三次数域上的天椭圆曲线

中文摘要 AI 辅助

天阿贝尔簇的研究源于Ihara的一个问题。当椭圆曲线$E/K$在$\ell$处为天时，扩张$K(E[\ell^\infty])/K(\mu_\ell^\infty)$是pro-$\ell$的，并且在$\ell$之外不分歧。这些算术条件与射影直线在$K$上除去三个点后的pro-$\ell$平展覆盖所对应的核的固定域的条件相同。本文研究定义在三次数域上的天椭圆曲线。继McLeman和Rasmussen在二次情形下的工作之后，我们发现三次情形下平衡椭圆曲线的迹可能出现更广泛的行为。在此背景下，我们进一步区分平衡曲线和完全平衡曲线。利用这一区分，我们证明了完全平衡椭圆曲线的$\ell$-挠表示之迹是非满射的。我们还计算了一个素数$\ell$的界，超过该界后，任何定义在三次数域上的天椭圆曲线必定是平衡的。最后，我们将完全平衡椭圆曲线的迹行为与CM椭圆曲线进行比较。

英文摘要

The study of heavenly abelian varieties is motivated by a question of Ihara. When an elliptic curve $E/K$ is heavenly at $\ell$, the extension $K(E[\ell^\infty])/K(μ_\ell^\infty)$ is pro-$\ell$ and unramified away from $\ell$. These are the same arithmetic conditions as the fixed field of the kernel attached to pro-$\ell$ étale covers of the projective line over $K$ minus three points. In this paper we study heavenly elliptic curves defined over cubic number fields. Following the work on McLeman and Rasmussen in the quadratic case, we find that there is a wider range of possible behaviors for the trace of a balanced elliptic curve in the cubic case. In this setting, we introduce a further distinction between balanced and totally balanced curves. With this, we show that trace of the representation on the $\ell$-torsion for totally balanced elliptic curves is non-surjective. We also compute a bound on primes $\ell$ after which any heavenly elliptic curve defined over a cubic number field must be balanced. Finally we compare the trace behavior of totally balanced elliptic curves with CM elliptic curves.

发表机构

• Wesleyan University（韦斯利安大学）

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