AI 中文总结
研究超奇异椭圆曲线与其共轭曲线间同构最小次数，通过新技术计算次数并给出界，此界渐近最优且在多情况下精确，还基于数据给出多个猜想。
AI 中文摘要
设\(E\)是定义在\(\bar{\mathbb{F}}_p\)上的超奇异椭圆曲线，\(E^{(p)}\)是其共轭曲线。我们给出了从\(E\)到\(E^{(p)}\)同构的最小次数关于\(p\)的一个界，并表明该界在渐近意义下是最优的且在许多情况下是精确的。此界通过开发一种新技术来计算从超奇异椭圆曲线到其共轭曲线的某些同构的次数得到，我们还给出了包含这些同构的格的相继极小值的大量计算。随后，我们根据所获得的数据给出了几个猜想，包括关于达到本文所给界的素数\(p\)的集合的一些猜想。
英文摘要
Let $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$ and $E^{(p)}$ be its conjugate. We give a bound on the minimal degree of an isogeny from $E$ to $E^{(p)}$ depending on $p$, and show that this bound is both asymptotically optimal as well as sharp in many cases. This bound is obtained by developing a new technique to compute the degree of certain isogenies from a supersingular elliptic curve to its conjugate, and we present extensive computations of the successive minima of the lattice containing these isogenies. Following this, we give several conjectures supported by the data we have obtained, including some on the set of primes $p$ for which the bound we give in this article is attained.
Commentsaccompanying code available at https://github.com/christellevincent/WISDE
