Non-simple principally polarised abelian varieties

The paper investigates the locus of non-simple principally polarised abelian $g$-folds. We show that the irreducible components of this locus are $\Is^g_{D}$, defined as the locus of principally polarised $g$-folds having an abelian subvariety with induced polarisation of type $D=(d_1,\ldots,d_k)$, where $k\leq\frac{g}{2}$. The main theorem produces Humbert-like equations for irreducible components of $\Is^g_{D}$ for any $g$ and $D$. Moreover, there are theorems which characterise the Jacobians of curves that are \'etale double covers or double covers branched in two or four points.


Introduction
A common approach to understand the geometry of the moduli space of abelian varieties is to use ideas coming from the geometry of curves. That is possible because of the Torelli theorem, which says that the Jacobian completely characterises the curve. Because of that, many geometric constructions from the theory of curves give rise to interesting constructions in the theory of Jacobians. One remarkable construction is the Prym construction, which gives a subvariety of a Jacobian for any finite cover of curves. More precisely, every cover of curves f : C −→ C ′ induces a pullback map f * : JC ′ −→ JC. Therefore JC is a non-simple abelian variety, as it contains im f * and the complementary abelian subvariety, called the Prym variety of the cover.
The motivation behind results of this paper is to understand the locus of non-simple abelian varieties itself. One can ask: (⋆) What does the locus of non-simple principally polarised abelian g-folds look like? For abelian surfaces, a non-simple abelian surface contains an elliptic curve, and Humbert [H] proved that the locus of non-simple principally polarised abelian surfaces is the union of countably many irreducible surfaces called Humbert surfaces, in the moduli space. The Humbert surfaces are indexed by the degree of the polarisation restricted to an elliptic curve.
In Section 3, we propose a definition of the generalised Humbert locus, denoted by Is g D , which is the locus of principally polarised abelian g-folds having an abelian subvariety with induced polarisation of type D = (d 1 , . . . , d k ), where k ≤ g 2 . The definition was stated by O. Debarre in [D, p 259], denoted by A δ g ′ ,g−g ′ . Then he proves irreducibility of Is g D , using irreducibility of some moduli space. The main result of Section 3 is Proposition 3.17 that also states that Is g D is irreducible. Both ideas of proofs are similar, but the proof presented in this paper is explicit.
Using the fact that every non-simple abelian g-fold belongs to some Is g D , we get that the only discrete invariants of the locus of non-simple principally polarised abelian varieties are the dimensions of subvarieties and the type of the induced polarisation on the smaller one. Moreover, all possibilities of k ≤ g 2 and D = (d 1 , . . . , d k ) can occur. The main results of this paper are contained in Section 4, where we find equations of preimage of Is g D in the Siegel space h g , for all g > 2. Because the codimension is bigger than 1, we could not find symplectic invariants similar to Humbert's discriminant, so Theorem 4.2 provides only a set of linear equations of one particular irreducible component of the preimage in h g of Is g D . The main result of the paper states that a principally polarised abelian g-fold (A, H) is nonsimple, thus containing an abelian subvariety M ⊂ A of dimension k ≤ g 2 such that H| M is of type D = (d 1 , . . . , d k ), if and only if one can find a period matrix Z A = [z ij ] ∈ h g satisfying the linear in such a way that Z belongs to the Siegel space Moreover, on such a variety we always choose a polarisation of type D, usually denoted by H.
For any polarisation H we define an isogeny φ H : A −→Â with analytic representation given by (Im Z) −1 . The exponent of φ H is denoted by e(H) and called the exponent of the polarisation. By K(H) we denote the kernel of φ H . Using a decomposition for H one proves that where v 1 , v 2 are any preimages of w 1 , w 2 in C g . We denote by A g = h g / Sp 2g (Z) the moduli space of principally polarised abelian g-folds and by A D the moduli space of D-polarised abelian varieties. Inside A g we denote the locus of Jacobians by J and the locus of hyperelliptic Jacobians by J H.

Complementary abelian subvarieties.
