On Frobenius' theta formula

Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Frobenius' theta formula, which derives from Riemann's theta formula. In a 2004 paper Grushevsky gives a different characterization in terms of cubic equations in second order theta functions. In this note we first show the connection between the methods by proving that Grushevsky's cubic equations imply Frobenius' theta formula and we then come to a new proof of Mumford's characterization via Gunning's multisecant formula.


Introduction
We denote by H g the Siegel upper half-space -the space of complex symmetric g × g matrices with positive definite imaginary part. An element τ ∈ H g is called a period matrix, and defines the complex abelian variety X τ := C g /Z g + τ Z g . It is a well-known fact that the moduli space of principally polarized abelian varieties (ppavs for short) can be identified with the quotient of H g by an action of the symplectic group Sp(2g, Z) which is a generalization of the standard action of SL(2, Z) on the complex upper-half plane, namely: where a, b, c, d are g × g blocks. A ppav is called irreducible if it is not isomorphic to a product of two lower-dimensional ppavs. For ε, δ ∈ Z g 2 and z ∈ C g we define the first order theta function with characteristic [ǫ, δ] to be Characteristics can be defined for ε, δ in Z g , but the reduction formula: (1) θ ε + 2δ ε ′ + 2δ ′ (τ, z) = (−1) <ε,δ ′ > θ ε δ (τ, z) (where the symbol < ·, · > stands for the standard inner product) shows that these functions are uniquely determined up to a sign by considering ε and δ as vectors of zeros and ones. Henceforward we shall work with reduced characteristics with the agreement that a theta function associated with a sum of characteristics is meant to be non-reduced. Characteristics are defined even or odd depending on whether respectively < ε, δ >= 0 or < ε, δ >= 1. A straightforward computation shows there are 2 g−1 (2 g + 1) even characteristics and 2 g−1 (2 g − 1) odd ones. It is easily seen that theta functions associated with even characteristics are even functions in the variable z, whereas those associated with odd characteristics are odd functions. Triplets of characteristics [ε, δ], [ε ′ , δ ′ ], [ε ′′ , δ ′′ ] are also called azygetic if < ε, δ > + < ε ′ , δ ′ > + < ε ′′ , δ ′′ > + < ε+ε ′ +ε ′′ , δ +δ ′ +δ ′′ >= 1. More generally, a k-tuple of characteristics is called azygetic if its subtriplets are azygetic. Theta functions satisfy an addition formula (cf. [Ig72] for a general formulation and [Gr] or [GSM] for the specific version we use here): For a given τ ∈ H g we denote by X τ [2] the group of points of order two on X τ and by L τ a symmetric line bundle on X τ defining the principal polarization. Note that: and the function z → θ ε δ (τ, z) defines the unique (up to scalar multiplication) section of t * x L τ , i.e the translate of the line bundle L τ by the point of order two x = (τ ε + δ)/2. Because of the duplication map, the functions z → θ ε δ (τ, 2z) are a basis of H 0 (X τ , L ⊗4 τ ).
For ε ∈ Z g 2 we also define the second order theta function with characteristic ε to be The functions z → Θ[ε](τ, z) are a basis of H 0 (X τ , L ⊗2 τ ). Since these functions are even, i.e Θ[ε](τ, −z) = Θ[ε](τ, z), they induce a map T h 2 : X τ → P 2 g −1 that factorizes along the Kummer variety K τ := X τ / ± 1 By evaluating first order theta functions at z = 0 we get the so-called theta constants θ ε δ (τ, 0); because of the parity of theta functions, theta constants associated with odd characteristics are clearly trivial. Analogously, second order theta functions evaluated at z = 0 yield second order theta constants Θ[ε](τ, 0). As functions of τ these are not well defined on the moduli space of ppavs but, as a consequence of how they transform under the action of Sp(2g, Z), they induce maps on finite covers of the moduli space. More precisely, if we introduce the following family of subgroups of finite index in Sp(2g, Z) first order theta constants are seen to transform as follows (cf. [Ig72] and [RF] for the general formula): where κ(γ) is an eighth root of the unity for any γ and χ ε,δ are characters of the group Γ g [2, 4]/Γ g [4,8]; in particular, an action of Γ g [2, 4]/Γ g [4, 8] is naturally induced on theta constants and can be described in terms of these characters χ ε,δ , which span the character group of Γ g [2, 4]/ ± Γ g [4, 8] (cf. [SM]). Thanks to the transformation formula, first order theta constants induce an immersion of H g / ± Γ g [4,8] into the projective space. Second order theta constants also induce a map on a finite cover of the moduli space of ppavs: which is generically injective for any g. Once a basis ω 1 , . . . , ω g for the cohomology of an algebraic curve of genus g is chosen together with a a symplectic basis of cycles δ 1 , . . . , δ g , δ ′ 1 , . . . , δ ′ g , the g × 2g matrix ( δ j ω i , δ ′ j ω i ) defines a complex torus which is isomorphic to a ppav X τ ; this complex torus is known as the Jacobian variety of the curve and the locus in the moduli space of ppavs defined by those ppavs that are isomorphic to Jacobians of curves is known as the Jacobian locus. Because of the existence of the map T , one knows that the Jacobian locus and the hyperelliptic locus (i.e. the locus defined by those ppavs that are isomorphic to Jacobians of hyperelliptic curves) can be described in terms of equations involving theta constants by taking the preimages of these loci under the covering map H g /Γ g [2, 4] → H g /Sp(2g, Z); these preimages have several irreducible components in H g /Γ g [2, 4].
