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Theta functions on the theta divisor

We show that the gradient and the hessian of the Riemann theta function in dimension n can be combined to give a theta function of order n+1 and modular weight (n+5)/2 defined on the theta divisor. It can be seen that the zero locus of this theta function essentially gives the ramification locus of the Gauss map. For jacobians this leads to a description in terms of theta functions and their derivatives of the Weierstrass point locus on the associated Riemann surface.

In the analytic theory of the Riemann theta function a natural place is taken by the study of the first and second order terms of its Taylor series expansion along the theta divisor. The first order term essentially gives the gradient, and hence the tangent bundle on the smooth locus, whereas the second order terms give rise to hessians, which are widely recognised to carry subtle geometric information along the singular locus of the theta divisor. For example, in the case of a jacobian these hessians define quadrics containing the canonical image of the associated Riemann surface. Or, in the general case one could investigate the properties of those principally polarised abelian varieties that have a singular point of order two on their theta divisor, such that the hessian of the theta function at that singular point has a certain given rank. This is a recent line of investigation begun by Grushevsky and Salvati Manni, with interesting connections to the Schottky problem [5] [6].
If one moves outside the singular locus of the theta divisor, it is not immediately clear whether the hessian of the theta function continues to have some geometric significance. In this paper we prove that it does. More precisely, we show that a certain combination of the gradient and the hessian gives rise to a well-defined theta function living on the theta divisor. We can compute its transformation behavior, i.e. its order and automorphy factor explicitly. As it turns out, the geometric meaning of our function is that on the smooth locus of the theta divisor it precisely gives the ramification locus of the Gauss map, i.e. the map sending a smooth point to its tangent space, seen as a point in the dual of the projectivised tangent space of the origin of the ambient abelian variety. We can study the situation in more detail for jacobians, where our theory leads to a global description of the locus of Weierstrass points on the corresponding Riemann surface in terms of theta functions and their derivatives. By way of example we state and prove an explicit formula for our theta function in genus two.

Definition and main theorem
Let H n denote the Siegel upper half space of degree n > 0. On C n × H n we have the Riemann theta function θ = θ(z, τ ) = m∈Z n e πi t mτ m+2πi t mz .
Here and henceforth, vectors are column vectors and t denotes transpose. For any fixed τ , the function θ = θ(z) on C n satisfies a transformation formula (1.1) θ(z + τ u + v) = e −πi t uτ u−2πi t uz θ(z) , The author acknowledges a grant from the Netherlands Organisation for Scientific Research (NWO).
for all z in C n and for all u, v in Z n . Moreover, it has a symmetry property for all z in C n . Equation (1.1) implies that div θ is well-defined as a Cartier divisor on the complex torus A = C n /(Z n + τ Z n ). We denote this divisor by Θ. In fact θ gives rise to a global section of O A (Θ), and we say that θ is a "theta function of order 1". By equation (1.2), the divisor Θ is symmetric: [−1] * A Θ = Θ. For τ varying through H n , it turns out that θ = θ(z, τ ) is a modular form of weight 1/2. More precisely, consider the group Γ 1,2 of matrices γ = a b c d in Sp(2n, Z) with a, b, c, d square matrices such that the diagonals of both t ac and t bd consist of even integers. We can let Γ 1,2 Under this action, the Riemann theta function transforms as for some 8-th root of unity ζ γ (cf. [10], p. 189). We write θ i for the first order partial derivative ∂θ/∂z i and θ ij for the second order partial derivative The function we want to study in this paper is then the following. Definition 1.1. Let (θ i ) be the gradient of θ in the C n -direction, and let (θ ij ) be its hessian. Then we put We want to consider this as a function on the vanishing locus θ −1 (0) of θ on C n × H n .
