Automorphic form twisted Shintani zeta functions over number fields

TL;DR

This work generalizes twisted Shintani zeta functions associated to binary cubic forms from $\mathbb{Q}$ to arbitrary number fields by embedding the problem into an adelic GL$_2$ framework. It defines and analyzes the zeta functional $Z(s, \Phi, \varphi)$ via a decomposition into $Z_+$, $I$, and $J$, and proves analytic continuation for cusp twists as well as meromorphic continuation for Eisenstein twists, with explicit pole-residue descriptions. The authors develop a uniform method using automorphic forms, Whittaker models, and explicit local zeta integrals to control the integrals $I(s, \Phi, \varphi)$ and $J(s, \Phi, \varphi)$, including detailed Mellin transform arguments and the Poisson summation on binary cubic forms. The results extend prior Q$-$based work (Hough–Bobin) to general number fields and provide a framework potentially applicable to the arithmetic of cubic field extensions and related shape problems. The approach is adelic, uniform across places, and does not require unramified twists, offering a robust analytic tool for understanding twisted Shintani zeta functions.

Abstract

In this paper we study the twisted Shintani zeta function over number fields.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Proposition 6: Proposition 2.1 of W
• Proposition 7: Corollary to Proposition 2.2 of W
• Lemma 8
• proof
• Lemma 9