Atkin-Serre Type Conjectures for Automorphic Representations on $GL(2)$

Let $H(z)$ be a newform of weight $k \geq 4$ without complex multiplication on $\Gamma_{0}(N)$ with normalized $L$-function $L(H,s) = \prod_{p} (1 - \alpha_{p} p^{-s})^{-1} (1 - \beta_{p} p^{-s})^{-1}$. A conjecture of Atkin and Serre states that for sufficiently large primes $p$, \begin{equation} \label{atkinserre} |\alpha_{p} + \beta_{p}| \gg p^{-1-\epsilon} \end{equation} for all $\epsilon > 0$. Let $\pi$ a genuine cuspidal automorphic representation on $GL_{2}(\A_{F})$, where $F$ is a totally real number field. Assuming GRH for the symmetric power $L$-functions associated to $\pi$, we prove that \[ |\alpha_{v} + \beta_{v}| \geq q_{v}^{-\delta} \] for all but $O(x^{1 - \delta}/\log x)$ places $v$ with $q_{v} \leq x$ provided $\delta \leq 1/8$. This implies a strong form of \eqref{atkinserre} for almost all primes $p$.

Full Text (PDF format)

Published 1 January 2007