Abhishek Saha

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

Local spectral equidistribution for Siegel modular forms and applications

We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k , subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L -functions (for restricted test functions) gives global evidence for a well-known conjecture of Bocherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

L-Functions for Holomorphic Forms on GSp(4) x GL(2) and Their Special Values

We provide an explicit integral representation for L-functions of pairs (F, g), where F is a holomorphic genus two Siegel newform and g a holomorphic elliptic newform, both of square-free levels and of equal weights. When F, g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well as an explicit formula for the Bessel model for GSp(4) in the Steinberg case; the latter is possibly of independent interest. As an application, we prove an algebraicity result for a critical value of L(s, F x g). This is in the spirit of known results on critical values of triple product L-functions, also of degree eight, though there are significant differences.

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

Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular L-functions

Given elliptic modular forms f and g satisfying certain condi- tions on their weights and levels, we prove (a quantitative version of the statement) that there exist infinitely many imaginary quadratic fields K and charactersof the ideal class group ClK such that L( 1 ,BCK(f) × �) 6 0 and L( 1 ,BCK(g) × �) 6 0. The proof is based on a non-vanishing result for Fourier coefficients of Siegel modular forms combined with the theory of Yoshida liftings.

Hybrid sup-norm bounds for Maass newforms of powerful level

Let f be an L^2-normalized Hecke--Maass cuspidal newform of level N, character \chi and Laplace eigenvalue \lambda. Let N_1 denote the smallest integer such that N|N_1^2 and N_0 denote the largest integer such that N_0^2 |N. Let M denote the conductor of \chi and define M_1= M/\gcd(M,N_1). In this paper, we prove the bound |f|_\infty \ll_{\eps} N_0^{1/6 + \eps} N_1^{1/3+\eps} M_1^{1/2} \lambda^{5/24+\eps}, which generalizes and strengthens previously known upper bounds for |f|_\infty. This is the first time a hybrid bound (i.e., involving both N and \lambda) has been established for |f|_\infty in the case of non-squarefree N. The only previously known bound in the non-squarefree case was in the N-aspect; it had been shown by the author that |f|_\infty \ll_{\lambda, \eps} N^{5/12+\eps} provided M=1. The present result significantly improves the exponent of N in the above case. If N is a squarefree integer, our bound reduces to |f|_\infty \ll_\eps N^{1/3 + \eps}\lambda^{5/24 + \eps}, which was previously proved by Templier. The key new feature of the present work is a systematic use of p-adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for GL_2(F) where F is a local field.

On sup-norms of cusp forms of powerful level

Final version, to appear in JEMS. Please also note that the results of this paper have been significantly improved in my recent paper arXiv:1509.07489 which uses a fairly different methodology

A note on fourier coefficients of poincaré series

We give a short and “soft” proof of the asymptotic orthogonality of Fourier coefficients of Poincare series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.

Hilbert modular forms of weight 1/2 and theta functions

Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result -— under certain mild restrictions on the level and character -— to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre–Stark; however we are forced to overcome technical difficulties which arise when we move out of Q.

Large values of newforms on GL(2) with highly ramified central character

We give a lower bound for the sup-norm of an