Subconvexity for Half Integral Weight $L$-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.

33 pages

References in corpus (3)

• On sup-norms of cusp forms of powerful level
• Triple Shifted Sums of Automorphic L-Functions
• Hecke operators in half-integral weight

Cited by in corpus (6)

• Second Moments in the Generalized Gauss Circle Problem
• On Some Variants of the Gauss Circle Problem
• The Second Moment of Sums of Coefficients of Cusp Forms
• The twisted second moment of modular half integral weight $L$--functions
• Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms
• Self-Correlations of Hurwitz Class Numbers