Endomorphism algebras of Jacobians

Jordan S. Ellenberg Princeton University

                     ellenber@math.princeton.edu

June 26, 2000

Abstract

Very little is known about the existence of curves, and families of curves, whose Jacobians are acted on by large rings of endomorphisms. In this paper, we show the existence of curves X with an injection K ↪ Hom(Jac(X), Jac(X)) ⊗_{ℤ} ℚ, where K is a subfield of even index at most 10 in a primitive cyclotomic field ℚ(ζ_{p}), or a subfield of index 2 in ℚ(ζ_{pq}) or ℚ(ζ_{p^{α}}). This result generalizes previous work of Brumer, Mestre, and Tautz-Top-Verberkmoes. Our curves are constructed as branched covers of the projective line, and the endomorphisms arise as quotients of double coset algebras of the Galois groups of these coverings. In certain cases, we show the existence of continuous families of Jacobians admitting the desired endomorphism algebras. At the end, we raise some questions about upper bounds for endomorphism algebras of the type described here, and ask about a “non-Galois version of Hurwitz’s theorem.”

2000 MSC classification: 14H40 (14H30)

Keywords: Jacobian, endomorphism, Hecke, double coset, Hurwitz.

Introduction

Let k be an algebraically closed field, A/k a principally polarized abelian variety, and End(A) the ℚ-algebra of endomorphisms of A. It has long been known which algebras E arise as End(A) for some principally polarized abelian variety A [20, §21]. If we restrict our attention to Jacobians of smooth curves of a given genus, however, the question is less well understood. Van der Geer and Oort remark:

“...one exepcts excess intersection of the Torelli locus and the loci corresponding to abelian varieties with very large endomorphism rings; that is, one expects that they intersect much more than their dimensions suggest.”[25, §5]

This expectation has been borne out by many examples of families of curves whose Jacobians are acted on by large algebras. Theorems of Tautz-Top-Verberkmoes [24], Mestre [16], and Brumer [2] produce, for each odd prime p, families of curves X of genus (p − 1)/2 such that End(Jac(X)) contains the real cyclotomic field ℚ(ζ_{p} + ζ_{p}^{−1}). Brumer also produces a family of genus 2 curves with ℚ(\sqrt{2}) ⊂ End(Jac(X)). Hashimoto-Murabayashi [13] and Bending [1] have produced families of genus 2 curves whose Jacobians are acted on by some quaternion algebras of small discriminant. De Jong and Noot give examples of infinitely many curves of genus 4 and 6 whose Jacobians have complex multiplication [4]. Ekedahl-Serre [8] produce many examples of curves of genus g with ℚ^{⊕g} ∈ End(Jac(X)). Finally, Shimada [23] uses methods similar to ours to produce examples of curves of large genus whose Jacobians are acted on by quadratic fields.

In the present article we present a general procedure for constructing curves whose Jacobians have large endomorphism algebras, and we show that the endomorphisms produced in [24],[16], and [2], and most of those in [8], arise via our construction. Furthermore, we show the existence of some new families of curves whose Jacobians are acted on by totally real fields.

If X is a curve whose Jacobian has real multiplication by a totally real field F, we say for short that X has real multiplication by F. We denote the index n subfield of ℚ(ζ_{p}) by ℚ(ζ_{p}^{(n)}). Our main theorem is the following.

Theorem. Let k be an algebraically closed field. Then

• If p > 5 is a prime not equal to char k, there exists a 3-dimensional family of curves over k with real multiplication by ℚ(ζ_{p} + ζ_{p}^{−1}).

• If p is a prime congruent to 1 mod 4, and char k does not divide 2p, then there exists a 1-dimensional family of curves over k with real multiplication by ℚ(ζ_{p}^{(4)}).

• If p is a prime congruent to 1 mod 6, and char k does not divide 6p, then there exists a 1-dimensional family of curves over k with real multiplication by ℚ(ζ_{p}^{(6)}).

• If p is a prime congruent to 1 mod 8, and char k does not divide 2p, then there exists a curve over k with real multiplication by ℚ(ζ_{p}^{(8)}).

• If p is a prime congruent to 1 mod 10, and char k does not divide 10p, then there exists a curve over k with real multiplication by ℚ(ζ_{p}^{(10)}).

• If p, q are odd primes not equal to char k, there exists a 1-dimensional family of curves over k with real multiplication by ℚ(ζ_{pq} + ζ_{pq}^{−1}).

• If p is an odd prime not equal to char k, and α > 1 is an integer, there exists a 2-dimensional family of curves over k with real multiplication by ℚ(ζ_{p^{α}} + ζ_{p^{α}}^{−1}).

Let X_{0}/k be a smooth curve, and Y/X_{0} a Galois cover with Galois group G. Let H be a subgroup of G and let X be the quotient curve Y/H. Then the Jacobian of X is acted on by the double coset algebra ℚ[H\G/H]. We say that the image of ℚ[H\G/H] in End(X) is of Hecke type. The study of endomorphism algebras of Hecke type is the main purpose of the present paper. In section 1, we explain how to compute the endomorphism algebra of Hecke type associated to a branched cover X → X_{0}. We construct the families of curves referred to above in sections 2 and 3, and show that these families have the stated dimensions in section 4.

The negative side of the question–that is, the question of which algebras cannot occur in End(Jac(X))–is more mysterious. One can also ask weaker questions about bounds on algebras which can occur as endomorphism algebras of Hecke type attached to smooth curves. We discuss the current state of knowledge about such questions in section 5.

Acknowledgments

It is my pleasure to thank Armand Brumer, who generously communicated his ideas to me and thereby inspired this paper. I also thank Johan de Jong, Robert Guralnick, Adam Logan, and Stefan Wewers for useful discussions.

Notation

If V is a vector space acted on by a group H, denote by V^{H} the subspace of V fixed by H.

A “curve” will always be a nonsingular projective algebraic curve over a field unless specifically declared otherwise.

If A is an abelian variety, End(A) is its ℚ-algebra of endomorphisms Hom(A, A) ⊗_{ℤ} ℚ.

If p is a prime and n a divisor of p − 1, we denote by ℚ(ζ_{p}^{(n)}) the subfield of index n in the cyclotomic field ℚ(ζ_{p}). We often write ℚ(ζ_{p}^{+}) for the real cyclotomic field ℚ(ζ_{p}^{(2)}).

1 Endomorphism algebras of Hecke type

Let Y be a nonsingular projective algebraic curve over an algebraically closed field k, and let G be a finite group which acts on Y. Write C for the quotient of Y by G (Precisely, C is the nonsingular curve associated to the G-invariant subfield of k(Y).)

The map

ϕ : G → Aut(Y)

induces a map

e : ℤ[G] → End(Jac(Y)).

Now let H be a subgroup of G, and let X be the quotient curve Y/H. Define π_{H} ∈ End(Jac(Y)) by

∑_{h∈H} e(h).

Then there is a natural action

ℚ[H\G/H] → End(π_{H} Jac(Y)).  (1.1)

The natural map

Jac(X) → Jac(Y)

restricts to an isogeny

Jac(X) → π_{H} Jac(Y).