As for the hyperelliptic locus, we know from the result in [M] that the irreducible components are defined in terms of vanishing and non-vanishing conditions for certain theta constants θ ε δ (τ, 0). Mumford's result is based on Neumann's dynamical system. In [Po] Poor used the cross ratio to prove that the vanishing conditions alone are sufficient to characterize the hyperellpitic locus, once the irreducibility of the principally polarized abelian variety is assumed. In both cases a fundamental tool is the so-called Frobenius' theta formula, which is a consequence of Riemann's theta formula (cf. [M]). A few years ago in [Gr], a characterization of the hyperelliptic abelian varieties was given in terms of cubic equations in the second order theta functions. These cubics were obtained by using the explicit coefficients for the addition formula from [BK]. The aim of this note is twofold: we first relate Grushevsky's approach to the others' by proving that the cubic equations imply Frobenius' theta formula; we then come to a new proof of Mumford's characterization by applying Gunning' s multisecant formula (cf. [Gu]).
The authors would like to thank Bert van Geemen for drawing their attention to the result in [EPW]. They are also grateful to Sam Grushevsky for many helpful discussions and explanations.

Theta Functions
Throughout the rest of the paper we shall omit the subscript τ whenever it is clear that τ is fixed. Once we set M : , the following group is well defined: and an irreducible action of G(M ) on the space of global sections of M is also defined by setting (x, φ)s := φ(t * x s). A theta structure is an isomorphism between G(M ) and the Heisenberg group, namely the set: H := C * × F g 2 × Hom(F 2 , C * ) g provided with the group law: (t, x, x * ) · (s, y, y * ) := (tsy * (x), x + y, x * + y * ) Now, let B n := Γ(X, M n ) for any n ≥ 1. The action of {±1} decomposes each of these spaces into the direct sum of two factors B + n and B − n . Note that B 1 = B + 1 and B 1 is an irreducible representation of H. Moreover, B 1 admits a basis {X σ } with σ ∈ F g 2 such that (t, x, x * )X σ = tx * (x + σ)X x+σ . The Heisenberg group naturally acts on the n-fold symmetric products S n B 1 as well, and we recall from [vG] the following: 2 , the symbol < ·, · > stands for the standard inner product and < ǫ, ǫ ′ >= 0.
The action of H on the elements of the basis is given by By evaluating at the theta functions X σ = Θ[σ](τ, z) and applying the addition formula, we get: Hence, a basis for the quadrics containing the Kummer variety is given by those Q[ǫ, ǫ ′ ] such that θ ǫ ǫ ′ (τ, 0) = 0. We will shortly discuss the case of the quartics; more details on the subject can be found in [vG].
The space S 4 B 1 decomposes as where χ : H → {±1} runs over the characters that factorize along (Z/2Z) 2g . The dimension of the eigenspace associated with the trivial character is equal to (2 g + 1)(2 g−1 + 1)/3, whereas the dimension of the other eigenspaces is (2 g−1 + 1)(2 g−2 + 1)/3. A similar decomposition holds for the space: In this case the dimensions of the eigenspaces are respectively equal to 2 g−1 (2 g + 1) and 2 g−2 (2 g−1 + 1). Obviously we have a surjective map According to [Fa] (cf. also [APS] or [FGS]) the relations are the so-called Riemann relations that are generated by those of the form Here the vector v = (v ε,δ ) varies in a suitable space of dimension (2 2g − 1)/3. To sum up, we can state the following: Proposition 1.2. All bi-quadratic Riemann relations induce trivial relations between second order theta functions.
We know there exist highly non-trivial relations like the equation defining the Kummer surface in genus 2 or the Coble quartic in genus 3. These are in the kernel of the map: Remark 1.3. More general Riemann relations are obtained by evaluating (2) at the points z + (τ x + y)/4, x, y ∈ Z g , which yields: (3) <ε,δ>=0 ).

