bigl((-R_*)-(R_*)\bigr) \quad\longleftrightarrow\quad -\frac{2n}{m}R_*=-\frac{a}{m}\quad\text{in }E(\mathbb C).\] This class has exact order \(m\), so that cover is connected. It is unramified of degree \(m\) over an elliptic curve and hence is itself an elliptic curve \(E'\). The exceptional cover is rational by Lemma 3, and its \(m\) attachments are the nodes just computed. ◻

Proof. For small \(s\), enclose the colliding sections in a small disk about \(O\). The model at \(R=O\) restricts outside that disk to \(m\) disjoint copies of its complement, by Proposition 4. If \(D\) is such a disk, the inclusion \(E\setminus\operatorname{int}(D)\hookrightarrow E\setminus I_s\) is a homotopy equivalence. The unramified cover of \(E\setminus I_s\) is therefore trivial. As \(s\) varies in \((0,1/2)\), the arcs \(I_s\) move by isotopy, and the topological isomorphism class of this unramified cover remains constant. The monodromy around either branch point is a generator of \(\Gamma_m\), since its valuation in \(f_R\) is \(1\) or \(-1\). This gives the stated gluing. ◻

Proof. The genus is \(m\), so \(H_1\) has rank \(2m\). Equation (13) shows that the \(2m\) displayed classes are linearly independent and span a sublattice with unimodular intersection pairing. If its index in \(H_1\) were \(d\), its intersection determinant would be \(d^2\) times the determinant on \(H_1\). Both determinants are \(1\), so \(d=1\). The deck action now gives the first isomorphism in (14).

The dual of the regular integral module is again regular: explicitly, the coefficient-of-identity pairing \[\mathbb Z[\Gamma_m]\times\mathbb Z[\Gamma_m]\longrightarrow\mathbb Z, \qquad (x,y)\longmapsto [1](xy)\] is perfect. With the pullback action on the dual, it identifies \(\operatorname{Hom}_{\mathbb Z}(\mathbb Z[\Gamma_m],\mathbb Z)\) with \(\mathbb Z[\Gamma_m]\). This proves the cohomological assertion without dividing by \(m\). Topological local triviality transports the result through the connected smooth parameter open, and comparison gives the étale assertion. ◻

Proof. Arrange the boundary of the disk to cross the slit transversely twice. A lifted boundary circle has its upper arc on one sheet and its lower arc on an adjacent sheet. Index the upper sheet by \(i\); the lower one is \(i+1\) or \(i-1\), by Lemma 7. The circle is disjoint from every \(a_j\). It crosses the two corresponding vertical loops once each, with opposite intersection signs, and no other \(b_j\). These intersections determine its class in the integral symplectic basis of Proposition 8, giving (15). See Figure 2. ◻

Proof. The Picard–Lefschetz formula for a positively oriented loop about a semistable fiber is the product of the Dehn twists at its nodes; see (Deligne and Katz 1973, Exposé XV). In the present family, every node has smoothing order one by (11), so every twist has exponent one. The vanishing cycles are disjoint. Their intersection pairings with one another therefore vanish, and the product acts on a class \(\gamma\) by \[ \gamma\longmapsto \gamma+\sigma\sum_i \langle\gamma,\delta_i\rangle[\delta_i], \tag{18}\] with one sign \(\sigma\) depending only on the intersection convention. Substituting (15) gives zero on the \(a_j\) and, up to this one sign, \[(a_j-a_{j+1})-(a_{j-1}-a_j) =2a_j-a_{j-1}-a_{j+1}\] on \(b_j\). This proves (16). Replacing the horizontal path by \(R_s=(s/n)b\) gives (17), initially in the basis obtained from the corresponding vertical slit.

It remains to compare the two bases; this matters because separate triangular matrices in unrelated bases would not give the stated conclusion. Choose the starting parameters of both paths sufficiently close to \(O\). The horizontal loop at \(y=1/2\) and the vertical loop at \(x=1/2\) avoid one fixed small disk about \(O\) and avoid both short slits. Outside this disk the covering extends across \(R=O\) as \(m\) disjoint sheets. Trivialize it over a small, simply connected parameter disk containing \(O\), and use this trivialization to label the lifts of the two fixed loops. Parallel transport between the starting parameters now identifies both sets of \(a_i,b_i\) with the same basis. There is no monodromy ambiguity from circling \(O\) on these classes: the sheet trivialization extends over \(O\) itself. A different initial labeling makes only a common cyclic shift, which leaves (16)–(17) unchanged.

Both loops therefore define the asserted operators in a single geometric monodromy group. Pullback along the open parameter curve in \(E\to U\) maps its fundamental group into that of \(U\), so these operators occur in the latter image too. Removing additional finitely many points from either smooth parameter open induces a surjection on fundamental groups and does not remove any of the operators. ◻

Proof. The rank-two regular-module statement of Proposition 8, followed by comparison between singular and \(\ell\)-adic cohomology, gives \(\dim V_{\theta,\ell}=2\).

