Projective p-adic Representations of the K-rational Geometric Fundamental Group

G. Frey and E. Kani

Herrn E. Lamprecht zum 75. Geburtstag gewidmet

1 Introduction

Let C/K be a smooth, geometrically connected curve over a finitely generated field K, and let F = κ(C) denote its function field. If C has a K-rational point P ∈ C(K), then the K-rational fundamental group π_{1}(C, P) of C/K with base point P is the Galois group

π_{1}(C, P) = Gal(F_{nr,P} /F)

of the maximal unramified extension F_{nr,P} of F in which P splits completely. Since the extension F_{nr,P} /F is regular, it is clear that π_{1}(C, P) is a quotient of the usual geometric fundamental group π_{1}(CK̄, P) of CK̄ = C ⊗ K̄ over the algebraic closure K̄ of K with base point P.

The group π_{1}(C, P), which is a special case of the groups Gal(F_{S}^{ur}/F) introduced by Ihara [Ih], was studied in some detail in [FKV]. It may be viewed as a natural analogue of the fundamental group π_{1}(F) of a number field F, as the results of section 2 of [FKV] indicate. For example, the maximal abelian subextension F_{nr,P}^{ab} of F_{nr,P}, which may be interpreted as an analogue of the Hilbert class field (cf. Rosen [Ro]), is always finite.

However, this picture changes if we look at p-adic representations if K has positive characteristic, particularly in view of the following consequence of Fontaine-Mazur Conjecture [FM]:

Conjecture 1 ([FM], Conjecture 5b) If F is a number field, then any p-adic representation

ρ : π_{1}(F) → GL_{n}(ℚ_{p})

factors through a finite quotient group. In particular, any quotient group of the maximal unramified pro-p-group π_{1}(F)^{(p)}, which is a p-adic analytic group, is finite.

Now the natural analogue of this conjecture for the group π_{1}(C, P) is false if K has positive characteristic. This can be deduced from the work of Oort[Oo1] (as will be explained in section 4), but this approach does not lead to explicit counterexamples. We therefore present here the following result:

Theorem 1.1 Suppose K is a field of characteristic char(K) ≡ 3 (mod 4) and that K contains the fourth roots of unity. Let b ∈ K^{×}, b^{4} ≠ ±1, and put c = 1 + b^{4} and a = \frac{2b^{2}}{c}. Let C/K be the curve defined by the equation

s^{4} = ct(t^{2} − 1)(t − a)g(t),

where g(t) ∈ K[t] is any polynomial with

g(a) = 1 and g(0)g(1)g(−1) ≠ 0,

and put P = (a, 0) ∈ C(K). Then the K-rational geometric fundamental group π_{1}(C, P) is infinite; more precisely, for every prime p ≡ 5 (mod 12) (with p ≠ char(K)), the group PSL_{3}(ℤ_{p}) is a factor of π_{1}(C, P), i.e. there is a surjection

ρ̃: π_{1}(C, P) → PSL_{3}(ℤ_{p}).

Corollary 1.2 In the above situation, let C_{p} denote the finite cover of C of degree p + 1 corresponding to a pro-p-Sylow subgroup U_{p} of PSL_{3}(ℤ_{p}). Then for any point P′ over P, the fundamental group π_{1}(C_{p}, P′) has a quotient which is isomorphic to the p-adic analytic group U_{p}.

It should also be mentioned here that Ihara [Ih] has constructed other explicit counterexamples in the case that K is a finite field (of characteristic 3 or 5).

To construct such representations ρ̃, we shall actually follow the spirit of the Main Conjecture of Fontaine-Mazur[FM] which asserts that all “reasonable” p-adic representations of G_{F} come from geometry, i.e. they are subrepresentations of an étale cohomology group H^{q}(XF̄, ℚ_{p}(r)) attached to some algebraic variety X/F.

In our case we shall take X to be a curve and q = 1, which means that we are considering representations which are closely related to the ℚ_{p}-Tate module V_{p}(A) := T_{p}(A) ⊗ ℚ_{p} of an abelian subvariety A/F of the Jacobian variety of X.

It is easily seen that the usual p-adic G_{F} -subrepresentations V ⊂ V_{p}(A) do not lead to representations of π_{1}(C, P) (cf. Proposition 2.2). We use instead the associated projective representations

ρ̃_{V} : G_{F} → PGL(V) = Aut(ℙ(V)).

Since it seems difficult to find a useful (geometric) criterion that such a single representation ρ̃_{V} factors over π_{1}(C, P), we work instead with a complete system {ρ̃_{V_{i}}} of subrepresentations belonging to a G_{F} -decomposition V_{p}(A) = ⊕V_{i}.

A first condition for the existence of such a system {ρ̃_{V_{i}}} is given by the criterion of Néron-Ogg-Shafarevich: A has to have good reduction everywhere, i.e. at all points of C.

A second condition is that the fixed fields of (finite quotients of) the kernels of the ρ̃_{V_{i}}’s are regular extensions of F.

Note that by using group theoretical methods it not difficult to find examples of abelian varieties A/F and corresponding representations {ρ̃_{V_{i}}} which satisfy these two conditions (cf. [FKV]).

The most difficult condition is the existence of a point P ∈ C(K) which splits completely in Fix(Ker(ρ̃_{V_{i}})). Of course this condition implies the second one but at present we have no “canonical” method for constructing abelian varieties A/F which satisfy the first and the third condition. (Actually, the “supersingular moduli constructions” discussed in section 4 do give such a method, but this method does not apply in characteristic 0.)

The aim of this paper is to analyze the arithmetical conditions for A which are imposed by these conditions.

In the case that the V_{i} are the eigenspaces of a suitable automorphism σ ∈ Aut(A) (this situation occurs in the constructions of [FKV]), we give in Theorem 2.5 a complete characterization of the case when all the ρ̃_{V_{i}} factor over π_{1}(C, P): under suitable hypotheses, this happens precisely when the reduction \overline{A}_{P} of A at P has complex multiplication and the centre Z(End_{K}^{0}(\overline{A}_{P})) of the endomorphism algebra of \overline{A}_{P} is contained in the algebra ℚ(σ̄) ⊂ End^{0}(\overline{A}_{P}) generated by the reduction σ̄ ∈ Aut(\overline{A}_{P}) of σ. The proof of this theorem is given in section 3.

In section 5 we apply this criterion to the example considered in the last section of [FKV]. Under the assumption that K contains the fourth roots of unity we obtain a three dimensional subvariety A of the Jacobian of a suitable curve of genus 4 and a point P on C such that \overline{A}_{P} is (isogenous to) the product of three copies of the elliptic curve

E : Y^{2} = X^{3} − X

(which has complex multiplication by ℚ(i)). However, when we compute the centre Z(End_{K}^{0}(\overline{A}_{P})) it turns out that it is not contained in ℚ(\overline{σ}) unless E is supersingular! Thus, we have to correct the statement of Theorem 5.22 of [FKV] by adding the condition that char(K) ≡ 3 (mod 4). It remains open whether there is a curve over a number field for which the analogue of Conjecture 1 is not true.

