The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Abstract

Let $f:S'\longrightarrow S$ be a cyclic branched covering of smooth projective surfaces over ${\mathbb {C}}$ whose branch locus $\Delta \subset S$ is a smooth ample divisor. Pick a very ample complete linear system $|{\mathcal {H}}|$ on S, such that the polarized surface $(S, |{\mathcal {H}}|)$ is not a scroll nor has rational hyperplane sections. For the general member $[C]\in |{\mathcal {H}}|$ consider the $\mu _{n}$-equivariant isogeny decomposition of the Prym variety ${{\,\mathrm{Prym}\,}}(C'/C)$ of the induced covering $f:C'{:}{=}f^{-1}(C)\longrightarrow C$:

$$\begin{aligned} {{\,\mathrm{Prym}\,}}(C'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C). \end{aligned}$$

We show that for the very general member $[C]\in |{\mathcal {H}}|$ the isogeny component ${\mathcal {P}}_{d}(C'/C)$ is $\mu _{d}$-simple with ${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map ${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S')$.

1 Introduction

For a cyclic cover $f:X\longrightarrow Y$ of smooth complex projective curves with $\deg (f)=n$, we fix a generator $\sigma \in {{\,\mathrm{Aut}\,}}(X/Y)$ of the automorphism group of f. The $\mu _{n}$-action of X induces a ${\mathbb {Q}}$-algebra homomorphism

$$\begin{aligned} \rho :{\mathbb {Q}}[\mu _{n}]\cong {\mathbb {Q}}[T]/(T^{n}-1)\rightarrow {{\,\mathrm{End}\,}}({{\,\mathrm{Jac}\,}}(X)), T\mapsto \sigma ^{*}, \end{aligned}$$

and we define ${\mathcal {P}}_{d}(X/Y){:}{=}\ker ^{0}(\Psi _{d}(\sigma ^{*}))$ for d|n, where $\Psi _{d}\in {\mathbb {Z}}[T]$ is the d-th cyclotomic polynomial. In what follows we freely use the following well-known results, which can be easily checked:

• (1) ${\mathcal {P}}_{1}(X/Y)=\ker ^{0}(\sigma ^{*}-{{\,\mathrm{id}\,}})=f^{*}({{\,\mathrm{Jac}\,}}(Y))\sim {{\,\mathrm{Jac}\,}}(Y)$

• (2) The addition map ${{\,\mathrm{Jac}\,}}(Y)\times {{\,\mathrm{Prym}\,}}(X/Y)\longrightarrow {{\,\mathrm{Jac}\,}}(X),(\alpha , \beta )\mapsto f^{*}(\alpha )+\beta$ is an isogeny.

• (3) Similarly, the addition map gives rise to the isogeny $\prod _{d|n,\ d\ne 1} {\mathcal {P}}_{d}(X/Y)\sim {{\,\mathrm{Prym}\,}}(X/Y).$

Then, we can state the main result of this paper, which is the following:

Theorem 1.1

Let S be a smooth projective surface over ${\mathbb {C}}$ with an ample line bundle ${\mathcal {L}}$. Assume $\Delta \in |{\mathcal {L}}^{\otimes n}|$ is smooth and consider the n-fold cyclic covering $f:{S}'\longrightarrow S$ branched along the divisor $\Delta$. Given a very ample complete linear system $|{\mathcal {H}}|$ on S, such that $(S,|{\mathcal {H}}|)$ is not a scroll nor has rational hyperplane sections. Then, for the very general member $[C]\in |{\mathcal {H}}|$ we have that

$$\begin{aligned} {{\,\mathrm{Prym}\,}}({C}'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C), \end{aligned}$$

with ${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$. Especially, each ${\mathcal {P}}_{d}(C'/C)$ is a $\mu _{d}$-simple abelian variety.

If we restrict to the case of double coverings, we note that the involution $\sigma$ of the covering f acts as $-{{\,\mathrm{id}\,}}$ on ${\mathcal {P}}_{2}(C'/C)={{\,\mathrm{Prym}\,}}(C'/C)$ and thus, ${{\,\mathrm{End}\,}}_{\mu _{2}}({{\,\mathrm{Prym}\,}}(C'/C))={{\,\mathrm{End}\,}}({{\,\mathrm{Prym}\,}}(C'/C))$. In particular, () can be stated as follows:

Corollary 1.2

Let S be a smooth projective surface over ${\mathbb {C}}$ with an ample line bundle ${\mathcal {L}}$. Assume $\Delta \in |{\mathcal {L}}^{\otimes 2}|$ is smooth and consider the double covering $f:{S}'\longrightarrow S$ branched along the divisor $\Delta$. Given a very ample complete linear system $|{\mathcal {H}}|$ on S, such that $(S,|{\mathcal {H}}|)$ is not a scroll nor has rational hyperplane sections. Then, for the very general member $[C]\in |{\mathcal {H}}|$ we have that

$$\begin{aligned} {{\,\mathrm{End}\,}}({{\,\mathrm{Prym}\,}}(C'/C))\cong {\mathbb {Z}}. \end{aligned}$$

The condition the line bundle ${\mathcal {L}}$ is ample in () implies that ${{\,\mathrm{Alb}\,}}(f):{{\,\mathrm{Alb}\,}}(S')\longrightarrow {{\,\mathrm{Alb}\,}}(S)$ is an isomorphism cf. page 11 and therefore the map ${\mathcal {P}}_{d}(C'/C)\longrightarrow {{\,\mathrm{Alb}\,}}(S')$ is trivial. For the general situation one needs to consider the abelian subvariety

$$\begin{aligned} {\mathcal {R}}_{d}(C',C,S'){:}{=}\ker ^{0}({\mathcal {P}}_{d}(C'/C)\longrightarrow {{\,\mathrm{Alb}\,}}(S')). \end{aligned}$$

Then, the result can be reformulated as follows:

Theorem 1.3

Let S be a smooth projective surface over ${\mathbb {C}}$ with a line bundle ${\mathcal {L}}$. Assume $\Delta \in |{\mathcal {L}}^{\otimes n}|$ is smooth and consider the n-fold cyclic covering $f:{S}'\longrightarrow S$ branched along the divisor $\Delta$. Given a very ample complete linear system $|{\mathcal {H}}|$ on S, such that $(S,|{\mathcal {H}}|)$ is not a scroll nor has rational hyperplane sections. Then, exactly one of the following assertions holds true:

• (i) For the general member $[C]\in |{\mathcal {H}}|$ we have that ${\mathcal {R}}_{d}(C',C,{S}')=0$.

• (ii) For the very general member $[C]\in |{\mathcal {H}}|$ we have that ${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {R}}_{d}(C',C,S'))\cong {\mathbb {Z}}[\zeta _{d}]$.