Let \(\langle\ ,\ \rangle\) be the cup-product pairing on \(H^1(C_n)\), with values in \(\overline\mathbb Q_\ell(-1)\). It pairs the \(\theta\)-space perfectly with the \(\theta^{-1}\)-space. The involution \(w\) is defined over \(F\), preserves cup product, and exchanges these two spaces. Consequently \[[x,y]_\theta=\langle x,w^*y\rangle, \qquad x,y\in V_{\theta,\ell},\] is a perfect \(G_F\)-equivariant pairing into \(\overline\mathbb Q_\ell(-1)\). It is alternating, because \[\langle y,w^*x\rangle =\langle w^*y,x\rangle =-\langle x,w^*y\rangle.\] A perfect alternating pairing on a two-dimensional space identifies its determinant with the target line. This proves \(\det V_{\theta,\ell}=\epsilon_\ell^{-1}\).

To compute the Hodge numbers, make any embedding of \(F\) into \(\mathbb C\). Let \(a_\theta\) be the multiplicity of \(\theta\) in the holomorphic differentials of the resulting complex curve. The holomorphic involution \(w\) gives \(a_\theta=a_{\theta^{-1}}\). Hodge decomposition and complex conjugation identify the dimension of the full \(\theta\)-space with \(a_\theta+a_{\theta^{-1}}\). Since this dimension is two, \(a_\theta=1\). This argument works for every embedding and every character. The comparison theorems therefore give the labelled weights \(0,1\). Potential semistability follows from semistable reduction and the semistable comparison theorem for smooth proper varieties; see (Tsuji 1999). These properties pass to the character summands.

Outside a finite set, \(C_n\) has good reduction and the rotations extend to its smooth proper model. Smooth proper base change gives unramifiedness. The projector onto the \(\theta\)-space is \[e_\theta=\frac1n\sum_{u\in\mu_n(F)} j_\ell(\theta(u))^{-1}u^*.\] This formula is used only in characteristic zero. The Lefschetz trace formula for a rotation composed with a positive power of Frobenius shows that its trace on \(H^1(C_n)\) is independent of \(\ell\). Applying the displayed projector gives traces in \(M\), compatible under the embeddings \(j_\ell\). Together with the already determined determinant, this gives the asserted common characteristic polynomial.

Finally, \(H^1(C_n)\) is the first cohomology of its Jacobian. Faltings’ semisimplicity theorem (Faltings 1983, sec. 5, Satz 3) makes this a semisimple \(G_F\)-representation. Extension of scalars and passage to a \(G_F\)-stable direct summand preserve semisimplicity, proving the remaining assertion. ◻

Proof. The quotient map \(C_n\to C_m\) sends the covering coordinate \(Y\) to \(Y^{n/m}\). Pullback is injective on rational cohomology, since its composition with trace is multiplication by the degree. Its image is the subspace fixed by the kernel of \(\mu_n\to\mu_m\). The rank-two calculation therefore identifies its order-\(m\) character summand with \(V_{\theta,\ell}\). The same argument for the quotient \(C_n\to B_t\) identifies the trivial character space.

On \(H^1(C_3)\), the trivial rotation space is \(H^1(B_t)\), and \(w\) acts there by \([-1]^*=-1\). It exchanges the two nontrivial rotation spaces, each of dimension two. Their direct sum consequently has a two-dimensional \(w\)-invariant subspace. Rational cohomology of a finite quotient is the invariant subspace, so \[H^1(D_t)\simeq H^1(C_3)^{w=1}\] has dimension two. Thus \(D_t\) has genus one and its Jacobian is an elliptic curve, whether or not \(D_t\) has an \(F\)-rational point. Projection of a vector \(x+w^*x\) onto the \(\eta\)-space is \(x\); therefore this projection is a \(G_F\)-equivariant isomorphism from the invariant subspace to \(V_{\eta,\ell}\). Finally, \(H^1(D_t)\simeq H^1(\mathop{\mathrm{Jac}}(D_t))\). The latter identification is independent of the geometric base point used in an Abel–Jacobi map, because translations act trivially on cohomology. ◻

Proof. By Proposition 8 and comparison, \[H^1_{\mathrm{\acute et}}(C_{n,\overline F},\mathbb Z_\ell) \simeq \mathbb Z_\ell[\mu_n]^2\] as group-ring modules. The Galois action commutes with this group ring because the rotations are defined over \(F\). Tensoring by evaluation at \(j_\ell\theta\) therefore gives a free rank-two \(\mathcal O\)-module with continuous Galois action. In characteristic zero the group algebra is semisimple; its quotient by character evaluation is precisely the corresponding character summand. This identifies its rational representation with \(V_{\theta,\ell}\).