Acknowledgment: The authors gratefully acknowledge receipt of partial funding by both the DFG (Forschergruppe and Graduiertenkolleg Essen) and the Natural Science and Engineering Research Council of Canada (NSERC) (through an operating grant held by the second author) which made this joint research possible.

2 Projective p-adic Representations

As was mentioned in the introduction, we shall construct p-adic representations of the K-rational geometric fundamental group π_{1}(C, P) by making use of the p-adic Galois representation

ρ_{A,p} : G_{F} = Gal(F^{sep}/F) → GL(V_{p}(A)) ≃ GL_{2d}(ℚ_{p})

induced by the natural action of G_{F} on the ℚ_{p}-Tate module V_{p}(A) := T_{p}(A) ⊗ ℚ_{p} of a suitable abelian variety A/F. (We shall always assume tacitly that p ≠ char(F).)

Since we want this representation to factor over π_{1}(C, P), the extension

F (V_{p}(A)) := (F^{sep})^{Ker(ρ_{A,p})} = ⋃_{n} F (A[p^{n}])

of all p^{n}-th torsion points has to lie in F_{nr,P} and hence in particular has to be unramified everywhere. Now by the criterion of Néron-Ogg-Shafarevich (cf. [ST] or [FKV]) this latter condition is equivalent to

Assumption 2.1 A has good reduction everywhere over C.

Assuming this, we thus see that F (V_{p}(A)) ⊂ F_{nr,P} if and only if P splits completely in F (V_{p}(A)), i.e. in each F (A[p^{n}]), for all n ≥ 1. Now this in fact can never happen, for otherwise all the p^{n}-torsion points of the reduction \overline{A}_{P} would be K-rational, which is impossible since \overline{A}_{P} (K) is finitely generated by the Theorem of Mordell-Weil-Néron-Lang.

Moreover, a similar conclusion holds for any subrepresentation ρ_{V} : G_{F} → GL(V) associated to a (non-zero) ℚ_{p}[G_{F}]-submodule V ⊂ V_{p}(A). Indeed, if we let

F (V) := (F^{sep})^{Ker(ρ_{V} )} = ⋃ F (M/p^{n}M), where M = T_{p}(A) ∩ V,

then the same argument shows that F (V) cannot lie in F_{nr,P}, and hence we have

Proposition 2.2 No non-trivial p-adic subrepresentation of ρ_{A,p} factors over π_{1}(C, P).

Thus, none of the representations ρ_{V} induces a representation of π_{1}(C, P). However, if we replace ρ_{V} by its associated projective representation

ρ̃_{V} : G_{F} → PGL(V) = Aut(ℙ(V)),

then it turns out in some cases that ρ̃_{V} does factor over π_{1}(C, P), or equivalently, that P splits completely in the subfield

F (ℙ(V)) := F (V)^{Z(GL(V))} ⊂ F (V)

consisting of those elements of F (V) which are fixed by all the Galois automorphisms lying in the centre Z(GL(V)) of GL(V).

For our purposes it is useful to characterize this splitting property in terms of the Galois action on the Tate module of the reduction \overline{A}_{P} /K of A at P. For this, let \overline{V} ⊂ T_{p}(\overline{A}_{P}) denote the image of V under the natural identification V_{p}(A) ≃ V_{p}(\overline{A}_{P}) induced by the reduction map; clearly \overline{V} is a ℚ_{p}[G_{K}]-module. Then we have

Proposition 2.3 The projective p-adic representation ρ̃_{V} : G_{F} → PGL(V) factors over π_{1}(C, P) if and only if G_{K} acts centrally on \overline{V}, i.e. if for every g ∈ G_{K} we have

ρ\overline{V} (g) = c_{g} id\overline{V} , for some c_{g} ∈ ℚ_{p}^{×}. (1)

Proof. By the discussion above, ρ̃_{V} factors over π_{1}(C, P) if and only if P splits completely in each F (ℙ(M_{n})), for n ≥ 1, where M_{n} = M/p^{n}M ≤ A[p^{n}] and M = V ∩ T_{p}(A). Now by the first part of Corollary 5.11 of [FKV] this holds if and only if K(ℙ(\overline{M}_{n})) = K ⇔ every σ ∈ Gal(K(\overline{M}_{n})/K) acts centrally on \overline{M}_{n} ⇔ every σ ∈ G_{K} acts centrally on \overline{M}_{n}, and so the assertion follows.

Remark 2.4 Note that condition (1) holds (for any V) in the case that the reduction \overline{A}_{P} /K of A/F at P is K-isogenous to a power of a supersingular elliptic curve E/K (with all endomorphisms defined over K), i.e. \overline{A}_{P} ∼ E^{g} and dim_{ℚ} End_{K}^{0}(E) = 4.

Indeed, in this case we have End_{K}^{0}(\overline{A}_{P}) ⊗ ℚ_{p} = End_{ℚ_{p}}(V_{p}(A)) ≃ M_{2g}(ℚ_{p}), and so Im(ρ_{\overline{A}_{P} ,p}) ⊂ C_{V_{p}(A)}(End_{K}^{0}(\overline{A}_{P})) = Z(V_{p}(A)) = ℚ_{p}, which means that (1) holds.

In fact, as we shall see below, the above situation is the only case for which condition (1) holds for the full Tate module V_{p}(A); cf. Corollary 2.6 below.

In order to be able to use the result of Proposition 2.3 for the construction of suitable projective representations, we need to translate condition (1) into a geometric property of the reduction \overline{A}_{P}. This seems to be difficult for a single subrepresentation ρ̃_{V} of ρ̃_{A,p}, but if we consider instead a complete collection {ρ̃_{V_{i}}} of subrepresentations attached to a ℚ_{p}[G_{F}]-decomposition

V_{p}(A) = ⨁_{i=1}^{r} V_{i}, (2)

then such a translation is possible, particularly if we assume that the decomposition is obtained as the eigenspace decomposition of an automorphism σ ∈ Aut(A).

Indeed, if condition (1) holds for each \overline{V}_{i}, then it follows in particular that G_{K} acts diagonally on V_{p}(\overline{A}_{P}), and this imposes a very strong restriction on the abelian variety \overline{A}_{P} /K, for it implies (as we shall see below) that \overline{A}_{P} /K has complex multiplication (CM) in the sense that for every K-simple abelian subvariety B ≤ \overline{A}_{P} we have

(3) dim_{ℚ} End_{K}^{0}(B) = \frac{4}{e}(dim B)^{2}, where e = [Z(End_{K}^{0}(B)) : ℚ]; cf. Mumford [Mu2], p. 183. (Note that this condition is equivalent to existence of a subfield L ⊂ End_{K}^{0}(B) of degree [L : ℚ] = 2 dim B, which is the definition used in Serre-Tate [ST].)

