Paramodular Group Research Articles

Bessel periods and anticyclotomic 𝑝-adic spinor 𝐿-functions

We construct the anticyclotomic p p -adic L L -function that interpolates a square root of central values of twisted spinor L L -functions of a quadratic base change of a Siegel cusp form of genus 2 2 with respect to a paramodular group of square-free level.

Paramodular groups and theta series

For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling or pullback of Eisenstein series method. For this we give a new geometric proof of Garrett’s double coset decomposition which works in our more general situation.

The Maaß space for paramodular groups

We describe a characterization for the Maaß space associated with the paramodular group of degree 2 and square-free level N . As an application we show that the Maaß space is invariant under all Hecke operators. As a consequence, we conclude that the associated Siegel–Eisenstein series belongs to the Maaß space.

Graded rings of integral Jacobi forms

We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of (0,2)-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi forms of weight zero and integral index with integral Fourier coefficients. These forms are the data for Borcherds products for the Siegel paramodular groups.

Symmetries in A-type little string theories. Part I. Reduced free energy and paramodular groups

We analyse the symmetries of a class of A-type little string theories that are engineered by N parallel M5-branes with M2-branes stretched between them. This paper deals with the so-called reduced free energy, which only receives contributions from the subset of the BPS states that carry the same charges under all the Cartan generators of the underlying gauge algebra. We argue (and check explicitly in a number of examples) that the former is invariant under the paramodular group ΣN ⊂ Sp(4, ℚ), which gets extended to a subgroup of Sp(4, ℝ) in the Nekrasov-Shatashvili-limit. This extension agrees with the observation made in [18] that these BPS states form a symmetric orbifold CFT. Furthermore, we argue that ΣN (along with other symmetries) places strong constraints on the BPS counting function that governs the intersection between the M5- and M2-branes.

Antisymmetric Paramodular Forms of Weights 2 and 3

Abstract We define an algebraic set in $23$-dimensional projective space whose ${{\mathbb{Q}}}$-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree 2 paramodular groups. Weight $3$ examples provide antisymmetric canonical differential forms on Siegel modular three-folds. Weight $2$ is the minimal weight and these examples, via the paramodular conjecture, give evidence for the modularity of some rank 1 abelian surfaces defined over $\mathbb{Q}$.

The Hecke algebras for the orthogonal group SO(2,3) and the paramodular group of degree 2

In this paper, we consider the integral orthogonal group with respect to the quadratic form of signature [Formula: see text] given by [Formula: see text] for square-free [Formula: see text]. The associated Hecke algebra is commutative and also the tensor product of its primary components, which turn out to be polynomial rings over [Formula: see text] in two algebraically independent elements. The integral orthogonal group is isomorphic to the paramodular group of degree [Formula: see text] and level [Formula: see text], more precisely to its maximal discrete normal extension. The results can be reformulated in the paramodular setting by virtue of an explicit isomorphism. The Hecke algebra of the non-maximal paramodular group inside [Formula: see text] fails to be commutative if [Formula: see text].

COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL

This article gives upper bounds on the number of Fourier- Jacobi coecients that determine a paramodular cusp form in degree two. The level N of the paramodular group is completely general throughout. Additionally, spaces of Jacobi cusp forms are spanned by using the theory of theta blocks due to Gritsenko, Skoruppa and Zagier. We combine these two techniques to rigorously compute spaces of paramodular cusp forms and to verify the Paramodular Conjecture of Brumer and Kramer in many cases of low level. The proofs rely on a detailed description of the zero dimensional cusps for the subgroup of integral elements in each paramodular group.

Paramodular cusp forms

We classify Siegel modular cusp forms of weight two for the paramodular group K(p) for primes p< 600. We find that weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian varieties defined over the rationals of conductor p. The arithmetic classification is in a companion article by A. Brumer and K. Kramer. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida, is a partial extension to degree 2 of the Shimura-Taniyama Conjecture. These nonlift Hecke eigenforms share Euler factors with the corresponding abelian variety $A$ and satisfy congruences modulo \ell with Gritsenko lifts, whenever $A$ has rational \ell-torsion.

Jacobi forms of higher index and paramodular groups in $ \mathcal{N} $ = 2, D = 4 compactifications of string theory

We associate a Jacobi form over a rank s lattice to N=2, D=4 heterotic string compactifications which have s Wilson lines at a generic point in the vector multiplet moduli space. Jacobi forms of index m=1 and m=2 have appeared earlier in the context of threshold corrections to heterotic string couplings. We emphasize that higher index Jacobi forms as well as Jacobi forms of several variables over more generic even lattices also appear and construct models in which they arise. In particular, we construct an orbifold model which can be connected to models that give index m=3, 4 or 5 Jacobi forms through the Higgsing process. Constraints from being a Jacobi form are then employed to get threshold corrections using only partial information on the spectrum. We apply this procedure for index m=3, 4 or 5 Jacobi form examples and also for Jacobi forms over A_2 and A_3 root lattices. Examples with a single Wilson line are examined in detail and we display the relation of Siegel forms over a paramodular group \Gamma_m to these models, where \Gamma_m is associated with the T-duality group of the models we study. Finally, results on the heterotic string side are used to clarify the linear mapping of vector multiplet moduli to Type IIA duals without using the one-loop cubic part of the prepotential on the Type II side, and also to give predictions for the geometry of the dual Calabi-Yau manifolds.

