Subconvex bound for Rankin-Selberg $L$-functions in prime power level

TL;DR

This paper establishes a subconvex bound for the central values of Rankin-Selberg L-functions associated with certain cusp forms of prime power level, advancing understanding of their growth and distribution.

Contribution

It provides a new subconvexity bound for Rankin-Selberg L-functions in the context of prime power levels with supercuspidal local representations at p.

Findings

Established a bound: $L(1/2, f imes g) \,\ll_{g,\epsilon} p^{\frac{23r}{12} + \epsilon}$

Applicable to cusp forms with supercuspidal local representations at p

Advances the understanding of L-function behavior at prime power levels

Abstract

Let $f$ be a $p$-primitive cusp form of level $p^{4r}$, where local representation of $f$ be supercuspidal at $p$, $p$ being an odd prime, $r\geq 1$ and $g$ be a Hecke-Maass or holomorphic primitive cusp form for $\mathrm{SL}(2,\mathbb{Z})$. A subconvex bound for the central values of the Rankin-Selberg $L$-functions $L(s, f \otimes g )$ is given by $$ L (\frac{1}{2}, f \otimes g ) \ll_{g,\epsilon}p^{\frac{23r}{12} +\epsilon}.$$