Juan Marcos Cerviño
We present a deterministic and explicit algorithm to compute the endomorphism rings of supersingular elliptic curves. As an example we compute the endomorphism rings of all supersingular elliptic curves defined over characteristic p=29,...,97.
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On the supersingular locus of Shimura varieties for quaternionic unitary groups
本文研究全实域上全不定四元数代数中斜埃尔米特形式所对应的酉相似群志村簇，给出了斜埃尔米特自对偶格存在的充要条件，证明了超特殊轨迹非空，并在p为奇数且非分歧时给出超奇异轨迹不可约分量数的显式公式。
We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree $2$ over a ramified odd prime $p$ if the level at $p$ is given by a special maximal compact open subgroup. More precisely, we show that such a locus has purely $2$-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport--Zink space for the q…
Let p>3 be a prime and let E, E' be supersingular elliptic curves over F_p. We want to construct an isogeny phi: E --> E'. The currently fastest algorithm for finding isogenies between supersingular elliptic curves solves this problem by performing a "meet-in-the-middle" breadth-first search in the full supersingular 2-isogeny graph over F_{p^2}. In this paper we consider the structure of the isogeny graph of supersingular elliptic curves over F_p. We give an algorithm to construct isogenies between such supersingular elliptic curves that works faster than the usual algorithm. We then discuss …
Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs
面向基于同源的密码学，本文将层结构引入超奇异椭圆曲线，建立其与四元数代数中 Eichler 序的对应，并研究相关同源图。
In this paper, we prove the Iwasawa main conjecture for elliptic curves at an odd supersingular prime p. Some consequences are the p-parts of the leading term formulas in the Birch and Swinnerton-Dyer conjectures for analytic rank 0 or 1.
On the anticyclotomic Mazur-Tate conjecture for elliptic curves with supersingular reduction
该短注针对具有超奇异约化的椭圆曲线，研究其Selmer群Fitting理想相关的Mazur-Tate“弱”主猜想的反分圆类似问题。
We describe the neighborhood of the vertex $[E_0]$ (resp. $[E_{1728}]$) in the $\ell$-isogeny graph $\mathcal{G}_\ell(\mathbb{F}_{p^2}, -2p)$ of supersingular elliptic curves over the finite field $\mathbb{F}_{p^2}$ when $p>3\ell^2$ (resp. $p>4\ell^2$) with $E_0: y^2=x^3+1$ (resp. $E_{1728}: y^2=x^3+x$) supersingular.
Supersingular elliptic curves and twisting endomorphisms
将扭曲自同态概念推广到O-定向超奇异椭圆曲线，提出算法寻找具有素数度ℓ扭曲自同态的曲线并计算其完整自同态环基。
Computation of Hilbert class polynomials and modular polynomials from supersingular elliptic curves
该研究针对素数模下类多项式与模多项式的计算问题，借助同源密码学推动下发展的超奇异曲线Deuring对应研究成果，提出多种基于超奇异椭圆曲线的新启发式算法，打破了此前超奇异情形效率低的固有认知。
We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over $\mathbb{Q}(\sqrt{2})$.
For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over $\overline{\mathbb{F}_p}$. We show that these supersingular zeros are in bijection with zeros modulo $p$ of an associated quaternionic modular form $v_E$. This allows us to prove that if the root number of $E$ is $-1$ then all supersingular $j$-invariants of elliptic curves defined over $\mathbb{F}_{p}$ are zeros of the corresponding divisor polynomial. If the roo…
This is a translation of a research announcement by Anas G. Nasybullin from 1976, in which he states formulas for the p-primary part of the Tate-Shafarevich group of an elliptic curve in cyclotomic $\Z_p$-extensions of number fields.
Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in F_p^2, determines the supersingularity of E in O(n^3 log^2 n) time and O(n) space, where n=O(log p). Both these complexity bounds are significant improvements over existing methods, as we demonstrat…
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.
Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D<0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $\order_{D}$ to supersingular elliptic curves in characteristic $p$. In the algorithm we first determine an explicit constant $D_p$ so that $|D|> D_p$ implies that the map is necessarily surjective and then we compute explicitly the cases $|D|<D_p$.
In 1987, B. Perrin-Riou formulated a Heegner point main conjecture for elliptic curves at primes of ordinary reduction. In this paper, we formulate an analogue of Perrin-Riou's main conjecture for supersingular primes. We then prove this conjecture under mild hypotheses, and deduce from this result a $Λ$-adic extension of Kobayashi's $p$-adic Gross-Zagier formula, new cases of B.-D. Kim's doubly-signed main conjectures, and a strengthened version of Skinner's converse to the Gross-Zagier-Kolyvagin theorem for supersingular primes.
At a prime of ordinary reduction, the Iwasawa ``main conjecture'' for elliptic curves relates a Selmer group to a $p$-adic $L$-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the $p$-adic $L$-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is similar in structure to the ordinary case. Namely, Kobayashi's conjecture relates modified Selmer groups, which he defined, with modified $p$-adic $L$-functions defined by the first author. In …
In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Z_p-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi and Perrin-Riou, we define restricted Selmer groups and λ^\pm, μ^\pm-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms of these invariants. To be able to work with non-cyclotomic Z_p-extensions, a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime …
For a fixed prime p, we consider the (finite) set of supersingular elliptic curves over $\bar{\mathbb{F}}$. Hecke operators act on this set. We compute the asymptotic frequence with which a given supersingular elliptic curve visits another under this action.
