Monday, 25 February 2019

Modular forms occur in many facets in even more different areas of mathematics. Beginning with classical aspects of these marvellous functions these lectures will introduce multiple Eisenstein series and their linkage to the theory of multiple zeta values and their q-analogues.

In this talk, we will present an integral representation of the standard L-function of holomorphic Siegel modular forms twisted by a character. In the genus 2 case, we will give several applications of this integral representation. We will obtain algebraicity of special values of the L-function proving instances of Deligne's conjecture in this context. This leads to the algebraicity of the twists of symmetric fourth power L-functions of elliptic cusp forms. A detailed study of the arithmetic and cuspidality of the restriction of the Eisenstein series involved in the integral representation leads to a result on congruence primes for the Siegel cusp forms. This is joint work with Abhishek Saha and Ralf Schmidt.

I explain the theory of Piatetski-Shapiro and Rallis for the construction of standard L-functions via the doubling method. Alongside the global theory I develop the local theory and prove meromorphic continuation and the functional equation of local zeta integrals. Though the doubling method works for all simple classical groups, I describe it for unitary groups as the unitary case attracts geometric or p-adic attentions. Keywords: Rankin-Selberg theory, orbit structure, Siegel Eisenstein series, generic uniqueness, Bernstein’s principle

Let p be an odd prime number and K/F be a quadratic extension of p-adic fields. Say that an irreducible complex representation of GL(n,K) is distinguished by GL(n,F) if its vector space carries a GL(n,F)-invariant non-zero linear form. Any distinguished representation is isomorphic to the contragredient of its Gal(K/F)-conjugate, but the converse is not true. We will explain how to canonically associate to any Gal(K/F)-selfcontragredient cuspidal representation of GL(n,K) a finite tamely ramified extension T of F and a quadratic character of the multiplicative group of T, by using Bushnell-Kutzko’s theory of types, and how to get a necessary and sufficient condition on this character for this cuspidal representation to be distinguished. Our approach is purely algebraic and local, and can be extended to representations with coefficients in a finite field of characteristic different from p. In that case, our necessary and sufficient condition still holds for supercuspidal representations. For cuspidal non-supercuspidal representations, there are new phenomenons appearing.

Tuesday, 26 February 2019

I explain the theory of Lapid and Rallies for gamma factors via local doubling method, emphasizing multiplicativity. Basic properties of zeta integrals follows from multiplicativity systematically. This technique is common for other integral representations. The L-factor can be defined as “g.c.d." of local zeta integrals. We will show that the zeta integral and local factors are compatible with parabolic induction. Keywords: Normalized intertwining operator, degenerate Whittaker model, Ten commandments, degenerate principal series, good sections

Wednesday, 27 February 2019

After I finish the local Rankin-Selberg theory, I apply it to local theta correspondence. A main goal is to derive the conservation relation, which relates the invariant functional obtained from the local zeta integral to local theta correspondence. Keywords: Langlands classification, Godement-Jacquet theory, Weil representations, persistence, first occurrence, theta dichotomy, seesaw dual pairs.

Friday, 01 March 2019

We apply the integral representation to the global theta correspondence. By combining the local and global results, I will prove a necessary and sufficient condition for non-vanishing of theta liftings in terms of poles or special values of complete standard L-functions. Keywords: regularized theta integral, Siegel-Weil formula, Rallis inner product formula.

Monday, 04 March 2019

It is a classical fact that Siegel modular forms possess so-called Fourier-Jacobi expansions. The question then arises, given such an expansion, when does it originate from a Siegel modular form. In the complex setting, J. Bruinier and M. Raum gave a necessary and sufficient criterion when Fourier-Jacobi expansions give rise to Siegel modular forms. In our talk we would like to revisit this problem however using the arithmetic compactifications of the moduli space of principally polarized abelian varieties established by G. Faltings and C.-L. Chai. In particular, this will allow us to generalize the result of J. Bruinier and M. Raum to the arithmetic setting.

Günter Harder pioneered the study of Eisenstein cohomology on GL(2) as a tool to understand the special values of various Hecke L-functions. Over the last decade, Harder and I developed his earlier ideas to study Eisenstein cohomology on GL(N) over a totally real field and proved rationality results for the critical values of Rankin-Selberg L-functions. In continuation, I have worked on Eisenstein cohomology on GL(N) over a CM field. In this talk, I will report on some recent results of mine on the rationality properties of critical values of Rankin-Selberg L-functions for GL(n) x GL(m) over a CM field

Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve L-functions. These conjectures relate to the asymptotic growth of the first and second moments of the modular symbols, as well as the distribution of of modular symbols. We explain these conjectures and how they may be addressed using analytic properties of Eisenstein series twisted by modular symbols and how they can be extended to other situations.

