Eötvös Loránd University（罗兰大学）；University of Arizona（亚利桑那大学）；University of Auckland（奥克兰大学）；University of Alberta – Augustana Campus（阿尔伯塔大学奥古斯塔纳学院）；University of French Polynesia（法属波利尼西亚大学）；University of Vermont（佛蒙特大学）
Chenfeng He, Gaurish Korpal, Ha T. N. Tran, Christelle Vincent
本文研究了超奇异椭圆曲线的格的极小值，探讨了第三极小值D_3与曲线j不变量在有限域上的关系，以及当p≡3 mod 4时的更细致的环结构信息。
摘要：
设p为素数，E为定义在ar{\mathbb{F}}_p上的超奇异椭圆曲线，\mathscr{O}为其（几何）自同构环。Chevyrev-Galbraith和Goren-Love之前的结果表明，Gross格的连续极小值刻画了\mathscr{O}的同构类。在本文中，我们扩展了这项工作，证明Gross格的第三连续极小值D_3的值给出了曲线j不变量在域\mathbb{F}_p或集合\mathbb{F}_{p^2} \setminus \mathbb{F}_p中的必要且充分条件，以及当j不变量属于\mathbb{F}_p且p≡3 mod 4时关于E的自同构环的更详细信息。我们以对超奇异椭圆曲线Gross格的几何进行调查作为本文的结尾。
英文摘要：
Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$, and $\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima of the Gross lattice of $\mathscr{O}$ characterize the isomorphism class of $\mathscr{O}$. In this paper, we extend this work and show that the value of the third successive minimum $D_3$ of the Gross lattice gives necessary and sufficient conditions for the curve to have its $j$-invariant in the field $\mathbb{F}_p$ or in the set $\mathbb{F}_{p^2} \setminus \mathbb{F}_p$, as well as finer information about the endomorphism ring of $E$ when its $j$-invariant belongs to $\mathbb{F}_p$ and $p \equiv 3 \pmod{4}$. We end our article with an investigation of the geometry of Gross lattices of supersingular elliptic curves.
Bleher et al. began studying higher codimension Iwasawa theory for classical Iwasawa modules. Subsequently, Lei and Palvannan studied an analogue for elliptic curves with supersingular reduction. In this paper, we obtain a vast generalization of the work of Lei and Palvannan. A key technique is an approach to the work of Bleher et al. that the author previously proposed. For this purpose, we also study the structure of $\pm$-norm subgroups and duality properties of multiply-signed Selmer groups.
We revisit theoretical background on OSIDH, that is an isogeny-based key-exchange protocol proposed by Colò and Kohel at NutMiC 2019. We give a proof of a fundamental theorem for OSIDH. The theorem was stated by Colò and Kohel without proof. Furthermore, we consider parameters of OSIDH, give a sufficient condition on the parameters that the protocol works, and estimate the size of the parameters for a certain security level.
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve $E$ that runs, under certain heuristics, in time $O((\log p)^2p^{1/2})$. The algorithm works by first finding two cycles of a certain form in the supersingular $\ell$-isogeny graph $G(p,\ell)$, generating an order $Λ\subseteq \operatorname{End}(E)$. Then all maxi…
Minimal degree of an isogeny between a supersingular elliptic curve and its conjugate
Université de Toulon（土伦大学）；Aix Marseille Univ, UMR 7373 CNRS（艾克斯-马赛大学，法国国家科学研究中心联合研究单位7373）；University of French Polynesia（法属波利尼西亚大学）；University of Vermont（佛蒙特大学）
In this article, we provide a relation between the $μ$-invariants of the dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from the cyclotomic $\mathbb{Z}_p$-extension to a $\mathbb{Z}_p^2$-extension over an imaginary quadratic field. Furthermore we show that the supersingular $\mathfrak{M}_H(G)$-conjecture is equivalent to the fact that the $μ$-invariants doesn't change as we go up the tower.
We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show -- under additional congruence assumptions on the CM order -- that the reductions are surjective (and even become equidistributed) on the product of supersingular loci when the discriminant of the order becomes large. This variant of the equidistribution theorems of Duke and Cornut-Vatsal is an(other) application of the recent work of Einsiedler and Lindenstrauss on the classification of joinings of higher-rank diagonalizable actions.
