Siegel Theta Series Research Articles

Siegel theta series for quadratic forms of signature (m − 1,1)

We investigate Siegel theta series for quadratic forms of signature (m−1,1). On the one hand, we construct a holomorphic series that does not transform like a modular form. On the other hand, we construct a non-holomorphic series that transforms like a Siegel modular form of weight m/2. Moreover, the holomorphic series describes almost everywhere the holomorphic part of the modular series.

Siegel modular forms of weight 13 and the Leech lattice

For g=8,12,16 and 24 , there is a nonzero alternating g -multilinear form on the {\rm Leech} lattice, unique up to a scalar, which is invariant by the orthogonal group of {\rm Leech} . The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight 13 for {\rm Sp}_{2g}(\mathbb{Z}) . We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard {\rm L} -functions. These forms are interesting since, by a recent work of the authors, they are the only nonzero Siegel modular forms of weight 13 for {\rm Sp}_{2n}(\mathbb{Z}) , for any n\geq 1 .

Siegel theta series for indefinite quadratic forms

The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.

Hecke operators on Hilbert–Siegel theta series

We consider the action of Hecke-type operators on Hilbert–Siegel theta series attached to lattices of even rank. We show that the average Hilbert–Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.

48 dimensional even unimodular nearly extremal lattices.

In this paper we introduce the notion of an even unimodular nearly extremal lattice in the case of 48 dimension. We prove two basic properties of such lattices $L$. First we prove that any nearly extremal lattice $L$ is generated by the vectors of norm 4 and norm 6 in $L$. Next we prove that the Siegel theta series of degree 2 associated with an even unimodular nearly extremal lattice is determined by the Fourier coefficients which are connected with the vectors of norm 4.

Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

We find nice representatives for the 0-dimensional cusps of the degree n Siegel upper half-space under the action of $$\Gamma _0(\mathcal N )$$ . To each of these, we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level $$\mathcal N $$ and Dirichlet character $$\chi _{_L}$$ modulo $$\mathcal N $$ as a linear combination of Siegel Eisenstein series.

A numerical study of Siegel theta series of various degrees for the 48-dimensional even unimodular extremal lattices

Salvati Manni showed that the difference of the Siegel theta series of degree 4 associated with the two even unimodular 48-dimensional extremal lattices is a constant multiple of the cube J3 of the Schottky modular form J, which is a Siegel cusp form of degree 4 and weight 8. His result implies that the Siegel theta series of degree up to 3 is unique. But apparently his method does not supply us the process to compute the Fourier coefficients of these series. In the present paper we show that the Fourier coefficients of the Siegel theta series associated with the even unimodular 48-dimensional extremal lattices of degrees 2 and 3 can be computed explicitly, and the Fourier coefficients of the Siegel theta series of degree 4 for those lattices are computed almost explicitly.

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.

Distinguishing Siegel theta series of degree 4 for the 32-dimensional even unimodular extremal lattices

In a previous paper we showed that if one particular Fourier coefficient of the Siegel theta series of degree 4 for a 32-dimensional even unimodular extremal lattice is known then the other Fourier coefficients of the series are in principle determined. In this paper we choose the quaternary positive definite symmetric matrix $$\mathfrak {T}_{40}$$ , and calculate the Fourier coefficient $$a(\mathfrak {T}_{40},\mathcal{L}_{32})$$ of the Siegel theta series of degree 4 associated with the five even unimodular extremal lattices which come from the five binary self-dual extremal [32,16,8] codes. As a result we can show that the five Siegel theta series of degree 4 associated with the five 32-dimensional even unimodular extremal lattices are distinct.

A modular invariance property of multivariable trace functions for regular vertex operator algebras

We prove an SL2(Z)-invariance property of multivariable trace functions on modules for a regular VOA. Applying this result, we provide a proof of the inversion transformation formula for Siegel theta series. As another application, we show that if V is a simple regular VOA containing a simple regular subVOA U whose commutant Uc is simple, regular, and satisfies (Uc)c=U, then all simple U-modules appear in some simple V-module.