Conversely, if \overline{A}_{P} has CM (and if ℚ_{p} splits End_{K}^{0}(\overline{A}_{P})), then G_{K} acts diagonally (cf. Proposition 3.1), although not necessarily via the given decomposition. However, if (2) is the eigenspace decomposition of an automorphism σ ∈ Aut(A), then we can give a complete characterization:

Theorem 2.5 Let A/F be an abelian variety with good reduction everywhere, and suppose that A has an automorphism σ ∈ Aut(A) of order N such that for some prime^{1} p ≡ 1 (mod N) the eigenspace decomposition of V_{p}(A) with respect to σ has the form

V_{p}(A) = ⨁_{(i,N)=1} V_{i}, (4)

where V_{i} = V_{p}(A)_{λ^{i}} = {v ∈ V_{p}(A) : σv = λ(σ)^{i}v} denotes the i-th eigenspace with respect to a fixed character λ : ⟨σ⟩ → ℚ_{p}^{×} of order N. In addition, assume that all these spaces have the same dimension dim_{ℚ_{p}} V_{i} = n. (5)

Then each projective p-adic subrepresentation ρ̃_{A,λ^{i}} := ρ̃_{V_{i}} : G_{F} → PGL(V_{i}) of ρ̃_{A,p} factors over π_{1}(C, P), i.e. ρ̃_{A,λ^{i}} induces a homomorphism

ρ̃_{V_{i}} : π_{1}(C, P) → PGL(V_{i}), for all i with (i, N) = 1,

if and only if the reduction \overline{A}_{P} has complex multiplication and if the algebra ℚ(σ̄) ⊂ 𝔼 := End_{K}^{0}(\overline{A}_{P}) generated by the reduction σ̄ of σ contains the centre Z(𝔼) of 𝔼, i.e. if ℚ(σ̄) ⊃ Z(𝔼).

Before proving this theorem in the next section, let us observe here that it also gives a complete answer to the question of when the “full” projective representation ρ̃_{A,p} factors over π_{1}(C, P). In particular, we see that ρ̃_{A,p} never factors over π_{1}(C, P) if char(F) = 0.

Corollary 2.6 The projective representation ρ̃_{A,p} factors over π_{1}(C, P) if and only the reduction \overline{A}_{P} is K-isogenous to a power of a supersingular elliptic curve E/K (with all endomorphisms defined over K).

Proof. If the reduction \overline{A}_{P} satisfies these conditions, then by Remark 2.4 we know that ρ̃_{A,p} factors over π_{1}(C, P). Conversely, if the latter condition holds, then by Theorem 2.5 (with σ = id_{A}) we conclude that \overline{A}_{P} has CM with centre Z(𝔼) = ℚ. Thus \overline{A}_{P} ≃ B^{r}, for some K-simple abelian variety B with CM. Since the centre of D = End_{K}^{0}(B) is Z(D) = ℚ, it follows from the classification theory (cf. [Mu2], p. 201) that either D = ℚ or that D is a quaternion algebra over ℚ. But since B has CM, equation (3) shows that this is only possible in the second case, and then dim B = 1. But this means that B is a supersingular elliptic curve (with all endomorphisms defined over K); cf. [Mu2], p. 217.

1Note that if (4) holds for one prime, then it holds for all primes p ≡ 1 (N) by [FKV], Corollary 5.4.

3 The Splitting Condition

We now analyze the meaning of the splitting condition (1) in terms of geometric properties of the reduction \overline{A}_{P}, particularly in the case that we have a decomposition (4), and thereby prove Theorem 2.5. Since this analysis only involves the reduction \overline{A}_{P} /K (and not A/F), we shall simplify the notation here by writing A in place of \overline{A}_{P}.

Thus, let A/K be an abelian variety over a finitely generated field K, and suppose that its ℚ_{p}-Tate module V_{p}(A) has a ℚ_{p}[G_{K}]-decomposition

V_{p}(A) = ⨁_{i=1}^{r} V_{i}. (6)

The aim here is to find necessary and sufficient conditions such that G_{K} acts centrally on each V_{i}; cf. (1). Since this condition is equivalent to requiring that V_{i} be contained in the χ_{i}-eigenspace

V_{p}(A)_{χ_{i}} = {v ∈ V_{p}(A) : v^{g} = χ_{i}(g)v, for all g ∈ G_{K}}

for some character χ_{i} ∈ Hom(G_{K}, ℚ_{p}), we see that this condition implies in particular that G_{K} acts diagonally on V_{p}(A), i.e. we have

V_{p}(A) = ⨁_{χ∈Hom(G_{K},ℚ_{p})}V_{p}(A)_{χ}. (7)

This latter condition can be characterized geometrically as follows.

Proposition 3.1 Let A/K be an abelian variety over a finitely generated field K, and let p ≠ char(K) be a prime. Then G_{K} acts diagonally on V_{p}(A) if and only if A/K has CM and 𝔼 := End_{K}^{0}(A) is split by ℚ_{p}, i.e. 𝔼 ⊗ ℚ_{p} is a direct sum of matrix rings over ℚ_{p}.

Proof. Since K is finitely generated, we have

𝔼_{p} := 𝔼 ⊗ ℚ_{p} = End_{G_{K}}(V_{p}(A)) (8) by the Tate Conjecture for abelian varieties which was proved by Faltings [Fa]; cf. also Schappacher [Sch].

Let us suppose first that G_{K} acts diagonally on V. Then by (8) and (7) we have

𝔼_{p} = End_{G_{K}}(V_{p}(A)) ≃^{(7)} ⨁_{χ} M_{n_{χ}}(ℚ_{p}), (9)

where n_{χ} = dim_{ℚ_{p}}(V_{p}(A)_{χ}). Thus ℚ_{p} splits 𝔼, and hence in particular, p splits completely in the centre Z = Z(𝔼), i.e. Z_{𝔭} ≃ ℚ_{p}, for every prime 𝔭|p in (the ring of integers of) Z.

Suppose now for the moment that A is simple, so 𝔼 is a skewfield (and hence is simple). In this case all the n_{χ} ≠ 0 are equal; in fact, we have

n_{χ}^{2} = dim_{Z} 𝔼, if n_{χ} ≠ 0; (10)

here Z = Z(𝔼) denotes the centre of 𝔼. To see this, note that since 𝔼⊗ℚ_{p} ≃ ∏_{𝔭|p} 𝔼⊗_{Z}Z_{𝔭}, the E ⊗_{Z} Z_{𝔭} are precisely the simple factors of 𝔼_{p}, and hence by (9) we have 𝔼 ⊗_{Z} Z_{𝔭} ≃ M_{n_{χ}}(ℚ_{p}). Since Z_{𝔭} ≃ ℚ_{p}, the assertion (10) follows by comparing dimensions.

Now from (10) (and (7)) it follows that 2 dim A = dim_{ℚ_{p}} V_{p}(A) = ne, where n^{2} = dim_{Z} 𝔼, and so by (9) we have

dim_{ℚ} 𝔼 = dim_{ℚ_{p}} 𝔼_{p} =^{(9)} n^{2}e = \frac{4}{e}(dim A)^{2},

which means that (3) holds, i.e. that A has CM.

