Atkin-Serre Type Conjectures for Automorphic Representations on $GL(2)$
Atkin-Serre Type Conjectures for Automorphic Representations on $GL(2)$
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$GL(2)$ 上自守表示的阿特金-塞尔类型猜想
DOI：
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发表时间：
2007
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通讯作者：
Jeremy A. Rouse
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作者：
Jeremy A. Rouse
Let $H(z)$ be a newform of weight $k geq 4$ without complex multiplication on $Gamma_{0}(N)$ with normalized $L$-function $L(H,s) = prod_{p} (1 - alpha_{p} p^{-s})^{-1} (1 - eta_{p} p^{-s})^{-1}$. A conjecture of Atkin and Serre states that for sufficiently large primes $p$, egin{equation} label{atkinserre} |alpha_{p} + eta_{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} + eta_{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 $p$.
Let $H(z)$ be a newform of weight $k geq 4$ without complex multiplication on $Gamma_{0}(N)$ with normalized $L$-function $L(H,s) = prod_{p} (1 - alpha_{p} p^{-s})^{-1} (1 - eta_{p} p^{-s})^{-1}$. A conjecture of Atkin and Serre states that for sufficiently large primes $p$, egin{equation} label{atkinserre} |alpha_{p} + eta_{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} + eta_{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 $p$.