After reduction, the displayed tensor product becomes \[H^1_{\mathrm{\acute et}}(C_{n,\overline F},\mathbb Z_\ell) \otimes_{\mathbb Z_\ell[\mu_n],\,\overline\theta} k.\] It depends only on the residual evaluation map \(\overline\theta\). This proves the asserted isomorphism without assuming that \(\ell\) is prime to \(n\). ◻

Proof. A root of unity of \(\ell\)-power order reduces to \(1\) in residue characteristic \(\ell\). Thus \(\chi\) and \(\eta\) have the same reduction at \(p\), \(\chi\) and \(\psi\) have the same reduction at \(3\), and \(\psi\) and \(1\) have the same reduction at \(p\). Lemma 14 gives the three congruences. The rational isomorphisms with the smaller covers and elliptic curves in Proposition 13 preserve semisimplified reduction, even if their natural integral pullbacks have index divisible by \(\ell\). ◻

Proof. The group scheme \(\mathcal A[N]\) is finite étale over \(X\), since the base has characteristic zero. Equivalently, the scheme of full level-\(N\) structures is a finite étale torsor over \(X\) under the constant group \(\mathop{\mathrm{GL}}_{2g}(\mathbb Z/N\mathbb Z)\), where \(g\) is the relative dimension. Choose a finite Galois extension \(L'/L\) over which the fiber of this torsor at \(x_0\) splits. The inverse function theorem for an étale morphism gives a local section through each point of the fiber over a neighborhood of \(x_0\) in \(X(L')\). Shrinking that neighborhood, these sections enumerate all points in every nearby fiber. Intersecting its finitely many Galois conjugates, and shrinking once more if necessary, makes the neighborhood Galois stable and preserves the Galois permutations of the sections. The right action of the finite level group likewise preserves the same permutations after shrinking. Restricting to \(X(L)\) therefore identifies the fibers as Galois sets with their level-group action, and hence identifies the underlying torsion modules. ◻

Proof. Let \(r\) be the toric rank of \(A\). The semistable filtration of Grothendieck identifies the graded pieces of \(V_d(A)\) with the rational Tate module of a torus of rank \(r\), the rational Tate module of an abelian variety of dimension \(g-r\) with good reduction, and a lattice of rank \(r\) (Grothendieck et al. 1972, Exposé IX). The character lattice of the torus and the lattice quotient have finite unramified Galois action. Consequently the arithmetic Frobenius eigenvalues on the toric and lattice graded pieces are respectively \(Q\) times roots of unity and roots of unity. The middle piece has the \(2(g-r)\) Frobenius eigenvalues of an abelian variety over \(\mathbb F_Q\), all of complex absolute value \(Q^{1/2}\).

Each graded piece has an integral characteristic polynomial independent of \(d\): for the toric and lattice pieces this follows from the finite-order integral actions just described, together with the toric Tate twist; for the abelian piece it is the usual finite-field Frobenius polynomial. Inertia acts trivially on the graded pieces, so the product of these three polynomials is also independent of the lift of Frobenius. Every root has complex absolute value at most \(Q\). Thus the coefficient of \(T^{2g-j}\) has absolute value at most \(\binom{2g}{j}Q^j\). Integral polynomials of fixed degree with these bounds form a finite set.

Exactly \(r\) roots have absolute value \(1\), exactly \(r\) have absolute value \(Q\), and the others have absolute value \(Q^{1/2}\). Since \(Q>1\), these three absolute values distinguish \(r\). If \(r=0\), semistability is good reduction, and the normalized \(\ell\)-adic valuations of the roots of \(P_A\) are its Newton slopes. ◻

Proof. Choose an odd prime \(d\) different from the residue characteristic. By semistable reduction, we may choose a finite extension \(L/L_0\) over which \(\mathcal A_{x_0}\) is semistable and all its \(d\)-torsion points are rational. Lemma 20, applied after base change to \(L\), shows that, in a neighborhood of \(x_0\), all nearby fibers also have rational \(d\)-torsion over \(L\).

These fibers are semistable over \(L\). Indeed, the matrix of any inertia element on their \(d\)-adic Tate module has the form \(1+dB\) with \(B\) integral. Each eigenvalue \(\lambda\) therefore satisfies \(v_d(\lambda-1)\geq 1\), where \(v_d(d)=1\). The local monodromy theorem makes \(\lambda\) a root of unity. For odd \(d\) the only root of unity satisfying this inequality is \(1\): a root with nontrivial prime-to-\(d\) part has \(v_d(\lambda-1)=0\), and a primitive \(d^a\)th root has valuation \(1/(d^{a-1}(d-1))<1\). Thus every inertia element acts unipotently, and Grothendieck’s criterion gives semistability (Grothendieck et al. 1972, Exposé IX).

Let \(\mathcal P\) be the finite set of polynomials in Lemma 21 for dimension \(g\) and the residue cardinality of \(L\). Choose \(j\) so large that reduction modulo \(d^j\) is injective on \(\mathcal P\). A second application of Lemma 20 gives, throughout a smaller neighborhood, \[\mathcal A_x[d^j](\overline L)\simeq \mathcal A_{x_0}[d^j](\overline L) \quad\text{as }G_L\text{-modules}.\] Their full Frobenius polynomials are congruent modulo \(d^j\), hence equal. Lemma 21 now gives the assertions about toric rank and Newton polygon. The inverse image of this neighborhood under \(X(L_0)\to X(L)\) is a neighborhood of \(x_0\), as required.