Now suppose that A/K is arbitrary abelian variety such that G_{K} acts diagonally on V_{p}(A). Then the same is true for every abelian subvariety B ≤ A, and so, if B is simple, then the above argument shows that B has CM, which means that A has CM.

Conversely, suppose that 𝔼 is split by ℚ_{p} and that A ∼ B_{1} × … × B_{r} has CM, where each B_{i} is simple. Then by Serre-Tate [ST], Theorem 5, Corollary 2, the image of G_{K} in each V_{p}(B_{i}) is abelian, and hence the same is true for the image of G_{K} on V_{p}(A) = V_{p}(B_{1}) ⊕ … ⊕ V_{p}(B_{r}). Let R_{p} ⊂ End(V_{p}(A)) denote the ℚ_{p}-algebra generated by G_{K}. Then by Faltings[Fa] R_{p} is semi-simple, so R_{p} ≃ K_{1} ⊕ … ⊕ K_{t} is a direct sum of fields (with [K_{i} : ℚ_{p}] < ∞). Now if any K_{i} ≠ ℚ_{p}, then End_{R_{p}}(V_{p}(A)) cannot be a sum of

matrix rings over ℚ_{p}, i.e. 𝔼_{p} =^{(8)} End_{R_{p}}(V_{p}(A)) is not split, contrary to hypothesis. Thus R_{p} ≃ ℚ_{p}^{e} (for some e), and so all the characters of R_{p} are 1-dimensional, which means that V_{p}(A) has a decomposition of the form (7).

Remark 3.2 a) As the above proof shows, we can also characterize the validity of (7) by the fact that we have an isomorphism R_{p} ≃ ℚ_{p}^{e}, for some e, where, as above, R_{p} ⊂ End(V_{p}(A)) denotes the ℚ_{p}-algebra generated by the image of G_{K}.

b) A similar characterization of CM abelian varieties is given (without proof) in Serre[Se2], (2.2.6).

c) Note that if K is a finite field, then by Tate’s theorem every abelian variety A/K has CM; cf. [Mu2], Theorem 3(b) on p. 256.

d) On the other hand, we observe that G_{K} acts centrally on V_{p}(A) if and only A is Kisogenous to a power E^{g} of a supersingular elliptic curve E/K all of whose endomorphisms are defined over K. Indeed, one direction was proved in Remark 2.4 and the other in the course of the proof of Corollary 2.6. (If K is finite, then further characterizations of this condition can be found in [Mu2], Theorem 3(d) on p. 256.)

Let us now assume that A has an automorphism σ ∈ Aut(A) of order N and that p ≡ 1 (mod N). Then there exists a character λ : ⟨σ⟩ → ℚ_{p}^{×} of order N, and so V_{p}(A) admits a decomposition V_{p}(A) = ⊕V_{i} into σ-eigenspaces V_{i} = V_{p}(A)_{λ^{i}}. We assume that this decomposition has the following form:

(11) V_{p}(A) = ⨁_{(i,N)=1} V_{i}, and n := dim_{ℚ_{p}} V_{i} does not depend on i (for (i, N) = 1).

Remark 3.3 The hypothesis (11) clearly implies that dim_{ℚ_{p}} V_{p}(A) = ϕ(N)n and hence we have dim A = \frac{1}{2}ϕ(N)n. In addition, it implies that

dim_{ℚ_{p}} End_{ℚ_{p}[σ]}(V_{p}(A)) = ϕ(N)n^{2}, (12)

for V_{i} ≃ W_{i}^{n}, where W_{i} is the 1-dimensional ℚ_{p}[σ]-module affording the character λ^{i}.

Furthermore, we observe that the subalgebra ℚ(σ) ⊂ 𝔼 = End_{K}^{0}(A) generated by σ is isomorphic to the N-th cyclotomic field ℚ(ζ_{N}). Indeed, Q := ℚ(σ) is a quotient algebra of the group ring ℚ[σ] with the property that in the faithful Q ⊗ ℚ_{p}-module V precisely the irreducible components W_{i} with (i, N) = 1 appear, and any such quotient is isomorphic to ℚ(ζ_{N}).

For abelian varieties A/K satisfying property (11), we have the following geometric characterizations of the splitting condition (1):

Theorem 3.4 If A/K and σ satisfy the above property (11), then the following conditions are equivalent:

(i) G_{K} acts centrally on each V_{i}.

(ii) Im(ρ_{K}) ⊂ ℚ_{p}(σ), where ρ_{K} : G_{K} → Aut(V_{p}(A)) denotes the associated Galois representation.

(iii) A has CM and Z(𝔼) ⊂ ℚ(σ), where 𝔼 = End_{K}^{0}(A).

(iv) The centralizer C_{𝔼}(ℚ(σ)) of ℚ(σ) in 𝔼 has dimension dim_{ℚ} C_{𝔼}(ℚ(σ)) = ϕ(N)n^{2}.

(v) End_{ℚ_{p}(σ)}(V_{p}(A)) ⊂ 𝔼 ⊗ ℚ_{p}.

Proof. (i) ⇒ (ii): Since p ≡ 1 (mod N), the idempotent ε_{i} := \frac{1}{N} ∑_{k} λ(σ^{k})^{i}σ^{−k} lies in ℚ_{p}(σ); clearly V_{i} = ε_{i}V. Thus, if g ∈ G_{K}, then by hypothesis we have ρ_{K}(g)_{|V_{i}} = c_{g,i}id_{V_{i}} for some c_{g,i} ∈ ℚ_{p}^{×}, and hence ρ_{K}(g) = ∑_{i} c_{g,i}ε_{i} ∈ ℚ_{p}(σ), i.e. condition (ii) holds.

(ii) ⇒ (i): This is trivial, since by hypothesis (11) the automorphism σ and hence also all x ∈ ℚ_{p}(σ) act centrally on each V_{i}.

(ii) ⇒ (iii): The hypothesis (ii) implies in particular that G_{K} acts diagonally on V, and hence by Proposition 3.1 we see that A has CM. In addition, the hypothesis (ii) implies that R_{p} := ℚ_{p}[Im(ρ_{K})] ⊂ ℚ_{p}(σ). Now by Faltings’ results [Fa] we have

Z(𝔼_{p}) = 𝔼_{p} ∩ R_{p}, (13)

because Z(𝔼_{p}) = C(𝔼_{p}) ∩ 𝔼_{p} =^{(8)} C(C(R_{p})) ∩ 𝔼_{p} = R_{p} ∩ 𝔼_{p} by the double centralizer theorem Curtis-Reiner [CR], Theorem 59.6 (p. 405) or Exercise 26.1 (p. 178)). From this we therefore obtain that Z(𝔼_{p}) ⊂ R_{p} ⊂ ℚ_{p}(σ), and hence it follows that Z(𝔼) ⊂ ℚ(σ).