Some congruences for Siegel theta series

We discuss an arithmetic approach to some congruence properties of Siegel theta series of even positive definite unimodular quadratic forms.

A formula for the action of Hecke operators on half-integral weight Siegel modular forms and applications

We introduce an alternate set of generators for the Hecke algebra, and give an explicit formula for the action of these operators on Fourier coefficients. Using this, we compute the eigenvalues of Hecke operators acting on average Siegel theta series with half-integral weight (provided the prime associated to the operators does not divide the level of the theta series), and show that some of the Hecke operators annihilate theta series. Next, we bound the eigenvalues of these operators in terms of bounds on Fourier coefficients. Then we show that the half-integral weight Kitaoka subspace is stable under all Hecke operators. Finally, we observe that an obvious isomorphism between Siegel modular forms of weight k + 1 / 2 and “even” Jacobi modular forms of weight k + 1 is Hecke-invariant (here the level and character are arbitrary).

Regularized theta lifts and Niebur-type Poincaré series on n -dimensional hyperbolic space

We study theta liftings from H n to H 2 and vice versa. The theta-lift is realized via an integral transform with a Siegel theta series as kernel function. Since this Siegel theta series fails to be square integrable it has to be regularized. The regularization is obtained by applying a suitable differential operator built from the Laplacian. The Siegel theta series is seen to be related to an automorphic Selberg kernel function on H n and therefore the Selberg transform applies to compute the Fourier coefficients of the lift. This gives formulas of Katok–Sarnak type for Fourier coefficients of positive as well as for negative index, involving geodesic cycles. As an application we can lift Poincaré series of Niebur type. The lifted series are again Poincaré series built with Whittaker functions.

A finiteness theorem for representability of quadratic forms by forms

In this article, Conway-Schneeberger's and Bhargava's results on representation positive integers by positive denite integral quadratric forms are fully generalized as follows : for any innite set S of positive denite integral quadratic forms of bounded rank, there is a nite subset S0 of S such that any positive denite integral quadratic form that represents every element of S0 represents all elements of S. Some interesting corollaries are obtained and applications to Fourier coecients of Siegel theta-series are introduced.

Hecke actions on certain strongly modular genera of lattices

We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of Siegel cusp forms.

Petersson Products of Singular and Almost Singular Theta Series

We evaluate Petersson products of pairs of Siegel theta series of degree n attached to even lattices of rank m≤n. This allows us to get nice expressions, involving Dirichlet series and standard L-functions, for Petersson products of Siegel modular forms that are linear combinations of such theta series. Some applications to the theory of theta-liftings are also discussed.

Siegel theta series of degree 2 of extremal lattices

It is shown that there is a unique Siegel theta series of degree 2 for extremal even unimodular lattices in dimension N for N = 32, 48, 56. For N = 40 a question of Ozeki ( J. Number Theory 28 (1988), 119–131) is answered.

Examples of even unimodular extremal lattices of rank 40 and their Siegel theta-series of degree 2

Using two newly discovered [40, 20] codes and a [40, 20] code, which is equivalent to a known one, we construct three non-isomorphic even unimodular extremal lattices of rank 40. Next we show these lattices are distinguished by their Siegel theta-series of degree 2.

Siegel's formula for genus 2

Let S be a semi-integral, symmetric, positive-definite m x m matrix; m⩾n⩾1. By the Siegel fundamental formula we mean the identity between the Siegel theta series of genus n, associated with the genus of the matrix S, and the correspond ing Eisenstein-Siegel series (C. L. Siegel, Lectures on the Analytical Theory of Quadratic Forms, 3rd rev. edition, Peppmüller, Göttingen, 1963). The validity of the mentioned formula for m/2⩽n+1 is an open problem in the general case. In this paper we prove Siegel's formula for n=2, m=6.

Identity between the fourth-degree Riemann theta-constant of genus 2 and the Eisenstein series

One proves the identity between the Siegel theta-series of genus 2 associated with the 4×4 identity matrix and the corresponding Eisenstein-Siegel series.