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 ${\pi }_2$, $$\mathcal{N}({\pi }_1,{\pi }_2)\le C_N.$$ 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 $f_1=c{f}_2$ for some constant $c$. Here, for each prime $p$, ${\lambda }_{f_i}(p)$ denotes the $p$-th Hecke eigenvalue of $f_i$.

A new paper preparing with other collaborators will include the results of this paper