For the final assertion, choose \(L\) so that \(E_0\) has good or split multiplicative reduction. In the good case an elliptic curve is either ordinary or supersingular; in the latter case both its Newton slopes are \(1/2\). In the multiplicative case \(E_0^g\) has toric rank \(g\). Enlarging \(L\) to rationalize its \(d\)-torsion preserves these properties, so the preceding argument applies. ◻

Proof. Monodromy rank does not change after finite extension. We may therefore assume that \(A\) has split toric uniformization \[A^{\mathrm{an}}\simeq (\mathbb G_m^g)^{\mathrm{an}}/\Lambda,\] where \(\Lambda\simeq\mathbb Z^g\) is rational over \(L\). Choosing bases of the torus and lattice writes the periods as a matrix \((q_{ij})\in(L^\times)^{g\times g}\). The valuation pairing is nondegenerate, so \[ \bigl(v_L(q_{ij})\bigr)_{1\leq i,j\leq g} \quad\text{is nonsingular.} \tag{23}\] This is the toric case of the monodromy pairing for a semistable abelian variety (Grothendieck et al. 1972, Exposé IX); its realization on de Rham cohomology is described in (Coleman and Iovita 1999).

If \(d\ne\ell\), the same pairing describes the monodromy map from the lattice quotient to the toric subspace of the Tate module. Its nondegeneracy gives rank \(g\). We now prove this rank assertion for \(d=\ell\).

Toric uniformization supplies an exact sequence \[ 0\longrightarrow\mathbb Q_\ell(1)^g\longrightarrow V_\ell(A) \longrightarrow\mathbb Q_\ell^g\longrightarrow0, \tag{24}\] whose extension classes are the Kummer classes of the \(q_{ij}\). The representation is semistable, by the semistable comparison theorem for abelian varieties (Coleman and Iovita 1999). Let \(L_{\mathrm{ur}}\) be the maximal unramified subfield of \(L\), and write \(D_{\mathrm{st}}\) for Fontaine’s semistable functor over \(L\). On the two outer terms of (24), monodromy is zero. Consequently it induces a map \[\nu:D_{\mathrm{st}}(\mathbb Q_\ell^g) \longrightarrow D_{\mathrm{st}}(\mathbb Q_\ell(1)^g).\] After the standard identifications, Frobenius on these two spaces is respectively \(\sigma\) and \(\ell^{-1}\sigma\), with \(\sigma\) the Frobenius of \(L_{\mathrm{ur}}/\mathbb Q_\ell\). The identity \(N\varphi=\ell\varphi N\) therefore makes \(\ker\nu\subset L_{\mathrm{ur}}^g\) stable under \(\sigma\).

Suppose that \(\nu\) has rank less than \(g\). Finite Galois descent for \(L_{\mathrm{ur}}/\mathbb Q_\ell\) then gives a nonzero vector \(a=(a_j)\in\mathbb Q_\ell^g\cap\ker\nu\). Pull (24) back along the line spanned by \(a\) in its trivial quotient. The resulting extension of \(\mathbb Q_\ell\) by \(\mathbb Q_\ell(1)^g\) is semistable and has zero monodromy, hence is crystalline. Under Kummer theory the Bloch–Kato crystalline subspace \(H_f^1(L,\mathbb Q_\ell(1))\) is the kernel of the valuation map (Bloch and Kato 1990, Example 3.9). Applying this to each coordinate of the pulled-back extension gives \[\sum_{j=1}^g a_jv_L(q_{ij})=0 \qquad(1\leq i\leq g),\] contrary to (23). Thus \(N\) has rank \(g\) on \(V_\ell(A)\). Dualizing gives the same rank on \(H^1_{\mathrm{et}}(A_{\overline L},\mathbb Q_\ell)\). ◻

Proof. Fix \(v_0\mid3p\). For each \(v\mid v_0\), apply Proposition 22 to the extended Jacobian family over \(F_v\), at the point \(O\) of its parameter curve. This gives a neighborhood of \(O\) in \(E(F_v)\) and a fixed finite extension of \(F_v\) on which the asserted reduction type holds. The inverse images of these finitely many neighborhoods in \(E(K_{v_0})\) have an intersection that is again a neighborhood of \(O\).

Choose \(R_0\ne O\) in this intersection, sufficiently close to \(O\) and away from the finitely many excluded parameters of the cover family. We may also avoid the ramification points of \(x:E\to\mathbf P^1\), since these form a finite set. The local inverse function theorem gives a nonempty neighborhood \(\Omega_{v_0}\subset U(K_{v_0})\) of \(x(R_0)\) and an analytic lift \[R(t)=(t,y(t))\in E(K_{v_0}),\qquad y(t)^2=h(t),\] whose image remains in the chosen intersection. Shrink the neighborhood so that \(y(t)\ne0\) and it avoids all excluded parameters.

The map \[B_t\longrightarrow E,\qquad (x,y)\longmapsto(x,y(t)y)\] is an isomorphism over \(K_{v_0}\) carrying the marked point \((t,1)\) to \(R(t)\). It carries the defining function for the cover to the function in the constant-\(E\) construction: their divisors agree, and both have value \(1\) at \(O\). Thus it identifies the covers and their Jacobians at every \(v\mid v_0\). The reduction conclusions follow from the choice of the neighborhoods. Carrying this out independently at the finitely many \(v_0\mid3p\) proves the proposition. ◻

Proof. Choose a finite extension \(L/F_v\) on which \(J_t\) has the reduction given by Proposition 24. In the two good-reduction cases, crystalline comparison makes \(H^1_{\mathrm{et}}(J_{t,\overline L},\mathbb Q_\ell)\), and hence each character summand, crystalline over \(L\).

