A general canonical curve X determines a finite set T(X) of hyperplanes, which is in bijective correspondence with the set of odd theta-characteristics of X. The definition of T(X) can be extended to certain singular curves, in a way that is compatible with degenerations. If X is a Deligne-Mumford stable curve (or a cuspidal curve), we describe the enumerative and local geometry of T(X); this is applied to show that some singular curves are uniquely determined by T(X). This work generalizes the preliminary steps used in AG/0008239 (with E. Sernesi) to show that a general curve of genus 3 can be recovered from its odd theta-characteristics.
We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.
We give a new method for constructing vector-valued modular forms from singular scalar-valued ones. As an application we prove the identity between two remarkable spaces of vector-valued modular forms which seem to be unrelated at a first look, since they are constructed in two very different ways. If $V_{grad}$ is the vector space generated by vector-valued modular forms constructed with gradients of odd theta functions and $V_Θ$ is the one generated by vector-valued modular forms arising from second order theta constants with our new construction, we will prove that $V_{grad}=V_Θ$. This resu…
We show that a general canonical curve is uniquely determined by the finite set of hyperplanes cutting theta-characteristics on it. Geometrical and combinatorial properties of the moduli space of stable spin curves are proved, which play an essential role.
Algebraic curves in Hilbert modular surfaces that are totally geodesic for the Kobayashi metric have very interesting geometric and arithmetic properties, e.g. they are rigid. There are very few methods known to construct such algebraic geodesics that we call Kobayashi curves. We give an explicit way of constructing Kobayashi curves using determinants of derivatives of theta functions. This construction also allows to calculate the Euler characteristics of the Teichmueller curves constructed by McMullen using Prym covers.
We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense are a generalization to ppavs of bitangents of plane quartics, and of hyperplanes tangent to a canonical curves of genus $g$ in $g-1$ points. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli …
We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions.
Hecke expected that an explicit set of theta series obtained from maximal orders of the definite quaternion algebra over Q which is ramified at a prime N will be a basis of the space of holomorphic modular forms of weight 2 and level N. However, later Eichler noticed that Hecke's conjecture does not hold in general. It is natural to ask for the dimension of the subspace of the modular forms spanned by the theta series. This question is called Hecke's basis problem. Boecherer and Schulze-Pillot have given an answer using the theory of theta liftings. In this paper we will give another proof of …
The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $Γ$ is located between $Γ_2(4,8)$ and $Γ_2(2,4)$ in such a way the map factors on the related level moduli space $A_Γ$, the new map being injective on $A_Γ$. Satake's compactification $\text{Proj}A(Γ)$ and the desingularization $\text{Proj}S(Γ)$ are also due to be investigated, since the map does not extend to the boundary of the compactification; to aim at this, an algebraic descript…
In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of singular and critical weights for root systems using Jacobi theta-series. We describe the Jacobi lifting for Jacobi forms of half-integral indices. In some case it gives additive lifting construction of new reflective modular forms.
We show that analytic torsion of smooth theta divisor is represented by a Siegel modular form characterizing the Andreotti-Mayer locus if its dimension is positive.
In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We construct the period mapping for a family $\mathcal{F}=\{S(X,Y)\}$ of $K3 $ surfaces with $2$ complex parameters $X$ and $Y$. The inverse correspondence of the period mapping gives a system of generators of Hilbert modular functions for $\mathbb{Q}(\sqrt{5})$. Moreover, we show an explicit expression of this inverse correspondence by theta constants.
Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular forms, defined on the Teichmuller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte χ_{68}, which is a proof of the recently rediscovered…
It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperellip…
Let L be the even unimodular lattice of signature (2,10), In the paper [FS] we considered the subgroup O(L)^+ of index two in the orthogonal group. It acts biholomorphically on a ten dimensional tube domain H_{10}. We found a 715 dimensional space of modular forms with respect to the principal congruence subgroup of level two O^+(L)[2]. It defines an everywhere regular birational embedding of the related modular variety into the 714 dimensional projective space. In this paper, we prove that this space of orthogonal modular forms is related to a space of theta series. The main tool is a modular…
We apply the homological mirror symmetry for elliptic curves to the study of indefinite theta series. We prove that every such series corresponding to a quadratic form of signature (1,1) can be expressed in terms of theta series associated with split quadratic forms and the usual theta series. We also show that indefinite theta series corresponding to univalued Massey products between line bundles on elliptic curve are modular.
We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the computation of divisor classes of Hilbert Jacobi forms on the universal abelian surface over the pseudo-Hilbert modular surface.
We develop two structure theorems for vector valued Siegel modular forms for Igusa's subgroup Γ_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular forms with rational tensors with easily handleable poles on P^3\C. It follows that the observed modules of modular forms are generated by the Rankin-Cohen brackets of the four theta series of the second kind.
In this paper we investigate the (classical) weights of mod $p$ Siegel modular forms of degree 2 toward studying Serre's conjecture for $GSp_4$. We first construct various theta operators on the space of such forms a la Katz and define the theta cycles for the specific theta operators. Secondly we study the partial Hasse invariants on each Ekedahl-Oort strata and their local behaviors. This enable us to obtain a kind of weight reduction theorem for mod $p$ Siegel modular forms of degree 2 without increasing level.
We describe a family of $q$-series generating the space of weight 1 modular forms coming from indefinite binary quadratic forms and study linear relations between these series.
Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL(2,Z) and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2.…
We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta characteristics and $2^{g-1}$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.
Theta functions of level n on the principally polarised Prym varieties of an algebraic curve are dual to sections of the orthogonal theta line bundle on the moduli space of Spin(n)-bundles over the curve. As a by-product of our computations we also note that when n is odd the pfaffian line bundle on moduli space has a basis of sections labelled by the even theta characteristics of the curve.
Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-…
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the μ-ordinary locus, when applied to scalar-valued forms.
We find that the equation of $E_8$-singularity possesses two distinct symmetry groups and modular parametrizations. One is the classical icosahedral equation with icosahedral symmetry, the associated modular forms are theta constants of order five. The other is given by the group $\text{PSL}(2, 13)$, the associated modular forms are theta constants of order $13$. As a consequence, we show that $E_8$ is not uniquely determined by the icosahedron. This solves a problem of Brieskorn in his ICM 1970 talk on the mysterious relation between exotic spheres, the icosahedron and $E_8$. Simultaneously, …
