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

Jeremy Rouse
2007 Mathematical Research Letters  
Let H(z) be a newform of weight k ≥ 4 without complex multiplication A conjecture of Atkin and Serre states that for sufficiently large primes p, for all > 0. Let π 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 π, we prove that |αv + βv| ≥ q −δ v for all but O(x 1−δ / log x) places v with qv ≤ x provided δ ≤ 1/8. This implies a strong form of (1) for almost all primes p.
doi:10.4310/mrl.2007.v14.n2.a3 fatcat:kba7oilblja2dfpuozjtvaygn4