In this paper we present a complete proof for the above results, inspired by Ciliberto and Van der Geer’s approach in [3]. We note that this method does not capture the étale situation, cf. (), () and (). In addition, if we rephrase the statement for $n > 2$ by requiring simplicity instead of $\mu _{d}$-simplicity to the isogeny components, we observe that this method cannot be adopted. Namely, the abelian variety B in () cannot be chosen in general to be $\mu _{d}$-invariant and for this reason the last combinatorial argument in () fails. Lastly, a result due to Ortega and Lange, cf. [6] may be used to find counter-example for the case the covering f is étale of degree 7.

Notations and Conventions. For $n\in {\mathbb {N}}$, $\mu _{n}$ is the constant group scheme over ${\mathbb {C}}$, which is associated to the abstract group ${\mathbb {Z}}/n{\mathbb {Z}}$. The symbol $\zeta _{n}$ stands for a primitive n-th root of unity. If A is an abelian variety over ${\mathbb {C}}$, which is endowed with a $\mu _{n}$-action, then ${{\,\mathrm{End}\,}}_{\mu _{n}}(A)$ is the ring of $\mu _{n}$-equivariant endomorphisms of A. A very general point of a given variety X is a closed point $x\in X$, that lies in the complement of a countable union of nowhere dense closed subvarieties.

2 Preliminaries

In this section, we state some well-known results, which are needed later.

Proposition 2.1

Let $\pi :{\mathcal {A}}\longrightarrow S$ be a projective abelian scheme over a Noetherian base S. Then, the endomorphism functor of ${\mathcal {A}}$ over S is representable by an S-scheme ${{\,\mathrm{End}\,}}_{{\mathcal {A}}/S}$, which is a disjoint union of projective and unramified S-schemes.

Proof

This is well-known, cf. [4, pp. 133]. $\square$

The following proposition relates the correspondences on $C\times C$ with the endomorphisms of the Jacobian ${{\,\mathrm{Jac}\,}}(C)$.

Proposition 2.2

Let $\pi :{\mathcal {X}}\longrightarrow S$ be a projective smooth morphism over a Noetherian base S, whose fibres are geometrically integral curves. Furthermore, assume that the morphism $\pi$ admits a section, i.e. ${\mathcal {X}}(S)\ne \emptyset$. Then, there is a natural and functorial isomorphism

$$\begin{aligned} {{\,\mathrm{Corr}\,}}_{S}({\mathcal {X}}){:}{=}{{\,\mathrm{Pic}\,}}({\mathcal {X}}\times _{S}{\mathcal {X}})/({{{\,\mathrm{pr}\,}}_{1}})^{*} {{\,\mathrm{Pic}\,}}({\mathcal {X}})\otimes ({{{\,\mathrm{pr}\,}}_{2}})^{*}{{\,\mathrm{Pic}\,}}({\mathcal {X}})\cong {{\,\mathrm{End}\,}}_{S}({{\,\mathrm{Pic}\,}}^{0}_{{\mathcal {X}}/S}). \end{aligned}$$

Proof

Consider the commutative diagram:

The first row is clearly exact: Indeed, the relative Picard functor is an fppf-sheaf, cf. [13, Tag 021L], [5, Thm. 2.5] and thus, the restriction map $({{\,\mathrm{pr}\,}}_{1})^{*}$ is injective. Furthermore, the map q is just the cokernel of $({{\,\mathrm{pr}\,}}_{1})^{*}$. Next, we give the definition of the map d. Fix $x\in {\mathcal {X}}(S)$ and let $\phi :{\mathcal {X}}\longrightarrow {{\,\mathrm{Pic}\,}}_{{\mathcal {X}}/S}$ be any S-morphism. Then, $d\phi$ is the unique endomorphism of ${{\,\mathrm{Pic}\,}}^{0}_{{\mathcal {X}}/S}$, making the diagram below commutative.

Note that under our assumptions the Albanese map $can:{\mathcal {X}}\longrightarrow {{\,\mathrm{Alb}\,}}_{{\mathcal {X}}/S}$ exists and has the desired universal property, cf. [1, Thm. 2.17], [1, Rem. 2.19] and [8], Thm. 10.2]. Moreover, the construction of the map d indicates that d is surjective and also that the second row in the diagram above is exact at the middle. Now, the existence of g and the fact that it is an isomorphism are clear, since the first two vertical maps are isomorphisms by [5, Thm. 4.8] and [5, Thm. 2.5]. $\square$

The following proposition is well-known.

Proposition 2.3

Suppose that the polarized surface $( S, |{\mathcal {H}}| )$ is not a scroll nor has rational hyperplane sections. Then, the following assertions hold true:

• (i) The discriminant divisor ${\mathcal {D}}$ is irreducible and has codimension one in $| {\mathcal {H}}|$, i.e. ${\mathcal {D}}$ is a prime divisor of $| {\mathcal {H}}|$.

• (ii) The general curve $[ C ]\in {\mathcal {D}}$ is irreducible and has a single ordinary double point as its only singularity.

Proof

Cf. [3, Lem. 3.1]. $\square$

We close this section by introducing the $\mu _{n}$-equivariant isogeny decomposition in (). Let $f:C'\longrightarrow C$ be a cyclic branched covering of smooth complex projective curves with $\deg (f)=n$ and let $\sigma$ stand for a generator of the Galois group of f. The $\mu _{n}$-action on $C'$ induces an action on ${{\,\mathrm{Jac}\,}}(C')$ and thus, it defines a ${\mathbb {Q}}$-algebra homomorphism

$$\begin{aligned} \rho :{\mathbb {Q}}[\mu _{n}]\cong {\mathbb {Q}}[T]/(T^{n}-1)\longrightarrow {{\,\mathrm{End}\,}}^{0}({{\,\mathrm{Jac}\,}}(C')),\ T\mapsto \sigma ^{*} . \end{aligned}$$

For any divisor d|n, we define ${\mathcal {P}}_{d}(C'/C){:}{=}\ker ^{0}(\Psi _{d}(\sigma ^{*}))$, where $\Psi _{d}(T)\in {\mathbb {Z}}[T]$ is the d-th cyclotomic polynomial. Then, the addition map

$$\begin{aligned} \mu :\prod _{d|n}{\mathcal {P}}_{d}(C'/C)\longrightarrow {{\,\mathrm{Jac}\,}}(C') \end{aligned}$$

is a $\mu _{n}$-equivariant isogeny. Lange and Recillas [7] have stated and proved the relation between ${\mathbb {Q}}$-representations and the G-equivariant isogeny decomposition of an abelian variety with G-action, in terms of the finite group G involved, cf. [7, Thm. 2.2]. The $\mu _{n}$-equivariant isogeny decomposition of ${{\,\mathrm{Jac}\,}}(C')$ given above is in fact identical with the one introduced by Lange and Recillas [7]. This can be seen for example by using [2, Rem. 5.5] and [2, Cor. 5.7]. Moreover, we also note that the isogeny components ${\mathcal {P}}_{d}(C'/C)$ are non-trivial as long as the genus $g(C)\ge 1$, cf. [7, Thm. 3.1], [11, Thm. 5.12] and [11, Thm. 5.13].