Suppose first that the reduction is ordinary. The connected–étale sequence of the ordinary \(\ell\)-divisible group, dualized to \(H^1\), gives a filtration whose subrepresentation is unramified and whose quotient is \(\epsilon_\ell^{-1}\) times an unramified representation. This filtration is functorial in endomorphisms and therefore stable under the rotations. It restricts to such a filtration on each \(V_{\theta,\ell}\). By Proposition 12, this summand has dimension two and labeled Hodge–Tate weights \(0,1\). Its unramified part contributes only weight \(0\), whereas the quotient contributes only weight \(1\); thus both parts have dimension one. This is the required ordinary filtration. After replacing \(L\) by a finite Galois extension, its weight-zero line is uniquely determined by its inertia action, since \(\epsilon_\ell\) has infinite image on the inertia of every finite extension. Conjugation by \(G_{F_v}\) preserves that line, so the line descends to the original decomposition group.

If the reduction has all slopes \(1/2\), the crystalline module of every summand has the same slope. These normalized slopes remain \(1/2\) after further finite extension. An ordinary filtration of weight \(0\) would provide a crystalline rank-one subrepresentation of parallel Hodge–Tate weight zero. For that character weak admissibility forces Newton slope zero, contradicting the slopes of the summand. Thus no such potential ordinary filtration exists.

Finally, suppose that \(J_t\) has potentially toric rank \(n\). Lemma 23 gives rank \(n\) for monodromy on its \(2n\)-dimensional first cohomology, at every coefficient prime. Monodromy commutes with the rotations, so preserves the \(n\) character summands. Each summand has dimension two. On first cohomology of a semistable abelian variety one has \(N^2=0\), so the rank on each summand is at most one. Their ranks sum to \(n\) and there are exactly \(n\) summands; every rank is therefore one. Nonzero monodromy persists after finite extension and excludes potential crystallinity. ◻

Proof. We first work with homology; duality will give the cohomological statements. On the character space with generator eigenvalue \(\lambda\in\overline\mathbb F_\ell^\times\), the transformations of Theorem 11 have the form \[T_a=\begin{pmatrix}1&\sigma c_\lambda\\0&1\end{pmatrix}, \qquad T_b=\begin{pmatrix}1&0\\\tau c_\lambda&1\end{pmatrix}, \qquad c_\lambda=2-\lambda-\lambda^{-1},\] where \(\sigma,\tau\in\{1,-1\}\). Since \(c_\lambda=-(\lambda-1)^2/\lambda\), it is nonzero when \(\lambda\ne1\). The two matrices have different unique invariant lines, even over the algebraic closure. Hence their common representation is absolutely irreducible.

The product \(T_aT_b^k\) has determinant one and trace \[ 2+\sigma\tau k c_\lambda^2. \tag{26}\] In odd characteristic, a determinant-one matrix is regular semisimple exactly when its trace is different from \(2\) and \(-2\). If \(m=3\), then \(\lambda+\lambda^{-1}=-1\), so \(c_\lambda=3\) for both nontrivial characters. Take \(k=2\). The two possible traces in (26) are \(20\) and \(-16\). Modulo \(5\) these are \(0\) and \(4\), and neither equals \(2\) or \(-2\); modulo \(p>100\) they again avoid \(2\) and \(-2\).

For \(m=p\) and \(\ell=3\), take \(k=1\). Failure of regular semisimplicity would force \(c_\lambda^2=4\) or \(-4\), since \(c_\lambda\ne0\). Multiplying either equation by \(\lambda^2\) gives a polynomial of degree four over \(\mathbb F_3\) satisfied by \(\lambda\). Its degree over \(\mathbb F_3\) is therefore some \(d\le4\), and its order divides \(3^d-1\le80\). This contradicts the fact that its order is the prime \(p>100\). Thus the same product works for every nontrivial character in this case as well.

The transformations come from loops in the constant-elliptic-curve parameter space. As explained in Theorem 11, that family is a pullback of the family over \(U\). Its monodromy image is therefore contained in the geometric monodromy image over \(U\). Duality preserves absolute irreducibility, nonidentity unipotents, and regular semisimplicity. This proves the assertions for cohomology. The geometric argument applies to every complex elliptic curve, so it applies equally to the conjugate of \(E\). ◻

Proof. Let \(\Gamma\) be the image of \(G_K\) on \(E[p]\). The abelian extension \(F/K\) implies that the image of \(G_F\) contains \([\Gamma,\Gamma]\). Since \(\Gamma\) contains \(\mathop{\mathrm{SL}}_2(\mathbb F_p)\) and this group is perfect for \(p>3\), its commutator subgroup is \(\mathop{\mathrm{SL}}_2(\mathbb F_p)\): one inclusion follows from perfectness, and the other from the determinant map. Moreover, the determinant is trivial on \(G_F\) because \(\mu_p\subset F\). We have therefore proved \[ \rho_{E,p}(G_F)=\mathop{\mathrm{SL}}_2(\mathbb F_p). \tag{27}\] The same equality holds for the conjugate curve \(E^c\), where \(c\) is the nontrivial automorphism of \(K/\mathbb Q\), because \(F/\mathbb Q\) is normal.

