On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case

TL;DR

The paper proves a Weyl-type subconvex bound for cube-free level Hecke characters over totally real fields by developing an explicit inverse transform for Motohashi’s formula and exploiting invariant weight distributions. It builds a real-place dual transform with a kernel $K(x,\tau)$ and constructs positive-type test functions that isolate short spectral families; non-archimedean places are handled via compact-open subgroup vectors, yielding precise dual-weight bounds. Globally, the authors formulate a spectral large sieve bound that couples cubic and quartic moments, then deduce $L(\tfrac{1}{2},\chi) \ll_{F,\epsilon} \mathbf{C}(\chi)^{1/6+\epsilon}$ by leveraging positivity and the Motohashi reciprocity between moments. The framework generalizes prior cube-free work to totally real fields and clarifies archimedean/non-archimedean dichotomies, offering explicit integral transforms and uniform hypergeometric bounds that may extend to broader Motohashi-variants.

Abstract

We prove a Weyl-type subconvex bound for cube-free level Hecke characters over totally real number fields. Our proof relies on an explicit inversion to Motohashi's formula. Schwartz functions of various kinds and the invariance of the relevant Motohashi's distributions discovered in a previous paper play central roles.

Theorems & Definitions (155)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Proposition 1.8
• Theorem 1.9
• Remark 1.10