3 Reduction to the generic fibre

Let S be a smooth projective surface over ${\mathbb {C}}$ with an ample line bundle ${\mathcal {L}}$. Assume $\Delta \in |{\mathcal {L}}^{\otimes n}|$ is smooth and consider the n-fold cyclic covering $f :{S}' \longrightarrow S$ branched along the divisor $\Delta$. Furthermore, fix a very ample complete linear system $| {\mathcal {H}}|$ on S, such that the polarized surface $( S,| {\mathcal {H}}| )$ is not a scroll nor has rational hyperplane sections. In this section we reduce the proof of Theorem 1.1 to showing that ${\mathcal {P}}_{d}(C'_{\eta }/C_{\eta })$ is a $\mu _{d}$-simple abelian variety, where $[C_{\eta }]$ is the generic member of $|{\mathcal {H}}|$.

Let $x\in S$ be a closed point of S. We denote by $|{\mathcal {H}}|_{x}$ the linear system of hyperplane sections in $|{\mathcal {H}}|$ passing through x. In the following we impose restrictions on the point x, i.e. $x\in S$ will be taken from some appropriate non-empty open subset of S.

Let $g:{\mathcal {X}}\subset S\times | {\mathcal {H}}|_{x}\longrightarrow | {\mathcal {H}}|_{x}$ denote the universal family of hyperplane sections and $h :{\mathcal {Y}}\subset {S}'\times |{\mathcal {H}}|_{x}\longrightarrow | {\mathcal {H}}|_{x}$ its pullback to ${S}'$, i.e. ${\mathcal {Y}}{:}{=}{\mathcal {X}}\times _{S} {S}'$. Note that over the non-empty open subset $U\subset |{\mathcal {H}}|_{x}$ of smooth curves which intersect the branch locus $\Delta$ transversally both g and h are smooth families of curves having a section. The latter allows us to consider their families of Jacobians over U, which we denote by $p :{{\,\mathrm{Pic}\,}}^{0}_{{\mathcal {X}}/U}\longrightarrow U$ and $q :{{\,\mathrm{Pic}\,}}^{0}_{{\mathcal {Y}}/U}\longrightarrow U$, respectively.

A generator $\sigma :S'\longrightarrow S'$ of the Galois group of the covering f induces an automorphism of ${\mathcal {Y}}$ over U and thus, an automorphism $\sigma ^{*}:{{\,\mathrm{Pic}\,}}^{0}_{{\mathcal {Y}}/U}\longrightarrow {{\,\mathrm{Pic}\,}}^{0}_{{\mathcal {Y}}/U}$. We define

$$\begin{aligned} {\mathcal {P}}_{d}{:}{=}\ker ^{0}(\Psi _{d}(\sigma ^{*}))\text { for any divisor}\ d|n. \end{aligned}$$

Then, $\varphi _{d}:{\mathcal {P}}_{d}\longrightarrow U$ is an abelian fibration with fibres ${({\mathcal {P}}_{d})}_{[C]}={\mathcal {P}}_{d}(C'/C)$ for $[C]\in U$.

As a first step we use the representability of the endomorphism functor of abelian schemes cf. () to reduce the proof of Theorem 1.1 to showing that ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\eta }}})\cong {\mathbb {Z}}[\zeta _{d}]$, where ${\bar{\eta }}$ is a fixed geometric generic point of $|{\mathcal {H}}|_{x}$. The proof of this is standard and so we omit it.

Lemma 3.1

Assume that ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\eta }}})\cong {\mathbb {Z}}[\zeta _{d}]$. Then, for the very general member $[C]\in U$, one has that ${{\,\mathrm{End}\,}}_{\mu _{d}} (({\mathcal {P}}_{d})_{[C]} )\cong {\mathbb {Z}}[\zeta _{d}]$.

Let $[C]\in |{\mathcal {H}}|_{x}$ be an irreducible member with a single ordinary double point as its only singularity and intersecting the branch locus $\Delta$ transversally. Then, $C'{:}{=}f^{-1}(C)$ is irreducible and has n ordinary double points as its only singularities. In this case the group variety ${\mathcal {P}}_{d}(C'/C)$ is semi-abelian. In particular, the result is the following:

Lemma 3.2

For an irreducible member $[C]\in |{\mathcal {H}}|_{x}$ with a single ordinary double point as its only singularity and intersecting the branch locus $\Delta$ transversally, there is an exact sequence:

where $\nu :\tilde{C}\longrightarrow C$ is the normalisation map and $\varphi (d)$ is the Euler’s totient function.

Proof

We have a commutative diagram

where $\tilde{f}$ is the cyclic covering branched along the divisor $\nu ^{*}\Delta |_{C}\in |\nu ^{*}{\mathcal {L}}|_{C}^{\otimes n}|$ and $\nu '$ is the normalisation of $C'$. Fix a generator $\sigma$ of ${{\,\mathrm{Aut}\,}}(C'/C)$ and let ${\tilde{\sigma }}$ be the corresponding generator of ${{\,\mathrm{Aut}\,}}(\tilde{C'}/\tilde{C})$, i.e. the one for which the diagram below commutes

Let $\{y,\sigma (y),\sigma ^{2}(y),\ldots ,\sigma ^{n-1}(y)\}$ be the set of ordinary double points of $C'$. Then, we find a commutative diagram with exact rows and columns

We show that $\beta$ induces a surjection ${\mathcal {P}}_{d}(C'/C)=\ker ^{0}(\Psi _{d}(\sigma ^{*}))\rightarrow \rightarrow {\mathcal {P}}_{d}(\tilde{C'}/\tilde{C})=\ker ^{0}(\Psi _{d}({\tilde{\sigma }}^{*}))$. Indeed, by Snake lemma we have the exact sequence

$$\begin{aligned} \ker (\Psi _{d}(\sigma ^{*}))\longrightarrow \ker (\Psi _{d}({\tilde{\sigma }}^{*}))\longrightarrow {{\,\mathrm{coker}\,}}(\gamma )\longrightarrow 0. \end{aligned}$$

Note that ${{\,\mathrm{coker}\,}}(\gamma )$ is an affine algebraic group, as it is the quotient of a commutative affine algebraic group by an algebraic subgroup. Since $\ker (\Psi _{d}({\tilde{\sigma }}^{*}))$ is a projective variety and the last arrow in the above sequence is surjective, [14, Cor. 12.67] shows that ${{\,\mathrm{coker}\,}}(\gamma )$ is finite. The latter provides the surjectivity of the map $\ker ^{0}(\Psi _{d}(\sigma ^{*}))\longrightarrow {\mathcal {P}}_{d}(\tilde{C'}/\tilde{C})=\ker ^{0}(\Psi _{d}({\tilde{\sigma }}^{*}))$, as claimed. $\square$

We are now in the position to prove the following:

Proposition 3.3

The abelian variety $({\mathcal {P}}_{d})_{{\bar{\eta }}}$ is $\mu _{d}$-simple if and only if ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\eta }}})\cong {\mathbb {Z}}[\zeta _{d}]$.

