Extremal primes for elliptic curves without complex multiplication
Extremal primes for elliptic curves without complex multiplication
复制标题
无需复数乘法的椭圆曲线的极值素数
DOI：
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发表时间：
2018
影响因子：
1
通讯作者：
C. Turnage
中科院分区：
文献类型：
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作者：
Chantal David;A. Gafni;Amita Malik;Neha Prabhu;C. Turnage
修复 Q 上的椭圆曲线 E。E 的极值素数是良好归约的素数 p，使得 E 模 p 上的有理点数相对于哈塞界而言最大或最小。假设与 E 相关的所有对称幂 L 函数都是自守的并且满足广义黎曼假设，当 E 是没有复数乘法的曲线时，我们给出此类素数数量的第一个非平凡上限。为了获得这个界限，我们对 Sato-Tate 测度使用显式等分布，如 Rouse 和 Thorner 的工作 (arXiv:1305.5283)，并利用极值素数具有非常小的 Sato-Tate 测度这一事实来细化某些中间估计。
Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound. Assuming that all the symmetric power L-functions associated to E are automorphic and satisfy the Generalized Riemann Hypothesis, we give the first non-trivial upper bounds for the number of such primes when E is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner (arXiv:1305.5283) and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.