Monday, 27 June 2022

A fundamental problem in analysis is understanding the distribution of mass of Laplacian eigenfunctions via bounds for their L^p-norms in terms of the size of their Laplacian eigenvalue. Number theorists are interested in the Laplacian eigenfunctions on the modular surface that are additionally joint eigenfunctions of every Hecke operator — namely the Hecke Maass cusp forms. In this talk, I will describe joint work with Peter Humphries in which we prove new bounds for L^p-norms in this situation. This is achieved by using L-functions and their reciprocity formulae: certain special identities between two different moments of central values of L-functions.

I will talk about the cubic moment of central L-values for Maass forms. It was studied by Aleksandar Ivi\'c at the beginning of this century, obtaining asymptotic on the long interval $[0, T]$ with error term $O(T^{8/7+\epsilon})$ and Lindel\"of-on-average bound on the short window $[T-M, T+M]$ for $M$ as small as $T^{\epsilon}$. Ivi\'c's results are improved in my recent work; in particular, Ivi\'c's conjectured error term $O (T^{1+\epsilon})$ is proven. Our proof follows the standard Kuznetsov--Voronoi approach stemmed from the work of Conrey and Iwaniec. Our main new idea is a combination of the methods of Xiaoqing Li and Young.

We will discuss on the zeros of $L$-functions attached to half-integral weight modular forms on the critical line.

This is joint work with Kamalakshya Mahatab and Łukasz Pańkowski.

Let $$S(N) = \sum_{n \sim N}^{\text{smooth}} \, \lambda_{f}(n) \, \chi(n),$$

where $\lambda_{f}(n)$'s are Fourier coefficients of Hecke-eigen form, and $\chi$ is a primitive character of conductor $p^{r}$. In this talk I will discuss about a sub-Weyl strength bounds for $S(N)$ as $r$ varying and $p$ being fixed.

Tuesday, 28 June 2022

Beyond Endoscopy" is a proposal of Langlands for studying poles of L-functions by an indirect method that uses the trace formula to convert the problem to that of estimating certain exponential sums. A modified version of this, due to Sarnak, has been used to prove analytic continuation of some L-functions (e.g., symmetric square of GL(2), Rankin Selberg convolution GL(2)xGL(2) etc.). However, this method seems to be quite versatile and one may attempt to estimate general sums of the type $\sum_{n<N} a(n) b(n)$, where $a(n)$ are coefficients of a fixed cusp form, by this method, possibly in conjunction with other methods such as the delta method (DFI, GL(2) etc.). It is not yet clear exactly when, i.e., for which sequences $b(n)$, this method is useful. I will give some examples where this method gives good bounds for the above sum with very little work, including one that has no direct connection with L-functions.

In this talk, we will discuss the quantum variances for families of automorphic forms on modular surfaces. The resulting quadratic forms are compared with the classical variance. The proofs depend on moments of central L-values and then the shifted convolution sums / non-split sums. The circle method can be used to estimate those sums. (Based on joint work with Stephen Lester.)

Getting `non-trivial' estimates for automorphic $L$-functions at the central point $1/2$ is one of the active area of research in analytic number theory. In this talk we will discuss such estimates for the Rankin-Selberg $\mathrm{GL(3)} \times \mathrm{GL(2)}$ $L$-functions in the spectral ($\mathrm{GL(2)}$ and $\mathrm{GL(3)}$) aspect using the delta method. In particular, we will give subconvexity for $\mathrm{GL(3)}$ forms having spectral parameter $(iT, iT^{3/4} , -i(T+T^{3/4}))$ which is a conductor dropping situation. The talk is based on the joint work with Prof. Mallesham and Prof. Saurabh Singh.

I will discuss some recent work on the subject in the title, with particular attention on a family of Eisenstein series on GL_2 with varying level.

Wednesday, 29 June 2022

Let $Q(x_1,...,x_n)$ be a quadratic form with integral coefficients and $n\ge 3$. The question of how the solutions $(x_1,...,x_n)$ of congruences of the form $Q(x_1,...,x_n)\equiv 0 \bmod{q}$ distribute modulo $q$ has received a lot of attention. Of particular interest are small solutions to such congruences. There are still unresolved problems in this connection. We review known results in this direction and report about recent work on small solutions to congruences for diagonal quadratic forms in three variables by Anup Halder and the speaker (both RKMVERI). We also indicate possible improvements by an application of the circle method.

