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09:30 to 10:30 |
Stephen Baier (RKMERI, Kolkata, India) |
Small Solutions of Quadratic Congruences 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.
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09:30 to 12:45 |
Jaban Mehar (NISER, Bhubaneswar, India) |
Session Chair |
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11:00 to 12:00 |
Karam Deo Shankhadhar (IISER, Bhopal, India) |
Simultaneous nonvanishing of central $L$-values with large prime level 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.
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12:15 to 12:45 |
Ratnadeep Acharya (RKMERI, Kolkata, India) |
A Modular Analogue of a Problem of Vinogradov 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.
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14:30 to 15:30 |
Gergely Harcos (Alfréd Rényi Institute of Mathematics, Budapest, Hungary) |
Beyond the Spherical Sup-norm Problem 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.
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14:30 to 17:00 |
Sneha Chaubey (Indraprastha Institute of Information Technology Delhi, India) |
Session Chair |
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16:00 to 17:00 |
Ian Petrow (University College London, UK) |
Generalized PGL(2) Kuznetsov Formulas 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.
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