Consider the joint image \[J\subset\mathop{\mathrm{SL}}_2(\mathbb F_p)\times\mathop{\mathrm{SL}}_2(\mathbb F_p)\] of \(G_F\) on \(E[p]\) and \(E^c[p]\). Both projections are surjective. In \(\mathop{\mathrm{SL}}_2(\mathbb F_p)\) there are exactly \(p^2\) matrices of trace \(2\). Indeed, writing a matrix as \(\left(\begin{smallmatrix}a&b\\c&d \end{smallmatrix}\right)\), the trace and determinant conditions give \(d=2-a\) and \(bc=-(a-1)^2\). The value \(a=1\) gives \(2p-1\) choices for \((b,c)\); the other \(p-1\) values of \(a\) each give \(p-1\) choices. Their sum is \(p^2\). Replacing a matrix by its negative gives the same count for trace \(-2\).

The order of \(\mathop{\mathrm{SL}}_2(\mathbb F_p)\) is \(p(p^2-1)\). Since each projection of \(J\) has equal-sized fibers, the proportion of elements of \(J\) whose image in at least one factor has trace \(2\) or \(-2\) is at most \[2\frac{2p^2}{p(p^2-1)} =\frac{4p}{p^2-1}<1.\] Thus some element of \(G_F\) acts regular semisimply on both \(E[p]\) and \(E^c[p]\).

Take a finite Galois extension of \(\mathbb Q\) containing \(F\) and the fields needed for these two residual actions. Conjugation by \(G_\mathbb Q\) either preserves the two curves or exchanges them, and conjugates their residual matrices. The selected element and all its conjugates therefore act regular semisimply on the representation of \(E\). Chebotarev supplies rational primes \(q\) splitting completely in \(F\) with this property at every \(v\mid q\) of \(F\). We exclude \(p\) and the finitely many primes of bad reduction. Since \(\mu_p\subset F\), these primes satisfy \(q\equiv1\pmod p\). Distinct Frobenius eigenvalues then give the ratio condition in the definition of decomposed genericity.

The curve \(B_t\) is a quadratic twist of \(E\). Twisting multiplies residual matrices by scalar signs and hence preserves absolute irreducibility and eigenvalue ratios. Excluding the finitely many ramified primes of this twist leaves the same Chebotarev argument available. Finally, \(\zeta_p\in F\), and taking the dual of \(E[p]\) gives the residual \(H^1\) realization. Proposition 13 now proves both assertions for \(\overline V_{1,p}\). ◻

Proof. The quotient \(H\) defines a connected finite \(\acute etale\) Galois cover of \(W\). Hilbert irreducibility with weak approximation gives a rational point in the prescribed local opens for which its fiber is connected; see (Serre 2008, Proposition 3.3.1 and Theorem 3.5.3). Connectedness of this Galois fiber is equivalent to surjectivity of \(s_x\). Geometric connectedness of the cover is not required. The composition of \(s_x\) with \(H\to\mathop{\mathrm{Gal}}(D/\mathbb Q)\) is the usual restriction map of absolute Galois groups. Consequently, an element of \(H\) in the displayed kernel has a preimage in \(G_D\), which proves (28). ◻

Proof. Put \[W=\mathop{\mathrm{Res}}_{K/\mathbb Q}U.\] Since \(U\) is an open subset of \(\mathbb A^1_K\), the scheme \(W\) is an open subset of \(\mathbb A^2_\mathbb Q\). Choose a finite Galois extension \(D/\mathbb Q\) containing \(F\) and \(\zeta_\ell\) for every residual characteristic in (25). Let \(c\) be the nontrivial automorphism of \(K/\mathbb Q\), and write \(U^c\) and \(\mathcal C_m^c\) for the conjugate parameter open and curve family. Then \[W_D\simeq U_D\times (U^c)_D.\] For all three pairs, pull back the finite local system \(H^1(C_m,\mathbb F_\ell)\) from the first factor and \(H^1(C_m^c,\mathbb F_\ell)\) from the second.

Their joint monodromy has finite image on \(\pi_1(W_D)\). Take the normal core, in \(\pi_1(W)\), of the common kernel of these actions. It is an open normal subgroup contained in \(\pi_1(W_D)\). The resulting finite quotient \(H\) of \(\pi_1(W)\) records every listed action and has a quotient \(\mathop{\mathrm{Gal}}(D/\mathbb Q)\).

Over \(\mathbb C\), a loop in either parameter factor acts trivially on local systems pulled back from the other factor. Thus the geometric monodromy transformations of Lemma 26 are available independently in both factors. In particular, for each fixed pair \((m,\ell)\) we can choose an element \(h_{m,\ell}\) of the image of geometric monodromy that is regular semisimple on every nontrivial character space in both factors. To obtain it, take a product of the two factor loops furnished by that lemma. Every such element maps trivially to \(\mathop{\mathrm{Gal}}(D/\mathbb Q)\). The quotient also contains the individual opposite unipotents on every character space.