Therefore, the action (1.1) yields an action

ℚ[H\G/H] → End(Jac(X)).

We refer to the image of ℚ[H\G/H] in End(Jac(X)) as an endomorphism algebra of Hecke type, and denote it by H_{X/C}. In general, we say an element α of End(Jac(X)) is of Hecke type if there exists some branched cover X → C such that α ∈ H_{X/C}. We note in passing that there exist Jacobians of smooth curves with endomorphisms which are not of Hecke type. For if α is of Hecke type, then ℚ(α) is a field contained in a double coset algebra ℚ[H\G/H], and is therefore an abelian extension of ℚ. Now the Jacobians of smooth genus 3 curves form an open subscheme of the moduli space of principally polarized abelian three-folds; in particular, there exist Jacobians of smooth genus 3 curves which have real multiplication by non-abelian cubic fields. The endomorphisms of these Jacobians are thus not of Hecke type.

Let ℓ be a a prime not equal to the characteristic of k. The natural map

End(Jac(X)) → End(H_{1}(X, ℚ_{ℓ}))

is injective. Thus, to compute H_{X/C} it suffices to study the representation of ℚ[H\G/H] on H_{1}(X, ℚ_{ℓ}) ≅ π_{H}H^{1}(Y, ℚ_{ℓ}). This representation, in turn, is determined by the action of G on H^{1}(Y, ℚ_{ℓ}). Refer to this representation of G as ρ_{Y}. One can compute ρ_{Y} from the branching data of the map Y → C, by means of the following proposition. Notation: for each g ∈ G, denote by χ_{g} the character of the representation of G induced from the trivial representation on the cyclic subgroup generated by g. Write χ_{triv} for the trivial character of G.

Proposition 1.1. Suppose the map Y → C is branched at r points p_{1}, … p_{r} ∈ C(k), with monodromy g_{1}, …, g_{r}, and suppose that each g_{i} has order prime to char k. Then

χ(ρ_{Y}) = 2χ_{triv} + 2(g(C) − 1)χ_{1} + ∑_{i}(χ_{1} − χ_{g_{i}}).  (1.2)

Remark 1.2. Let U be the open curve C − {p_{1}, …, p_{r}}. If t ∈ k(C) is a uniformizer at p_{i}, then we have a morphism

Spec k((t)) → U

which induces a map of étale fundamental groups

Gal((k((t)))^{sep}/k((t))) → π_{1}(U),

defined only up to conjugacy, since we have neglected to specify basepoints. The étale G-cover Y ×_{C} U → U allows us to extend this diagram to

Gal((k((t)))^{sep}/k((t))) → π_{1}(U) → G.

When we say “Y → C has monodromy g_{i} at p_{i},” we mean that the composite homomorphism above factors through the tame quotient of Gal((k((t)))^{sep}/k((t))), and that the image of a generator of tame inertia lies in the conjugacy class of g_{i}. In case k = ℂ, this definition coincides with the natural topological one.

Remark 1.3. It seems natural to call a character χ of a finite group a Hurwitz character if χ is of the form (1.2) for some set of generators g_{1}, …, g_{r} with g_{1} … g_{r} = 1. It follows from Proposition 1.1 and the Riemann Existence theorem that a Hurwitz character is the character of a representation of G; a purely group-theoretic proof of this fact is given by Scott in [22]. The question of which characters of a finite group are Hurwitz characters is purely combinatorial, and seems quite mysterious; we will discuss this question further in section 5.

Proof. The proposition follows from the Lefschetz fixed point formula in étale cohomology; see for instance [18, V, Cor. 2.8]. The formula given there for χ(ρ_{Y}) is

χ(ρ_{Y}) = 2χ_{triv} + 2(g(C) − 1)χ_{1} + ∑_{i}(a_{p_{i}}),

where a_{p_{i}} is the Artin representation associated to the branch point p_{i} ∈ C(k). It thus suffices to show that

a_{p_{i}} = χ_{1} − χ_{g_{i}}.

Let y_{1}, …, y_{m} be the points of Y (k) lying over p_{i}. Take g ∈ G with g ≠ 1. Then −(χ_{1}(g) − χ_{g_{i}}(g)) is the number of the y_{i} which are fixed by g. On the other hand,

a_{p_{i}}(g) = − ∑_{j} i_{j}(g)

where i_{j}(g) is the multiplicity of the fixed point of g at y_{j}. Since char k is prime to the order of g_{i}, the map Y → C is tamely ramified at y, so i_{j}(g) = −1 for each y_{j} which is a fixed point. ([18, p. 188].) We have thus shown that a_{p_{i}}(g) = χ_{1}(g) − χ_{g_{i}}(g) for all g ≠ 1; since a_{p_{i}} and χ_{1} − χ_{g_{i}} are both orthogonal to the trivial character, they are equal.

We recall for future reference Grothendieck’s computation of the prime-to-p fundamental group of a curve, which we use here as a replacement for the Riemann existence theorem in characteristic p.

Proposition 1.4. Let k be an algebraically closed field. Let G be a finite group whose order is prime to char k, or an arbitrary finite group if char k = 0. If g_{1}, …, g_{r} are generators for G satsifying g_{1} … g_{r} = 1, there exists an Galois covering Y → ℙ^{1} with Galois group G, branched at r points with monodromy g_{1}, …, g_{r}.

Proof. Immediate from [10, XIII,2.12].

We will give examples in the following sections to show that judicious choices of {G, H, g_{1}, …, g_{r}} yield interesting endomorphism algebras of Hecke type.

2 Metacyclic groups

Let F be a totally real number field. We say an abelian variety A has real multiplication by F if dim A = [F : ℚ] and there exists an injection F ↪ End(A). If the Jacobian of a curve has real multiplication by F, we will say for short that the curve has real multiplication by F. In this section, we use covers of ℙ^{1} with metacyclic Galois groups in order to produce curves with real multiplication by certain totally real number fields.

Let p be a prime, and n be an integer such that n|p − 1. Let G = G_{p,n} be the metacyclic group

< σ, α : σ^{p} = α^{n} = 1, ασα^{−1} = σ^{k} >

where k is an element of order n in (ℤ/pℤ)^{∗}. Let H be the subgroup of G generated by α. The irreducible representations of G fall into two types:

• n one-dimensional representations V_{a}, indexed by a ∈ ℤ/nℤ; these arise as compositions

G → ℤ/nℤ → ℂ^{∗}

where the second map sends 1 to e^{2πia/n}.

• (p − 1)/n representations of dimension n.

Denote by W the direct sum of all the n-dimensional irreducible representations of G.

Let g_{1}, …, g_{r} be non-trivial elements of G, and for each i in 1, …, r let d_{i} be either 0 (if g_{i} has order p) or n/ord(g_{i}) (if g_{i} has order dividing n.) Note that d_{i} determines the cyclic group generated by g_{i} up to conjugacy. We can compute

χ_{g_{i}} = d_{i}χ(W) + ∑_{a∈ℤ/nℤ:ad_{i}=0} χ(V_{a}).