In Section 3, we will try to understand the locus of non-simple abelian varieties. The idea is to improve the statement of uniqueness in Poincaré's Complete Reducibility Theorem. Therefore we need to recall the following definitions and Poincaré's Reducibility Theorems. For details we refer to [BL].
Definition 2.1. [BL,p.132] Let ι : M −→ A be an abelian subvariety of a principally polarised abelian variety (A, H). Then ι * H is a polarisation on M , denoted also by H| M . Define the exponent of M by e(M ) = e(ι * H). Moreover, we define the norm-endomorphism of A associated to M by is called the associated symmetric idempotent. Conversely for any symmetric idempotent ε ∈ End Q (A) there exists n ∈ N such that nε ∈ End(A), and we can define the abelian subvariety A ε = im(nε).
The next theorem is the main tool in proving Poincaré's Reducibility Theorems.
Theorem 2.2. [BL,Thm 5.3.2] The assignments M → ε M and ε → A ε are inverse to each other and give a bijection between the sets of abelian subvarieties of A and symmetric idempotents in End Q (A) The main advantage of translating the existence of subvarieties into symmetric idempotents is that the latter have an obvious canonical involution ε → 1−ε. This leads to the following definition.
Definition 2.3. [BL,p.125] Let A be a polarised abelian variety. Then the polarisation induces a canonical involution on the set of abelian subvarieties of A: We call N the complementary abelian subvariety of M in A, and (M, N ) a pair of complementary abelian subvarieties.
In this paper we often consider products of principally polarised abelian varieties, and therefore we introduce the following notation.
If (M, H M ) and (N, H N ) are polarised abelian varieties of types D andD then, even if not written explicitly, we will treat the product M × N as the (D,D)-polarised variety with the canonical product polarisation. Strictly speaking (D,D) is not a polarisation type, so one needs to permute the coordinates. The following proposition shows that indeed the complementary abelian subvariety in the principally polarised abelian variety has a complementary polarisation type (1) there exists M ⊂ A such that H| M is of type D.
(4) ⇒ (3) Let us denote the inclusions by ι M = ρ| M ×{0} and ι N = ρ| {0}×N Then ρ(m, n) = ι M (m) + ι N (n) and so Theorem 2.6. [BL,Thm 5 Theorem 2.7. [BL,Thm 5.3.7] For an abelian variety A there is an isogeny 2.2. Symplectic forms on finite abelian groups. Later, we shall be interested in isotropic subspaces of K (H) and their behaviour under the action of the symplectic group. Therefore we need some basic facts from the theory of finite symplectic Z-modules. We will use the fundamental theorem of finite abelian groups.
Definition 2.9. We say that X is of dimension k and a basis of X is an image of any basis of Z k by an epimorphism for some m on an abelian variety A and consider a decomposition V 1 ⊕ V 2 for H. Then It is convenient to work under the assumption that the domain and codomain are of the same type. Therefore, we define Definition 2.11. Let (X, ω X ) and (Y, ω Y ) be symplectic Z-modules of the same type. Then a Z-linear map f : X −→ Y is called an antisymplectic map if for all x, y ∈ X, we have Proposition 2.12. Every antisymplectic map is a bijection and the inverse map is also antisymplectic. Moreover, the space of antisymplectic maps is modelled on Sp(X, Z), i.e. for every antisymplectic f, g : X −→ Y , we have g −1 • f ∈ Sp(X, Z) and for all s X ∈ Sp(X, Z), the maps f • s X are antisymplectic. By symmetry it is also modelled on Sp(Y, Z).
Proof. By equation (1), if f (x) = 0, then ω X (x, y) = 0 for every y, so x = 0, which means f is injective. Bijectivity comes from the domain and codomain having the same type. The rest of the proposition comes from the fact that (−1) · (−1) = 1.
. Then the set of graphs of antisymplectic maps is the set of maximal isotropic subspaces of X ⊕ Y intersecting X and Y only in {0}.
In particular, all maximal isotropic subspaces of X ⊕ Y intersecting X and Y only in {0} are equivalent under the actions of symplectic groups on X and on Y .