The prescribed local conditions are compatible with this restriction of scalars. For a rational prime \(r\) one has \[W(\mathbb Q_r)=\prod_{v_0\mid r}U(K_{v_0}).\] The sets \(\Omega_{v_0}\) therefore specify nonempty open subsets of \(W(\mathbb Q_r)\) for the finitely many relevant \(r\); at unprescribed factors impose no additional condition. Apply Lemma 28. It gives a point \(x\in W(\mathbb Q)\), equivalently \(t\in U(K)\), satisfying all these local conditions and inducing a surjection \(G_\mathbb Q\to H\). Over \(D\), this point has coordinates \((t,t^c)\) in the displayed product. Smooth proper base change identifies the specialized local systems with the residual cohomology of \(C_m(t)\) and \(C_m^c(t^c)\).

By (28), every geometric monodromy element used above is the image of an element of \(G_D\). The opposite unipotents therefore survive on each specialized character space already over \(G_D\), and prove its absolute irreducibility. Since \(G_D\subset G_{F(\zeta_\ell)}\), this also proves absolute irreducibility on the latter group. Because \(\ell\nmid m\), taking rotation eigenspaces in integral cohomology of \(C_m\) commutes with reduction after adjoining the character values. These specialized character spaces are thus reductions of \(V_{\theta,\ell}\). The rational identifications and independence of semisimplified reduction in Section 5 show that they give precisely \(\overline V_{\theta,\ell}\), including when this system is defined using \(C_n\).

To establish decomposed genericity, fix a pair \((m,\ell)\) and choose a preimage \(g\in G_D\) of \(h_{m,\ell}\). An element \(\sigma\in G_\mathbb Q\) preserves or exchanges the two specialized covers \(C_m(t)\) and \(C_m^c(t^c)\), according to its restriction to \(K\); it also permutes the rotation characters through its action on the roots of unity. The action of \(\sigma g\sigma^{-1}\) on a character space of the first cover is therefore conjugate to the action of \(g\) on some character space of one of the two covers. The construction imposed regular semisimplicity on all these spaces, so the property holds for every such conjugate.

The kernel of the surjective specialization \(G_\mathbb Q\to H\) defines a finite Galois extension of \(\mathbb Q\). It contains \(D\). Chebotarev applied to the conjugacy class of \(h_{m,\ell}\) in this extension gives rational primes \(q\) whose Frobenius class is that class. They split completely in \(D\), since \(h_{m,\ell}\) is trivial on \(D\). Discard \(q=\ell\) and the finitely many primes where the curves or residual representations are ramified. For every \(v\mid q\) of \(F\), its Frobenius is represented by a conjugate of the selected element, and acts regular semisimply on each required character space. Here the residue degree is one because \(q\) splits in \(F\). Moreover, \(q\equiv1\pmod\ell\) because \(\zeta_\ell\in D\). Thus the ratio of the two Frobenius eigenvalues is different from \(q\) in characteristic \(\ell\), at every \(v\mid q\). This is exactly decomposed genericity. A different \(q\) may be used for each pair, as the definition permits. Assertion (2) follows from Lemma 27.

It remains to exclude CM for \(A_t\). For \((m,\ell)=(3,p)\) the preceding specialization places two opposite nonidentity unipotents in the residual image of \(V_{\eta,p}\). Proposition 13 identifies this with the residual cohomology of \(A_t\): both are irreducible, so their common semisimplification is the representation itself. If \(A_t\) had CM, all its endomorphisms would be defined over an extension of \(F\) of degree at most two. Over that extension, the characteristic-zero Galois image lies in the commutative centralizer of the CM algebra. Its reduction is abelian as well. Consequently the full residual image would have an abelian normal subgroup of index at most two.

Each of the two unipotents has the odd prime order \(p\), so both would belong to that abelian subgroup. But \[\begin{pmatrix}1&a\\0&1\end{pmatrix} \begin{pmatrix}1&0\\b&1\end{pmatrix} \ne \begin{pmatrix}1&0\\b&1\end{pmatrix} \begin{pmatrix}1&a\\0&1\end{pmatrix} \qquad(a b\ne0).\] This contradiction proves (3). Finally, the local assertions follow by choosing the prescribed opens as in Proposition 24; the Hilbert specialization was made inside all those opens. ◻

Proof. At all places outside a finite set and away from the coefficient prime, the characteristic polynomial of Frobenius on \(V_\theta\) is independent of that prime, as an algebraic polynomial with the fixed embeddings of the character values. The good-place characteristic polynomials of \(r_\iota(\pi)\) have the same compatibility. Equality at the original prime therefore identifies the polynomials at every other prime. Both representations are semisimple: for \(V_\theta\) this follows from Faltings’ theorem as in Proposition 12, and the automorphic Galois representations are taken semisimple. Chebotarev density and the Brauer–Nesbitt theorem give the asserted isomorphism. ◻

Proof. Choose a coefficient prime \(a\) different from the residue characteristic \(l\). Lemma 30 identifies \(r_{\iota_a}(\pi)\) with \(V_{\theta,a}\). Varma’s local-global compatibility theorem identifies the underlying semisimplified Weil representations of its local Weil–Deligne parameter and \(\mathop{\mathrm{rec}}^T_{F_v}(\pi_v)\) (Varma 2024, Theorem 1). We use this statement about the Weil representations first; equality of the monodromy operators is not assumed.