Suppose now that k is an algebraically closed field of characteristic prime to |G|. Let Y be a Galois cover of ℙ^{1}, with Galois group G, branched at r points with monodromy g_{1}, …, g_{r}. (Note that the product g_{1} … g_{r} must then be the identity.) It follows from Proposition 1.1 that

χ(ρ_{Y}) = mχ(W) + ∑_{a∈ℤ/nℤ} c_{a}χ(V_{a})

where

c_{a} = \{ \begin{matrix}−2 + |\{i : ad_{i} ≠ 0\}| & a ≠ 0 \\ 0 & a = 0 \end{matrix}

and

m = −2n + ∑_{i}(n − d_{i}).

Let X be the quotient curve Y/H. Since no non-trivial element of V_{a} is fixed by H for a ≠ 0, we have

H_{1}(X, ℚ) ≅ (W^{H})^{⊕m}

as ℚ[H\G/H]-modules. The image of ℚ[H\G/H] in End(W^{H}) is isomorphic to the subfield of index n in ℚ(ζ_{p}). The dimension of W^{H} is (p − 1)/n. Therefore, if n is even and m = 2, we have a real field of degree (p − 1)/n acting on Jac(X), which is of dimension (p − 1)/n. So Jac(X) has real multiplication. An elementary computation shows that the only possibilities for n and {d_{i}} yielding m = 2 are:

1. n = 2; {d_{i}} = {0, 0, 1, 1}

2. n = 2; {d_{i}} = {0, 1, 1, 1, 1}

3. n = 2; {d_{i}} = {1, 1, 1, 1, 1, 1}

4. n = 4; {d_{i}} = {0, 1, 1}

5. n = 4; {d_{i}} = {1, 1, 2, 2}

6. n = 6; {d_{i}} = {2, 2, 3, 3}

7. n = 6; {d_{i}} = {1, 1, 2}

8. n = 8; {d_{i}} = {1, 1, 4}

9. n = 10; {d_{i}} = {1, 2, 5}.

The first three cases give examples of curves whose Jacobians have real multiplication by real cyclotomic fields ℚ(ζ_{p}^{+}). The families of curves described by Tautz, Top, and Verberkmoes [24] fall under cases 1 and 2 above. In case 3, the curve Y is a cover of ℙ^{1} whose Galois group is the dihedral group D_{2p}. The subcover corresponding to ℤ/pℤ ∈ D_{2p} is a double cover of ℙ^{1} branched over six points–that is, a genus 2 curve C′. The curves X in case 3 are exactly those produced by Brumer [2] as quotients of étale covers of genus 2 curves. The curves produced by Mestre in [16] are special cases of case 3, arising when C′ admits a non-constant morphism to an elliptic curve.

In cases 5 and 6, the argument above results in the following two propositions.

Proposition 2.1. Let g_{2}, g′_{2}, g_{4}, g′_{4} be elements of G_{p,4}, of orders 2, 2, 4, 4 respectively, satisfying g_{2}g′_{2}g_{4}g′_{4} = 1. Let H be an order-4 subgroup of G_{p,4}. Let k be an algebraically closed field of characteristic greater than 2.

Suppose Y/k is a Galois cover of ℙ^{1}/k, with Galois group G_{p,4}, branched at four points with monodromy g_{2}, g′_{2}, g_{4}, g′_{4}, and let X = Y/H.

Then X has real multiplication by the index-4 subfield of ℚ(ζ_{p}).

Proposition 2.2. Let g_{2}, g′_{2}, g_{3}, g′_{3} be elements of G_{p,6}, of orders 2, 2, 3, 3 respectively, satisfying g_{2}g′_{2}g_{3}g′_{3} = 1. Let H be an order-6 subgroup of G_{p,6}. Let k be an algebraically closed field of characteristic greater than 3.

Suppose Y/k is a Galois cover of ℙ^{1}/k, with Galois group G_{p,6}, branched at four points with monodromy g_{2}, g′_{2}, g_{3}, g′_{3}, and let X = Y/H.

Then X has real multiplication by the index-6 subfield of ℚ(ζ_{p}).

Case 4 provides further examples of a curve with real multiplication by ℚ(ζ_{p}^{(4)}); likewise, case 7 provides further examples of real multiplication by ℚ(ζ_{p}^{(6)}). Cases 8 and 9 bring new fields of real multiplication into the picture; we record the results in the following propositions.

Proposition 2.3. Let g_{1}, g_{2}, g_{3} be elements of G_{p,8}, of orders 2, 8, 8 respectively, satisfying g_{1}g_{2}g_{3} = 1. Let H be an order-8 subgroup of G_{p,8}. Let k be an algebraically closed field of characteristic greater than 2.

Suppose Y/k is a Galois cover of ℙ^{1}/k, with Galois group G_{p,8}, branched at three points with monodromy g_{1}, g_{2}, g_{3}, and let X = Y/H.

Then X has real multiplication by the index-8 subfield of ℚ(ζ_{p}).

Proposition 2.4. Let g_{1}, g_{2}, g_{3} be elements of G_{p,10}, of orders 2, 5, 10 respectively, satisfying g_{1}g_{2}g_{3} = 1. Let H be an order-10 subgroup of G_{p,10}. Let k be an algebraically closed field of characteristic prime to 10.

Suppose Y/k is a Galois cover of ℙ^{1}/k, with Galois group G_{p,10}, branched at three points with monodromy g_{1}, g_{2}, g_{3}, and let X = Y/H.

Then X has real multiplication by the index-10 subfield of ℚ(ζ_{p}).

It follows from Proposition 1.4 that covers satisfying the conditions of Propositions 2.3 and 2.4 exist whenever the characteristic of k is prime to 2p or 10p respectively.

3 More dihedral groups

In this section, we produce some curves whose Jacobians have real multiplication using dihedral groups of order 2m, where m is odd but not necessarily prime. Let G = D_{m} be the dihedral group of order 2m:

< σ, τ : σ^{m} = τ^{2} = 1, τστ^{−1} = τ^{−1} >

The group G has two irreducible complex representations of dimension 1, and (m−1)/2 of dimension 2. Denote by V_{1} and V_{−1} the representations of dimension 1, and by W the direct sum of the 2-dimensional representations. Note that W splits over ℚ into a direct sum of irreducible ℚrepresentations

W = ⊕_{d|m:d≠1}W_{d}

where dim W_{d} = ϕ(d).

Let H be the subgroup of G generated by τ. Then the image of ℚ[H\G/H] in End(W_{m}) is the field ℚ(ζ_{m}^{+}) generated by ζ_{m} + ζ_{m}^{−1}, where ζ_{m} is a primitive mth root of unity.

Suppose m = pq, where p and q are odd primes. Let g_{1}, g_{2}, g_{3}, g_{4} be elements of G conjugate to τ, τ, σ^{p}, σ^{q} respectively, and satisfying

g_{1}g_{2}g_{3}g_{4} = 1.

Let k be an algebraically closed field of characteristic prime to |G|, and let Y be a Galois cover of ℙ^{1}, branched at 4 points with monodromy g_{1}, g_{2}, g_{3}, g_{4}. It then follows from Proposition 1.1 that

(ρ_{Y})^{H} = (W_{pq}^{H})^{⊕2}.

