Seminar Announcement
Date: Monday, 7 July 2025
Time: 12.00 PM
Venue: Seminar Hall
Small solution of generic ternary quadratic congruences
Aishik Chattopadhyay
Ramakrishna Mission Vivekananda Educational and Research Institute.
07-07-25

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Abstract

We study small solutions of quadratic congruences of the form $x_1^2+\alpha_2 x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is odd prime power. We show that for "almost all" $\alpha_3$ modulo $q$ which are coprime to $q$, an asymptotic formula for the number of solutions $(x_1, x_2, x_3) $ to the above congruence in a box of a side length $N$ holds if $N\geq q^{11/24+\varepsilon}$. It is of significance to break the barrier $1/2$ in above exponent. Later we managed to reduce the exponent $11/24$ above to $11/25$ by using p-adic exponent pairs by Mili\'{c}evi\'{c}. Furthermore, we derive a general asymptotic formula that applies to arbitrary square-free moduli. Under the Lindel\"{o}f hypothesis for Dirichlet L-functions, we are able to replace the exponent $11/24$ above by $1/3$. This is a joint work with Prof. Stephan Baier.