Proof

The one direction is clear: Indeed, if ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\eta }}})\cong {\mathbb {Z}}[\zeta _{d}]$, then every non-zero $\mu _{d}$-equivariant endomorphism of $({\mathcal {P}}_{d})_{{\bar{\eta }}}$ is an isogeny and thus, $({\mathcal {P}}_{d})_{{\bar{\eta }}}$ is a $\mu _{d}$-simple abelian variety. Conversely, assume that $({\mathcal {P}}_{d})_{{\bar{\eta }}}$ is $\mu _{d}$-simple. We divide the proof into steps.

Step 1. There is a closed subscheme ${{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}(0)\subset {{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}$ whose points parametrise the $\mu _{d}$-equivariant endomorphisms of ${\mathcal {P}}_{d}$, which are not isogenies, i.e. the ones, which are of degree 0. $\square$

Proof of Step 1

Observe that the functor of $\mu _{d}$-equivariant endomorphisms of ${\mathcal {P}}_{d}$ denoted by ${{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}$ is representable by a closed subscheme of ${{\,\mathrm{End}\,}}_{{\mathcal {P}}_{d}/U}$, since the equivariant condition is closed. It follows that we have a universal endomorphism $\alpha$, such that every other $\mu _{d}$-equivariant endomorphism of ${\mathcal {P}}_{d}$ over some scheme T is obtained by pulling-back $\alpha$ along a morphism $T\longrightarrow {{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}$. By [14, Prop. 12.93] the set

$$\begin{aligned} {\mathcal {V}}{:}{=}\{ x\in {{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}| \ \alpha _{x}{:}{=}\alpha \times {{\,\mathrm{id}\,}}_{\kappa ( x )} \ \text {is an isogeny} \} \end{aligned}$$

is open. Therefore, ${{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}(0){:}{=}{{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}\setminus {\mathcal {V}}$ with the reduced induced closed subscheme structure has the desired property. $\square$

Step 2. The fibre $({\mathcal {P}}_{d})_{[C]}$ for the very general member $[C]\in |{\mathcal {H}}|_{x}$ is a $\mu _{d}$-absolutely simple abelian variety.

Proof of Step 2

Recall that the U-scheme ${{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}(0)$ is unramified cf. (). It follows that a geometric fibre of this U-scheme is a disjoint union of points, corresponding to the $\mu _{d}$-equivariant endomorphisms of ${\mathcal {P}}_{d}$, which are not isogenies cf. Step 1. Since $({\mathcal {P}}_{d})_{{\bar{\eta }}}$ is a $\mu _{d}$-simple abelian variety, the only $\mu _{d}$-equivariant endomorphism of $({\mathcal {P}}_{d})_{{\bar{\eta }}}$, that is not an isogeny is the zero-morphism. In particular, this means that the geometric generic fibre of the U-scheme ${{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}(0)$ is connected and therefore, we can determine countably many non-empty open subsets $U_{i}\subset U$, such that the U-scheme ${{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}(0)$ has (geometrically) connected fibres for all points lying in the intersection of the $U_{i}$’s, cf. [13, Tag 055C]. Thus, for the very general member $[C]\in |{\mathcal {H}}|_{x}$, the only $\mu _{d}$-equivariant endomorphism of $({\mathcal {P}}_{d})_{[C]}$, which is not an isogeny is the zero-morphism. The latter is equivalent to the $\mu _{d}$-simplicity of $({\mathcal {P}}_{d})_{[C]}$, proving the claim. $\square$

Pick a Lefschetz pencil $(C_{t})_{t\in {\mathbb {P}}^{1}}\subset |{\mathcal {H}}|_{x}$. We may assume that all its singular members are irreducible and intersect the branch locus $\Delta$ transversally, cf. ().

Step 3. Given a Lefschetz pencil $(C_{t})_{t\in {\mathbb {P}}^{1}}$ as above, we construct a homomorphism:

$$\begin{aligned} \rho :{{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\mu }}}) \longrightarrow {{\,\mathrm{End}\,}}({\mathbb {G}}^{\varphi (d)}_{m}), \end{aligned}$$

where ${\bar{\mu }}$ is a fixed geometric generic point of ${\mathbb {P}}^{1}$.

Proof of Step 3

Since the endomorphism ring of any abelian variety is finitely generated, cf. [9], Thm. 12.5], we find a finite field extension $L\supset \kappa (\mu )$, such that every endomorphism of ${\mathcal {P}}_{d}$ over $\kappa ({\bar{\mu }})$ is defined over L, i.e. ${{\,\mathrm{End}\,}}(({\mathcal {P}}_{d})_{{\bar{\mu }}})={{\,\mathrm{End}\,}}(({\mathcal {P}}_{d})_{L})$. Consider the smooth projective model E of L together with the morphism $E\longrightarrow {\mathbb {P}}^{1}$ induced by this field extension and fix a closed point $y\in E$ lying over a point of the pencil that corresponds to a nodal curve. The map $\rho :{{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\mu }}}) \longrightarrow {{\,\mathrm{End}\,}}({\mathbb {G}}^{\varphi (d)}_{m})$ is constructed as follows: Let $f\in {{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{L})$. Then, f extends to an endomorphism over the local ring R of E at the point y, cf. [12, Prop. 7.4.3]. The restriction of the first projection of ${\mathcal {P}}_{d} \times _{R}{\mathcal {P}}_{d}$ to the graph of f is an isomorphism. We set $\alpha {:}{=}{{\,\mathrm{pr}\,}}_{1}|_{(\Gamma _{f})_{y}}$. By pulling back $\alpha$ along ${\mathbb {G}}^{\varphi (d)}_{m}\hookrightarrow {({\mathcal {P}}_{d})}_{y}$, we get an isomorphism $\alpha :\alpha ^{-1}({\mathbb {G}}^{\varphi (d)}_{m})\longrightarrow {\mathbb {G}}^{\varphi (d)}_{m}$. We claim that $\alpha ^{-1}$ is the graph of a homomorphism ${\mathbb {G}}^{\varphi (d)}_{m}\longrightarrow {\mathbb {G}}^{\varphi (d)}_{m}$. Indeed, it suffices to show that ${{\,\mathrm{pr}\,}}_{2}(\alpha ^{-1}({\mathbb {G}}^{\varphi (d)}_{m}))\subset {\mathbb {G}}^{\varphi (d)}_{m}$. To see this, observe that the composite