In particular, if X is the quotient curve Y/H, then X is a curve of genus (1/2)(p − 1)(q − 1) whose Jacobian has real multiplication by ℚ(ζ_{pq}^{+}).

Now suppose m = p^{α}, with p an odd prime and α > 1. Let g_{1}, g_{2}, g_{3}, g_{4}, g_{5} be elements of G conjugate to τ, τ, τ, τ, σ^{p^{α−1}} repsectively, and satisfying

g_{1}g_{2}g_{3}g_{4}g_{5} = 1.

Let k be an algebraically closed field of characteristic prime to |G|, and let Y be a Galois cover of ℙ^{1}, branched at 4 points with monodromy g_{1}, g_{2}, g_{3}, g_{4}, g_{5}.

Another application of Proposition 1.1 gives

(ρ_{Y})^{H} = (W_{p^{α}}^{H})^{⊕2}.

Again setting X = Y/H, we have X a curve of genus (1/2)(p^{α} − p^{α−1}) whose Jacobian has real multiplication by ℚ(ζ_{p^{α}}^{+}).

One can check that the dihedral groups of orders 2p, 2pq, and 2p^{α} are the only ones which produce curves with real multiplication via the above strategy.

4 Families of Jacobians with real multiplication

In the sections above, we have exhibited curves with real multiplication by various totally real abelian number fields. These curves arose as branched covers X → ℙ^{1} with specified ramification above the branch points. By moving the branch points, we naturally expect to obtain continuously varying families of curves X with real multiplication.

The problem is that, a priori, we do not know that moving the branch points alters the curve. For instance, it might be the case that there is a single curve X such that every 4-branched cover of ℙ^{1} with Galois group G_{p,4} and monodromy elements of order 2, 2, 4, 4 is isomorphic to X! Our goal in this section is to prove that such annoying circumstances do not, in fact, arise.

For the length of this section, we fix:

• a finite group G with a subgroup H;

• an algebraically closed field K with characteristic prime to |G|;

• an r-branched Galois cover Y/K → ℙ^{1}/K with Galois group G and monodromy elements g_{1}, …, g_{r}.

Let χ_{Y} the character of G acting on H^{1}(Y, ℚ_{ℓ}), as computed in Proposition 1.1.

We let X = Y/H be the quotient curve, and π : X → ℙ^{1} the projection map. In X × X we have a divisor D = X ×_{ℙ_{1}} X; concretely, the closed points of D are the points (x_{1}, x_{2}) : π(x_{1}) = π(x_{2}). The irreducible components of D are in bijection with the double cosets HgH of H in G. Write D_{g} for the component of D corresponding to HgH. An important step in the argument below is the observation that, in most of the cases we consider, D_{g} has negative self-intersection. The self-intersection is computed by means of the following proposition.

Proposition 4.1. The self-intersection of D_{g} is

|H|^{−2} ∑_{g_{1},g_{2}∈HgH} 2 − χ_{Y} (g_{1}g_{2}^{−1})

Remark 4.2. The summand 2 − χ_{Y} (g_{1}g_{2}^{−1}) can be thought of as the number of fixed points of the action of g_{1}g_{2}^{−1} on Y, whenever g_{1} ≠ g_{2}.

Proof. Let D̃_{g} be the divisor in Y × Y whose k-points are all those of the form (y, g(y)). It follows from the Lefschetz fixed point theorem in étale cohomology that

D̃_{g_{1}} · D̃_{g_{2}} = 2 − χ_{Y} (g_{1}g_{2}^{−1}).

Moreover, if f : Y → X is the quotient map, we have

f^{∗}D_{g} = ⋃_{g_{1}∈HgH} D̃_{g_{1}}.

The result now follows.

Suppose now that K has an algebraically closed subfield k, such that K is finitely generated over k; we may think of k as a field of constants and K as the algebraic closure of a function field over k. In particular, we will say that the branched cover π : X/K → ℙ_{K}^{1} is non-varying if it fits into a diagram

X  X ⟶ X_{0}
π↓  ↓  ↓^{π_{0}}
ℙ_{K}^{1} ⟶^{α} ℙ_{K}^{1} ⟶^{g} ℙ_{k}^{1}

where the right-hand square is Cartesian, g is the base change of the structure map Spec K → Spec k, and α is some automorphism of ℙ_{K}^{1}.

We show below that, under a certain numerical condition on the ramification of π, that “a branched cover from a curve defined over k to ℙ^{1} is constant.”

Theorem 4.3. Suppose X = X_{0} ×_{k} K for some curve X_{0}/k. Suppose furthermore that there exists a double coset HgH with cardinality |H|^{2} such that

∑_{g_{1},g_{2}∈HgH} 2 − χ_{Y} (g_{1}g_{2}^{−1}) < 0.  (4.3)

Then π : X → P_{K}^{1} is non-varying.

Proof. The basic algebro-geometric ingredient of the proof is the fact that divisors with negative self-intersection cannot vary continuously. More precisely, we need the following lemma.

Lemma 4.4. Let S/k be a smooth projective algebraic surface over an algebraically closed field. Let K/k be a algebraically closed extension of k of finite transcendence degree. Suppose that C ↪ S_{K} = S ×_{k} K is an irreducible effective Weil divisor with negative self-intersection. Then C = C_{0} ×_{k} K for some irreducible closed curve C_{0} ↪ S.

Proof. Let U be a variety over k, the algebraic closure of whose function field is K. Let C_{U} be a closed subscheme of S ×_{k} U whose geometric generic fiber is C_{K}. Replacing U by an open subscheme of U, we may assume that U is affine and that the fibers of C_{U} → U are irreducible curves.

Now let P be a k-point of U. Write C_{0} for the restriction of D to P; define C′ = C_{0} ×_{P} K and C′_{U} = C_{0} ×_{P} U. Note that C′, being smooth and connected, is an irreducible curve in S_{K}. We claim that C = C′.

To prove the claim, observe that the intersection product can be expressed in terms of Euler characteristics [19, Lecture 12]:

(C_{x} · C′_{x}) = χ(O_{S}) − χ(O_{S}(−C_{x})) − χ(O_{S}(−C′_{x})) + χ(O_{S}(−C_{x} − C′_{x}))

where x is either P or Spec K. Since S is projective, we may think of O_{S}, O_{S}(C_{x}), O_{S}(C′_{x}), and χ(O_{S}(−C_{x} − C′_{x})) as coherent sheaves on ℙ_{U}^{N}. Now the fact that Hilbert polynomials of coherent sheaves are locally constant in flat families implies that

(C_{K} · C′_{K}) = (C_{P} · C′_{P}) = (C_{P} · C_{P}) = (C_{K} · C_{K}) < 0.

So C_{K} and C′_{K} must share an irreducible component; since both divisors are irreducible curves, they must coincide.

We first consider the case in which H is trivial–that is, X = Y is a Galois cover of ℙ_{K}^{1}. Note that in this case the numerical condition (4.3) says precisely that g(X) ≥ 2.