Let p be an odd prime number and K/F be a quadratic extension of p-adic fields. Say that an irreducible complex representation of GL(n,K) is distinguished by GL(n,F) if its vector space carries a GL(n,F)-invariant non-zero linear form. Any distinguished representation is isomorphic to the contragredient of its Gal(K/F)-conjugate, but the converse is not true. We will explain how to canonically associate to any Gal(K/F)-selfcontragredient cuspidal representation of GL(n,K) a finite tamely ramified extension T of F and a quadratic character of the multiplicative group of T, by using Bushnell-Kutzko’s theory of types, and how to get a necessary and sufficient condition on this character for this cuspidal representation to be distinguished. Our approach is purely algebraic and local, and can be extended to representations with coefficients in a finite field of characteristic different from p. In that case, our necessary and sufficient condition still holds for supercuspidal representations. For cuspidal non-supercuspidal representations, there are new phenomenons appearing.

The Generalized Injectivity Conjecture of Casselman-Shahidi states that the unique irreducible generic subquotient of a (generic) standard module is necessarily a subrepresentation. It was proven for classical groups (SO(2n+1), Sp_{2n}, SO(2n)) by M.Hanzer in 2010. In my work, soon to be submitted, I am aiming at proving it for any quasi-split reductive group.

In this talk, I will describe some techniques more amenable to prove this conjecture for all quasi-split groups; I will also explain the architecture and key ideas going into the proof.

Tuesday, 05 March 2019

Let E/F be a finite Galois extension of non-archimedean local fields. Let G(E) be the group of E-rational points a split connected reductive group G defined over E such that the local Langlands correspondence is known for G(E). Let M denote the Weil restriction of G with respect to E/F. We will describe a candidate for the local Langlands correspondence for M(F), compare the depths of the corresponding L-parameters, and give an application to the Asai lift.

In this talk, we will present an integral representation of the standard L-function of holomorphic Siegel modular forms twisted by a character. In the genus 2 case, we will give several applications of this integral representation. We will obtain algebraicity of special values of the L-function proving instances of Deligne's conjecture in this context. This leads to the algebraicity of the twists of symmetric fourth power L-functions of elliptic cusp forms. A detailed study of the arithmetic and cuspidality of the restriction of the Eisenstein series involved in the integral representation leads to a result on congruence primes for the Siegel cusp forms. This is joint work with Abhishek Saha and Ralf Schmidt.

A representation of a group G has a model with respect to a subgroup H if it has a unique, up to scalar, imbedding into the representation induced from the trivial representation of H. This talk will discuss several new observations for the case where G is the general linear group GL(2n) over a p-adic field and H is the symplectic subgroup Sp(2n). If an irreducible representation of G admits an H-model its restriction to SL(2n) has a unique component that admits an H-model. A conjecture of Dijols and Prasad predicts the maximal unipotent orbit satisfied by this component. We prove their conjecture for the class of ladder representations (and for many other representations). Along the way, we observe that for a large family of irreducible representations of GL(2n), if they admit a symplectic model then the same is true for the highest derivative of their highest derivative.

A standard version of the circle method can be used to obtain strong bounds for GL(3)xGL(2) Rankin-Selberg L-functions in various aspects. As a consequence several known cases of sub-convexity can be obtained from one general set-up.

Wednesday, 06 March 2019

In the lecture I shall report on joint work with Ernst Kani on closed geodesics on the triaxial ellipsoid, both defined over a fixed number field. Geodesics on the ellipsoid lead by the work of Knoerrer to geodesics on an abelian surface which is canonically attached to the ellipsoid. Via the analytic subgroup theorem this determines an elliptic subcover of the hyperelliptic curve which determines the abelian surface. Such objects are classified in certain cases by Teichmüller curves.The subcover gives a rational point on such a Teichmüller curve. If the curve has genus at least 2 one concludes via Faltings theorem that there are only finitely many such closed geodesics.

TBA

It is well known that classical modular forms are uniquely determined from its Fourier expansion. However for applications, one often needs to consider only special subsets of all Fourier coefficients, like those indexed by square-free integers. We would give a survey of known results on this topic along with very recent developments.

The global A-packets appear in Arthur's conjectural description of the discrete automorphic spectrum of a reductive group over the number field. Their local components, also called local A-packets, are conjectured to be finite sets of irreducible admissible representations of the reductive group over the local field. On the other hand, one has the L-packets from the conjectural local Langlands correspondence. So it is a natural problem to understand the relation between the two. In this talk, I will consider non-endoscopic global A-packets of a symplectic/orthogonal group and present the corresponding results for their nonarchimedean components.

Thursday, 07 March 2019

Unitary cohomological representations of real reductive groups were classified in the early 1980s by Vogan and Zuckerman and their Arthur packets were defined in 1987 by Adams and Johnson. We will show that cohomological Arthur packets are functorial for certain morphisms of Langlands dual groups and draw some consequences of this. This is based on joint work with D. Prasad.

Multiple q-zeta values are q-series that specialize in q=1 to multiple zeta values. They form an algebra, which contains the quasi-modular forms as a subalgebra. We report on a conjecture on the dimensions of its weight- and depth-graded pieces and its relation to period polynomials.