Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

TL;DR

This paper investigates the analytic properties of adelic Igusa zeta functions associated with elliptic curves, establishing the existence of a natural boundary beyond which these functions cannot be meromorphically continued.

Contribution

It demonstrates the meromorphic continuation of global Igusa zeta functions for elliptic curves up to a natural boundary, using results from L-functions and the Sato-Tate conjecture.

Findings

Global Igusa zeta functions have a natural boundary for meromorphic continuation.

Meromorphic continuation is possible up to this boundary, but not beyond.

Results connect Igusa integrals with deep number-theoretic conjectures.

Abstract

We study the analytic behaviour of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato-Tate conjectures we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.