(iii) ⇒ (iv): Since ℚ(σ) is a field (cf. Remark 3.3), so is Z(𝔼), and hence 𝔼 is a simple algebra. This means that A ∼ B^{r}, for some r, where B is K-simple. Thus 𝔼 ≃ M_{r}(D), where D = End_{K}^{0}(B) is a skewfield with centre Z(D) ≃ Z(𝔼). Thus, by Remark 3.3 we have r dim B = dim A = \frac{1}{2}ϕ(N)n; (14)

Furthermore, since B/K has CM by hypothesis, we have by (3) and (14) that

(15) dim_{ℚ} 𝔼 = r^{2} dim_{ℚ} D =^{(3)} \frac{1}{e}4r^{2}(dim B)^{2} =^{(14)} \frac{1}{e}ϕ(N)^{2}n^{2},

where as before e = [Z(D) : ℚ].

Now since 𝔼 is a simple algebra and ℚ(σ) is a simple subalgebra containing the centre Z = Z(𝔼) ≃ Z(D), we have (cf. Huppert [Hu], p. 542) that

dim_{Z} C_{𝔼}(ℚ(σ)) = dim_{Z} 𝔼/ dim_{Z} ℚ(σ) = \frac{1}{e}ϕ(N)n^{2},

where the second equality follows from (15) (and from the fact that dim_{Z} ℚ(σ) = [ℚ(σ) : Z] = ϕ(N)/e). Thus dim_{ℚ} C_{𝔼}(ℚ(σ)) = e dim_{Z} C_{𝔼}(ℚ(σ)) = ϕ(N)n^{2}, as desired.

(iv) ⇔ (v): By definition we have C_{𝔼}(ℚ(σ)) ⊗ ℚ_{p} = End_{ℚ_{p}(σ)}(V) ∩ 𝔼_{p}, where V = V_{p}(A), and so End_{ℚ_{p}(σ)}(V) ⊂ 𝔼_{p} ⇔ dim_{ℚ}(C_{𝔼}(ℚ(σ))) = dim_{ℚ_{p}} End_{ℚ_{p}(σ)}(V) = ϕ(N)n^{2}, where the latter equality holds by (12).

(v) ⇒ (ii): Since End_{ℚ_{p}(σ)}(V) = C(ℚ_{p}(σ)) is the centralizer of ℚ_{p}(σ) in End_{ℚ_{p}}(V) and 𝔼_{p} = C(R_{p}) is that of R_{p} (cf. (8)), we see that hypothesis (v) implies that

C(C(ℚ_{p}(σ))) ⊃ C(E_{p}) = C(C(R_{p})).

Now since ℚ_{p}(σ) and R_{p} are both semi-simple, the double centralizer theorem (cf. [CR], loc. cit.) shows that C(C(ℚ_{p}(σ))) = ℚ_{p}(σ) and C(C(R_{p})) = R_{p}, and so R_{p} ⊂ ℚ_{p}(σ), which means that (ii) holds.

Proof of Theorem 2.5. The G_{F} -decomposition (1) of V_{p}(A) induces (via the reduction map) a G_{K}-decomposition (6) of the ℚ_{p}-Tate module \overline{V} = V_{p}(\overline{A}_{P}) reduction \overline{A}_{P}. Clearly, the image \overline{V}_{i} of the i-th σ-eigenspace V_{i} is the i-th σ̄-eigenspace of \overline{V}, and so the hypotheses imply that condition (11) holds for \overline{A}_{P} /K and σ̄.

Now by the splitting condition (Proposition 2.2) we know that all the ρ̃_{A,λ^{i}} factor over π_{1}(C, P) if and only if G_{K} acts centrally on each \overline{V}_{i}, and by the equivalence (i) ⇔ (iv) of Theorem 3.4 this happens if and only if \overline{A}_{P} /K has CM and Z(𝔼) ⊂ ℚ(σ̄).

4 Constructions via Moduli Spaces

Returning to the situation studied in section 2, we shall now use the geometry of the moduli spaces A_{g,d} and the results of Oort[Oo1] to construct (in every positive characteristic) examples of abelian varieties A/F = κ(C) whose projective p-adic representations ρ̃_{A,p} : G_{F} → PGL(V_{p}(A)) factor over π_{1}(C, P) (for suitable C and P).

As is explained below, all such examples arise from the existence of suitable morphisms f : C → A_{g,d} to the (coarse) moduli space A_{g,d} which classifies isomorphism classes of pairs (A, λ) of g-dimensional abelian varieties A/\overline{K} with a polarization λ : A → A^{t} of degree d^{2} (cf. Mumford [Mu1] or Oort[Oo1]). To make this more precise, it is convenient to introduce the following terminology:

Definition. As usual, a point x ∈ A_{g,d}(\overline{K}) is called supersingular if the abelian variety A/\overline{K} corresponding to x is supersingular (i.e. A is \overline{K}-isogenous to a power of a supersingular elliptic curve E/\overline{K}).

Furthermore, if (C, P) is a pointed curve over K (i.e. C/K is a curve and P ∈ C(K) is a K-rational point), then a non-constant ss-morphism f : (C, P) → A_{g,d} is a morphism f : C → A_{g,d} such that (a) f(P) is a supersingular point and (b) the generic point of f(C) is not supersingular (over \overline{κ(C)}).

Remark 4.1 In the above definition we could replace condition (b) by the (stronger) condition that some point x′ ∈ f(C) ⊂ A_{g,d}(\overline{K}) is not supersingular; this follows from a result of Oort (and Grothendieck/Katz) that the set of supersingular points in A_{g,d} is closed (cf. Oort[Oo2], (2.4)). Other equivalent conditions can be derived from the following proposition (which is required in the proof of Theorem 4.3 below).

Proposition 4.2 If A/F is an abelian variety over a finitely generated field F, then the following conditions are equivalent:

(i) A ⊗ \overline{F} (or A ⊗ F^{sep}) is not supersingular;

(ii) for some (or for every) p ≠ char(K), the image G̃_{F,p} := Im(ρ̃_{A,p}) of the associated projective p-adic representation ρ̃_{A,p} : G_{F} → PGL(V_{p}(A)) is infinite;

(iii) for some (or for every) p ≠ char(K), the image G̃_{F,p} contains an infinite pro-p-subgroup of finite index.

Proof. (i) ⇒ (ii): If (for some p) the image Im(ρ̃_{A,p}) were finite, then there is a finite separable extension F′/F such that the restriction ρ̃_{A⊗F′,p} of ρ̃_{A,p} to G_{F′} ≤ G_{F} is trivial. This means that G_{F′} acts centrally on V_{p}(A), and so by Remark 3.2(d) we see that A⊗F′ is supersingular, contradiction.

(ii) ⇒ (i): Suppose A ⊗ \overline{F} is supersingular; then there is a finite extension F′/F such that A ⊗ F′ is F′-isogenous to a power of a supersingular elliptic curve E/F′ with all endomorphisms defined over F′. Then by Remark 3.2(d) we see that (for each p ≠ char(K)) the image G̃_{F′,p} of G_{F′} is trivial, and hence the same is true for G̃_{F′_{sep},p}, where F′_{sep} is the separable closure of F in F′. Now G_{F′_{sep}} ≤ G_{F} is a subgroup of finite index, hence it follows that G̃_{F′_{sep},p} = {1} is a subgroup of finite index in G̃_{F,p}, i.e. G̃_{F,p} is finite.

