Sub-Weyl subconvexity for Dirichlet L-functions to prime power moduli

TL;DR

This paper establishes a sub-Weyl bound for Dirichlet L-functions at the central point for prime power moduli, introducing a new p-adic exponential sum estimation method that surpasses previous barriers.

Contribution

It develops a novel p-adic analogue of exponent pair methods, enabling sub-Weyl bounds for Dirichlet L-functions to prime power moduli, advancing analytic number theory techniques.

Findings

Achieved a sub-Weyl exponent theta < 1/6 for L(1/2, chi)

Developed a new p-adic exponential sum estimation theory

Applicable to a broad class of exponential sums involving p-adic phases

Abstract

We prove a subconvexity bound for the central value L(1/2, chi) of a Dirichlet L-function of a character chi to a prime power modulus q=p^n of the form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx 0.1645 < 1/6, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving p-adically analytic phases, which can be naturally seen as a p-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including many that arise from a stationary phase analysis of hyper-Kloosterman and other complete exponential sums.