A Note on the Existence of Absolutely Simple Jacobians

Ching-Li Chai and Frans Oort

Centre Emile Borel - UMS 839 IHP (CNRS/UPMC) - Paris

version 2.1, 5/03/99

The purpose of this note is to answer a question to us by M. Saïdi:

Question. Let g ≥ 2 be a positive integer. Does there exist a projective nonsingular curve of genus g over a finite field whose Jacobian is absolutely simple? If so, are there infinitely many such curves?

There is similar but simpler question, with “finite fields” replaced by “number fields”. The answer for both questions is “Yes”, see Remark 5 (v).

It is easy to show that there exist infinitely many curves over ℂ whose Jacobian is simple: the locus in the moduli space ℳ_{g} of curves of genus g, consisting of those curves whose Jacobians are not simple, is the union of a countable family of algebraic subvarieties of ℳ_{g} of lower dimension. So the union is not equal to ℳ_{g}.

Of course this naive argument does not work for the countable fields \overline{ℚ} and \overline{𝔽_{p}}.

In this note we answer this question affirmatively. We show that for any family of curves C → S over a base scheme S of finite type over 𝔽_{p} (or over ℚ) such that the monodromy is “maximal”, there exist infinitely many closed points s ∈ S such that the Jacobian of C_{s} is absolutely simple. The proof uses results on ℓ-adic representations, especially about Frobenius tori, due to Serre, Chi, Larsen and Pink. The proof also shows that the assumption on the monodromy can be weakened; this is discussed in Remark 5 (i). Also, the existence of closed points s ∈ ℳ_{g}(\overline{𝔽_{p}}) with Jac(C_{s}) absolutely simple follows from a Chebotarev density argument, so the set of all such points s has positive Dirichlet density in the set |ℳ_{g}| of all closed points of ℳ_{g,𝔽_{p}}.

Here is a sketch of the basic argument. We are given a family of abelian varieties A → S in characteristic p with an assumption on its monodromy. The goal is to show that there are many closed fibres in the family which are absolutely simple. If for a closed point s ∈ S, the ℚ_{ℓ}-Tate module T_{ℓ}(As̄) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} of the fibre A_{s} is ℚ_{ℓ}-irreducible under the action of Fr_{s}^{n} for each power of the Frobenius Fr_{s}, then A_{s} is absolutely simple. Let G^{♮} be the set of all semisimple conjugacy classes of the ℓ-adic monodromy group G for A → A. Let Ir^{♮} ⊂ G^{♮} be the set of semisimple conjugacy classes [γ] such that T_{ℓ}(As̄) ⊗_{ℤ_{ℓ}} ℚ_{ℓ} is irreducible as a ℚ_{ℓ}[γ] module. If Ir^{♮} contains an open subset of G^{♮}, then we can apply the Chebotarev density theorem to produce fibres A_{s} in the family which are absolutely simple. In general, whether the subset Ir^{♮} is empty is a property of the representation of G on the Tate module. In the exposition below, we assume for simplicity that the monodromy representation is the standard representation of the group of symplectic similitudes. In this case Ir^{♮} contains the conjugacy classes of all generators of elliptic maximal tori, and Krasner’s lemma implies the openness we need. Notations: Let S be a geometrically connected scheme smooth over a finite field 𝔽_{q} ⊇ 𝔽_{p}. Let (π : A → S, λ : A → A^{t}) be a polarized abelian scheme over S of relative dimension g. Let η (resp. η̄) be the generic point of S (resp. a geometric generic point of S). Let ℓ be a prime number different from p. Let ⟨, ⟩_{λ} be the symplectic pairing on the Tate module T_{ℓ}(Aη̄) ⊗ ℚ_{ℓ} attached to the polarization λ. Let ∗_{A,λ} be the symplectic involution on the endomorphism algebra End(T_{ℓ}(Aη̄) ⊗ ℚ_{ℓ}) defined by ⟨, ⟩_{λ}. Let Sp_{A,λ} (resp. GSp_{A,λ}) be the symplectic group (resp. the group of symplectic similitudes) in 2g variables defined by ∗_{A,λ}. Let ρ : π_{1}(S, η̄) → GSp_{A,λ}(ℚ_{ℓ}) be the ℓadic representations attached to T_{ℓ}(Aη̄). Recall that the fundamental group π_{1}(S, η̄) fits canonically into an exact sequence

1 ⟶ π_{1}(\overline{S}, η̄) ⟶ π_{1}(S, η̄) ⟶ Gal(\overline{𝔽_{q}}/𝔽_{q}) ⟶ 1 ,

where \overline{S} = S ×_{Spec 𝔽_{q}} Spec \overline{𝔽_{q}}.

