Multiplicity one bound for cohomological automorphic representations with a fixed level

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 $f_i$.