(ii) ⇔ (iii) Since G̃_{F,p} is an infinite compact p-adic Lie group, this equivalence follows from standard facts of p-adic Lie groups (cf. [Bou], §III.7, particularly Th. 4 and Prop. 5, Cor.). Alternatively, we can deduce this directly here by observing that Gal(F (A[p^{∞}])/F (A[p^{2})]) is always a pro-p-group of finite index in Gal(F (A[p^{∞}])/F).

As was suggested above, there is a close connection between non-constant ss-morphisms f : (C, p) → A_{g,d} and projective p-adic representations ρ̃_{A,p} which factor over π_{1}(C, P); more precisely, we have

Theorem 4.3 Let K be a finitely generated field of positive characteristic and let C/K be a (smooth, projective) curve with function field F and K-rational point P.

(a) If A/F is an abelian variety of dimension g with good reduction everywhere such that the projective p-adic representation ρ̃_{A,p} factors over π_{1}(C, P) and has infinite image, then there is a non-constant ss-morphism f : (C, P) → A_{g,d} for some d ≥ 1.

(b) Conversely, if we have a non-constant ss-morphism f : (C, P) → A_{g,d}, then there is a finite extension K′/K, a finite cover φ : C′ → C ⊗ K′ with φ^{−1}(P) ⊂ C′(K′), and an abelian variety A/F′ = κ(C′) with good reduction everywhere such that the associated projective p-adic representation ρ̃_{A,p} has infinite image and factors over π(C′, P′), for all P′ ∈ φ^{−1}(P).

(c) For every g ≥ 3 and d ≥ 1 there exists a pointed curve (C, P) (defined over some finite extension K′/K) and a non-constant ss-morphism f : (C, P) → A_{g,d}.

Proof. (a) Since A/F has good reduction everywhere, the Néron model A/C is an abelian scheme. Furthermore, since A/C is projective, there exists a polarization λ on A/C of some degree d. Thus, by the modular interpretation of A_{g,d}, the pair (A, λ) gives rise to a unique morphism f : C → A_{g,d}. Since the projective p-adic representation ρ̃_{A,p} factors over π_{1}(C, P), we know by Corollary 2.6 that the reduction \overline{A}_{P} is supersingular, and so (by the modular interpretation), f(P) is a supersingular point. Furthermore, since the ρ̃ has infinite image, we have by Proposition 4.2 that A ⊗ \overline{F} is not supersingular; i.e. the generic point of f(C) is not supersingular. Thus f is a non-constant ss-morphism as desired.

(b) Let A_{g,d}^{(n)} be the fine moduli space with level n-structure (where n ≥ 3 and char(K) ∤ n), and let p_{n} : A_{g,d}^{(n)} → A_{g,d} be the associated (finite) covering map. Let C\overline{K} = C ⊗ \overline{K}, and let C_{0} be an irreducible component of p_{n}^{−1}(f(C\overline{K})) and let C_{1} be a component of the fibre product C\overline{K} ×_{f,p_{n}} C_{0}. Thus, if ν : C′ → C_{1} is the desingularization of C_{1}, then C′ is a smooth, projective curve over \overline{K} which has surjective morphisms φ : C′ → C\overline{K} and f′ : C′ → C_{0} such that π_{n} ∘ f′ = f ∘ φ, and there is a finite extension K′/K such that C′ and f′ are defined over K′ and such that all points P′ ∈ φ^{−1}(P) are K′-rational.

Since A_{g,n}^{(n)} is a fine moduli space, there exists a universal abelian scheme (with level structure) A^{univ}/A_{g,d}^{(n)}, and so its pullback A = A^{univ} ×_{f′} C′ via f′ is an abelian scheme over C′ or, equivalently, the generic fibre A = A⊗F′ is an abelian variety over F′ = κ(C′) of dimension g which has good reduction everywhere. Furthermore, since p_{n}(f′(P′)) = f(P) is supersingular for each P′ ∈ φ^{−1}(P), it follows from the modular interpretation (and by enlarging K′ if necessary) that the reduction \overline{A}_{P′} of A at P′ satisfies \overline{A}_{P′} ∼ E^{g}, for some supersingular elliptic curve E/K′ with dim End_{K}^{0}(E) = 4. Thus, by the above Remark 2.4 and Proposition 2.3 it follows that the projective p-adic representation ρ̃_{A,p} factors over π_{1}(C′, P′). Furthermore, the image G̃_{F′,p} := Im(ρ̃_{A,p}) is infinite, for otherwise A ⊗ \overline{F} would be supersingular by Proposition 4.2, which contradicts the fact that f is a non-constant ss-morphism.

(c) Let x ∈ A_{g,d}(\overline{K}) be a supersingular point. Then x ∈ W_{g,d}, where (as in Oort [Oo1]) W_{g,d} denotes the subset of points of A_{g,d}(\overline{K}) of p-rank 0. Furthermore, since g ≥ 3, there exists an abelian variety A/\overline{K} which has p-rank 0 but which is not supersingular; for example, one can choose A such that its formal group Â ∼ G_{g−1,1} + G_{1,g−1}; cf. Lenstra/Oort [LO]. (Such an A has p-rank 0 since G_{1,0} does not occur as an isogeny factor of Â (cf. Manin [Ma], Th. 2) but is not supersingular since Â ≁ gG_{1,1}; cf. [Oo1], Th. 4.2.) Furthermore, by replacing A by an isogenous abelian variety if necessary, we may assume that A has a polarization λ of degree d^{2}, and so (A, λ) corresponds to a point x′ ∈ W_{g,d} ⊂ A_{g,d}(\overline{K}). By the proof of Theorem (1.1) of [Oo1] (cf. p. 96ff), W_{g,d} is a projective subvariety of A_{g,d} of dimension dim W_{g,d} ≥ \frac{1}{2}g(g − 1) ≥ 1, so there exists a projective irreducible curve C_{0} ⊂ W_{g,d} through x and x′(cf. [Mu2], p. 56). Let f : C → C_{0} denote the normalization/desingularization of C_{0}/\overline{K}; then f and C are defined over some extension field K′/K; by construction, f : C → C_{0} ⊂ A_{g,d} is a non-constant ss-morphism.

(Note that the generic point of f(C) = C_{0} cannot be supersingular, for otherwise every specialization (i.e. every point of C_{0}) would be supersingular.)

Corollary 4.4 For any finitely generated field K of positive characteristic and any g ≥ 3 there exists a finite extension K′/K, a curve C/K′ and an abelian variety A/F = κ(C) of dimension g with good reduction everywhere such that the projective p-adic Galois representation ρ̃_{A,p} : G_{F} → P GL_{2}(V_{p}(A)) factors over π_{1}(C, P) and has an infinite image G̃_{F,p} := Im(ρ̃_{A,p}) (which therefore contains an infinite pro-p-subgroup of finite index).