Proof. It is obvious that the graph of an antisymplectic map is an isotropic subspace with the desired properties. For the converse, let Z be a maximal isotropic subspace. Then the projections π X : Z −→ X and π Y : Z −→ Y are bijections. Moreover π Y • π −1 X is antisymplectic and Z is the graph of π Y • π −1 X . The second part of the proposition is a direct application of Proposition 2.12.

Generalised Humbert locus
3.1. Background -Humbert surfaces. We begin the study of moduli of non-simple abelian varieties with the surface case, by recalling the Humbert surfaces of square discriminant. (1) there exists an elliptic curve E ⊂ A such that H| E is of type p; (2) there exists an exact sequence and therefore its dual contains a symmetric endomorphism f with analytic and rational representations given by (6) (A, H) is isomorphic to an abelian surface defined by a period matrix with elliptic curves defined by period matrices [t 2 1] and [pt 3 − t 2 1] embedded as s → (ps, s) and s → (0, s).
For the second implication, to simplify notation, we write t ′ = Im(t) for any t ∈ C. Then Define ι E : s → (ps, s). Its analytic representation is given by the matrix so H| E is of type p.
Condition (6) of Theorem 3.1 implies that in A 2 , the locus of all principally polarised abelian surfaces satisfying the above conditions is the image of the surface given by the equation t 1 = pt 2 in h 2 , and therefore it is an irreducible surface in A 2 .
Definition 3.2. The locus in A 2 of all principally polarised abelian surfaces that satisfy the conditions of Theorem 3.1 is called the Humbert surface of discriminant p 2 .
Humbert showed more in [H]. He found the equations defining the preimage in h 2 of all Humbert surfaces. To be precise, any 5-tuple of integers without common divisor (a, b, c, d, e) with the same discriminant ∆ = b 2 − 4ac − 4de gives us the so-called singular relation In other words, the period matrix Z = t 1 t 2 t 2 t 3 ∈ h 2 is a solution to a singular relation with ∆ = p 2 if and only if the abelian surface A Z = C 2 /(ZZ 2 + Z 2 ) contains an elliptic curve with induced polarisation of type p. If we recall that A 2 = h 2 / Sp(4, Z), then it means that all matrices which satisfy the singular equation for some ∆ = p 2 form a symplectic orbit. Then condition 6 of Theorem 3.1 says that there always exists a normalised period matrix i.e. such that 3.2. Generalised Humbert locus. We would like to generalise the notion of Humbert surface to higher dimensions. There are a few immediate problems that arise. Firstly, Humbert surfaces are divisors in A 2 globally defined by one equation in h 2 , whereas in higher dimensions that is not the case. Secondly, all elliptic curves are essentially canonically principally polarised whereas in higher dimensions polarisations are much richer.
If we consider an abelian subvariety of an abelian variety, then the two obvious discrete invariants are the dimension of the subvariety and the type of the induced polarisation. So we define Definition 3.3. The generalised Humbert locus of type D = (d 1 , . . . , d k ) in dimension g, denoted by Is g D , is the locus in A g of principally polarised g-folds X such that there exists a k-dimensional subvariety Z of X such that the restriction of the polarisation from X to Z is of type D. If d 1 = d k then we say it is of principal type.
Remark 3.4. The same definition was proposed by O.Debarre in [D,p 259], denoted by A δ g ′ ,g−g ′ . Firstly some obvious remarks and connections with previously known notions: (1) Every non-simple principally polarised abelian variety belongs to a generalised Humbert locus for some g and D.
(2) The name comes from the fact that for surfaces, it gives back Humbert surfaces of discriminant D 2 . The word Is is an abbreviation of (polarised) isogenous to a product.
(3) Note that principal type does not mean d k = 1. If d k = 1 then the isogeny from Proposition 2.5 is actually an isomorphism and we get only the locus of products of principally polarised abelian varieties. If we restrict the generalised Humbert locus of principal type to Jacobians of smooth curves then we get Jacobians containing Prym-Tyurin varieties.