Let C be the divisor X ×_{ℙ^{1}} X ↪ X ×_{K} X. Then every component of C is a translate of the diagonal by an automorphism of X ×_{K} X; since g(X) ≥ 2, every component of C has negative self-intersection. From Lemma 4.4 we have that C is the base change of a divisor C_{0} ∈ X_{0} × X_{0}.

Now the intersection of X_{0} with a fiber x × X_{0} is a set S_{x} of n = |G| points in X_{0} (counted with multiplicity). One can thus define a morphism

η_{0} : X_{0} → Sym^{n} X_{0}

which sends x to S_{x}. Let F_{0} be the image of η_{0}, and write F, η for F_{0} ×_{k} K, η ×_{k} K. Now η(x) = η(y) if and only if π(x) = π(y), which is to say that η factors as

X →^{π} ℙ^{1} →^{α} F,

where α is an isomorphism. Then the diagram

X  X ⟶ X_{0}
π↓  ^{η}↓  ↓^{η_{0}}
ℙ_{K}^{1} ⟶^{α} F ⟶^{g} F_{0}

yields the desired result.

We now return to the general case. Let f_{g} : D̃_{g} → D_{g} be the restriction to D̃_{g} of the map f × f : Y × Y → X × X. We claim that f_{g} is generically one-to-one. It suffices to consider closed points. Suppose

(f(y), f(g(y))) = (f(y′), f(g(y′))).

Then y′ = h_{1}(y) and g(y′) = h_{2}(g(y)) for some h_{1}, h_{2} ∈ H. So g(h_{1}(y)) = h_{2}(g(y)).

Suppose y (whence also y′) does not lie over a branch point of π. Then y is not fixed by any nontrivial element of G, so h_{2}^{−1}gh_{1} = g. Since HgH has cardinality |H|^{2}, this implies that h_{1} = h_{2} = 1, so y′ = y. We have shown that f_{g} is one-to-one on closed points away from a closed subset of D̃_{g}; it follows that f_{g} is generically one-to-one, as claimed. In particular, since D̃_{g} is isomorphic to the smooth curve Y, we have that Y is a nonsingular model for the curve D_{g}.

It follows from Proposition 4 that D_{g} has negative self-intersection. Applying Lemma 4.4 with S = X_{0} × X_{0} and C = D_{g}, we find that D_{g} = D_{0} ×_{k} K for some curve D_{0} ∈ X_{0} ×_{k} X_{0}. Since Y is the normalization of D_{g}, we also have Y = Y_{0} × k. Write p = π ∘ f : Y → ℙ_{K}^{1}. Then p is a Galois cover, and it follows from the case treated above that p is non-varying. After composing p with an automorphism of ℙ_{K}^{1}, we may therefore assume that p is the base change of a Galois cover p_{0} : Y_{0} → ℙ_{k}^{1}. This means that the action of G on Y is the base change of an action of G on Y_{0}. Thus, p_{0} factors through a map π_{0} : Y_{0}/H → ℙ_{k}^{1}, and π is evidently isomorphic to the base change of π_{0}. This yields the desired result.

We will now prove that the theorems of the previous sections actually produce infinite families of curves with real multiplication. Our main tool will be Theorem 4.3; however, in the cases where the field of real multiplication is ℚ(ζ_{p}^{(4)}) or ℚ(ζ_{p}^{(6)}), we instead use a degeneration argument which yields a slightly stronger result than the application of Theorem 4.3 would.

Suppose K/k is an extension of algebraically closed fields of transcendence degree n. For each g ≥ 2, let M_{g} be the moduli space of smooth genus g curves. We say a genus g curve X/K is an n-dimensional family of curves over k if the map Spec K ↪ M_{g}(K) induced by X does not factor through any Spec L, where L is a subextension of K/k of transcendence degree less than n over k.

Corollary 4.5. Let k be an algebraically closed field. Then

1. If p > 5 is a prime not equal to char k, there exists a 3-dimensional family of curves with real multiplication by ℚ(ζ_{p}^{+}).

2. If p is a prime congruent to 1 mod 4, and char k does not divide 2p, then there exists a 1-dimensional family of curves with real multiplication by ℚ(ζ_{p}^{(4)}).

3. If p is a prime congruent to 1 mod 6, and char k does not divide 6p, then there exists a 1-dimensional family of curves with real multiplication by ℚ(ζ_{p}^{(6)}).

4. If p, q are odd primes not equal to char k, there exists a 1-dimensional family of curves with real multiplication by ℚ(ζ_{pq}^{+}).

5. If p is an odd prime not equal to char k, and α > 1 is an integer, there exists a 2-dimensional family of curves with real multiplication by ℚ(ζ_{p^{α}}^{+}).

Proof. Let K_{r} be the algebraic closure of the function field of the moduli space M_{0,r}/k of genus 0 curves with r marked points, where r ≥ 4. Then K_{r} has transcendence degree r − 3 over k. Moreover, the inclusion

Spec K_{r} → M_{0,r}

gives us a set of r points p_{1}, …, p_{r} on ℙ_{K_{r}}^{1}. Suppose π : X → ℙ_{K_{r}}^{1} is a cover branched at p_{1}, …, p_{r}; such a cover exists by Proposition 1.4. As usual, let G be the Galois group of the cover and H the subgroup corresponding to X via the Galois correspondence. Suppose that G, H, χ_{Y} satisfy the numerical condition (4.3) of Theorem 4.3. Then X is an (r − 3)-dimensional family of curves over k. For suppose otherwise; then X is isomorphic to X_{0} ×_{L} K_{r}, where L/k is a subextension of K_{r}/k with transcendence degree less than r − 3. It then follows from Theorem 4.3 that π is non-varying, which means in particular that the branch locus of π is isomorphic to the base change of a set of r points on ℙ_{L}^{1}. But that would imply that

Spec K_{r} → M_{0,r}

factors through Spec L, which contradicts the hypothesis that this map is a geometric generic point of M_{0,r}.

It follows that, in each case of the corollary, we only need to demonstrate that there exists a branched cover π : X → ℙ_{K_{r}}^{1} branched at p_{1}, …, p_{r} which has the desired real multiplication and which satisfies (4.3).

We will now work case by case. Let F be the field of real multiplication.

Case 1: F = ℚ(ζ_{p}^{+}). We use the branched cover of Brumer [2]; in the notation of section 3, we are in the case n = 2, {d_{i}} = {1, 1, 1, 1, 1, 1}. Here r = 6, the Galois group G is the dihedral group G_{p,2}, the subgroup H is generated by an involution τ, and the monodromy at each branch point is conjugate to τ. It follows from Proposition 1.1 that

• χ_{Y} (1) = 2p + 2;

• χ_{Y} (τ) = −4;

• χ_{Y} (σ) = 2.

The double coset HσH has cardinality 4, and one checks that

∑_{g_{1},g_{2}∈HσH} 2 − χ_{Y} (g_{1}g_{2}^{−1}) = 4(−2p) + 8 · 6 = 48 − 8p,

which is negative once p > 5. Note that the assertion is in fact false for p = 5, since the moduli space of abelian surfaces with real multiplication by ℚ(\sqrt{5}) is only 2-dimensional.