Cubic equations
As soon as the genus is greater than 2, there are also cubic equations defining the Kummer variety. To focus on cubic relations, we first note that the spaces S 3 B 1 and B + 3 decompose under the action of the Heisenberg group into 2 g irreducible representations that are all isomorphic to B 1 (see again [vG]); we are interested in studying the kernel of the map Ψ 3 : S 3 B 1 → B + 3 . A simple way of constructing cubic relations is considering quadratic relations. Quadratic relations exist whenever we have the vanishing of some even theta constants i.e θ ε δ (τ, 0) = 0 with < ε, δ >= 0. If none of the theta constants vanish, we can prove cubic relations always exist by means of a dimensional argument; indeed, we have: In the genus 3 case there are exactly 8 cubics: in the non-hyperelliptic case (no vanishing theta constants) these cubics can be obtained as derivatives of the Coble quartic, while in the hyperelliptic case (one vanishing theta constant) they are induced by the quadric. When the genus is higher, we need to build up a set of theta constants θ ǫ ǫ ′ (τ, 0) that vanish at the point τ in the hyperelliptic locus; to do this, we start with a hyperelliptic point to which we can associate a fundamental system of characteristics, namely an azygetic collection of 2g + 2 characteristics m 1 , . . . , m g , m g+1 , . . . , m 2g+2 such that the first g are odd and the last g + 2 are even. Once we denote by {e k } k=1,...g the elements of the natural basis in (Z/2Z) g , and set e g+1 = 0 and s k = e 1 + · · · + e k , we can choose the following as fundamental system: To be consistent with Mumford's notation in [M], page 106, we set: . We then know from the characterization of the hyperelliptic locus (cf. [T] and [M]) that the period matrix of a hyperelliptic curve always admits a conjugate τ ∈ H g under the action of the group Sp(2g, Z), for which the following vanishing and non-vanishing conditions for theta constants hold: The case g = 4.
To obtain all the even characteristics m T corresponding to vanishing theta constants we can choose T such that #T = 4 and Note that in [Ig] the fundamental system is given by a set of characteristics which is obtained from the above collection by switching the ε with the δ. Nevertheless, we prefer to use our form, since it is compatible with the notation in [Gr]. For this collection of vanishing theta constants the 10 induced quadratic relations Q[ǫ, ǫ ′ ] = 0 produce 160 generators for KerΨ 3 that are linearly dependent, as the dimension of the space is 144 (computed with Mathematica).
Another known example of a point τ ∈ H 4 with a set of 10 vanishing even theta constants is given by the so-called Varley-Debarre abelian variety. For the period matrix associated with this variety the set of characteristics corresponding to vanishing theta constants can be deduced from [Be] and chosen as follows (cf. Prop. 4.5 in [EPW]): For such a collection the dimension of the kernel of Ψ 3 turns out to be equal to 160 (all the generators are linearly independent). Hence, we could not have original cubics.
In general, cubic equations in the hyperelliptic case that turn out to be independent from the quadrics are given in [Gr]; we will discuss these cases in detail and deduce interesting results. First of all, we recall the following result from [Gr]: Theorem 2.1. An irreducible period matrix τ ∈ H g is the period matrix of a hyperelliptic Jacobian with the basis of cycles suitably chosen in such a way that the corresponding fundamental system is the one given in (4) if and only if the following cubic identity for second order theta functions is satisfied for all σ ∈ (Z/2Z) g and for all z ∈ C g : where we assume that not all the coefficients appearing in front of Θ[σ](τ, z) and Θ[σ + s k ](τ, z) are identically zero in z.
We recall that the result is obtained by computing the coefficients in the addition formula that appears in [BK]. Using the addition formula for theta functions, Equation (8) can be rewritten as Hence, the condition on the coefficients of Θ[σ](τ, z) and Θ[σ + s k ](τ, z) stated in Theorem 2.1 turns into the assumption that not all θ ε δ (τ, 0) are , . . . , s g 0 .
We observe that this assumption is satisfied in Mumford's proof; actually, the non-vanishing assumption (7) is stronger. As proved in [Gr], these equations in genus 3 are equivalent to the vanishing of one theta constant, yet when it comes to the genus 4 case they appear to be new; we try to explain the reason. First of all, we need to recall from [M] Theorem 7.1, the so-called generalized Frobenius' theta formula. For U ⊂ B as in (5) we set ε U (j) := ±1 according as j ∈ U or j / ∈ U ; then we have Theorem 2.2. Let τ ∈ H g satisfy the non-vanishing conditions in (7). Then: In [M] extra variables a i actually appear, but, as shown in [Po], they are redundant. It is easily checked that all Grushevsky's relations generate a 2 g -dimensional space that is an irreducible representation W ⊂ S 3 B 1 w.r.t. the Heisenberg group H. We shall consider the image, say V , of W ⊗B 1 in S 4 B 1 , decompose the space w.r.t. the characters and consider the eigenspaces V χ associated with the characteristics [ε 1 , δ 1 ].