The first example of generalised Humbert locus arises in dimension three.
Proposition 3.5. A variety A ∈ Is 3 n is either a product of an elliptic curve with an abelian surface or the Jacobian of a smooth genus 3 curve which is an n : 1 cover of an elliptic curve branched in 4 points and all such Jacobians are contained in Is 3 n . In other words, the only non-simple Jacobians are Jacobians of covers of elliptic curves.
Proof. By [BL,Cor 11.8.2] every principally polarised abelian threefold is either a product or a Jacobian so we restrict our attention to Jacobians. If JC ∈ Is 3 n , then it contains an elliptic curve, say E. Taking the Abel-Jacobi map composed with the dual of the inclusion, we get a map C −→ E. As H| E is of type n, it is an n : 1 cover. Using the Hurwitz formula, we get that it has to be branched in 4 points. Conversely, if we have an n : 1 cover π : C −→ E, then we can have the norm map Nm π : JC −→ JE = E, given by Nm π (P − Q) = π(P ) − π(Q). Moreover Nm π (π −1 (P ′ − Q ′ )) = n(P ′ − Q ′ ) is a multiplication by n, so the induced polarisation is of type n.
Before stating Proposition 3.6, Theorem 3.10 and Corollary 3.11, I would like to note that the results are based on well-known ideas of Prym construction (see [M] and [BL]) and the generalised Torelli Theorem (see [W]), but I was not able to find any exact references in the literature.
The first result is a characterisation of a 3-dimensional family of Jacobians ofétale double covers of genus 2 curves.
Proposition 3.6. The locus of Jacobians ofétale double covers of genus 2 curves is Is 3 2 ∩J H. Proof. Let f : C −→ C ′ be anétale double cover. It is defined by a 2-torsion point in JC ′ , say η. Then ker(f * ) = {0, η} ([ACGH, Ex. B.14]). Therefore f * JC ′ = JC ′ / η is a (1, 2)-polarised abelian surface, which is an abelian subvariety of JC. Hence JC ∈ Is 3 2 . To finish the implication, let us note that C ′ , being of genus 2, has to be hyperelliptic and anyétale double cover of a hyperelliptic curve is hyperelliptic. This implication can be easily deduced from the proof of part (a) of [M,Thm 7.1], or from [O].
As for the other implication, let JC ∈ Is 3 2 ∩J H. Denote by E an elliptic curve in JC. Denote by ι E the involution of C which defines the double cover and by i E its extension to JC. From construction, im(1 − i E ) = E, and therefore ǫ E = 1−i E 2 . On the other hand, if we denote by ι the hyperelliptic involution on C, then its extension to JC is (−1). This is because for a branch point Q, we have (P − Q) + (ι(P ) − Q) = 0, being the principal divisor of a pullback of a meromorphic function on P 1 . Now, ι • ι E is an automorphism on C and its extension is −i E . Denoting by Z = im(1 − (−i E )) and ǫ Z = 1+i E 2 , we immediately get that ǫ Z + ǫ E = 1 and so (E, Z) is a pair of complementary abelian subvarieties of JC.
It is obvious that dim JC ′ = dim Z = 2, so C ′ is of genus 2 and by the Hurwitz Formula f has to be anétale double cover.

Corollary 3.7. Proposition 3.6 says that a genus 3 curve is anétale double cover of a genus 2 curve if and only if it is both hyperelliptic and a double cover of an elliptic curve branched in 4 points.
Proof. Immediate from Proposition 3.6 Now, we would like to recall the tools which we use in the proof of Theorem 3.10.