THE CUSP STRUCTURE OF THE PARAMODULAR GROUPS FOR DEGREE TWO

We describe the one-dimensional and zero-dimensional cusps of the Satake compactification for the paramodular groups in degree two for arbitrary levels. We determine the crossings of the one-dimensional cusps. Applications to computing the dimensions of Siegel modular forms are given.

THE CHIRAL SUPERSTRING SIEGEL FORM IN DEGREE TWO IS A LIFT

We prove that the Siegel modular form of D'Hoker and Phong that gives the chiral superstring measure in degree two is a lift. This gives a fast algorithm for computing its Fourier coefficients. We prove a general lifting from Jacobi cusp forms of half integral index t=2 over the theta group 1(1 ; 2) to Siegel modular cusp forms over certain subgroups para ( t; 1 ; 2) of paramodular groups. The theta group lift given here is a modication of the Gritsenko lift.

Siegel modular forms of genus 2 with the simplest divisor

We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus 2 vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference ‘Black holes, Black Rings and Modular Forms’ (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998.

On classical Saito-Kurokawa liftings

There exist two different generalizations of the classical Saito-Kurokawa lifting to modular forms with (square-free) level; one lifting produces modular forms with respect to Γ 0 ( m ), the other one with respect to the paramodular group Γ para ( m ). We shall give an alternative and unified construction of both liftings using group theoretic methods. The construction shows that a single elliptic modular form may in fact have many Saito-Kurokawa liftings. We also obtain precise information about the spin L -function of the resulting Siegel modular forms.

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].

An extension of the level-2 paramodular group, and the Barth-Nieto quintic

In this note we construct a maximal discrete extension of Γˆ1,p(2), the paramodular group with a full level-2 structure. The corresponding Siegel modular variety parametrizes (birationally) the space of Kummer surfaces associated to (1, p)-polarized abelian surfaces with a level-2 structure. In the case p=3 this is related to the Barth-Nieto quintic and in this case we also determine the space of cusp forms of weight 3.

CHARACTERS OF PARAMODULAR GROUPS AND SOME EXTENSIONS

ABSTRACT For P a rational symmetric positive-definite matrix of dimension let be the paramodular group of degree g and level P. We determine , the group of abelian characters of . In all cases is a subgroup of . The characters can be realized via homomorphisms for some finite groups G, such as for , and certain new finite analogs of paramodular groups. The characters are explicitly known (on generators at least). All characters arise as multiplier-systems of certain paramodular forms also. Moreover in the case of degree 2 we use the characters of , to determine all the characters of the well-known maximal normal discrete extension.

The moduli space of bilevel-6 abelian surfaces

AbstractWe show that the moduli space of abelian surfaces with polarisation of type (1,6) and a bilevel structure has positive Kodaira dimension and indeed pg ≥ 3. To do this we show that three of the Siegel cusp forms with character for the paramodular symplectic group constructed by Gritsenko and Nikulin are cusp forms without character for the modular group associated to this moduli problem. We then calculate the divisors of the corresponding differential forms, using information about the fixed loci of elements of the paramodular group previously obtained by Brasch.

The modular form of the Barth-Nieto quintic

Barth and Nieto have found a remarkable quintic threefold which parametrizes Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a (1,3)-polarization and a lecel 2 structure. A double cover of this quintic, which is also a Calabi-Yau variety, is birationally equivalent to the moduli space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2 structure. As a consequence the corresponding paramodular group \Gamma_3(2) has a unique cusp form of weight 3. In this paper we find this cusp form which is \Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a character with respect to the paramodular group \Gamma_3. It has several interesting properties. One is that it admits an infinite product representation, the other is that it vanishes of order 1 along the diagonal in Siegel space. In fact \Delta_1 is an element of a short series of modular forms with this last property. Using the fact that \Delta_1 is a weight 3 cusp form with respect to the group \Gamma_3(2) we give an independent construction of a smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).

Commutator coverings of Siegel threefolds

We investigate the existence and non-existence of modular forms of low weight with a character with respect to the paramodular group $\Gamma_t$ and discuss the resulting geometric consequences. Using an advanced version of Maa\ss lifting one can construct many examples of such modular forms and in particular examples of weight 3 cusp forms. Consequently we find many abelian coverings of low degree of the moduli space ${\Cal A}_t$ of (1,t)-polarized abelian surfaces which are not unirational. We also determine the commutator subgroups of the paramodular group $\Gamma_t$ and its degree 2 extension $\Gamma^+_t$. This has applications for the Picard group of the moduli stack ${\underline{\Cal A}}_t$. Finally we prove non-existence theorems for low weight modular forms. As one of our main results we obtain the theorem that the maximal abelian cover ${\Cal A}_t^{com}$ of ${\Cal A}_t$ has geometric genus 0 if and only if t=1, 2, 4 or 5. We also prove that ${\Cal A}_t^{com}$ has geometric genus 1 for t=3 and 7.