Siegel Modular Forms Of Degree Research Articles

RAMANUJAN CONGRUENCES FOR SIEGEL MODULAR FORMS

We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.

On the Spezialschar of Maass

Let be the space of Siegel modular forms of degree n and even weight k. In this paper firstly a certain subspace Spez the Spezialschar of , is introduced. In the setting of the Siegel threefold, it is proven that this Spezialschar is the Maass Spezialschar. Secondly, an embedding of into a direct sum Sym2Mk+2υ is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the nonvanishing of certain special values of L‐functions related to the Gross‐Prasad conjecture. This is illustrated by a significant example in the paper.

Quaternionic theta constants

We investigate the six quaternionic theta constants introduced by Freitag and Hermann. More precisely we investigate their restrictions to the Hermitian resp. Siegel half-space of degree 2. It turns out that these theta constants generate the graded ring of symmetric Hermitian modular forms for the principal congruence subgroup of level 1 + i over the Gaussian number field resp. of Siegel modular forms for the principal congruence subgroup of level 2 and even weight. As an application we obtain a simple construction of Igusa’s Siegel modular form of degree 2 and weight 30 with respect to the non-trivial character.

Integral representation for L-functions for [formula omitted

Let π be a cuspidal, automorphic representation of GSp 4 attached to a Siegel modular form of degree 2. We refine the method of Furusawa [M. Furusawa, On L-functions for GSp ( 4 ) × GL ( 2 ) and their special values, J. Reine Angew. Math. 438 (1993) 187–218] to obtain an integral representation for the degree-8 L-function L ( s , π × τ ) , where τ runs through certain cuspidal, automorphic representation of GL 2 . Our calculations include the case of any representation with unramified central character for the p-adic components of τ, and a wide class of archimedean types including Maaß forms. As an application we obtain a special value result for L ( s , π × τ ) .

Siegel modular forms of degree 2 over rings

Based on moduli theory of abelian varieties, extending Igusa's result on Siegel modular forms over C , we describe the ring of Siegel full modular forms of degree 2 over any Z -algebra in which 6 is invertible.

Bessel Models for Lowest Weight Representations of

We prove uniqueness and give precise criteria for existence of split and nonsplit Bessel models for a class of lowest and highest weight representations of the groups and including all holomorphic and antiholomorphic discrete series representations. Explicit formulas for the resulting Bessel functions are obtained by solving systems of differential equations. The formulas are applied to derive an integral representation for a global L-function on GSp(4) × GL(2) involving a vector-valued Siegel modular form of degree 2.

Separators of Siegel modular forms of degree two

We prove that cuspidal Siegel modular forms of degree two and weight 2k are uniquely determined by their Fourier coefficients on small subsets of matrices of content one. This generalizes results of Breulmann, Kohnen, Katsurada, Scharlau and Walling. We give applications to the space of Saito-Kurokawa lifts.

On Petersson products of not necessarily cuspidal modular forms

We show that Petersson products of any pair of elliptic modular forms of weight 1 2 and of any pair of Siegel modular forms of degree 2 and weight 1 have an expression as the residue of a Dirichlet series of Rankin–Selberg type. This generalizes a well-known property of cuspidal Petersson products.

On Rankin-Cohen brackets for Siegel modular forms

We determine an explicit formula for a Rankin-Cohen bracket for Siegel modular forms of degree n on a certain subgroup of the symplectic group. Moreover, we lift that bracket via a Poincare series to a Siegel cusp form on the full symplectic group.

The graded ring of quaternionic modular forms of degree 2

In the 1960's Igusa [I] described the graded ring of Siegel modular forms of degree 2. In the sequel a few more concrete examples of graded rings of modular forms with respect to paramodular groups or Hermitian modular groups were given, e.g. by Freitag [F], Aoki and Ibukiyama [AI] or Dern and Krieg [DK].

Note on mod p Siegel modular forms II

The structure of the ring of mod p Siegel modular forms of degree two is determined in the cases where the prime p is 2 or 3.

