My research interest broadly comes under the Langlands
Program.
In particular, I am interested in understanding classical
modular forms and automorphic representations associated to them.
Below
is the list of my papers and preprints. Also, see my CV for more details.
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(pr), r > 0, has infinitely many sign changes.
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.
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.
In this paper, we give a new definition for the space of non-holomorphic Jacobi Maass forms (denoted by Jnhk,m) of weight k and index m as eigenfunctions of a degree three differential operator Ck,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 Jnhk,m. We construct new examples of cuspidal Jacobi Maass forms Ff of even weight k and index 1 from weight k-1/2 Maass forms f with respect to Γ0(4) and show that the map f → Ff 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 show that all of Jnhk,m can be "essentially" obtained from scalar or vector valued half integer weight Maass forms.
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.
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.
We characterize the irreducible, admissible, spherical representations of GSp(4,F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-Kurokawa lifting. We show how the classical Maass relations can be deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito-Kurokawa lifts in terms of Fourier coefficients of half-integral weight modular forms or Jacobi forms. As an additional application of our methods, we give a new characterization of Saito-Kurokawa lifts involving a certain average of Fourier coefficients.
Let F be a nearly holomorphic vector-valued Siegel modular form of weight ρ with respect to some congruence subgroup of Sp(2n,Q). In this note, we prove that the function on Sp(2n,R) obtained by lifting F has the moderate growth (or "slowly increasing") property.
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 GSp(4,R) and Sp(4,R) 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.
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, quadratic twists of the Steinberg representation π of GSp(4, 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 GL(2, F). Together with results in [4] and [10], we derive an integral representation for the global L-function of an irreducible, cuspidal automorphic representation of GSp(4, A) obtained from a Siegel cuspidal Hecke newform, with respect to the Borel congruence subgroup of square-free level, twisted by any irreducible, cuspidal, automorphic representation of GL(2, A). A special value result for this L-function in the spirit of Deligne's conjecture is obtained.
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.
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.
In this semi-expository note, we give a new proof of a structure theorem due to Shimura for nearly holomorphic modular forms on the complex upper half plane. Roughly speaking, the theorem says that the space of all nearly holomorphic modular forms is the direct sum of the subspaces obtained by applying appropriate weight-raising operators on the spaces of holomorphic modular forms and on the one-dimensional space spanned by the weight 2 nearly holomorphic Eisenstein series. While Shimura's proof was classical, ours is representation-theoretic. We deduce the structure theorem from a decomposition for the space of n-finite automorphic forms on SL(2, R). To prove this decomposition, we use the mechanism of category O and a careful analysis of the various possible indecomposable submodules. It is possible to achieve the same end by more direct methods, but we prefer this approach as it generalizes to other groups. This note may be viewed as the toy case of our paper [6], where we prove an analogous structure theorem for vector-valued nearly holomorphic Siegel modular forms of degree two.
We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G, K), where G = Sp(4, R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight detl symm with respect to an arbitrary congruence subgroup of Sp(4, Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight detl' symm' with (l', m') varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight det3 symm' that cannot be obtained from holomorphic forms. As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.
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).
Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk → ∞. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions of Watson's formula to certain triple product integrals involving forms of non-squarefree level. By a theorem of Ichino and a lemma of Michel-Venkatesh, this amounts to a detailed study of Rankin-Selberg integrals ∫ |f|2E attached to newforms f of arbitrary level and Eisenstein series E of full level. We find that the local factors of such integrals participate in many amusing analogies with global L-functions. For instance, we observe that the mass equidistribution conjecture with a power savings in the depth aspect is equivalent to knowing either a global subconvexity bound or what we call a "local subconvexity bound"; a consequence of our local calculations is what we call a "local Lindeof hypothesis".
Let π be the automorphic representation of GSp(4, A) generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and τ be an arbitrary cuspidal, automorphic representation of GL(2, A). Using Furusawa’s integral representation for GSp(4) × GL(2), combined with a pullback formula involving the unitary group GU(3, 3), we prove that the L-functions L(s, π × τ ) are “nice”. The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations π have a functorial lifting to a cuspidal representation of GL(4, A). Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of π to a cuspidal representation of GL(5, A). As an application, we obtain analytic properties of various L-functions related to full level Siegel cusp forms. We also obtain special value results for GSp(4) × GL(1) and GSp(4) × GL(2).
We determine local test vectors for Waldspurger functionals for GL(2), in the case where both the representation of GL(2) and the character of the degree two extension are ramified, with certain restrictions. We use this to obtain an explicit version of Waldspurger’s formula relating twisted central L-values of automorphic representations on GL(2) with certain toric period integrals. As a consequence, we generalize an average value formula of Feigon and Whitehouse, and obtain some nonvanishing results.
We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients satisfy a partial Ramanujan bound and do not differ by too much. Additionally, we prove a number of multiplicity one type results for the number-theoretic objects attached to L-functions. These results follow from our main result, which has slightly weaker hypotheses than previous multiplicity one theorems for L-functions.