Proof. The first statement follows by combining parts (c) and (a) of Theorem 4.3, and the last assertion follows from Proposition 4.2.

Remark 4.5 It seems more difficult (if not impossible) to characterize the more general situation studied in section 2 in terms of moduli spaces. Clearly, if the conditions of Theorem 2.5 hold, then we have a morphism f : C → A_{g,d} (for some d) such that f(P) is a CM-point (i.e. the reduction \overline{A}_{P} ⊗ \overline{K} is an abelian variety with CM), but the converse need not hold, even if we assume the existence of an automorphism σ ∈ Aut(A/F); cf. Remark 5.2 below.

5 Explicit Constructions via Cyclic Coverings

We now apply the results of section 2 to the abelian varieties A/F associated to certain cyclic coverings of curves which were studied in [FKV].

For this, we first observe that the conditions of Theorem 2.5 are applicable to the general situation considered in [FKV], which therefore yields a general criterion for factorizability of the projective p-adic representations {ρ̃_{λ_{i}}} over π_{1}(C, P).

Proposition 5.1 Let F_{0} = K(t), where t is transcendental over K and K contains the N-th roots of unity. Suppose that t_{1}, …, t_{n+1} are n + 1 distinct elements in K, and put t_{n+2} = t. In addition, suppose that m_{1}, …, m_{n+2} are integers with 1 ≤ m_{i} < N such that gcd(m_{1}, …, m_{n+2}, N) = 1, m_{1} + … + m_{n+2} ≡ 0 mod N and m_{i} ≠ N − m_{n+2}, for 1 ≤ i ≤ n + 1. Let π : X → ℙ_{F_{0}}^{1} denote the cyclic covering of degree N defined by the equation y^{N} = c(x − t_{1})^{m_{1}} ⋯ (x − t_{n+2})^{m_{n+2}}, (16)

where c ∈ K^{∗}, and let σ ∈ Aut(π) be a generator, which we view as an automorphism of the Jacobian J_{X} of X.

a) There is a σ-stable abelian subvariety A_{0} := J_{X}^{new} ≤ J_{X} of dimension dim A_{0} = \frac{1}{2}ϕ(N)n such that condition (4) holds for every prime p ≡ 1 (mod N).

b) A_{0} has potentially stable reduction over F_{0}; more precisely, for every cyclic covering F/K(t) which is ramified of order N at t_{1}, …, t_{n+1}, the abelian variety A = A_{0} ⊗ F has good reduction everywhere.

c) If A/F is as in b) and P ∈ C(K), where C/K is the smooth curve with function field κ(C) = F, then all the projective representations {ρ̃_{A,λ^{i}}}_{(i,N)=1} factor over π_{1}(C, P) if and only if the reduction \overline{A}_{P} has CM and if Z(End^{0}(\overline{A}_{P})) ⊂ ℚ(σ̄).

d) Suppose that the condition of d) holds, and that in addition we have p > 3 and (n, p − 1) = 1 (and also N ∤ 6 if n = 3). Then for each i with (i, N) = 1 the projective representation ρ̃_{A,λ^{i}} yields a surjection

ρ̃_{A,λ^{i}} : π_{1}(C, P) ↠ PGL_{n}(ℤ_{p}).

Proof. a) The abelian subvariety A_{0} = J_{X}^{new} defined in section 5.1 of [FKV] has this property (cf. [FKV], Theorem 5.5(a)).

b) This is (a special case of) Theorem 5.15 of [FKV].

c) In view of a) and b), this follows directly from Theorem 2.5.

d) By construction, Im(ρ̃_{A,λ^{i}}) ⊂ PGL_{n}(ℤ_{p}) = PSL_{n}(ℤ_{p}). If r_{p} : PSL_{n}(ℤ_{p}) → PSL_{n}(𝔽_{p}) denotes the homomorphism induced by reduction modulo p, then by Theorem 5.5(d) of [FKV], the map r_{p} ∘ ρ̃_{A,λ^{i}} : π_{1}(C, P) → PGL_{n}(𝔽_{p}) = PSL_{n}(𝔽_{p}) is surjective, and hence the same is true for ρ̃_{A,λ^{i}} by Serre [Se1], Lemma 3 (and exercise 1) of section IV.3.4 (pp. IV-23 and IV-27).

Proof of Theorem 1.1. By the results of [FKV], section 5.3, we know that there is an abelian variety A/F of dimension 3 which has good reduction everywhere and which has an automorphism σ ∈ Aut(A) of order 4 such that (11) holds. Furthermore, we know by that its reduction \overline{A}_{P} ∼ E^{3}, where E/K is an elliptic curve with ℚ(i) ⊂ End_{K}^{0}(E), and hence \overline{A}_{P} has CM. Furthermore, since the assumption char(K) ≡ 3 (mod 4) means that E/K is supersingular, we have Z(End_{K}^{0}(\overline{A}_{P})) ≃ End_{K}^{0}(E) = ℚ, and so we have automatically that ℤ(End_{K}^{0}(\overline{A}_{P})) ≤ ℚ(σ̄). Thus, by Theorem 2.5 we see that ρ̃_{A,λ} and ρ̃_{A,λ^{−1}} both factor over π_{1}(C, P), and by the same argument as in Proposition 5.1 we have that their image is PSL_{3}(ℤ_{p}) = PGL_{3}(ℤ_{p}), if p ≡ 5 (12).

Remark 5.2 Unfortunately, if we assume that char(K) ≢ 3 (4), which means (in the situation of Theorem 1.1) that End_{K}^{0}(E) = ℚ(i), then the condition Z(End_{K}^{0}(\overline{A}_{P})) ⊂ ℚ(σ̄) does not hold, and so ρ̃_{A,λ_{i}} does not factor over π_{1}(C, P), even though the reduction \overline{A}_{P} is a CM-variety (in fact, \overline{A}_{P} is isogenous to the third power of an elliptic curve with CM by ℚ(i), as was mentioned in the above proof).

Indeed, in this case we have that Z(\overline{A}_{P}) ≃ ℚ(i), so the desired containment holds if and only if Z(\overline{A}_{P}) = ℚ(σ̄), i.e. if and only if σ̄ ∈ Z(\overline{A}_{P}). But the following lemma shows that this is never the case:

Lemma 5.3 The reduction \overline{A}_{P} contains three isomorphic elliptic curves E_{i} ≤ \overline{A}_{P} such that \overline{A}_{P} ∼ E_{1} × E_{2} × E_{3}. Furthermore, for each i = 1, 2, 3, the automorphism σ̄ restricts to an automorphism σ̄_{i} = σ̄_{|E_{i}} ∈ Aut(E_{i}) with the property that there exist isomorphisms

f_{i} : E_{1} →^{∼} E_{i} with f_{i} ∘ σ̄_{1} = σ̄_{i}^{−1} ∘ f_{i}, for i = 2, 3.