Let G = G_{A} = ρ(π_{1}(S)) (resp. G_{1} = G_{A,1} = ρ(π_{1}(\overline{S}, η̄)) ) be the image of the fundamental group (resp. the geometric fundamental group) in GSp_{A,λ}(ℚ_{ℓ}). Let G^{alg} = G_{A}^{alg} (resp. G_{1}^{alg} = G_{A,1}^{alg}) be the Zariski closure of G_{A} (resp. G_{A,1}) in GSp_{A,λ}. Construction/Definition: For each closed point s ∈ S, there is a Frobenius element Frob_{s} in π_{1}(S, η̄) attached to s, well-defined up to conjugation.

The neutral component of the ℚ_{ℓ}-Zariski closure of the subgroup of GSp_{A,λ} generated by a Frobenius element ρ(Frob_{s}) is a subtorus of GSp_{A,λ}; therefore it is also a subtorus of the Zariski closure G_{A}^{alg}. This subtorus T_{s} of G_{A}^{alg} is well-defined up to conjugation in G_{A}^{alg}(ℚ_{ℓ}). We call it “the” Frobenius torus attached to the closed point s, and denote it by T_{s}.

Remark 1. The notion of Frobenius torus is due to Serre, see [4, §3], [7, §4, §7]. The Frobenius torus T_{s} can be defined over ℚ; here we use its “ℓ-adic realization”.

In the next lemma we show that under suitable conditions on the monodromy of A → S, there are many Frobenius tori T_{s} which are elliptic maximal tori; in other words T_{s} is anisotropic modulo the center of GSp_{2g,λ}. We denote by |S| the set of all closed points of S, and by |S|_{emFt} the subset of all s ∈ |S| such that the Frobenius torus T_{s} is an elliptic maximal torus of GSp_{A,λ} over ℚ_{ℓ}.

Lemma 2. Notations as above. Assume that the Zariski closure of the image ρ(π_{1}(\overline{S}, η̄)) of the geometric fundamental group is equal to Sp_{A,λ}. Then |S|_{emFt} has positive Dirichlet density in |S|. In particular |S|_{emFt} is an infinite set.

Proof. This statement is certainly known. We reproduce a proof for the convenience of the reader. As a first step, the assumption implies that G is an open subgroup in G^{alg}(ℚ_{ℓ}) = GSp_{A,λ}(ℚ_{ℓ}). Admit this for the moment.

The ℚ_{ℓ}-rational maximal tori in GSp_{A,λ} are given by maximal commutative semisimple subalgebras B in End(T_{ℓ}(Aη̄) ⊗ ℚ_{ℓ}) which are stable under the involution ∗_{A,λ}. For any such subalgebra B, the ℚ_{ℓ}-subspace of elements fixed under ∗_{A,λ} has dimension g = \frac{1}{2} dim_{ℚ_{ℓ}}(B). Among these maximal tori, the elliptic ones correspond to subfields of dimension 2g over ℚ_{ℓ}, and they are known to exist. For any element x ∈ GSp_{A,λ}(ℚ_{ℓ}), the neutral component of the Zariski closure of the subgroup generated by x is an elliptic maximal torus if and only if the algebra ℚ_{ℓ}[x] ⊂ End(T_{ℓ}(Aη̄) ⊗ ℚ_{ℓ}) is a field. Let Em be the subset of G consisting of all elements of x ∈ G such that ℚ_{ℓ}[x] ⊂ End(T_{ℓ}(Aη̄) ⊗ ℚ_{ℓ}) is a field of degree 2g over ℚ_{ℓ}. By Krasner’s lemma [6, Prop. 3, p. 43], Em is an open subset of G. More concretely, if ℚ_{ℓ}[x_{0}] is a field of degree 2g over ℚ_{ℓ}, then for all x ∈ GSp_{A,λ}(ℚ_{ℓ}) sufficiently close to x_{0} in the ℓ-adic topology, the characteristic polynomial and the eigenvalues of x are close to those for x_{0}. So ℚ_{ℓ}[x] is a field of degree at least 2g, therefore the degree is exactly 2g. It is known that Em is not empty; so there exists an open normal subgroup N ⊂ G such that Em is a union of cosets for N.