Our main result is that the function η transforms well with respect to both lattice translations and the action of the congruence symplectic group Γ 1,2 . Theorem 1.3. The function η = η(z, τ ) is a theta function of order n + 1 and weight (n + 5)/2 on the theta divisor. In other words, for any fixed τ in H n , the function η gives rise to a global section of the line bundle O Θ (Θ) ⊗n+1 on Θ in A = C n /(Z n + τ Z n ). Furthermore, when viewed as a function of two variables (z, τ ), the function η transforms under the action of Γ 1,2 with an automorphy factor det(cτ + d) (n+5)/2 on θ −1 (0). It follows that for any fixed τ , the zero locus of η is well-defined on Θ. This zero locus contains Sing Θ, the singular locus of Θ, as well as the set of 2-division points Θ ∩ A [2] if n ≥ 2. Moreover this zero locus is stable under the involution z → −z of Θ.
For τ varying through H n it follows that the function η gives rise to a global section of a line bundle on the canonical symmetric theta divisor Θ of a universal principally polarised abelian variety with level structure π : U n → A (1,2) n = Γ 1,2 \ H n . Here M is a certain line bundle on A (1,2) n . It follows from general principles that M is a power of λ, the determinant of the Hodge bundle on A (1,2) n . By counting weights we find that M ∼ = λ ⊗2 . Example 1.4. When n = 1, the theorem states that dθ dz is of order 1 and of modular weight 3/2 on the theta divisor. Both statements follow directly from (1.1) and (1.3). Alternatively, the statement on the modular weight can be seen using Jacobi's derivative formula (cf. [10], p. 64). This formula says that e πiτ /4 dθ dz where θ 0 0 , θ 0 1/2 and θ 1/2 0 are the usual elliptic theta functions with even characteristic.
A proof of Theorem 1.3 will be given in Section 4.

Properties
In this section we collect some further properties of η.
Proposition 2.1. Assume that Θ = div θ on C n /(Z n + τ Z n ) is non-singular, and denote its canonical bundle by K Θ . Then η gives rise to a global section η of K ⊗n+1 Θ , locally given by Proof. We view η as a global section of the line bundle O Θ (Θ) ⊗n+1 on Θ. By the adjunction formula and the fact that the canonical bundle K A of A is trivial we have an identification This identification can be represented locally by the Poincaré residue map From this the corollary follows.
By way of illustration, we make the multidifferential η more explicit in the case that n = 2. We need the notions of a Wronskian differential and of Weierstrass points, which we recall briefly. Let X be a compact Riemann surface of genus n > 0, and let ζ = (ζ 1 , . . . , ζ n ) be any basis of H 0 (X, K X ). The Wronskian differential ω ζ of ζ is then given as follows: let t be a local coordinate, and write ω j (t) = f j (t)dt with the f j holomorphic. We put This local definition gives, in fact, rise to a global section of the line bundle K ⊗n(n+1)/2 X . It can be proved that this section is non-zero. If the basis ζ is changed, the differential ω ζ changes by a non-zero scalar. Hence the divisor W = div ω ζ is independent of the choice of ζ. We call this divisor the classical divisor of Weierstrass points of X. It has degree n 3 − n.
is an indecomposable abelian surface with theta divisor Θ = div θ. Then Θ is a compact Riemann surface of genus 2, and there is a canon- corresponding under this identification to the standard basis (dz 1 , dz 2 ) of H 0 (A, Ω 1 ). We claim that η(z) is equal to the Wronskian differential ω ζ of ζ. This amounts to a small computation: choose an open subset of Θ where z 1 is a local coordinate. In this local coordinate we can write We find, indeed, . A similar computation can be done on the locus where z 2 is a local coordinate, giving ω ζ (z) = η(z) globally on Θ, as required. Note that our identification ω ζ (z) = η(z) gives, as a corollary, that div η = W , the divisor of Weierstrass points of Θ. In Section 5 we will prove a generalisation of this result.
It is interesting to know when η is identically zero on the theta divisor. Proposition 2.3. If A is a decomposable principally polarised abelian variety, then η is identically zero on the theta divisor.