In this talk, we discuss simultaneous nonvanishing of the central $L$-values associated to cusp forms. More precisely, for a given normalized newform $f$ of large prime level, we estimate the cardinality of the set of those normalized newforms $g$ of the same weight and level as of $f$ such that $L(1/2, f\otimes \chi_{D_0}) L(1/2, g\otimes \chi_{D_0}) \neq 0$ for some fundamental discriminant $D_0$. This is based on a joint work with M. Manickam and B. Kumar.

For a non-CM, primitive holomorphic cusp form f, λf (n) is called the normalized n-th Fourier coefficient. With this normalization, the Ramanujan Conjecture predicts λf (p) ∈ [−2, 2] for all primes p (which is known by the work of Deligne). The Sato-Tate Conjecture for distribution of the angles θp, defined by λf (p) = 2 cos θp, as p runs over primes, (which is now a theorem of Clozel, Harris, Shepherd-Barron and Taylor) implies that any interval of positive measure within [−2, 2] contains infinitely many values of λf (p). In this talk, given an interval I ⊂ [−2, 2], we discuss the least prime p such that λf (p) ∈I. This can be considered as an analogue of Vinogradov’s problem of estimating, given a modulus q ≥ 1, the size of the least quadratic non-residue modulo q. We exhibit strong explicit bounds on p, depending on the analytic conductor of f for some specific choices of I. The quality of our bounds will be measured in terms of the analytic conductor of the form f. This is a joint work with S. Drappeau, S. Ganguly and O. Ramare.

I will discuss non-trivial bounds for the sup-norm of non-spherical Hecke–Maass forms on SL2(Z[i])\SL2(C). The term “non-spherical” refers to the fact that the form to be bounded is not invariant under the right action of SU2(C), and “non-trivial” means that we achieve a power saving in the dimension of the SU2(C)-representation generated by the form. The proof involves analytic theory of independent interest, such as localization estimates for generalized spherical functions on SL2(C) and a Paley–Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions. Joint work with Valentin Blomer, P ́eter Maga, and Djordje Mili ́cevi ́c.

We give a generalized Kuznetsov fromula arising from the relative trace formula perspective, and discuss potential applications to spectral large sieve inequalities and subconvexity. This is work in progress with M.P.\ Young and Y.\ Hu.

Thursday, 30 June 2022

Let \chi_q be real characters of large conductor q. Let f be a fixed GL(2) cusp form of trivial central character. In 2000, by using a moment method Conrey and Iwaniec obtained the best known subconvexity bounds for self-dual L-functions L(\chi_q,1/2) and L(f x\chi_q,1/2) in the large q-aspect. This approach was later adapted by Xiaoqing Li to give the first subconvexity bounds for GL(3)xGL(2) self-dual L-functions in the large GL(2) spectral aspect. In this talk, we revisit Li's work and explain how her bounds can be improved to the limit of this method. Some Motohashi-type formulae surrounding these problems will also be discussed. Joint work with Ramon Nunes and Zhi Qi.

Inspired from Spieser's and Spira's results on the relation between zeros of the Riemann zeta function and it's derivatives, Levinson and Montgomery deduced several results concerning the zeros of the k-th derivatives of the Riemann zeta. In one such result, they show that the RH implies that all the k-th derivatives of the Riemann zeta function have at most a finite number of non-real zeros in the left of the critical line. In this talk, we discuss this for L functions belonging to the Selberg class and prove that an analogous result holds true for this class as well. This is a joint work with Suraj Singh Khurana and Ade Irma Suriajaya.

We develop a two dimensional version of the delta symbol method and apply it to establish quantitative Hasse principle for a smooth pair of quadrics defined over Q defined over at least 10 variables. This is a joint work with Simon Myerson (warwick) and Junxian Li (Bonn).

Friday, 01 July 2022

I will use the spectral theory of $GL(3)$ automorphic forms to evaluate a second moment of $GL(3)\times GL(2)$ L-functions and deduce subconvexity for the same in the $GL(3)$ spectral aspect.