Recall that Chebotarev’s density theorem states that for any finite étale Galois covering X → Y between normal integral schemes of finite type over ℤ, the Frobenius conjugacy classes in the covering group attached to the closed points of Y is equidistributed, see [9, Thm. 7, p. 91]. Applying this density theorem to the finite étale covering of S corresponding to the finite quotient G → G/N of the monodromy group G, we see that there exist infinitely many Frobenius tori which are elliptic maximal tori in GSp_{A,λ} over ℚ_{ℓ}. This finishes the proof of Lemma 2, except for the fact that G is open in G^{alg}(ℚ_{ℓ}). This is done in the next two paragraphs. The reader may wish to skip them because it is a little technical and not central to this note; cf. Remark 3 (i).

Both G and G_{1} are closed subgroups of GSp_{A,λ}(ℚ_{ℓ}); hence they are analytic Lie subgroups of GSp_{A,λ}(ℚ_{ℓ}). Let 𝔤 (resp. 𝔤_{1}) be the Lie algebra of G (resp. G_{1}), and let 𝔤^{alg} (resp. 𝔤_{1}^{alg}) be the Lie algebra of G^{alg} (resp. G_{1}^{alg}). The assumption of the theorem means that 𝔤_{1}^{alg} is equal to the Lie algebra 𝔰𝔭_{2g,λ} of Sp_{A,λ}. Therefore 𝔤^{alg} is equal to the Lie algebra 𝔤𝔰𝔭_{2g,λ} of GSp_{A,λ}.

By [3, §7, Cor. 7.9], we have [𝔤_{1}^{alg}, 𝔤_{1}^{alg}] ⊆ 𝔤_{1}. We reproduce a proof here. The algebraic subgroup of GSp_{A,λ} consisting of all elements x ∈ GSp_{A,λ} such that Ad(x)(𝔤_{1}) ⊆ 𝔤_{1} clearly contains G, hence [𝔤_{1}^{alg}, 𝔤_{1}] ⊆ 𝔤_{1}. Repeating this argument again, one concludes that [𝔤_{1}^{alg}, 𝔤_{1}^{alg}] ⊆ 𝔤_{1}.

From [𝔤_{1}^{alg}, 𝔤_{1}^{alg}] ⊆ 𝔤_{1}, we deduce that 𝔤_{1} = 𝔰𝔭_{2g,λ} and 𝔤 = 𝔤𝔰𝔭_{2g,λ}. Especially G is an open subgroup of GSp_{A,λ}(ℚ_{ℓ}).

Remarks 3. (i) The argument to show that 𝔤_{1} is big since it contains the derived algebra of its algebraic envelope appeared in [2].

(ii) It is easy to find examples which satisfy the assumption in Lemma 2. For instance, one can take S to be equal to the moduli space ℳ_{g,n} of curves of genus g over 𝔽_{p} with the principal symplectic level-n structure, or a generic curve in ℳ_{g}, with n ≥ 3 and (n, pℓ) = 1. In this case the monodromy group G is equal to GSp_{2g}(ℚ_{ℓ}); this is equivalent to [5, Thm. 5.15, p. 108], that the moduli spaces ℳ_{g,nℓ^{i}} are geometrically connected. Then Proposition 4 below answers the Question at the beginning of this note.

Proposition 4. Notation and assumptions as in Lemma 2. Then for each s ∈ |S|_{emFt}, the fiber A_{s} is an absolutely simple abelian variety.

Proof. Suppose that s ∈ S is a closed point such that the Frobenius torus T_{s} is an elliptic maximal torus in GSp_{A,λ} over ℚ_{ℓ}. We know that each elliptic maximal torus T in GSp_{A,λ} is given by a subfield F in End(T_{ℓ}(Aη̄) ⊗ ℚ_{ℓ}) of degree 2g stable under ∗_{A,λ}, hence T acts irreducibly on T_{ℓ}(Aη̄) ⊗ ℚ_{ℓ}. So the action of any open subgroup of Gal(s̄/s) on the ℓ-adic Tate module of A_{s} is irreducible over ℚ_{ℓ}. This implies that A_{s} is not isogenous to a non-trivial product over any finite extension of the residue field κ(s) of s.