Proof. Let us suppose that A = A 1 × A 2 with A 1 given by a matrix τ 1 in H k and A 2 given by a matrix τ 2 in H n−k where k is an integer with 0 < k < n. We can write θ(z) = F (z 1 , . . . , z k )G(z k+1 , . . . , z n ) where F, G are the Riemann theta functions for A 1 and A 2 , respectively. Let Θ 1 ⊂ A 1 be the divisor of F , and Θ 2 ⊂ A 2 the divisor of G. By symmetry, it suffices to prove that η is zero on The subset Θ 1 × A 2 ⊂ Θ is given by the vanishing of F ; there we find Note that both F i G j and F j G i , being a product of a vector and a covector, have rank ≤ 1. Hence, a minor at (i, j) with 1 ≤ i, j ≤ k in (θ ij ) has rank ≤ 1 + (k − 1) = k. If k < n − 1, the determinant of this minor vanishes, and the cofactor matrix of (θ ij ) has the shape We find then that If k = n − 1, the last row of (θ ij ) and the vector t (θ i ) are linearly dependent. If 1 ≤ i ≤ n − 1 we see that the i-th entry of (θ ij ) c (θ j ) is just the determinant of the matrix obtained from (θ ij ) by removing the i-th row and adding t (θ i ) in its place. But this matrix then contains two linearly dependent vectors, and its determinant vanishes. So we obtain in this case as well.

Interpretation
The contents of the present section are based on kind suggestions made by Prof. Ciro Ciliberto. As we observed in Section 1, for any complex principally polarised abelian variety the zero locus of η is well-defined on the theta divisor. Given the simple description of η one expects that this zero locus has some intrinsic geometric interpretation. This indeed turns out to be the case. Proof. It follows from formula (1.4) that a point x on Θ s is in the zero locus of η if and only if the quadric Q in P(T 0 A) defined by the hessian is tangent to the projectivised tangent hyperplane P(T x Θ) defined by the gradient. The latter condition is equivalent to the condition that Q when restricted to P(T x Θ) becomes degenerate. Now note that Q when viewed as a linear map The locus where this map is degenerate is precisely the ramification locus of Γ.
Using the above interpretation, a converse to Proposition 2.3 can be readily proved. Proof. We need to prove that if Θ is irreducible, then the Gauss map on Θ has a proper ramification locus. But according to [9], Corollary 9.11 the Gauss map is generically finite and dominant in this case, and the result follows. Remark 3.3. It follows from Theorem 1.3 that in the indecomposable case the divisor of η belongs to the linear system defined by (n + 1)Θ on Θ. This fact can also be explained as follows. For simplicity, let us assume that Θ is non-singular. The Hurwitz formula applied to the Gauss map Γ : Θ → P = P(T 0 A) * gives that K Θ = Γ * K P + R, where R is the ramification locus of Γ. By definition Γ * O P (1) ∼ = K Θ and hence Γ * K P ∼ = K ⊗−n Θ . We find that R is in the linear system belonging to K ⊗n+1 Θ . As we have seen at the end of the previous section, this is the same as O Θ (Θ) ⊗n+1 . Alternatively, this remark shows that the equations θ = η = 0 define the right scheme structure on R.
Remark 3.4. The above description of the vanishing locus of η can also be found in the paper [6] of Grushevsky and Salvati Manni. In that paper they give an application of the form η to the study of certain codimension-2 cycles in the moduli space of principally polarised abelian varieties. In particular they are able to give a moduli interpretation of a certain cycle R g introduced by Debarre in [2], Section 4.

