Research

(1) Lifting from SL̃̃(2) to GSpin(1,4) , Int. Math. Res. Not. 63, 3919-3966 (2005) (pdf)

Abstract: This is based on my PhD thesis of the same title. In this paper, we construct a lift from half integer weight Maass forms to automorphic forms on GSpin(1,4). The main tool is the converse theorem due to Maass. We also construct the automorphic representation of GSpin(1,4)(A) corresponding to the lift and show that it is a CAP representation.

(2) (with Ralf Schmidt) Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2, Proc. of Amer. Math. Soc. 136, 3831-3838 (2008) (pdf)

Abstract: 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 Maass space, then there exist infinitely many primes p for which the sequence μ(p^k), k >0, has infinitely many sign changes.

(3) (with Ralf Schmidt) Ramanujan type results for Siegel cusp forms of degree 2, J. of Ramanujan Math. Soc., 24, No. 1, 87-111 (2009) (pdf)

Abstract: A result of Chai--Faltings on Satake parameters of Siegel cusp forms together with the classification of unitary, unramified, irreducible, admissible representations of GSp(4) over a p-adic field, imply that the local components of the automorphic representation of GSp(4) attached to a cuspidal Siegel eigenform of degree 2 must lie in certain families. Applications include estimates on Hecke eigenvalues, an improved domain of convergence of the standard L-function, and a new characterization of the Maass space.

(4) Jacobi Maass Forms, Abh. Math. Sem. Univ. Hamburg, 79, 87-111 (2009) (pdf)

Abstract: In this paper, we give a new definition for the space of non-holomorphic Jacobi Maass forms (denoted by J^nh_k,m) of weight k in Z and index m in N as eigenfunctions of a degree three differential operator C^k,m. We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J^nh_k,m. We construct new examples of cuspidal Jacobi Maass forms F_f of weight k in 2Z and index 1 from weight k-1/2 Maass forms f with respect to Γ₀(4) and show that the map f → F_f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we obtain a theta expansion for Jacobi forms and show that all of J^nh_k,m can be ``essentially" obtained from scalar or vector valued half integer weight Maass forms.

(5) (with Ralf Schmidt) Integral representation for L-functions for Gsp(4)xGL(2), J. Number Theory, 129, 1272-1324 (2009) (pdf)

Abstract: Let π be a cuspidal, automorphic representation of GSp₄ attached to a Siegel modular form of degree 2. We refine the method of Furusawa to obtain an integral representation for the degree-8 L-function L(s,π, τ), where τ runs through certain cuspidal, automorphic representation of GL₂. 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).

(6) (with Ralf Schmidt) Bessel models for lowest weight representations of GSp(4,R), Int. Math. Res. Not., 7, 1159-1212 (2009) (pdf)

Abstract: We prove uniqueness and give precise criteria for existence of split and non-split Bessel models for a class of lowest and highest weight representations of the groups GSp₄(R) and Sp₄ (R) including all holomorphic and anti-holomorphic 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 GSp₄ x GL₂ involving a vector-valued Siegel modular form of degree 2.

(7) Classical interpretation of the Ramanujan conjecture for Siegel cusp forms of genus n, Manuscr. Math., 130, Issue 2, 225-231 (2009) (pdf)

Abstract: We obtain a classical interpretation of the representation theoretic statement of the Generalized Ramanujan Conjecture for Siegel cusp forms of genus n in terms of estimates on Hecke eigenvalues.

(8) (with Ralf Schmidt) Integral representation of L-functions for GSp(4)xGL(2), II, submitted, 39 pages (pdf), available at arXiv:0908.1611

Abstract: Based on Furusawa's theory, we present an integral representation for the L-function L(s,π, τ), where π is a cupidal automorphic representation on GSp₄ related to a holomorphic Siegel modular forms, and where τ is an arbitrary cuspidal automorphic representation of GL₂. As an application, a special value result for this L-function in the spirit of Deligne's conjecture is proved.

(9) Steinberg representation of GSp(4): Bessel models and integral representation of L-functions, to appear in the Pacific Journal of Mathematics, 30 pages (pdf). A longer version is available here, available at arXiv:0909.4273

Abstract: We obtain explicit formulas for the test vector in the Bessel model and derive the criteria for existence and uniqueness for Bessel models for the unramified, quadrtic twists of the Steinberg representation π of GSp₄(F), where F is a non-archimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in π to be a test vector. We apply the formulas for the test vector to obtain an integral representation of the local L-function of π twisted by any irreducible, admissible representation of GL₂(F). We derive an integral representation for the global L-function of the irreducible, cuspidal, automorphic representation of GSp₄(A) obtained from a Siegel cuspidal Hecke newform, with respect to a Borel congruence subgroup of square-free level, twisted by any irreducible, cuspidal, automorphic representation of GL₂(A). A special value result for this L-function in the spirit of Deligne's conjecture is obtained.

(10) (with Ralf Schmidt and Abhishek Saha) Pullback of Eisenstein series from GU(3, 3), in preparation