Solvable Base Change and Rankin-Selberg Convolutions

TL;DR

This paper establishes a prime number theorem for Rankin-Selberg L-functions associated with automorphic representations over solvable algebraic number fields, under specific symmetry and invariance conditions.

Contribution

It extends prime number theorem results to automorphic L-functions over solvable fields using automorphic induction and self-contragredient assumptions.

Findings

Proves a prime number theorem for specific Rankin-Selberg L-functions.

Uses automorphic induction for solvable number fields.

Requires self-contragredient and Galois invariance conditions.

Abstract

Given unitary automorphic cuspidal representations $\pi$ and $\pi'$ defined on $GL_n(\mathbb{A}_E)$ and $GL_m(\mathbb{A}_F)$, respectively, with $E$ and $F$ solvable algebraic number fields we deduce a prime number theorem for the Rankin-Selberg L-function $L(s,AI_{E/\mathbb{Q}}(\pi)\times AI_{F/\mathbb{Q}}(\pi'))$ under a self-contragredient assumption and a suitable Galois invariance condition on the representations, where $AI_{K/\mathbb{Q}}$ denotes the automorphic induction functor for any number field $K/\mathbb{Q}$.