Ameya Pitale

Transfer of Siegel cusp forms of degree 2

Introduction Notation Distinguished vectors in local representations Global L-functions for GSp? X GL? The pullback formula Holomorphy of global L-functions for GSp? X GL? Applications Bibliography

Bounds For Rankin-Selberg Integrals And Quantum Unique Ergodicity For Powerful Levels

Reference EPFL-ARTICLE-202508doi:10.1090/S0894-0347-2013-00779-1View record in Web of Science Record created on 2014-10-23, modified on 2017-12-10

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Let μ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree 2. It is proved that if F is not in the Maaβ space, then there exist infinitely many primes p for which the sequence fi(p r ), r > 0, has infinitely many sign changes.

Irreducibility criteria for local and global representations

It is proved that certain types of modular cusp forms generate irreducible automorphic representations of the underlying algebraic group. Analogous Archimedean and non-Archimedean local statements are also given. Introduction One of the motivations for this paper was to show that full level cuspidal Siegel eigenforms generate irreducible, automorphic representations of the adelic symplectic similitude group. Such a result is well known for the case of classical elliptic modular forms. Given an elliptic cusp form f (of some weight and some level), an adelic function Φf can be constructed (see [Ge], §5), which is a cuspidal automorphic form on the adelic group GL(2,A); here, A denotes the ring of adeles of Q. Let Vf be the space of automorphic forms generated by all right translates of Φf . In this classical situation it turns out that Vf is irreducible precisely when f is an eigenform for the Hecke operators Tp for almost all primes p. The proof uses the strong multiplicity one property for cuspidal automorphic representations of GL(2). For most other types of modular forms, strong multiplicity one, or even weak multiplicity one, is not available. The goal of this paper is to show that, under certain circumstances, the automorphic representation Vf is still irreducible, even if multiplicity one is not known. Loosely speaking, this is the case whenever f is a holomorphic type of modular form and is an eigenfunction for all Hecke operators. See Corollary 3.2 for a precise statement. We stress that this result does not prove multiplicity one for full level automorphic forms; for example, under our current state of knowledge, it is still conceivable that two holomorphic Siegel cusp forms of degree n > 1 have the same weight and the same Hecke eigenvalues for all primes p, yet are linearly independent. While our results are applicable mainly in a reductive setting, we keep the definitions general enough to include certain non-reductive situations, such as Jacobi forms. For completeness we also include analogous local Archimedean and nonArchimedean irreducibility criteria. All of this is well known to experts, but for Received by the editors June 6, 2011. 2010 Mathematics Subject Classification. Primary 11F46, 11F50, 11F70; Secondary 22E50, 22E55.

Test vectors and central L-values for GL(2)

We determine local test vectors for Waldspurger functionals for GL2, in the case where both the representation of GL2 and the character of the degree two extension are ramied, with certain restrictions. We use this to obtain an explicit version of Waldspurger’s formula relating twisted central L-values of automorphic representations on GL2 with certain toric period integrals. As a consequence, we generalize an average value formula of Feigon and Whitehouse, and obtain some nonvanishing results. 1

Survey article: Characterizations of the Saito-Kurokawa lifting

There are a variety of characterizations of Saito-Kurokawa lifts from elliptic modular forms to Siegel modular forms of degree 2. In addition to giving a survey of known characterizations, we apply a recent result of Weissauer to provide a number of new and simpler characterizations of Saito-Kurokawa lifts.

Bessel models for GSp(4): Siegel vectors of square-free level

We determine test vectors and explicit formulas for all Bessel models for those Iwahori-spherical representations of GSp(4) over a p-adic field that have non-zero vectors fixed under the Siegel congruence subgroup.

Special values of L-functions for Saito-Kurokawa lifts

In this paper we obtain special value results for L-functions associated to classical and paramodular Saito–Kurokawa lifts. In particular, we consider standard L-functions associated to Saito–Kurokawa lifts as well as degree eight L-functions obtained by twisting with an automorphic form defined on GL(2). The results are obtained by combining classical and representation theoretic arguments.

A Note on the Growth of Nearly Holomorphic Vector-Valued Siegel Modular Forms

Let F be a nearly holomorphic vector-valued Siegel modular form of weight ρ with respect to some congruence subgroup of \(\mathrm {Sp}_{2n}({{\mathbb Q}})\). In this note, we prove that the function on \(\mathrm {Sp}_{2n}({\mathbb R})\) obtained by lifting F has the moderate growth (or “slowly increasing”) property. This is a consequence of the following bound that we prove: \(\|\rho (Y^{1/2})F(Z) \| \ll \prod _{i=1}^n (\mu _i(Y)^{\lambda _1/2} + \mu _i(Y)^{-\lambda _1/2})\) where λ1 ≥… ≥ λ n is the highest weight of ρ and μ i (Y ) are the eigenvalues of the matrix Y .

Bessel models for : Siegel vectors of square-free level

We determine test vectors and explicit formulas for all Bessel models for those Iwahori-spherical representations of GSp(4) over a p-adic field that have non-zero vectors fixed under the Siegel congruence subgroup.