Case 4: F = ℚ(ζ_{pq}^{+}). Take r = 4 and let X be the branched cover described in Section 3 with real multiplication by F. In this case, Proposition 1.1 shows that ρ_{Y} = W_{pq}^{⊕2}, and so

• χ_{Y} (1) = 2(p − 1)(q − 1)

• χ_{Y} (τ) = 0;

• χ_{Y} (σ) = 2.

As g_{1}, g_{2} run through the double coset HσH, the quotient g_{1}g_{2}^{−1} is trivial in 4 cases and conjugate to τ in 8 cases. The sum in (4.3) is therefore

4(2 − 2(p − 1)(q − 1)) + 8 · 2 = 24 − 8(p − 1)(q − 1),

which is negative for all odd primes p, q.

Case 5: F = ℚ(ζ_{p^{α}}^{+}). Take r = 5 and let X be the branched cover described in Section 3 with real multiplication by F. Here, we find that ρ_{Y} = (V_{−1} + W_{p^{α}})^{⊕2}, and so

• χ_{Y} (1) = 2 + 2p^{α} − 2p^{α−1}

• χ_{Y} (τ) = −2;

• χ_{Y} (σ) = 2.

It follows that the sum in (4.3) is

4(2p^{α−1} − 2p^{α}) + 8 · 4 = 32 − 8(p − 1)p^{α},

which is negative for any odd prime p and any α > 1.

We now turn our attention to the cases F = ℚ(ζ_{p}^{(n)}), where n = 4, 6. The methods above will show the existence of 1-dimensional families of curves with real multiplication by F for all sufficiently large p congruent to 1 mod n. In order to prove the desired result for all such p, we introduce a different approach.

Note that to prove our family of curves is 1-dimensional, it suffices to prove it is non-constant. In turn, to prove the family is non-constant, it suffices to prove that it degenerates to a nodal curve (since the generic member of the family is smooth.) In order to study the degeneration of our family of branched covers, we need to recall some facts about compactifications of Hurwitz moduli spaces.

Let C/k be a connected stable r-pointed curve; that is, X has only nodal singularities, and every rational component of X has at least three points which are either marked or nodal. An admissible cover of a stable r-pointed curve C is a finite morphism Y → C which is étale over all smooth unmarked points of C, and which displays a certain local behavior over the nodes of C; the precise definition will not concern us here. If C is a smooth curve, an admissible cover of C is just a smooth r-branched cover in the usual sense. The central fact we need is that the moduli stack of admissible covers is a compactification of the moduli stack of smooth branched covers.

Lemma 4.6. Let A be a strictly henselian discrete valuation ring with fraction field K. Let B → A be a stable r-pointed curve whose generic fiber B_{K} is smooth, and let π_{K} : X_{K} → B_{K} be an admissible cover whose Galois group G has order in A^{∗}. Then there exists a tamely ramified extension A′ of A, with fraction field K′, such that π_{K′} extends to an admissible cover π : X/A′ → B/A′.

Proof. Argue as in [17, §3.13]. The only difference is that the hypothesis |G| ∈ A^{∗} replaces Mochizuki’s hypothesis d! ∈ A^{∗}.

Let G be either G_{p,4} or G_{p,6}, let g be an element of order 4 or 3, respectively, and let F be the field ℚ(ζ_{p}^{(4)}) or ℚ(ζ_{p}^{(6)}) respectively. Let h be an element of G of order 2. We think of G as embedded in the symmetric group S_{p} by means of the action of G on the cosets of the cyclic group generated by α.

Let K be the algebraic closure of k(t). By Proposition 1.4, there exists a degree p cover X → ℙ_{K}^{1} branched at the points 0, 1, ∞, and t, with monodromy conjugate to g, g^{−1}, h^{−1}, h respectively. In fact, this cover has a model X_{U} → ℙ_{U}^{1}, where U is an open subscheme of a finite étale cover of Spec k[t], and X_{U} → ℙ_{U}^{1} is smooth. Note that X has real multiplication by F, by the results of section 2.

Let R be the integral closure of k[t] in the fraction field of U, and let S = Spec R. First of all, by the properness of the moduli space \overline{M}_{0,4} of 4-pointed stable genus 0 curves, there is a finite cover S′ of S and a stable 4-pointed genus 0 curve B → S′ whose geometric generic fiber is P_{K}^{1} marked at 0, 1, ∞, and t. Let U′ = S′ ×_{S} U. Then X_{U′} → B_{U′} is a a smooth 4-branched cover. It follows from Lemma 4.6 that, possibly after another finite extension of S′, the cover X_{U′} can be extended to an admissible cover X_{S′} → B. For simplicity of notation, take S = S′ and U = U′.

Let s ∈ S be a closed point of S lying over the point t = 0 of Spec k[t]. Then B_{s} is a nodal 4-pointed curve of genus 0; any such curve consists of two rational curves meeting at the node, with two marked points lying on each of the irreducible components. Let X_{s} be the fiber of X_{S} over s, so that π_{s} : X_{s} → B_{s} is an admissible cover. Let C, C′ be the two irreducible components of B_{s}. Then π_{s}^{−1}(C) → C is a branched degree p cover (not necessarily connected) of a smooth genus 0 curve, which is étale away from the two marked points on C and the node C ∩ C′. Call these points m_{0}, m_{t}, and n.

Now consider an infinitesimal neighborhood of s ∈ S, which is isomorphic to Spec k[[u]]. Since m_{0} is a smooth point of C, the completed local ring of B at m_{0} is isomorphic to Spec k[[u, v]]. The marking divisor 0 is a section from S to B, which is locally given by the map of rings k[[u, v]] → k[[u]] sending v to 0. Let Spec A be the scheme completing the Cartesian square

Spec A ⟶ X
↓  ↓
Spec k[[u, v]] ⟶ B.

Then it follows from Abhyankar’s Lemma that

A ≅ ⨁_{i=1}^{d} k[[u, v]][V]/(V^{m_{i}} − v)

where the m_{1}, …, m_{d} are the lengths of the orbits of the element g of S_{p}. Now, restricting Spec A to the fiber u = 0, we see that the monodromy around m_{0} in the cover π_{C} : π_{s}^{−1}(C) → C is conjugate to g. Similarly, π_{C} has monodromy conjugate to h arond m_{t}. So the monodromy of π_{C} around n is g′h′, where g′ is some conjugate of g^{−1} and h some conjugate of h^{−1}. It is now an easy Riemann-Hurwitz computation that every component of π_{s}^{−1}(C) has genus 0 or 1, and that the same is true of π_{s}^{−1}(C′). In particular, the stable model of X_{s} is not a smooth curve of genus (p − 1)/n. We conclude that the generically smooth family of curves X has a fiber X_{s} whose stable model is non-smooth; therefore, X is the desired non-constant family of curves with real multiplication by F.

