# L-functions and sharp resonances of infinite index congruence subgroups of $SL_2(\mathbb{Z})$

## TL;DR

This paper investigates the spectral properties of convex co-compact subgroups of SL2(Z) by analyzing L-functions and Selberg zeta functions related to congruence subgroups, establishing a factorization formula and proving the existence of non-trivial resonances.

## Contribution

It introduces a novel factorization formula for the Selberg zeta function involving L-functions of Galois representations and demonstrates the existence of non-trivial resonances in a low frequency range.

## Key findings

- Factorization formula for Selberg zeta function in terms of L-functions
- Bounds and analytic continuation for these L-functions
- Existence of non-trivial resonances in a low frequency strip

## Abstract

For convex co-compact subgroups of SL2(Z) we consider the "congruence subgroups" for p prime. We prove a factorization formula for the Selberg zeta function in term of L-functions related to irreducible representations of the Galois group SL2(Fp) of the covering, together with a priori bounds and analytic continuation. We use this factorization property combined with an averaging technique over representations to prove a new existence result of non-trivial resonances in an effective low frequency strip.