$$\begin{aligned} {\mathbb {G}}^{\varphi (d)}_{m}\overset{\cong }{\longrightarrow }\alpha ^{-1}({\mathbb {G}}^{\varphi (d)}_{m})\subset (\Gamma _{f})_{y}\overset{{{\,\mathrm{pr}\,}}_{2}}{\longrightarrow }({\mathcal {P}}_{d})_{y}\longrightarrow {\mathcal {P}}_{d}(\tilde{C'_{y}}/\tilde{C_{y}}) \end{aligned}$$

is the zero map by [9], Cor. 3.9] and hence, ${{\,\mathrm{pr}\,}}_{2}|_{{\mathbb {G}}^{\varphi (d)}_{m}}$ factors through the kernel of $({\mathcal {P}}_{d})_{y}\longrightarrow {\mathcal {P}}_{d}(\tilde{C'_{y}}/\tilde{C_{y}})$ which is ${\mathbb {G}}^{\varphi (d)}_{m}$. Finally, we define $\rho (f)$ to be this endomorphism of ${\mathbb {G}}^{\varphi (d)}_{m}$. One checks that $\rho$ is a homomorphism of rings. $\square$

Conclusion Eventually, we are in the position to complete the proof. Suppose ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\eta }}})\ne {\mathbb {Z}}[\zeta _{d}]$ and choose a $\mu _{d}$-equivariant endomorphism f not in ${\mathbb {Z}}[\zeta _{d}]$. The endomorphism f can be described as a $\kappa ({\bar{\eta }})$-point of ${{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}$ and we let $Z\subset {{\,\mathrm{End}\,}}^{\mu _{d}}_{{\mathcal {P}}_{d}/U}$ be the irreducible component containing this point. Then, the generic point $\theta \in Z$ corresponds to a $\mu _{d}$-equivariant endomorphism not in ${\mathbb {Z}}[\zeta _{d}]$. Consider the finite set

$$\begin{aligned} \Gamma {:}{=}\{ n{:}{=}(n_{0},n_{1},\ldots ,n_{\varphi (d)-1})\in {\mathbb {Z}}^{\varphi (d)}\ | \ {{\,\mathrm{im}\,}}([n]^{1})\cap Z\ne \emptyset \}. \end{aligned}$$

Each ${{\,\mathrm{im}\,}}( [n] )\cap Z$ is a proper closed subset of Z. Setting¹

$$\begin{aligned} Z_{n}{:}{=}\pi ({{\,\mathrm{im}\,}}( [n] )\cap Z), \end{aligned}$$

for $n\in \Gamma$, we get finitely many nowhere dense closed subsets of U, such that for every point $u\in U\setminus \bigcup _{n\in \Gamma }Z_{n}$ the fibre $\pi ^{-1}(u)$ contains a point, which is not in ${\mathbb {Z}}[\zeta _{d}]$. We can choose a Lefschetz pencil as above, such that $({\mathcal {P}}_{d})_{{\bar{\mu }}}$ is $\mu _{d}$-simple, cf. Step 2 and ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\mu }}})\ne {\mathbb {Z}}[\zeta _{d}]$. By Step 3 this leads to a contradiction. Indeed, using that every non-zero element of ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\mu }}})$ is invertible in ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\mu }}})\otimes {\mathbb {Q}}$, it is readily checked that the composition of the map $\rho$ constructed in Step 3 with $\psi {:}{=}{{\,\mathrm{pr}\,}}_{1}\circ -:{{\,\mathrm{End}\,}}({\mathbb {G}}_{m}^{\varphi (\delta )})\longrightarrow {{\,\mathrm{Hom}\,}}({\mathbb {G}}_{m}^{\varphi (\delta )},{\mathbb {G}}_{m})\cong {\mathbb {Z}}^{\varphi (\delta )}$ is injective. It follows that ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\mu }}})\otimes {\mathbb {Q}}\cong {\mathbb {Q}}(\zeta _{d})$. Since ${\mathbb {Z}}[\zeta _{d}]$ is a maximal order in ${\mathbb {Q}}(\zeta _{d})$, we also obtain ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{{\bar{\mu }}})\cong {\mathbb {Z}}[\zeta _{d}]$. The proof is complete. $\square$

The next lemma consists of the final reduction step.

Lemma 3.4

The abelian variety $({\mathcal {P}}_{d})_{\eta }$ is $\mu _{d}$-simple if and only if it is $\mu _{d}$-absolutely simple.

Proof

Clearly, if $({\mathcal {P}}_{d})_{\eta }$ is $\mu _{d}$-absolutely simple, then it is $\mu _{d}$-simple. Conversely, assume that $({\mathcal {P}}_{d})_{\eta }$ is $\mu _{d}$-simple but not $\mu _{d}$-absolutely simple. Then, there is a finite field extension $L\supset \kappa (\eta )$ and a non-zero and proper $\mu _{d}$-simple abelian subvariety B of $({\mathcal {P}}_{d})_{L}$, such that $({\mathcal {P}}_{d})_{L}$ can be written up to isogeny as a product $\prod B^{\tau }$, where $B^{\tau }$ stands for a Galois conjugate of B and $\tau$ runs through a finite subset $J\subset {{\,\mathrm{Gal}\,}}(L/\kappa (\eta ))$ of cardinality greater equal to 2. The field extension $L\supset \kappa (\eta )$ gives rise to a morphism $g:U'\longrightarrow U$, which we may assume is étale. For $\tau \in J$, we let $\varphi _{\tau }$ be the endomorphism of $({\mathcal {P}}_{d})_{L}$ whose image is $B^{\tau }$. More explicitly, $\varphi _{\tau }$ is given by

$$\begin{aligned} ({\mathcal {P}}_{d})_{L}\overset{\sim }{\longrightarrow }\prod B^{\tau }\overset{proj}{\longrightarrow }B^{\tau }\subset ({\mathcal {P}}_{d})_{L}. \end{aligned}$$