Proof of the main theorem
In this section we give a proof of Theorem 1.3. We start with the statement on the order of η. We fix an element τ in H n . The case n = 1 being discussed already in Example 1.4, we assume here that n ≥ 2. Recall from equation (1.1) that for all z in C n and all u, v in Z n , where We need to prove that η(z + τ u + v) = p(z, u) n+1 η(z) for all z in C n with θ(z) = 0. Denote by p i the first order partial derivative ∂p/∂z i . From (4.1) we have, for z with θ(z) = 0, and We are done, therefore, if we can prove, formally in some domain R containing the symbols p, p i , θ i , θ ij , the identity At this point we introduce some notation. Let h = (h i k j l ) be an R-valued matrix with rows and columns indexed by finite length ordered integer tuples I = (. . . , i k , . . .) and J = (. . . , j l , . . .), respectively. If I ′ ⊂ I and J ′ ⊂ J are proper subtuples, we denote by h I ′ J ′ the submatrix obtained from h by deleting the rows indexed by I ′ and the columns indexed by J ′ . If h is a square R-valued matrix with both the row index set I = (i 1 , . . . , i m ) and the column index set J = (j 1 , . . . , j m ) subtuples of (1, 2, . . . , n), we define η(h) to be the element of R. It can be written more elaborately as On the other hand, the identity we need to prove can be written more compactly as The function η can be defined recursively.
Proof. The double sum on the right hand side can be expanded as Expanding the determinant of h {i k ′ } {j l ′ } along each of its rows, one finds Combining both formulas one gets (k,l) Denote by a = (a ij ) the n-by-n R-valued matrix (a ij ) = (p i θ j + p j θ i ) .
It has the property that the η's of all its square submatrices vanish. Proof. By Lemma 4.1, the statement follows by induction once we prove the special case that h is a square submatrix of size 2. In this case h has the shape and η(h) has an expansion This is identically equal to zero.
Denote by b = (b ij ) the n-by-n R-valued matrix and by b = ( b ij ) the n-by-n R-valued matrix We need to prove that Here the second sum is over all pairs (I, J) of subtuples of (1, 2 . . . , n) of length m, and ε I,J is a sign. In order to see this, expand in the cofactor matrix b c of b each minor as a sum of (n − 1)! terms. A product p m θ i1j1 · · · θ imjm occurs as a factor in such a term exactly at all entries (k, l) of b c for which (k, l) is not in I × J. If (k, l) is such an entry, at that entry the product p m θ i1j1 · · · θ imjm is multiplied, up to a sign ε I,J depending only on I, J, by det a I∪{k} J∪{l} . This determinant is understood to be equal to 1 if m = n − 1. It follows that the entry (k, l) contributes to η( b) with a term Summing over all possible (k, l) we obtain, if m < n−1, a contribution p m θ i1j1 · · · θ imjm ε I,J η(a I J ), and if m = n − 1 the contribution η(b).
By Lemma 4.2, every η(a I J ) with I, J of size smaller than n − 1 is zero. Therefore all terms in the summation on the right hand side in (4.2) vanish. This proves the first half of Theorem 1.3.
In order to prove the statement on the modular weight, we recall from equation (1.3) that for all z in H n and all γ = a b c d in Γ 1,2 , where q(z, γ, τ ) = e πi t z(cτ +d) −1 cz and where ζ γ is an 8-th root of unity. We claim that for all (z, τ ) satisfying θ(z, τ ) = 0. This is just a calculation. For (z, τ ) with θ(z, τ ) = 0 we have by (4.3) Furthermore, for such (z, τ ) we have We can write, at least for the purpose of this proof, This gives that z, τ )) .

This simplifies to
which, in turn, is equal to det(cτ + d) (n+5)/2 ζ n+1 γ q n+1 η(z, τ ) by the same methods as above when we dealt with the order of η. This completes the proof of Theorem 1.3.