Theorem 3.8 (Generalised Torelli Theorem [W]). Let C be a smooth curve. Then Proof. The idea of the proof is that the inclusion ⊃ comes from the Prym construction and ⊂ comes from the generalised Torelli Theorem. We will prove the odd dimension case in detail. Let C ∈ D 2g+1 0 ∪ D 2g+1 4 . Let f be the quotient map and C ′ be the quotient curve. If C ∈ D 2g+1 4 then C ′ is of genus g and f * is injective, so JC ′ is embedded in JC with the induced polarisation of type 2, whereas for C ∈ D 2g+1 0 we have that f isétale defined by the two torsion point, say η, and C ′ is of genus g + 1, so JC ′ /η is embedded in JC with the induced polarisation of type (1, 2, . . . , 2). The complementary polarisation type is 2, so in both cases JC ∈ Is From the Hurwitz formula we get that 2(2g + 1) − 2 = 2(2(g + 1) − 2) + b, so b = 0 and therefore f is anétale double cover, so JC ∈ J D 2g+1 0 Analogously, if i N is the extension of the involution on C then f is branched in 4 points, so JC ∈ J D In particular when C is not hyperelliptic and JC ∈ Is 2g 2 , then for a pair of complementary subvarieties (M, N ) of type 2 exactly one of them is the Jacobian JC ′ of genus g curve such that C is a double cover of C ′ and the other is the Prym variety of the double cover. If C is hyperelliptic, then both subvarieties are Jacobians and Pryms for each other.
The idea of this proof leads to an interesting observation.
Corollary 3.11. Let D = (1, . . . , 2) be a g-tuple with a positive number of 1's and 2's. Then (1) Is 2g Proof. If either of those was non-empty, we would find a pair of complementary subvarieties (M, N ) and an involution on C which induces one of the involutions (1 − Nm M ) or (1 − Nm N ). Taking the quotient curve C ′ , with quotient map f , we would find that M = f * (JC ′ ) or N = f * (JC ′ ).
In the first case both subvarieties are of dimension g, so the Hurwitz formula tells us that f has to be a double cover branched in 2 points, which means that the induced polarisation on f * JC ′ is twice a principal polarisation, a contradiction.
In the second case the Hurwitz formula states that 2(2g + 1) − 2 = 2(2g(C ′ ) − 2) + b, where b is the number of branch points, and gives two possibilities. If g(C ′ ) = g + 1, then b = 0 and we have anétale double cover and from the Prym construction ( [BL,Thm 12.3.3]) we get a contradiction. If g(C ′ ) = g, then b = 4 and again we get a contradiction because the induced polarisation on N has to be of type 2 ( [BL,Prop 11.4.3]).
There is one more result related to Is 3 2 . Proposition 3.12. There is a 1 to 1 correspondence between the set of smooth genus 3 hyperelliptic curves (up to translation) on a general abelian surface A and the set of degree 2 polarised isogenies A −→ B, where B is the Jacobian of a smooth genus 2 curve. In particular, there are exactly three hyperelliptic curves in the linear system of a (1, 2) polarising line bundle on a (very) general abelian surface.
Proof. Let (JC ′ , Θ) be the Jacobian of a smooth genus 2 curve. Let ρ : A −→ JC ′ be a degree 2 isogeny. Then ρ −1 (Θ) is a genus 3 hyperelliptic curve on A, which is anétale double cover of C ′ .
Conversely, let C be a hyperelliptic genus 3 curve on A. Then O(C) is a (1, 2) polarising line bundle. From the universal property of Jacobians there exists a surjective map f : JC −→ A. By [BL2,Prop 4.3],f is an embedding ofÂ with restricted polarisation of type (1, 2). Therefore JC ∈ Is 3 2 . As C is also hyperelliptic, Proposition 3.6 tells us that there exists anétale double cover C −→ C ′ . It is defined by a 2-torsion point, say η, and there is an embedding of JC ′ / η to JC with the restricted polarisation of type (1, 2). As A is general, we haveÂ = JC ′ / η and by dualising the quotient map, we obtain a degree 2 polarised isogeny A −→ JC ′ . The last part follows from the fact that there are exactly three non-zero 2-torsion points in the kernel K(O(C)). A (very) general surface means one for which the resulting principally polarised abelian surface is the Jacobian of a smooth curve.
3.3. Irreducibility. The aim of this section is to show that Is g D is irreducible. This will be an indication that the choice of definition is a good one.