One would like to have explicit equations, as in [24],[16] and [2], for the families of branched covers π : X → ℙ_{K_{r}}^{1} discussed above. These families of branched covers are parametrized by certain Hurwitz spaces, which are smooth schemes of dimension r − 3 over k.

To give a rationally parametrized family of branched covers would be to give a branched cover X → ℙ_{U}^{1} with the specified monodromy g_{1}, …, g_{r}, where U is an open subscheme of a rational variety. Such a cover would produce a map from U to a Hurwitz space. In case r = 4, the Hurwitz space is a union of curves, whose genera can be computed by passing to characteristic 0 and using combinatorial methods. A computer calculation shows that in the cases F = ℚ(ζ_{13}^{(4)}), ℚ(ζ_{13}^{(6)}), the relevant Hurwitz spaces are already genus 1 curves. Our methods thus cannot produce rationally parametrized families of curves whose Jacobians have real multiplication by ℚ(ζ_{13}^{(4)}) and ℚ(ζ_{13}^{(6)}), and indeed, we have no reason to expect that such families exist.

The situation is different in the case F = ℚ(ζ_{pq}^{+}). In fact, one can rather easily construct a rationally parametrized family of curves with real multiplication by F.

Proposition 4.7. Let p and q be odd primes, let k be an algebraically closed field of characteristic prime to 2pq, and let X/k be the non-singular curve with affine model

T_{pq}(y/2) = P (x)/2(x − 1)^{p}(x − λ)^{q}

where T_{pq} is the degree pq Tchebyscheff polynomial satisfying

T_{pq}((z + z^{−1})/2) = (z^{pq} + z^{−pq})/2

and P (x) is the degree p + q polynomial such that

P (w^{2}) = (w − 1)^{2p}(w − λ)^{2q} + (w + 1)^{2p}(w + λ)^{2q}.

Then X is a curve of genus (1/2)(p − 1)(q − 1) with real multiplication by ℚ(ζ_{pq}^{+}).

Proof. Let Y be the curve with affine model

z^{pq} = \frac{(w − 1)^{p}(w − λ)^{q}}{(w + 1)^{p}(w + λ)^{q}} .

One easily checks that the map Y → ℙ^{1} sending (z, w) to w^{2} realizes Y as a cover of the line with Galois group D_{pq}, branched over 0, 1, ∞, and λ^{2} with ramification of order 2, q, 2, p respectively. The action of D_{pq} on Y is generated by

τ : (z, w) → (z^{−1}, −w)

and

σ : (z, w) → (ζ_{pq}z, w).

Let H be the subgroup of D_{pq} generated by τ. Then Y/H is evidently isomorphic to X; it now follows from the arguments of section 3 that X has the desired real multiplication.

Remark 4.8. By choosing λ in a field k_{0} which is not algebraically closed, one can arrange for X to have a model over k_{0}; however, the endomorphisms of Jac(X) will typically be defined only over an extension of k_{0}. The problem of determining fields of definition and fields of moduli for the families of curves constructed here is part of the active program of constructing models of branched covers over non-algebraically closed fields. The interested reader should consult the papers of Dèbes, Douai, and Emsalem [6],[7].

5 Questions

The main goal of this paper has been to exhibit smooth curves whose Jacobians had many endomorphisms of Hecke type. The negative version of this question–what endomorphism algebras cannot act on the Jacobian of a smooth curve?–seems substantially more difficult. In fact, we do not know of a single example of an algebra E which embeds in the endomorphism algebra of a principally polarized abelian variety but which can be proven not to embed in the endomorphism algebra of the Jacobian of a smooth curve. In this section, we frame some questions in the direction of bounds on endomorphisms of Hecke type.

5.1 A non-Galois version of Hurwitz’s theorem?

Hurwitz showed [14] that a curve of genus g > 1 over a field k of characteristic 0 has an automorphism group of size at most 84(g − 1). In other words, if Y/k → C/k is a Galois cover with Galois group G, the dimension of ℚ[G] is bounded linearly in g(Y).

Now let Y, C, G, H, X be defined as in section 1. Let H_{X/C} be the endomorphism algebra of Hecke type; recall that H_{X/C} is a homomorphic image of ℚ[H\G/H] and a subalgebra of End(Jac(X)). We will always suppose that X/C is a tamely ramified cover, so that Proposition 1.1 is in effect.

Question A: Is there a γ > 0 such that

dim H_{X/C} ≤ γg(X)

for all X, C?

Hurwitz’s theorem implies that the answer to question A is yes, with γ = 84, when H is trivial. Question A can be rephrased purely in terms of combinatorial group theory. First of all, if

π : X → C is a branched cover, we may compose π with a map ψ : C → ℙ^{1}. The endomorphism algebra of Hecke type associated to the branched cover ψ∘π : X → ℙ^{1} is easily seen to contain H_{X/C}. So Question A does not change if we replace C by ℙ^{1}. When C = ℙ^{1}, the action of G on H^{1}(Y, ℚ_{ℓ}) is described by a certain Hurwitz character, and the dimension of H_{X/C} can be determined from that character. We arrive at the following equivalent formulation for Question A.

Question A’: Let χ be a Hurwitz character of a finite group G (see Remark 1.3), and let H be a subgroup of G. We can decompose χ as

χ = ∑_{ψ} a_{ψ}ψ

where ψ ranges over the irreducible characters of G. If ψ is a character of G (trivial or not), write ψ(H) for the inner product of ψ with the character χ_{H} induced from the trivial character on H.

Now let d(G, H, χ) be the sum of ψ(H)^{2} over all irreducible characters ψ such that a_{ψ} > 0. Is there a constant γ, independent of G, H, χ, such that

d(G, H, χ) ≤ γχ(H)?

The value d(G, H, χ) is precisely the dimension of H_{X/ℙ^{1}}, where X → ℙ^{1} is a branched cover with Galois group G giving rise to the Hurwitz character χ on H^{1}(Y, ℚ_{ℓ}).

Results of Aschbacher, Guralnick, Neubauer, and Thompson give partial results in the direction of question A′ in case H is a maximal subgroup of G, the intersection of whose conjugates is trivial. Such subgroups are classified by a theorem of Ashbacher and Scott; the results of the four above authors show that, in all but one case of the Aschbacher-Scott classification, [G : H] is bounded above by a linear function in χ(H) [12, §1]. Plainly, d(G, H, χ) ≤ [G : H], so in these cases Question A′ is answered in the affirmative. For example, if G is solvable and H is subject to the above restrictions, then γ can be taken to be 2^{10} [21]. More results in this direction will appear in [11].

5.2 Decomposable Jacobians

In [8], Ekedahl and Serre ask whether there are curves over ℂ of every genus, or even of arbitrarily large genus, whose Jacobians are isogenous to products of elliptic curves. Let X/k be a curve over an algebraically closed field of characteristic 0, and let d(X) be the supremum of dim(A) over all simple isogeny factors of Jac(X). Then we can rephrase Ekedahl and Serre’s question to read: what conditions on the genus of X are imposed by the condition d(X) = 1? More generally, we might ask:

Question B: Is there an increasing, unbounded function f : ℕ → ℕ such that, for all smooth genus g curves X/k,

d(X) ≥ f(g)?

