Burgess-like subconvexity for ${\mathrm{GL}}_1$

Abstract

We generalize our previous method on subconvexity problem for ${\mathrm{GL}}_2\times {\mathrm{GL}}_1$ with cuspidal representations to Eisenstein series, and deduce a Burgess-like subconvex bound for Hecke characters, i.e., the bound $|L(1/2,\chi )|{\ll }_{\mathbf{F},\epsilon }\mathbf{C}(\chi {)}^{1/4-(1-2\theta )/16+\epsilon }$ for varying Hecke characters $\chi$ over a number field $\mathbf{F}$ with analytic conductor $\mathbf{C}(\chi )$. As a main tool, we apply the extended theory of regularized integral due to Zagier developed in a previous paper to obtain the relevant triple product formulas of Eisenstein series.

Final version, to appear in Compositio Mathematica

References in corpus (2)

• A generalized cubic moment and the Petersson formula for newforms
• A Note on Spectral Analysis for ${\mathrm{GL}}_2$: I

Cited by in corpus (4)

• Explicit subconvexity for ${\mathrm{GL}}_2$
• Weyl-type bounds for twisted $GL(2)$ short character sums
• Explicit Burgess-like subconvex bounds for ${\mathrm{GL}}_2\times {\mathrm{GL}}_1$
• Arithmetic Progressions of Squares and Multiple Dirichlet Series