In the proof of this fact we will use condition (4) of Proposition 2.5, so we define
(2) ker(ρ) is an isotropic subgroup of K(L) with respect to e L .
This leads to an obvious corollary. Let us state the main result of this section.
Proof. Proposition 2.5 tells us that A belongs to Is g D if and only if there exists an allowed isogeny to A. Therefore the idea of the proof is to show that there exists one map from h k × h g−k which covers all possible allowed isogenies and so Is g D is the image of an irreducible variety. The sketch of the proof is as follows. Take polarised abelian varieties (M, H M )  To make this more precise, we need to recall that a period matrix of an abelian variety is a choice of symplectic basis of a lattice in its universal cover.
Let l ≤ k be the number of integers bigger than 1 in D.
Then K B is an allowed isotropic subgroup. Moreover, if we change bases using the symplectic action, then im(K) will always define an allowed isotropic subgroup and by Proposition 2.13, every allowed isotropic subgroup arises in this way.
When we take the universal cover V of M × N , in order to write the period matrix we need to choose a symplectic basis of V . The obvious choice is to enlarge the symplectic basis B to a symplectic basis B. We need to enlarge the matrix K by zero-blocks to a matrix K, such that its image is equal to K B .
From this discussion, we have found and a matrix K, such that im(K) is an allowed isotropic subgroup of M × N . The data defining K is discrete, so im(K) will be allowed isotropic for any matrices Z k ∈ h k , Z g−k ∈ h g−k . Moreover, the symplectic action on h k × h g−k gives all possible period matrices hence all possible symplectic bases and therefore all possible allowed isotropic subgroups.
Thus we have proved that there exists a global map which covers all possible allowed isogenies; that is, for any allowed isogeny M × N −→ A, there exist period matrices Z(M ) and Z(N ) such that Ψ(Z(M ) × Z(N )) = A. From the construction, it is obvious that Is g D is the image of the above map and as the domain is irreducible, it follows that Is g D is an irreducible variety.
Remark 3.18. Proposition 3.17 is stated as a fact in [D, (9.2)] and proved using irreducibility of some moduli space. Both constructions are similar, but the proof presented in this paper is explicit.
There is a generalisation of Humbert surfaces to the moduli of non-principally polarised abelian surfaces. However, in that case, the generalised Humbert surface is no longer irreducible. For details, see [vdG].
One can also generalise further Proposition 3.17 to non-principally polarised abelian varieties. Let D be a polarisation type of an abelian g-fold. The idea is to define for any polarisation types D 1 ∈ Z k , D 2 ∈ Z g−k the locus Is g,D D 1 ,D 2 of D-polarised abelian g-folds which have a pair of complementary subvarieties of types D 1 and D 2 . The obvious question is whether Is g,D D 1 ,D 2 is non-empty. Using Proposition 3.15 one can translate the question into one about the existence of isotropic subgroups analogous to the allowed ones. The proof of Proposition 3.17 can be easily generalised, but one must have in mind that the number of irreducible components of Is g,D D 1 ,D 2 will be equal to the number of orbits of such isotropic subgroups. To sum up, the problem can be solved if one can deal with the combinatorics related to special isotropic subgroups in finite symplectic groups. Certainly this is possible in many cases, such as (1, p)-polarised surfaces (see [vdG]).

Equations in the Siegel space
As in the Humbert surface case, we would like to find equations for a locus in h g which maps to Is g D in A g . Ideally, we would like to find the equations of the whole preimage of Is g D which would involve understanding the action of Sp(2g, Z) on h g and finding good symplectic invariants.
We start by proving an obvious, yet important lemma. It remains to compute the restricted polarisation using analytic representations written in block matrices. We have so the induced polarisation is of type D. is the g × (g − k) block matrix of the last (g − k) columns of Z A with the last k columns of Z A multiplied by D from the right and having subtracted the first k columns. Therefore, images of generators of Λ N are primitive so ι N is an embedding. Checking that the restricted polarisation is of typeD is completely analogous.
Using Propositions 2.5 and 3.17, we can summarise the discussion into the following theorem.