By taking C to be a genus g_{0} curve and X to be the maximal elementary 2-abelian étale cover of C, one finds that such an f would have to be of order at most c log g. Serre has proved that, if k is replaced by a finite field 𝔽, and if d(X) is the maximal dimension of an 𝔽-simple isogeny factor of Jac(X) over 𝔽, then the answer to question B is yes. In fact, de Jong has shown [5] that f(g) can be chosen on the order of (log log g)^{1/2} in this case. On the other hand, if k is taken to be an algebraically closed field of characteristic p, the existence of supersingular curves of arbitrarily large genus shows that the answer to question B is no.

The methods of the present paper allow us to speak only of endomorphisms of Jac(X) which are of Hecke type. Let X, Y, G, H, C be defined as in section 1. Then H^{1}(X, ℚ_{ℓ}) is a Q[H\G/H]-module. If π is an element of Q[H\G/H], we refer to the subspace πH^{1}(X, ℚ_{ℓ}) as a factor of H^{1}(X, ℚ_{ℓ}). Finally, we define d(X/C) to be the maximal dimension among all minimal non-trivial factors of H^{1}(X, ℚ_{ℓ}). If π ∈ ℚ[H\G/H] (or some integral multiple of π) is considered as an endomorphism of Jac(X), then (1/2) dim πH^{1}(X, ℚ_{ℓ}) is the dimension of π Jac(X), which is an isogeny factor of Jac(X). Moreover, every simple isogeny factor of Jac(X) is contained in some such π Jac(X). It follows that d(X) ≤ d(X/C).

A slight modification of d(X/C) is sometimes useful; we take d′(X/C) to be half the maximal dimension of a minimal nontrivial factor other than H^{1}(C, ℚ_{ℓ}) = π_{H}π_{G}π_{H}H^{1}(X, ℚ_{ℓ}). Then the argument above shows that

d(X) ≤ max(d(C), d′(X/C).

Ekedahl and Serre give many examples of curves X with large genus such that d(X) = 1. With the exception of some modular curves, their examples, like ours, are produced by analyzing the action of a finite group on the Jacobian of a Galois cover of ℙ^{1}. In our notation, the Ekedahl-Serre examples are given as branched covers of low-genus curves C with d(C) = 1, and all have d′(X/C) ≤ 2. In the cases where d′(X/C) = 2, the authors observe that that the abelian surfaces corresponding to the 4-dimensional factors of H^{1}(X, ℚ_{ℓ}) are acted on by dihedral groups of order 6, 8, or 12, and are therefore decomposable.

We are now motivated to ask whether this strategy has any chance of providing a negative answer to Question B.

Question C: Is there an increasing, unbounded function f : ℕ → ℕ such that, for all smooth genus g curves X/k and all branched covers X → C,

d′(X/C) ≥ f(g)?

Note that question C can be rephrased as a question in combinatorial group theory. By composing the map X → C with an arbitrary map C → ℙ^{1}, we can think of X as a branched cover of the projective line; since H_{X/C} is contained in H_{X/ℙ^{1}}, we may assume C = ℙ^{1}. The following question is then equivalent to question C.

Question C’: Let G be a finite group, χ a Hurwitz character of G, and H a subgroup of G. Write

χ = ∑_{ψ} a_{ψ}ψ,

where ψ ranges over the characters of irreducible ℚ-rational representations of G. For each ψ, let V_{ψ} be a ℚ-rational representation of G realizing ψ, and let s(ψ) be the dimension of a minimal non-trivial factor of V_{ψ}. Then

s(ψ) = r(ψ)m(ψ)

where r(ψ) is the degree of the field extension ℚ(ψ)/ℚ and m(ψ) is the Schur multiplier of ψ. Is there an increasing, unbounded function f : ℕ → ℕ, independent of G, H, χ, such that

max_{ψ} a_{ψ}s(ψ) ≥ f(χ(H))

where ψ ranges over the characters of irreducible ℚ-rational representations of G such that ψ(H) is positive?

This problem seems rather difficult even for the case of Galois covers, where H is trivial.

We will discuss one special case, from which Ekedahl and Serre derive many examples of curves with decomposable Jacobians([8, §4]). Let X/ℙ^{1} be a Galois cover with group G, and suppose that G admits a homomorphism to ℤ/2ℤ, with kernel K. We thus obtain a diagram

X → X_{0} → ℙ^{1},

where X_{0} is a hyperelliptic curve. Let χ be the Hurwitz character of G corresponding to this cover, and suppose that the ℚ[G]-module H^{1}(X, ℚ_{ℓ}) has no minimal factor of dimension greater than some bound A. Then the same is evidently true if we replace ℚ[G] with Q̄[G] and H^{1}(X, ℚ̄_{ℓ}). It follows that χ can be written as ∑ a_{i}χ_{i}, where the χ_{i} are irreducible complex characters of G, and where all the a_{i} are less than A. Now if χ_{K} is an irreducible complex character of K, we have

< χ_{K}, χ|K >=< Ind_{K}^{G} χ_{K}, χ >

and since Ind_{K}^{G} χ_{K} is a sum of either one or two irreducible characters of G, we find that < χ_{K}, χ|K >≤ 2A.

Suppose furthermore that g(X_{0}) ≥ 2 and that the cover X → X_{0} is etale, as is the case in Ekedahl and Serre’s examples. One has by Proposition 1.1 that the character χ|K of K acting on H^{1}(X, ℚ_{ℓ}) is

2χ_{triv} + 2(g(X_{0}) − 1)χ_{1}.

In particular, every non-trivial χ_{K} appears with multiplicity 2(g(X_{0}) − 1)(χ_{K}(1)) in χ|K.

So K has no irreducible complex representations of dimension greater than 2B, where B = A/(2g(X_{0}) − 1). Hence, the irreducible complex representations of G have dimension at most 4B. A theorem of Isaacs and Passman [15, (12.23)] now tells us that K has an abelian subgroup H of index at most ((2B)!)^{2}.

Let τ be the character of G induced from ℤ/2ℤ. Let g be an element of G of exact order n, and let χ_{i} be an irreducible complex representation of G in which < g > acts faithfully. Then the field of definition of χ_{i} must have degree at least ϕ(n)/4B over ℚ. Moreover, since χ_{i}|K occurs with positive multiplicity in χ|K, either χ_{i} or χ_{i} ⊗ τ occurs with positive multiplicity in χ. In particular, the minimal ℚ-representation ψ containing χ_{i} has a_{ψ} ≥ 1 and r(ψ) ≥ ϕ(n)/4B. We conclude that ϕ(n) < 4AB.

Since K is the Galois group of a connected étale cover of X_{0}, it can be generated by 2g(X_{0}) elements, which in turn yields an upper bound on the number of generators of H. Since we have bounded the order of an element of H, we now have a bound on |H|, whence also on |G|. Note that the genus of X is just 1 + (g(X_{0}) − 1)|G|. Putting all this together yields a bound of the form

d′(X/ℙ^{1}) > c\frac{log log g(X)}{log log log g(X)}

for some absolute constant c.

References

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