In the potentially good case, pass to an extension where the Jacobian has good reduction. All Frobenius eigenvalues on its \(H^1\), and hence on the character summand, have complex absolute value the square root of the residue cardinality. Taking a positive power of Frobenius shows that the two eigenvalues before this extension also have the same absolute value. A rank-two Weil–Deligne representation with nonzero monodromy has Frobenius eigenvalues differing by the norm character. They cannot have the same absolute value. Thus the automorphic parameter has zero monodromy. The Tate twist in (2) does not change this conclusion.

In the totally toric case, Corollary 25 gives nonzero monodromy on \(V_{\theta,a}\). Its semisimplified Weil representation is therefore a sum of two characters differing by the norm. For this Weil representation, the two possibilities under the local Langlands correspondence for \(\mathop{\mathrm{GL}}_2\) are the special parameter with nonzero monodromy and the zero-monodromy parameter of a one-dimensional local representation. The latter is not generic. Since every local component of a cuspidal automorphic representation of \(\mathop{\mathrm{GL}}_2\) is generic, \(\pi_v\) is a twist of Steinberg. Its central character is trivial, so the twisting character has square one.

The local calculation in the proof of (Caraiani and Newton 2025, Lemma 6.1.3(4)), which invokes the proof of (Geraghty 2019, Lemma 5.6), shows that a quadratic twist of Steinberg is \(\iota_l\)-ordinary of weight zero. This calculation uses only the stated local representation and trivial central character. In particular, it also applies when the quadratic character is ramified: the ordinary condition is imposed at a sufficiently deep level, and a finite-order twist multiplies the relevant normalized Hecke eigenvalues by units. ◻

Proof. The order-three system \(V_\eta\) is \(H^1(A)\) for the elliptic curve \(A/F\) of Proposition 13. It is non-CM by Proposition 29. Its mod-\(5\) representation is decomposed generic and absolutely irreducible over \(F(\zeta_5)\) by the \((m,l)=(3,5)\) part of that proposition. Since \(\zeta_5\notin F\), Theorem 18 supplies a cuspidal weight-zero representation of \(\mathop{\mathrm{PGL}}_2(\mathbb A_F)\) realizing \(V_\eta\).

Apply Theorem 17 in the following order:

In each row Corollary 15 identifies the residual representations, and Proposition 29 gives decomposed genericity and absolute irreducibility over the cyclotomic extension. Proposition 12 supplies the determinant, finite ramification, potential semistability, and labeled Hodge–Tate weights. Lemma 30 makes the previously obtained automorphy available at the coefficient prime of the next row.

It remains to check the local hypotheses at that prime. At a place of potentially good ordinary type, both systems in the row are potentially crystalline and potentially ordinary of weight zero. At a place of potentially good supersingular type, they are potentially crystalline and neither is potentially ordinary of weight zero. These assertions follow from Corollary 25. In either case Lemma 31(i) gives zero monodromy in the input automorphic parameter. At a place of potentially totally toric type, neither Galois representation is potentially crystalline, and Lemma 31(ii) gives automorphic ordinarity. Thus the relevant alternative of Theorem 17 holds at every place above the coefficient prime.

The theorem therefore applies in all three rows. Its conclusion preserves cuspidality, weight zero, and trivial central character, so each row provides the input required for the next. The last row gives automorphy of \(V_1=H^1(B)\). ◻

Proof. For \(\sigma\in\mathop{\mathrm{Gal}}(L'/L)\), the representations \(\Pi^\sigma\) and \(\Pi\) have the same good-place Hecke polynomials, because the associated Galois representation extends to \(G_L\). Strong multiplicity one makes \(\Pi\) Galois invariant. The cyclic base-change theorem therefore gives a cuspidal automorphic representation \(\pi\) of \(\mathop{\mathrm{GL}}_2(\mathbb A_L)\) whose base change is \(\Pi\) (Langlands 1980; Arthur and Clozel 1989). Every archimedean place is complex, so its local extension is trivial. Consequently \(\pi\) is regular algebraic of weight zero.

Good-place compatibility with base change and Chebotarev imply \[\bigl(r_\iota(\pi)|_{G_{L'}}\bigr)^{\mathrm{ss}} \simeq r|_{G_{L'}}.\] The right-hand side is absolutely irreducible. The restriction on the left is therefore already irreducible and isomorphic to it, without semisimplification. After choosing this isomorphism, the two representations agree on \(G_{L'}\). For \(g\in G_L\), their matrices at \(g\) differ by a scalar, since they induce the same conjugation on the absolutely irreducible representation of \(G_{L'}\). Schur’s lemma shows that these scalars form a character \(\delta:\mathop{\mathrm{Gal}}(L'/L)\to\overline\mathbb Q_l^{\times}\) and \[r_\iota(\pi)\simeq r\otimes\delta.\] The finite-order Hecke character corresponding to \(\delta^{-1}\) removes this discrepancy. Twisting preserves cuspidality and weight zero, and the compatibility with Galois twisting follows at good places and hence everywhere by Chebotarev. ◻