Experimental Mathematics ( IF 0.5 ) Pub Date : 2019-12-02 , DOI: 10.1080/10586458.2019.1655816 Harris B. Daniels 1 , Enrique González-Jiménez 2
Abstract
Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre’s constant associated to E, that gives necessary conditions to conclude that the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying then is non-surjective. Conditionally under Serre’s Uniformity Conjecture, we determine all the Serre’s constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre’s constants.
中文翻译:
有理数上的 Serre 椭圆曲线常数
摘要
令E是一条椭圆曲线,没有在有理数上定义复数乘法。本文的目的是定义一个正整数A ( E ),我们称其为与 E 相关的 Serre 常数,它给出了得出以下结论的必要条件与E相关的 mod m 伽罗瓦表示是非满射的。特别是,如果存在一个满足m的素因子p然后是非满射的。有条件地在塞尔均匀性猜想下,我们确定了椭圆曲线的所有塞尔常数,而不需要对无限频繁出现的有理数进行复乘。此外,我们给出了 mod p Galois 表示的所有可能组合,这些表示出现在无限多个非同构类的非 CM 椭圆曲线上以及仅出现有限的已知案例。对于最后,我们推测这些组合的所有可能性,特别是这些 Serre 常数的所有可能性。