Pick a Lefschetz pencil $(C_{t})_{t\in {\mathbb {P}}^{1}}$, such that its singular members are irreducible and intersect the branch locus $\Delta$ transversally. Let X be any irreducible component of $g^{-1}({\mathbb {P}}^{1}\cap U)$. Then, X dominates ${\mathbb {P}}^{1}\cap U$ and if $\theta \in X$ is its generic point, then each $\varphi _{\tau }$ determines an endomorphism of ${\mathcal {P}}_{d}$ over $\theta$, e.g. using the Néron mapping property, such that if $B^{\tau }{:}{=}{{\,\mathrm{im}\,}}(\varphi _{\tau })$, then $\prod B^{\tau }\sim ({\mathcal {P}}_{d})_{\theta }$. Let ${\bar{X}}$ be a smooth compactification of X and ${\bar{X}}\longrightarrow {\mathbb {P}}^{1}$ the extension of $g:X\longrightarrow {\mathbb {P}}^{1}\cap U$. Fix a point $y\in {\bar{X}}$ lying over a point of the pencil which corresponds to a nodal curve and consider the local ring R of ${\bar{X}}$ at y. Since ${\mathcal {P}}_{d}$ admits a semi-abelian reduction over R , cf. () the same is true for all $B^{\tau }$, cf. [12, Cor. 7.1.6]. We denote by $\tilde{B}^{\tau }$ the identity component of the Néron model of $B^{\tau }$. Then, the isogeny of the generic fibre extends to an isogeny $\prod \tilde{B}^{\tau }\sim {\mathcal {P}}_{d}$ over R, cf. [12, Prop. 7.3.6]. Since $({\mathcal {P}}_{d})_{y}$ is an extension of an abelian variety by a torus of rank $\varphi (d)$, cf. (), it follows that the toric part of $\tilde{B}^{\tau }_{y}$ has rank $\delta ,\ 1\le \delta \le \varphi (d)$, such that $\delta |J|=\varphi (d)$. As in Step 3, one constructs a homomorphism $\rho _{\tau } :{{\,\mathrm{End}\,}}_{\mu _{d}}(B^{\tau }) \longrightarrow {{\,\mathrm{End}\,}}({\mathbb {G}}^{\delta }_{m})$. Since the restriction of $\psi \circ \rho _{\tau }$ to ${\mathbb {Z}}[\zeta _{d}]\subset {{\,\mathrm{End}\,}}_{\mu _{d}}(B^{\tau })$ is injective, where $\psi {:}{=}{{\,\mathrm{pr}\,}}_{1}\circ -:{{\,\mathrm{End}\,}}({\mathbb {G}}^{\delta }_{m})\longrightarrow {{\,\mathrm{Hom}\,}}({\mathbb {G}}^{\delta }_{m}, {\mathbb {G}}_{m})\cong {\mathbb {Z}}^{\delta }$ and ${\mathbb {Z}}[\zeta _{d}]$ has rank $\varphi (d)$ as a free abelian group, we conclude that $\delta =\varphi (d)$. But then $|J|=1$, which is absurd.$\square$

4 The Proof of Theorem 1.1

According to the results of Sect. 3, our task to prove Theorem 1.1 is reduced to showing $({\mathcal {P}}_{d})_{\eta }$ is a $\mu _{d}$-simple abelian variety. Recall, that we have an isogeny

$$\begin{aligned} {{\,\mathrm{Jac}\,}}(C'_{\eta })\sim {{\,\mathrm{Jac}\,}}(C_{\eta })\times \prod _{d|n,\ d\ne 1} ({\mathcal {P}}_{d})_{\eta }. \end{aligned}$$

Given a non-zero endomorphism $\varepsilon \in {{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {P}}_{d})_{\eta })$. Then, by considering the composite

$$\begin{aligned} { \varepsilon }':{{\,\mathrm{Jac}\,}}(C'_{\eta })\overset{\sim }{\longrightarrow } {{\,\mathrm{Jac}\,}}(C_{\eta })\times \prod _{d|n,\ d\ne 1}({\mathcal {P}}_{d})_{\eta }\overset{{{\,\mathrm{pr}\,}}_{d}}{\longrightarrow }({\mathcal {P}}_{d})_{\eta } \overset{\varepsilon }{\longrightarrow }({\mathcal {P}}_{d})_{\eta }\hookrightarrow {{\,\mathrm{Jac}\,}}(C'_{\eta }), \end{aligned}$$

we get an endomorphism of ${{\,\mathrm{Jac}\,}}(C'_{\eta })$ whose restriction to $({\mathcal {P}}_{d})_{\eta }$ is simply $\varepsilon \circ [n]$. Hence, it suffices to show that that the restriction of $\varepsilon '$ to $({\mathcal {P}}_{d})_{\eta }$ lies in ${\mathbb {Z}}[\zeta _{d}]$. Recall, that abelian schemes satisfy a stronger Néron mapping property, cf. [10, Sec. 3.1.5]. Thus, the endomorphism $\varepsilon '$ extends to an endomorphism

$$\begin{aligned} \varepsilon ':{{\,\mathrm{Pic}\,}}^{0}_{{\mathcal {Y}}/U}\longrightarrow {\mathcal {P}}_{d}\subset {{\,\mathrm{Pic}\,}}^{0}_{{\mathcal {Y}}/U}. \end{aligned}$$

Let $[T]\in {{\,\mathrm{Corr}\,}}_{U}({\mathcal {Y}})$ be the class of a correspondence T on ${\mathcal {Y}}\times _{U}{\mathcal {Y}}$ associated to the endomorphism $\varepsilon '$, cf. (). We write $T=\sum n_{i}T_{i}$, where $T_{i}$ are prime divisors. Let $\Sigma$ be a general two dimensional linear system in $|{\mathcal {H}}|_{x}$, i.e. the general member of $\Sigma$ is smooth and intersects the branch locus $\Delta$ transversally. Then, the correspondences $T_{i}$ are all defined over a non-empty open subset of $\Sigma$ and we can construct a rational map

, cf. [3, pp. 38]. Especially, we get a rational map

Let $[ C ]\in | {\mathcal {H}}|_{x}$ be a general member and choose a general two-dimensional linear system $\Sigma$ containing [C]. Consider the rational map $\phi _{\Sigma ,T}$. Then, for a general point $y\in {C}'$ we get a divisor $\Gamma _{y}=\phi _{\Sigma ,T}(y)$ on ${S}'$. Set $w=f(y)\in C$, $f^{-1}(w)= \{ y,\sigma (y),\ldots , \sigma ^{n-1}(y)\}$ and $f^{-1}(x)= \{ z,\sigma (z),\ldots ,\sigma ^{n-1}(z)\}$, where $\sigma$ is a generator of the Galois group of the covering f. The following lemma computes the divisor $E_{y}$ in ${C}'$ corresponding to the intersection of ${C}'$ with $\Gamma _{y}$.

Lemma 4.1

We have that $E_{y}={\alpha }_{0} z+{\alpha }_{1}\sigma (z)+\ldots +{\alpha }_{n-1}\sigma ^{n-1}(z)+{\beta }_{0} y+ {\beta }_{1}\sigma (y)+\ldots +{\beta }_{n-1}\sigma ^{n-1}(y)+\gamma {{\mathcal {B}}}'_{x,w}+T_{{C}'}(y)$, where $\alpha _{i} ,\beta _{i} ,\gamma \in {\mathbb {Z}}$ and ${{\mathcal {B}}}'_{x,w}$ is the pull-back of the divisor of base points different from x and w of $\Sigma _{w}$ under the covering f.