Remarks 5. (i) In the proof above, what we really need about the monodromy of π : A → S is that the neutral component G^{alg} of the Zariski closure of ρ(π_{1}(S)) has an elliptic maximal torus T over ℚ_{ℓ} such that the restriction to T of the representation of G^{alg} on the ℓ-adic Tate module is irreducible. For instance, if G^{alg} is isomorphic to the derived group (Res_{E/ℚ_{ℓ}}GL_{2})^{der} of Res_{E/ℚ_{ℓ}}GL_{2}) for a finite extension E of ℚ_{ℓ} and the representation is isomorphic to the standard representation of Res_{E/ℚ_{ℓ}}GL_{2}), the argument still works and one gets absolutely simple fibers in the abelian scheme in question.

(ii) On the other hand, suppose π : A → S is the universal family of abelian surfaces over the Shimura curve S attached to a quaternion division algebra B over ℚ, such that B is split at p, and ℓ is a prime number different from p. Then the representation of every Frobenius torus T_{s} on the ℓ-adic Tate module is reducible. The group G^{alg} is the group attached to B^{opp,×} where B^{opp} is the opposite algebra of B, and the representation on the ℓ-adic Tate module comes from the regular representation of B^{opp}. In this example one can show that none of the fibres A_{s} is absolutely simple.

(iii) If the representation of the Frobenius torus T_{s} on the ℓ-adic Tate module is reducible, then End(As̄) ⊗ ℚ_{ℓ} contains a nontrivial idempotent. But this in general does not imply that End(As̄) ⊗ ℚ contains a nontrivial idempotent, so A_{s} over the closed point s may or may not be absolutely simple.

(iv) If A_{K} is an abelian variety of dimension g over a number field such that the Zariski closure of the image of the ℓ-adic Galois representation attached to A is equal to GSp_{2g}, the same argument shows that there exist finite places v of K such that the reduction A_{v} of A at v is an absolutely simple abelian variety; such places v form a subset of positive density. One can also change GSp_{2g} to a group satisfying the property specified in (i).

By [8, 1.4], due to Serre, for any abelian variety A_{L} over a field L of finite type over ℚ, there exist specializations B_{K} to number fields K, such that the images of the ℓ-adic Galois representations for A_{L} and B_{K} are equal.

Therefore there exist examples satisfying the assumption in the previous paragraph.

(v) Proposition 4 gives an affirmative answer to the Question at the beginning of this note. The similar question, with \overline{𝔽_{p}} replaced by \overline{ℚ}, was answered by [8, Cor. 1.5]. and also by [1, Thm. 0.6.1, p. 8]. One can also deduce it from the \overline{𝔽_{p}}-version because of Lemma 6 below.

Lemma 6. Suppose that A_{K} is an abelian variety over a number field K with good reduction at a finite place v of K. If the reduction A_{v} of A is absolutely simple, then the abelian variety A is also absolutely simple.

Proof. Since the hypothesis does not change if one extends the base field K to a bigger number field L, it suffices to prove that the abelian variety A is simple over K. Recall that whether the abelian variety A is simple or not is a question on the existence of an endomorphism T of A such that T^{2} = n · T for some non-zero integer n, while Ker(T) is not finite. Since every endomorphism of A over K extends to an endomorphism of the abelian scheme A over 𝒪_{K,v} which extends A, we are done.

Acknowledgement: The authors would like to thank B. Moonen, R. Noot and Y. Zarhin for suggestions. The work was done at Institut Henri Poincaré during the 1999 Diophantine Geometry Trimester. We thank Centre Emile Borel for hospitality and for providing a stimulating environment.

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Ching-Li Chai Department of Mathematics University of Pennsylvania 209 South 33rd St. Philadelphia, PA 19104-6395 email: chai@math.upenn.edu

Frans Oort Mathematisch Instituut Budapestlaan 6 NL: 3508 TA Utrecht The Netherlands email: oort@math.uu.nl