Jacobians
The purpose of this section is to study η for jacobians. Assume that (A, Θ) = (C n /(Z n + τ Z n ), div θ) with n ≥ 2 is the jacobian belonging to a compact Riemann surface X marked with a symplectic basis B = (A 1 , . . . , A n , B 1 , . . . , B n ) of homology. This situation determines uniquely a basis (ζ 1 , . . . , ζ n ) = ζ = ζ B of H 0 (X, K X ) such that The isomorphism of C-vector spaces H 0 (X, K X ) * ∼ −→ C n given by ζ gives an isomorphism of complex tori H 0 (X, K X ) * /H 1 (X, Z) ∼ −→ C n /(Z n + τ Z n ) = A . By a theorem of Abel-Jacobi, the natural map AJ : Pic 0 X −→ C n /(Z n + τ Z n ) , is a bijection. By a theorem of Riemann, there is a unique element ∆ = ∆ B of Pic 1−n X such that under the composition of bijections From now on, we will take this identification of (A, Θ) with (Pic n−1 X, Θ 0 ) for granted. We have a natural surjection Σ : X (n−1) → Θ which is an isomorphism above Θ s . If we let this isomorphism be followed by the Gauss map Γ : Θ s → P(T 0 A) * = P(H 0 (X, K X )) we get the map sending a non-special divisor D of degree n − 1 on X to the linear span of its points on the canonical image in P(H 0 (X, K X ) * ). By Riemann-Roch, this span is indeed a hyperplane.
We denote by κ : X −→ Θ ⊂ A the map sending x to the class of (n − 1) · x. It is natural to study the pullback of η along κ. We claim the following result: Theorem 5.1. The section κ * η is not identically zero. The divisor of κ * η is equal to (n − 1)W , where W is the divisor of Weierstrass points of X.
We proceed in a few steps, starting with some notation. For any (m, i) in Z × Z ≥0 we put interpreted as a global section of the bundle κ * O A (Θ) ⊗ K ⊗i X of differential i-forms with coefficients in κ * O A (Θ). For example, for i = 0 we get f m,0 (x) = 0 identically, for i = 1 we get and for i = 2: For m = 0, . . . , n − 1 all f m,i are identically equal to zero. Of particular interest for us will be the section X is not identically zero, and we have div F = W .
Proof. Consider the map Φ : X × X → Pic n−1 X given by (x, y) → nx − y. From [3], p. 31 we obtain that Φ * Θ = W × X + n · ∆ X . Restricting to the diagonal we get κ * Θ = Φ * Θ| ∆X = (W × X + n · ∆ X ) | ∆X and by the adjunction formula it follows that Here 1 W denotes the tautological section of O X (W ). Lemma 5.3. We have κ * η(x) = 0 if and only if x is a Weierstrass point. In particular κ * η is not identically zero.
Proof. According to [2], p. 691 the ramification locus of the Gauss map is precisely given by the set of divisors D + x with D effective of degree g − 2 and x a point of X such that D + 2x is dominated by a canonical divisor, i.e. such that K X − D − 2x is linearly equivalent to an effective divisor. Thus, to say that η(κ(x)) = 0 means precisely that h 0 (K X − n · x) > 0 or equivalently, by Riemann-Roch, that h 0 (n · x) > 1. But this means precisely that x is a Weierstrass point.
Proof of Theorem 5.1. It follows from Section 1 that κ * η is a global section of κ * O A (Θ) ⊗n+1 ⊗ λ ⊗2 where λ is the trivial bundle det H 0 (X, K X ) ⊗ O X . By Lemma 5.3 we know that κ * η is non-zero. It is stated in Lemma 5.2 that F is a non-zero global section of κ * O A (Θ) ⊗ K ⊗n X . As was observed by Arakelov (cf. [1], Lemma 3.3), the bundle has a non-zero section given by with ω ζ the Wronskian differential on ζ. Combining, we find that ω 2 ζ ⊗ κ * η ⊗ F ⊗−(n+1) is a non-zero global section of O X . Hence it is a non-zero constant. We find div κ * η = (n + 1)div F − 2div ω ζ = (n + 1)W − 2W = (n − 1)W as required.