Proof

Cf. [3, Lem. 3.6]. $\square$

4.1 Regular case

The branched locus $\Delta$ of the covering f is a smooth ample divisor and thus, the canonical map ${{\,\mathrm{Alb}\,}}(f):{{\,\mathrm{Alb}\,}}(S')\longrightarrow {{\,\mathrm{Alb}\,}}(S)$ induced by f is an isomorphism. Indeed, since $f_{*}{\mathcal {O}}_{S'}\cong \bigoplus ^{n-1}_{i=0}{\mathcal {L}}^{-i}$, the Kodaira Vanishing theorem gives $H^{1}({\mathcal {O}}_{S'})=H^{1}({\mathcal {O}}_{S})$ and hence, ${{\,\mathrm{Alb}\,}}(f)$ is an isogeny. From this one immediately sees that the induced action on ${{\,\mathrm{Alb}\,}}(S')$ is trivial, i.e. ${{\,\mathrm{Alb}\,}}(\sigma )={{\,\mathrm{id}\,}}$. Consider the Albanese map ${{\,\mathrm{Alb}\,}}_{\xi _{o}}:S'\longrightarrow {{\,\mathrm{Alb}\,}}(S')$, where the point $\xi _{o}\in S'$ lies over a point of the branch locus $\Delta \subset S$ and observe that the map is invariant under the $\mu _{n}$-action. Therefore, we find a homomorphism ${{\,\mathrm{Alb}\,}}(S)\longrightarrow {{\,\mathrm{Alb}\,}}(S')$ that is inverse to ${{\,\mathrm{Alb}\,}}(f)$, proving the claim. In particular, we deduce that $q(S)=q(S')$. Here, we give the proof for the case S is regular, i.e. $q(S)=0$.

Proof of Theorem 1.1 for the regular case

If S is regular, then ${{\,\mathrm{Pic}\,}}({S}')$ is discrete and thus, the rational map $\phi _{\Sigma ,T}$ is constant. Hence, for a general point $y\in {C}'$, the curves $\Gamma _{y}$ and $\Gamma _{\sigma (y)}$ are linearly equivalent. It follows that $E_{y}$ and $E_{\sigma (y)}$ are also linearly equivalent and so, $E _{y}-E_{\sigma (y)}={\beta }_{0} (y-\sigma (y))+ {\beta }_{1}\sigma (y-\sigma (y))+\ldots +{\beta }_{n-1}{\sigma }^{n-1}(y-\sigma (y)) +T_{{C}'}(y-\sigma (y))\sim 0$. Since ${{\,\mathrm{Prym}\,}}(C'/C)={{\,\mathrm{im}\,}}({{\,\mathrm{id}\,}}-\sigma ^{*})$, the latter forces $T_{{C}'}(y)=(-{\beta }_{0})y+\ldots +(-{\beta }_{n-1})\sigma ^{n-1}(y)$ for all $y\in {{\,\mathrm{Prym}\,}}(C'/C)$. Eventually, we see that the restriction of $T_{C'}$ to ${\mathcal {P}}_{d}(C'/C)$ takes the desired form. This yields that the restriction of ${\varepsilon }'$ to $({\mathcal {P}}_{d})_\eta$ lies in ${\mathbb {Z}}[\zeta _{d}]$, as claimed.$\square$

4.2 Irregular case

The closed embedding $i:C'\hookrightarrow S'$ defines the natural map $i^{*}:{{\,\mathrm{Pic}\,}}^{0}(S')\longrightarrow {{\,\mathrm{Pic}\,}}^{0}({C}')$ whose kernel is finite, since $H^{1}(S',{\mathcal {O}}_{S'}(-C'))=0$. In what follows we view ${{\,\mathrm{Pic}\,}}^{0}(S')$ as an abelian subvariety of ${{\,\mathrm{Jac}\,}}(C')$ by identifying it with ${{\,\mathrm{im}\,}}(i^{*})$. We shall use the following lemma.

Lemma 4.2

Let $a:{{\,\mathrm{Jac}\,}}({C}')\longrightarrow {\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}({C}')$ be a homomorphism and let $T_a$ be a correspondence associated to it, cf. (). Assume that there exist $\alpha _{0},\ldots ,\alpha _{n-1} \in {\mathbb {Z}}$, such that for general $y\in {C}'$ the divisor class $T_{a}(y-\sigma (y))+ {\alpha }_{0}(y-\sigma (y))+\ldots +{\alpha }_{n-1}{\sigma }^{n-1}(y-\sigma (y))$ lies in ${{\,\mathrm{Pic}\,}}^{0}({S}')$. Then, the restriction of a to ${\mathcal {P}}_{d}(C'/C)$ lies in ${\mathbb {Z}}[\zeta _{d}]\subset {{\,\mathrm{End}\,}}( {\mathcal {P}}_{d}(C'/C))$.

Proof

Recall that ${{\,\mathrm{Prym}\,}}(C'/C)={{\,\mathrm{im}\,}}({{\,\mathrm{id}\,}}-\sigma ^{*})$ and for this reason the closed points of ${{\,\mathrm{Prym}\,}}(C'/C)$ are generated by elements of the form $y-\sigma (y)$, where $y\in C'$. Hence, the assumption clearly implies that $\eta (y){:}{=}a(y)+ {\alpha }_{0}y+\ldots +{\alpha }_{n-1}{\sigma }^{n-1}(y) \in {{\,\mathrm{im}\,}}(i^{*})\cap {{\,\mathrm{Prym}\,}}(C'/C)$ (note that ${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Prym}\,}}(C'/C)$) for all $y\in {{\,\mathrm{Prym}\,}}(C'/C)$, where $i^{*}:{{\,\mathrm{Pic}\,}}^{0}({S}')\longrightarrow {{\,\mathrm{Pic}\,}}^{0}({C}')={{\,\mathrm{Jac}\,}}({C}')$ is the natural pull-back induced by ${C}'\hookrightarrow {S}'$. We show that the intersection ${{\,\mathrm{im}\,}}(i^{*})\cap {{\,\mathrm{Prym}\,}}(C'/C)$ is finite. Indeed, consider the commutative square:

