Multiplicity one bound for cohomological automorphic representations with a fixed level

TL;DR

This paper establishes a uniform bound on the level at which two distinct cohomological automorphic representations of GL(n) over a totally real field differ, improving previous bounds related to their analytic conductors, with applications to classical modular forms.

Contribution

The authors prove a uniform bound on the minimal level where two distinct cohomological automorphic representations differ, refining previous bounds depending on the analytic conductor.

Findings

Existence of a constant C_N bounding the level difference for automorphic representations.

Application of the bound to classical modular forms showing equality of forms under certain Hecke eigenvalue conditions.

Improvement over previous bounds of the form O(Q^A) for the level difference.

Abstract

Let $F$ be a totally real field, and $\mathbb{A}_F$ be the adele ring of $F$. Let us fix $N$ to be a positive integer. Let $\pi_1=\otimes\pi_{1,v}$ and $\pi_2=\otimes\pi_{2,v}$ be distinct cohomological cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A}_{F})$ with levels less than or equal to $N$. Let $\mathcal{N}(\pi_1,\pi_2)$ be the minimum of the absolute norm of $v \nmid \infty$ such that $\pi_{1,v} \not \simeq \pi_{2,v}$ and that $\pi_{1,v}$ and $\pi_{2,v}$ are unramified. We prove that there exists a constant $C_N$ such that for every pair $\pi_1$ and $\pi_2$, $$\mathcal{N}(\pi_1,\pi_2) \leq C_N.$$ This improves known bounds $$ \mathcal{N}(\pi_1,\pi_2)=O(Q^A) \;\;\; (\text{some } A \text{ depending only on } n), $$ where $Q$ is the maximum of the analytic conductors of $\pi_1$ and $\pi_2$. This result applies to newforms on $\Gamma_1(N)$. In particular, assume that $f_1$ and $f_2$ are Hecke eigenforms of weight $k_1$ and $k_2$ on $\mathrm{SL}_2(\mathbb{Z})$, respectively. We prove that if for all $p \in \{2,7\}$, $$\lambda_{f_1}(p)/\sqrt{p}^{(k_1-1)} = \lambda_{f_2}(p)/\sqrt{p}^{(k_2-1)},$$ then $f_1=cf_2$ for some constant $c$. Here, for each prime $p$, $\lambda_{f_i}(p)$ denotes the $p$-th Hecke eigenvalue of $f_i$.