Subconvexity for Half Integral Weight $L$-functions

TL;DR

This paper establishes a subconvexity bound for certain half-integer weight modular form L-functions, providing evidence towards a Lindelöf-type hypothesis despite the absence of an Euler product.

Contribution

It proves a subconvexity bound for half-integer weight modular form L-functions in the conductor aspect, a novel result in this area.

Findings

Subconvexity bound established for half-integer weight L-functions

Supports the plausibility of a Lindelöf-type hypothesis for these L-functions

Highlights differences due to lack of Euler product in such L-functions

Abstract

We prove a subconvexity bound in the conductor aspect for $L(s,f,\chi)$ where $f$ is a half integer weight modular form. This $L$-function has analytic continuation and functional equation, but no Euler product. Due to the lack of an Euler product, one does not expect a Riemann hypothesis for half integer weight modular forms. Nevertheless one may speculate a Lindelof-type hypothesis, and this current subconvexity result is an indication towards its truth.