We also remark that if we apply the addition formula for theta functions to Grushevsky's cubic equations, we are given similar equations that are also consequence of the generalized Frobenius' theta formula; in fact, we have: Proposition 2.3. If τ ∈ H g satisfies Equations (9), we have : Hence, in the RHS we obtain g k=0 Q[s k , e k+1 ](τ, z)Q[s k + ε 1 , e k+1 + δ 1 ](τ, z) Remark 2.4. We observe that in the above formula (11) signs appear if we consider the characteristics as reduced, i.e. with entries equal to 0 or 1; these signs are actually induced by the reduction formula (1).
As a particular yet fundamental case, we obtain Corollary 2.5. If τ ∈ H g satisfies Equations (9) we have: The important fact is that this formula is exactly the one obtained in [M] (Corollary 7.4 , p. 112) when S = ∅. It is therefore a special case of Frobenius' theta formula, which is fundamental in Mumford's investigation of Neumann's dynamical system (see Lemma 9.7 in [M]). To obtain all Frobenius' relations described by Mumford in the above cited Corollary, we can evaluate Equation (12) at the points z + (τ x + y)/4 for x, y ∈ Z g , in the same way we did with Riemann's relations to obtain (3); this leads to the following Corollary 2.6. If τ ∈ H g is a period matrix satisfying Equation (12), then: One is led to suppose that Frobenius' formula might imply the vanishing of the suitable theta constants. Let us discuss this formula in low genera. In the genus 3 case let us assume θ 0 0 (τ, 0) = 0, and evaluate Formula (13) at x = (1, 0, 1) t , y = (1, 1, 1) t and z = 0. Because of the vanishing of theta constants corresponding to odd characteristics we obtain: It is a well-known fact that this equation characterizes a component of the hyperelliptic locus in genus 3. In genus 4 we can use the same method and evaluate x, y at ten points so as to get the previous ten characteristics.
Thus, we are left with a system of 6 equations in the six variables θ ε k δ k (τ, 0) 2 for k = 0, 5, 6, 7, 8, 9. The matrix of the coefficients is: which unfortunately has rank 5, since the determinant is Hence, a priori, the system can admit a non-trivial solution; one expects this solution to be ruled out by using Frobenius' formula in its full generality, i.e the equation in (10). As we are dealing with the genus 4 case, we obviously know from [Igu] that the vanishing of four azygetic theta constants implies that the point is hyperelliptic. We have to mention that Igusa's proof makes use of Riemann's relations of the form described in (3).

Characterization of the Hyperelliptic locus
Whenever τ is the period matrix of a hyperelliptic curve, a classic result (cf. [T]) states that a system of characteristics as in (4) with the vanishing and non-vanishing properties (6) and (7) can be associated with τ or with a conjugate of τ via the action of Sp(2g, Z).
It was not until 1984 that Mumford proved the reverse statement in [M]. We will now give a new proof of Mumford's result with a different approach that has the merit of involving Gunning's multisecant formula, as expressed in the following statement (cf. [Gu]): Theorem 3.1. Let X be an irreducible principally polarized abelian variety of dimension g, and let A 0 , ..., A g+1 be distinct points of X. Suppose that ∀z ∈ X the g+2 points T h 2 (A i +z) are linearly dependent. Assume moreover the following general position condition: that there exist some k and l such that for y = (−A k + A l )/2 the linear span of the points T h 2 (A i + y) for i = 0, . . . , g + 1 has dimension precisely equal to g + 1, and not less. Then X is the Jacobian of some curve C, and all the points A i are in the image of the Abel-Jacobi map.
We are now in a position to prove that the above conditions are satisfied in the hyperelliptic case; indeed, we have the following: Theorem 3.2. Let τ be a point whose vanishing and non-vanishing theta constants are those induced by the fundamental system (4); then τ is the period matrix of a hyperelliptic jacobian.
We are yet to prove the general position condition.
Thus, the matrix B turns into Therefore, the matrix B has rank g + 1, hence g + 1 ≥ rk(A) ≥ rk(B) = g + 1, which means the matrix A has the required rank.
Remark 3.3. This result also gives some evidence to the conjecture formulated in [Gr], according to which the evaluation of the cubics at z = 0 is expected to determine the corresponding irreducible component of the hyperelliptic locus.