Iwahori-spherical representations of GSp(4) and Siegel modular forms of degree 2 with square-free level

A theory of local old- and newforms for representations of GSp(4) over a p-adic field with Iwahori-invariant vectors is developed. The results are applied to Siegel modular forms of degree 2 with square-free level with respect to various congruence subgroups.

On the graded ring of Siegel modular forms of degree 2, level 3

In general, it is difficult to determine the dimension of the space of Siegel modular forms with low weight. In this paper, we consider the spaces of modular forms belonging to the principal congruence subgroup of level 3 as the representation spaces of the finite symplectic group to calculate their dimensions.

Graded rings of Hermitian modular forms of degree 2

At first it is shown that any Siegel modular form of degree 2 and even weight can be lifted to a Hermitian modular form of degree 2 over any imaginary-quadratic number field K. In the cases \(\) and \(\) we describe the graded rings of Hermitian modular forms with respect to all abelian characters. Generators are constructed as Maaslifts or as Borcherds products. The description allows a characterization in terms of generators and relations. Moreover we show that the graded rings are generated by theta constants by analyzing the associated five-dimensional representation of PSp2(ℤ/3ℤ). As an application the fields of Hermitian modular functions are determined.

Cycles in hyperbolic manifolds of non-compact type and Fourier coefficients of Siegel modular forms

Using the theta correspondence, we study a lift from (not necessarily rapidly decreasing) closed differential (p−n)-forms on a non-compact arithmetic quotient of hyperbolic p-space to Siegel modular forms of degree n. This generalizes earlier work of Kudla and the second named author (in the case of hyperbolic space). We give a cohomological interpretation of the lift and analyze its Fourier expansion in terms of periods over certain cycles. For Riemann surfaces, i.e., the case p= 2, we obtain a complete description using the theory of Eisenstein cohomology.

Motives and Siegel modular forms

For a motive M over Q , we define the fundamental periods of M using invariant theory. Our definition generalizes Deligne's periods. We show that if a motive M is constructed from motives M 1 , M 2 ,..., M n by a standard algebraic operation, then the fundamental periods of M can be expressed as monomials of the fundamental periods of M 1 , M 2 ,..., M n . Applying this theory, we discuss two (hypothetical) motives attached to a Siegel modular form. We show that a Siegel modular form of degree m has at most m + 1 period invariants.

Siegel modular forms and representations

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γ n =Sp(2n,ℤ), generalizing the well-known procedure for n=1. This will show that the so-called “standard” and ldquo;spinor”L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F <4, is then used to establish meromorphic continuation for the spinor L-function when n=3.

Symmetric Square L-Functions and Shafarevich-Tate Groups

We use Zagier's method to compute the critical values of the symmetric square L-functions of six cuspidal eigenforms of level one with rational coefficients. According to the Bloch-Kato conjecture, certain large primes dividing these critical values must be the orders of elements in generalised Shafarevich-Tate groups. We give some conditional constructions of these elements. One uses Heegner cycles and Ramanujan-style congruences. The other uses Kurokawa's congruences for Siegel modular forms of degree two. The first construction also applies to the tensor product L-function attached to a pair of eigenforms of level one. Here the critical values can be both calculated and analysed theoretically using a formula of Shimura.

On certain L-series of Rankin-Selberg type associated to Siegel modular forms of degree 2

We study certain L-series of Rankin-Selberg type associated to Siegel modular forms of degree 2: twists by Dirichlet characters, relation to zeta functions of spinor type, construction of Siegel cusp forms whose Fourier coefficients involve special values of such L-series, etc.

Rankin–Cohen Operators for Jacobi and Siegel Forms

For any non-negative integer v we construct explicitly ⌊v/2⌋+1 independent covariant bilinear differential operators fromJk,m×Jk′,m′toJk+k′+v,m+m′. As an application we construct a covariant bilinear differential operator mappingS(2)k×S(2)k′toS(2)k+k′+v. HereJk,mdenotes the space of Jacobi forms of weightkand indexmandS(2)kthe space of Siegel modular forms of degree 2 and weightk. The covariant bilinear differential operators constructed are analogous to operators already studied in the elliptic case by R. Rankin and H. Cohen and we call them Rankin–Cohen operators.