TY - JOUR

T1 - Almost all primes satisfy the Atkin–Serre conjecture and are not extremal

AU - Gafni, Ayla

AU - Thorner, Jesse

AU - Wong, Peng Jie

N1 - Funding Information:
The authors would like to thank Amir Akbary, Po-Han Hsu, and Wen-Ching Winnie Li for helpful comments. The third author is currently an NCTS postdoctoral fellow; he was supported by a PIMS postdoctoral fellowship and the University of Lethbridge during part of this research.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2021/6

Y1 - 2021/6

N2 - Let f(z)=∑n=1∞af(n)e2πinz be a non-CM holomorphic cuspidal newform of trivial nebentypus and even integral weight k≥ 2. Deligne’s proof of the Weil conjectures shows that |af(p)|≤2pk-12 for all primes p. We prove for 100% of primes p that 2pk-12loglogp/logp<|af(p)|<⌊2pk-12⌋. Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin–Serre conjecture is satisfied for 100% of primes, and the upper bound shows that | af(p) | is as large as possible (i.e., p is extremal for f) for 0% of primes. Our proofs use the effective form of the Sato–Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of f due to Newton and Thorne.

AB - Let f(z)=∑n=1∞af(n)e2πinz be a non-CM holomorphic cuspidal newform of trivial nebentypus and even integral weight k≥ 2. Deligne’s proof of the Weil conjectures shows that |af(p)|≤2pk-12 for all primes p. We prove for 100% of primes p that 2pk-12loglogp/logp<|af(p)|<⌊2pk-12⌋. Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin–Serre conjecture is satisfied for 100% of primes, and the upper bound shows that | af(p) | is as large as possible (i.e., p is extremal for f) for 0% of primes. Our proofs use the effective form of the Sato–Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of f due to Newton and Thorne.

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U2 - 10.1007/s40993-021-00258-w

DO - 10.1007/s40993-021-00258-w

M3 - Article

AN - SCOPUS:85104547442

VL - 7

JO - Research in Number Theory

JF - Research in Number Theory

SN - 2363-9555

IS - 2

M1 - 31

ER -