We consider an Eisenstein series on GL(2,L), where L is a real quadratic extension of Q, and study its restriction to GL(2,Q). In a special case, the Eisenstein series corresponds to a weight (1,1) Hilbert Eisenstein series. Given an elliptic cusp form of weight 2 and squarefree level, we obtain an explicit formula for the integral of the restriction of the Hilbert Eisenstein series against the cusp form. The answer is in terms of central twisted L-values. We obtain an application towards a conjecture due to Tonghai Yang regarding generation of elliptic cusp forms of weight 2 from restrictions of Hilbert Eisenstein series of weight (1,1).
L-functions can be viewed axiomatically, such as in the formulation due to Selberg, or they can be seen as arising from cuspidal automorphic representations of GL(n), as first described by Langlands. Conjecturally these two descriptions of L-functions are the same, but it is not even clear that these are describing the same set of objects. We propose a collection of axioms that bridges the gap between the very general analytic axioms due to Selberg and the very particular and algebraic construction due to Langlands. Along the way we prove theorems about L-functions that satisfy our axioms and state conjectures that arise naturally from our axioms.
We formulate an explicit refinement of Bocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L- functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.
We prove an explicit integral representation - involving the pullback of a suitable Siegel Eisenstein series - for the twisted standard L-function associated to a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case n = 2 we are able to prove a reciprocity law - predicted by Deligne's conjecture - for the critical special values of the twisted standard L-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan-Shahidi lift, we obtain a reciprocity law for the critical special values of the symmetric fourth L-function of a classical newform.
We study nearly holomorphic Siegel Eisenstein series of general levels and characters on H2n, the Siegel upper half space of degree 2n. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of Qp for all sufficiently large primes p. We also prove that the pullbacks of these Eisenstein series to HnxHn are cuspidal under certain assumptions.
We prove the expected algebraicity property for the critical values of character twists of the standard L-function associated to vector-valued holomorphic Siegel cusp forms of archimedean type (k1, k2,...,kn), where kn≥2n + 2 and all ki are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach.
In this paper, we construct an explicit lifting from half-integral weight Maass form with respect to Γ0(4), to Maass forms on the 4-dimensional hyperbolic space. We show that this construction is Hecke equivariant. This gives us a map from cuspidal representations of SL̃(2) to cuspidal representations of GSpin(1,4) which are nearly equivalent to induced representation on GSp(4). This allows us to show that the resulting representations violate the Ramanujan conjecture.
The aim of this paper is to carry out an explicit construction of CAP representations of GL(2) over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup Γ0(2).We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.
We explicitly construct non-holomorphic cusp forms on the orthogonal group of signature (1,8n+1) for an arbitrary natural number n as liftings from Maass cusp forms of level one. In our previous works the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maass cusp forms are Hecke eigenforms. Our non-archimedean local theory relates Sugano's local theory to non-tempered automorphic forms or representations of a general orthogonal group in a transparent manner.
We explicitly construct non-tempered cusp forms on the orthogonal group O(1,5) of signature (1+,5-). Given a definite quaternion algebra B over Q, the orthogonal group is attached to the indefinite quadratic space of rank 6 with the anisotropic part defined by the reduced norm of B. Our construction can be viewed as a generalization of the previous work by the first two authors joint with Masanori Muto to the case of any definite quaternion algebras, for which we note that the work just mentioned takes up the case where the discriminant of B is two. Unlike the previous work the method of the construction is to consider the theta lifting from Maass cusp forms to O(1,5), following the formulation by Borcherds. The cuspidal representations generated by our cusp forms are studied in detail. We determine all local components of the cuspidal representations and show that our cusp forms are CAP forms.
The world of a mathematician, with all its creativity and precision is fascinating to most people. This study is an account of collaboration between mathematicians and mathematics educators. In order to examine a mathematician’s daily activities, we have primarily employed Schoenfeld’s goal-orientated decision making theory to identify his Resources, Orientations and Goals (ROGs) in teaching an abstract algebra class. Our preliminary results report on a healthy and positive atmosphere where all involved freely express their views on mathematics and pedagogy.
Abstract algebra is a fascinating field of study among mathematics topics. Despite its importance, very little research has focused on the teaching of abstract algebra. In response to this deficiency, in this study we present an abstract algebra professor’s daily activities and thought processes as shared through his teaching diaries with a team of two mathematics educators and another abstract algebraist over the period of two semesters. We examined how he was able to live in the formal world of mathematical thinking while also dealing with the many pedagogical challenges that were set before him during the lectures.
We used continuous glucose monitoring to test the hypothesis that mean amplitude of glycemic excursions (MAGE) is associated with circulating markers of oxidative and vascular stress in adolescents with habitually low physical activity classified as healthy weight, healthy obese, or obese with type 2 diabetes mellitus (T2DM). We conclude that MAGE is highest in obese youth with T2DM. The associations between MAGE and oxidative stress markers support the proposed contribution of glycemic variability to risk for future cardiovascular disease.