Abstract
For an elliptic curve \(E/{\mathbb {Q}}\), let \(a_p\) denote the trace of its Frobenius endomorphism over \({\mathbb {F}}_p\), where p is a prime of good reduction for E. Hasse’s theorem asserts that \(|a_p| \le 2\sqrt{p}\). In this paper we establish average asymptotics for primes p for which \(a_p \in \left( 2\sqrt{p} - f(p), 2\sqrt{p}\right) \) or \(a_p \in \left( c\sqrt{p} - f(p), c\sqrt{p}\right) \), where \(f(x) = o(\sqrt{x})\) is a function satisfying mild growth conditions and \(0< c < 2\) is a constant.
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Acknowledgements
The authors are grateful to Joe Silverman for helpful discussion and to Clemson University for hosting the REU at which this work begun. We thank the anonymous referee for helpful suggestions and comments.
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This work was supported by an NSF Research Training Group (RTG) Grant (DMS # 1547399) promoting Coding Theory, Cryptography, and Number Theory at Clemson University and by Clemson University.
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Agwu, A., Harris, P., James, K. et al. Frobenius distributions of elliptic curves in short intervals on average. Ramanujan J 58, 75–120 (2022). https://doi.org/10.1007/s11139-021-00449-0
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DOI: https://doi.org/10.1007/s11139-021-00449-0
Keywords
- Elliptic curve
- Trace of Frobenius
- Frobenius distributions
- Sato–Tate conjecture
- Lang– Trotter conjecture
- Elliptic champion prime
- Elliptic trailing prime