Proof. As we shall see, we can use the elliptic curves E_{i} constructed in the proof of Lemma 5.21 of [FKV]. However, since we have to keep track of the induced σ-action, we need to review their construction from the very beginning.

Thus, let F = K(t, s) be as in Theorem 1.1, and let X/F denote (the normalization of) the curve defined by the equation

y^{4} = cx(x − 1)(x + 1)(x − a)^{3}(x − t)^{2}.

Recall that A = J_{X}^{new} is the new part of the Jacobian J_{X} of X with respect to the automorphism σ_{J} ∈ Aut(J_{X}) which is induced by the curve automorphism σ ∈ Aut_{F} (X) defined by σ^{∗}(x) = x, σ^{∗}(y) = iy, where i ∈ K is a fixed primitive fourth root of unity.

Now by [FKV], Proposition 5.18, the reduction \overline{X}_{P} at P = (a, 0) (of the associated stable model) of X/F has precisely two (smooth) irreducible components \overline{X}_{1} and \overline{X}_{2} given by the equations

y_{1}^{2} = x_{1}(x_{1}^{2} − d) and y_{2}^{4} = cx_{2}(x_{2}^{2} − 1)(x_{2} − a), (17)

respectively, where d = c/g(a). Since \overline{X}_{1} is an elliptic curve and \overline{X}_{2} is a curve of genus 3, it is clear that the reduction σ̄ of σ maps each component \overline{X}_{k} into itself, and so the induced automorphism σ̄_{J} on the reduction (\overline{J}_{X})_{P} ≃ J_{\overline{X}_{1}} ×J_{\overline{X}_{2}} has the form σ̄ = σ̄_{1} ×σ̄′_{2}.

We now claim that these automorphisms (viewed as automorphisms of \overline{X}_{k}) satisfy the relations

σ̄_{1}^{∗}x_{1} = −x_{1}, σ̄_{1}^{∗}y_{1} = −iy_{1}, and (σ̄′_{2})^{∗}(x_{2}) = x_{2}, (σ̄′_{2})^{∗}y_{2} = iy_{2}. (18)

Indeed, if x̄_{2} and ȳ_{2} denote the images of x and y under the second reduction map (corresponding to the component \overline{X}_{2}), then the proof of [FKV], Proposition 5.18, shows that x_{2} = x̄_{2} and y_{2} = ȳ_{2}/(x̄_{2} − a). Since (σ̄′_{2})^{∗}x̄_{2} = x̄_{2} and (σ̄′_{2})^{∗}ȳ_{2} = iȳ_{2}, the relation (18) is clear for σ̄′_{2}. On the other hand, if x̄_{1} and ȳ_{1} denote the images of (x − a)/s^{4} and y/s^{5} with respect to the first reduction map, then we have by the proof of Proposition 5.18 that x_{1} = ȳ_{1}^{2}/(x̄_{1}(x̄_{1} − b_{0})) and y_{1} = a_{0}ȳ_{1}/x_{1}, where a_{0} = ca(a^{2} − 1) and b_{0} = d/a_{0}. Thus, since σ̄_{1}^{∗}x̄_{1} = x̄_{1} and σ̄_{1}^{∗}ȳ_{1} = iȳ_{1}, we see that σ̄_{1}^{∗}x_{1} = (iȳ_{1})^{2}/(x̄_{1}(x̄_{1} − b_{0})) = −x_{1} and σ̄_{1}^{∗}y_{1} = a_{0}(σ̄_{1}^{∗}ȳ_{1})/(σ̄_{1}^{∗}x_{1}) = a_{0}(iȳ_{1})/(−x_{1}) = −iy_{1}, which means that (18) holds for σ̄_{1} as well.

Next, by Lemma 5.20 of [FKV] we know that J_{\overline{X}_{2}} ∼ E_{2} × E_{3} × E′, where the elliptic curves belong respectively to the three elliptic subfields F_{2}, F_{3} and F′ of index 2 of the function field K(x_{2}, y_{2}) of \overline{X}_{2}, and that these subfields are given by

F_{2} = K(z_{2}, v_{2}), F_{3} = K(z_{3}, v_{3}) and F′ = K(x_{2}, y_{2}^{2}),

where z_{2} = y_{2}(1 − b^{2}x_{2})/(x_{2}^{2} − 1), z_{3} = z_{2}(x_{2} − b^{2})/(b^{3}(1 − b^{2}x_{2})), v_{2} = z_{2}^{2}/(u − 1), and v_{3} = v_{1}/b^{2} with u := cx_{2}(x_{2} − 1)). These elements satisfy the relation

z_{k}^{2} = v_{k}(v_{k}^{2} − 1) for k = 2, 3,

and so we have an isomorphism f : E_{2} →^{∼} E_{3} such that f^{∗}v_{3} = v_{2} and f^{∗}z_{3} = z_{2}.

Furthermore, if σ̄_{k}^{∗} = (σ̄′_{2})_{|F_{k}}^{∗} denotes the restriction of (σ̄′_{2})^{∗} to the subfield F_{k} for k = 2, 3, then we have from (18)

σ̄_{k}^{∗}z_{k} = iz_{k}, σ̄_{k}^{∗}v_{k} = −v_{k}, for k = 2, 3, (19)

and so we have f ∘ σ̄_{2} = σ̄_{3}.

On the other hand, since d = d_{1}^{4} is a fourth power in K (by hypothesis), we have an isomorphism f_{2} : E_{1} := \overline{X}_{1} →^{∼} E_{2} such that f_{2}^{∗}v_{1} = d_{1}^{2}x_{1}, f_{2}^{∗}z_{1} = d_{1}^{3}y_{1}. Comparing (18) with (19) shows that f_{2} ∘ σ̄_{1} = σ̄_{2}^{−1} ∘ f_{2}, and hence a similar relation holds for

f_{3} := f ∘ f_{2} : E_{1} →^{∼} E_{3}.

Since σ̄^{2} acts non-trivially on E_{k}, we see that E_{k} ≤ \overline{A}_{P} = (\overline{J}_{X}^{new})_{P}, for k = 1, 2, 3, and so \overline{A}_{P} = E_{1} + E_{2} + E_{3} since dim \overline{A}_{P} = 3 and by the above E_{1}, E_{2}, E_{3} generate an abelian subvariety of dimension 3. Thus \overline{A}_{P} ∼ E_{1} × E_{2} × E_{3}.

Remark 5.4 Note that the above lemma also shows that Lemma 5.21(b) of [FKV] is false if char(K) ≢ 3 (mod 4), and hence the same is true for Theorem 5.22 (and for the consequence on p. 87). Thus, in both cases one has to add the condition that char(K) ≡ 3 (mod 4).

Explicitly, the error occurs in the second paragraph of the proof (of part (b)) where it is asserted (without proof) that there exist isomorphisms f_{1i} : E_{1} → E_{i} such that f_{1i} ∘ α_{1} = α_{i} ∘ f_{1i}, where the α_{i} ∈ End(E_{i}) are a certain endomorphisms of degree p.

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