Remark 5.4. An elaborate computation shows that actually Let us prove this relation in the case that n = 2. So we look at indecomposable (A, Θ) with A = C 2 /(Z 2 + τ Z 2 ) and Θ = X = div θ. For z on Θ = X put Then according to what we have said before Lemma 5.2 we have We conclude that and writing as before We have seen in Example 2.2 that ω ζ (z) = −(η(z)/θ 2 (z) 3 )(dz 1 ) ⊗3 . Combining we find ω 2 ζ ⊗ κ * η = F ⊗3 as required. We note in passing that our formula (for general n) leads to an alternative description of one of the analytic invariants studied in [7].

Explicit formula
In the case that n = 2 it is possible to give a closed formula for η using a more familiar theta function. This formula can be viewed as a generalisation of Jacobi's derivative formula (cf. Example 1.4), which gives η in the case that n = 1 as a product of even Thetanullwerte. Recall that for a, b column vectors of dimension n with entries in {0, 1/2} we have on C n × H n the theta function with characteristic a b given by θ a b (z, τ ) = m∈Z n e πi t (m+a)τ (m+a)+2πi t (m+a)(z+b) .
The choice a = b = 0 gives the Riemann theta function, and it follows from the definition that θ[ a b ](−z, τ ) = e 4πi t ab θ[ a b ](z, τ ). We call [ a b ] an even or odd theta characteristic depending on whether θ a b (z, τ ) is an even or odd function of z. If n = 2, there are ten even theta characteristics, and six odd ones. The product ε even θ[ε](0, τ ) 2 of Thetanullwerte is a modular form of weight 10 and level 1 and can be related to the discriminant of a hyperelliptic equation. holds.
Proof. It suffices to prove the formula for τ corresponding to an indecomposable abelian surface. We write A = C 2 /(Z 2 + τ Z 2 ) and Θ = div θ and assume that Θ is irreducible. By Theorem 1.3, the section η(z) is a theta function of order 3 on Θ, and by Example 2.2 or Theorem 5.1 it has zeroes exactly at Θ ∩ A [2], the Weierstrass points of Θ, all of multiplicity 1. On the other hand, the function θ(3z) gives rise to a global section of O A (Θ) ⊗9 , i.e. is a theta function of order 9 on A, and has zeroes on Θ exactly at Θ ∩ A [2], with multiplicity 3. It follows that η(z) 3 = c · θ(3z) on the zero locus of θ(z) in C 2 , where c is a constant only depending on τ . In order to compute c, we recall from Remark 5.4 that ω ζ (z) 2 η(z) = F (z) 3 , with ζ the basis of H 0 (Θ, K Θ ) given by (dz 1 , dz 2 ), so that (6.1) c = F (z) 9 ω ζ (z) 6 θ(3z) for z on Θ \ (Θ ∩ A [2]). We find c by letting z approach a point Q of Θ ∩ A [2], along Θ, and computing Taylor expansions of the numerator and denominator in (6.1). Note that the leading coefficient of a Taylor expansion of F (z) around Q is the same as the leading coefficient of a Taylor expansion of θ(2z)| Θ around Q. We start with the standard euclidean coordinates z 1 , z 2 , but now translated suitably so as to have them both vanish at Q on A. According to [4], formula (1.6) there exist a constant b 3 and an invertible 2-by-2 matrix µ such that in the coordinates (u 1 , u 2 ) = (z 1 , z 2 ) t µ one has a Taylor expansion (6.2) θ(z) = θ(µ −1 u) = γ e G(u) u 1 + 1 24 b 3 u 3 1 − 1 12 u 3 2 + higher order terms , with γ some non-zero constant and with G(u) some holomorphic function that vanishes at u = 0. This gives u 2 as a local coordinate around Q on Θ, as well as an expansion Writing out ω ζ ′ with respect to u 2 gives We find c = F (z) 9 ω ζ (z) 6 θ(3z) = 2 −9 γ 9 2 −6 (det µ) −6 · 2γ = 2 −4 (det µ) 6 γ 8 .
Substituting this in our formula for c we get the result.