The canonical map ${{\,\mathrm{Alb}\,}}({S}')\longrightarrow {{\,\mathrm{Alb}\,}}(S)$ induced by f is an isomorphism and so, is its dual, which is $f^{*}$. Hence, the latter yields that ${{\,\mathrm{im}\,}}(i^{*}:{{\,\mathrm{Pic}\,}}^{0}({S}')\longrightarrow {{\,\mathrm{Pic}\,}}^{0}({C}'))\subset f^{*}({{\,\mathrm{Pic}\,}}^{0}(C))$. By the definition of ${{\,\mathrm{Prym}\,}}(C'/C)$, we know that $f^{*}({{\,\mathrm{Pic}\,}}^{0}(C))\cap {{\,\mathrm{Prym}\,}}(C'/C)$ is finite and so, is the intersection ${{\,\mathrm{im}\,}}(i^{*})\cap {{\,\mathrm{Prym}\,}}(C'/C)$, as claimed. From the latter one deduces that the endomorphism $\eta$ of ${{\,\mathrm{Prym}\,}}(C'/C)$ defined above is the zero-map, simply because $\eta ({{\,\mathrm{Prym}\,}}(C'/C))$ is irreducible subvariety of ${{\,\mathrm{im}\,}}(i^{*})\cap {{\,\mathrm{Prym}\,}}(C'/C)$, which is a finite union of points. Finally, by restricting to ${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Prym}\,}}(C'/C)$, we conclude that a lies in the image of the map ${\mathbb {Z}}[\zeta _{d}]\subset {{\,\mathrm{End}\,}}( {\mathcal {P}}_{d}(C'/C)),\ \zeta _{d}\mapsto \sigma$. The proof is complete.$\square$

Proof of Theorem 1.1 for the irregular case

Using the curves $\Gamma _{y}$ we find that $E _{y}-E_{\sigma (y)}$ lies in the image of ${{\,\mathrm{Pic}\,}}({S}')\longrightarrow {{\,\mathrm{Pic}\,}}({C}')$. Therefore, we have that $T_{{C}'}(y-\sigma (y))+{\beta }_{0} (y-\sigma (y))+ {\beta }_{1}\sigma (y-\sigma (y))+\ldots +{\beta }_{n-1}{\sigma }^{n-1}(y-\sigma (y)) \in {{\,\mathrm{im}\,}}(i^{*}:{{\,\mathrm{Pic}\,}}({S}')\longrightarrow {{\,\mathrm{Pic}\,}}({C}'))$ for general $y\in C'$. It follows that ${\varepsilon }'\in {\mathbb {Z}}[\zeta _{d}]\subset {{\,\mathrm{End}\,}}(({\mathcal {P}}_{d})_{\eta })$, cf. ().$\square$

5 The proof of Theorem 1.3

The proof is similar to the case of (). First, we need to replace our earlier family $\varphi _{d}:{\mathcal {P}}_{d}\longrightarrow U$. In particular, we consider the abelian fibration

$$\begin{aligned} {\mathcal {R}}_{d}{:}{=}\ker ^{0}({\mathcal {P}}_{d}\longrightarrow {{\,\mathrm{Alb}\,}}(S')\times U). \end{aligned}$$

Assume that the abelian fibration $\varphi _{d}:{\mathcal {R}}_{d}\longrightarrow U$ is non-zero, i.e. ${\mathcal {R}}_{[C]}\ne 0$ for $[C]\in U$. Then, we show that for the very general member $[C]\in U$, we have that ${{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {R}}_{d})_{[C]})\cong {\mathbb {Z}}[\zeta _{d}]$. One checks that the results (3.3) and (3.4) still hold true for the family $\varphi _{d}:{\mathcal {R}}_{d}\longrightarrow U$.

We proceed as in the proof of Theorem 1.1. A non-zero endomorphism $\varepsilon \in {{\,\mathrm{End}\,}}_{\mu _{d}}(({\mathcal {R}}_{d})_{\eta })$ gives rise to an endomorphism $\varepsilon '\in {{\,\mathrm{End}\,}}({{\,\mathrm{Jac}\,}}(C'_{\eta }))$ and it is enough to check that the restriction of $\varepsilon '$ to $({\mathcal {R}}_{d})_{\eta }$ lies in ${\mathbb {Z}}[\zeta _{d}]$. The following lemma is needed.

Lemma 5.1

Let $a:{{\,\mathrm{Jac}\,}}({C}')\longrightarrow {\mathcal {R}}_{d}(C',C,{S}')\subset {{\,\mathrm{Jac}\,}}({C}')$ be a homomorphism and let $T_a$ be a correspondence associated to it, cf. (). Assume that there exist $\alpha _{0},\ldots ,\alpha _{n-1} \in {\mathbb {Z}}$, such that for general $y\in {C}'$ the divisor class $T_{a}(y-\sigma (y))+ {\alpha }_{0}(y-\sigma (y))+\ldots +{\alpha }_{n-1}{\sigma }^{n-1}(y-\sigma (y))$ lies in ${{\,\mathrm{Pic}\,}}^{0}({S}')$. Then, the restriction of a to ${\mathcal {R}}_{d}(C',C,{S}')$ lies in ${\mathbb {Z}}[\zeta _{d}]\subset {{\,\mathrm{End}\,}}( {\mathcal {R}}_{d}(C',C,{S}'))$.

Proof

Clearly, we have that $a(y)+{\alpha }_{0}y+\ldots +{\alpha }_{n-1}{\sigma }^{n-1}(y)\in {{\,\mathrm{im}\,}}(i^{*})$ for all $y\in {{\,\mathrm{Prym}\,}}(C'/C)$, where $i^{*}:{{\,\mathrm{Pic}\,}}^{0}({S}')\longrightarrow {{\,\mathrm{Pic}\,}}^{0}({C}')={{\,\mathrm{Jac}\,}}({C}')$ is the pull-back induced by ${C}'\hookrightarrow {S}'$. Let ${\mathcal {K}}(C',S'){:}{=}\ker ({{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S'))$ and observe that the intersection ${{\,\mathrm{im}\,}}(i^{*})\cap {\mathcal {K}}(C',S')$ is finite. Since ${\mathcal {R}}_{d}(C',C,{S}')\subset {\mathcal {K}}(C',S')$, we find that $a(y)+{\alpha }_{0}y+\ldots +{\alpha }_{n-1}{\sigma }^{n-1}(y)=0$ for all $y\in {\mathcal {R}}_{d}(C',C,{S}')$. Therefore, the restriction of a to ${\mathcal {R}}_{d}(C',C,{S}')$ belongs to ${\mathbb {Z}}[\zeta _{d}]$, as claimed.$\square$

Proof of Theorem 1.3

Using the curves $\Gamma _{y}$ one sees that $E _{y}-E_{\sigma (y)}$ lies in the image of ${{\,\mathrm{Pic}\,}}({S}')\longrightarrow {{\,\mathrm{Pic}\,}}({C}')$. It follows that $T_{{C}'}(y-\sigma (y))+{\beta }_{0} (y-\sigma (y))+ {\beta }_{1}\sigma (y-\sigma (y))+\ldots +{\beta }_{n-1}{\sigma }^{n-1}(y-\sigma (y)) \in {{\,\mathrm{im}\,}}(i^{*}:{{\,\mathrm{Pic}\,}}({S}')\longrightarrow {{\,\mathrm{Pic}\,}}({C}'))$. Now, the result is an immediate consequence of ().$\square$