A recent paper of Shekhar compares the ranks of elliptic curves $E_1$ and $E_2$ for which there is an isomorphism $E_1[p] \simeq E_2[p]$ as $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$-modules, where $p$ is a prime of good ordinary reduction for both curves. In this paper we prove an analogous result in the case where $p$ is a prime of good supersingular reduction.
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.
Asymptotic formula for Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic extensions
本文在广义Heegner假设下，研究了p-超奇异椭圆曲线在反圆塔Z_p-扩张中数域上的Tate-Shafarevich群的p-准素子群，并给出了其渐近公式。
The Classification of Supersingular Elliptic Curves in Characteristic 3
本文在特征3上对奇异椭圆曲线进行显式分类，并提供点计数公式，以支持Hecke.jl中的实现。
For small odd primes $p$, we prove that most of the rational points on the modular curve $X_0(p)/w_p$ parametrize pairs of elliptic curves having infinitely many supersingular primes. This result extends the class of elliptic curves for which the infinitude of supersingular primes is known. We give concrete examples illustrating how these techniques can be explicitly used to construct supersingular primes for such elliptic curves. Finally, we discuss generalizations to points defined over larger number fields and indicate the types of obstructions that arise for higher level modular curves.
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…
Constructing Cycles in Isogeny Graphs of Supersingular Elliptic Curves
本文利用可嵌入自同态环的虚二次序及素数分裂条件，在超奇异同源图中构造自环和环，分析其长度并给出意外2-环对应的素数上界。
In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the "weak main conjecture" à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the "strong main conjecture" suggested by the second named author under the validity of the $\pm$-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among "finite layer objects", "$\pm$-objects",…
On the cohomology of plus/minus Selmer groups of supersingular elliptic curves in weakly ramified base fields
本文推广了Meng Fai Lim的结果，证明了在弱分歧基域的分圆$\mathbb{Z}_p$-扩张上，超奇异椭圆曲线带符号Selmer群特征元素的Kida公式及整性结果。
Let $p$ be an odd prime number, $E$ an elliptic curve defined over a number field. Suppose that $E$ has good reduction at any prime lying above $p$, and has supersingular reduction at some prime lying above $p$. In this paper, we construct the plus and the minus Selmer groups of $E$ over the cyclotomic $\mathbb Z_p$-extension in a more general setting than that of B.D. Kim, and give a generalization of a result of B.D. Kim on the triviality of finite $Λ$-submodules of the Pontryagin duals of the plus and the minus Selmer groups, where $Λ$ is the Iwasawa algebra of the Galois group of the $\mat…
The Endomorphism Rings of Supersingular Elliptic Curves over $\mathbb{F}_p$ and the Binary Quadratic Forms
本文建立了 $\mathbb{F}_p$ 上超奇异椭圆曲线同构类与特定判别式的本原既约二元二次型间的一一对应，证明同源运算与二次型复合相容，并借此将 CSIDH 密码系统的安全性归约到显式计算该对应关系上。
An Asymptotic formula for Tate-Shafarevich groups of CM elliptic curves at supersingular primes
本文在Kobayashi符号Selmer群的附加假设下，对具有复乘的椭圆曲线在反圆$\mathbb{Z}_p$-扩张中间层上的Tate-Shafarevich群证明了渐近公式。
Iwasawa theory of CM elliptic curves at potentially supersingular primes
本文为潜在超奇异素数处的CM椭圆曲线建立正负Iwasawa理论，构造局部点以研究Selmer群，构造p-adic L-函数，证明主猜想，并推导Tate-Shafarevich群的渐近公式。
Minimal degree of an isogeny between a supersingular elliptic curve and its conjugate
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。
Cycles of supersingular elliptic curves for pairing-based proof systems
该研究针对无界递归基于配对的证明系统，构造了超奇异椭圆曲线循环，其可通过Bröker算法高效生成无限个循环，还可与小有限域上的配对友好型曲线连接。
We generalize a theorem of Ogg on supersingular $j$-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke …
We extend Kobayashi's formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case $a_p \neq 0$, where $a_p$ is the trace of Frobenius. To do this, we algebraically construct $p$-adic $L$-functions $L_p^{\sharp}$ and $L_p^{\flat}$ with the good growth properties of the classical Pollack $p$-adic $L$-functions that in fact match them exactly when $a_p=0$ and $p$ is odd. We then generalize Kobayashi's methods to define two Selmer groups $\Sel^{\sharp}$ and $\Sel^{\flat}$ and formulate a main conjecture, stating that each characteristic ideal of the duals of these…
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.
Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders
本文研究了寻找超奇异椭圆曲线定向的困难度，证明判断曲线可定向性的预言机能为计算对应自同态提供非平凡信息，并提供了显式算法、复杂度分析及四元数代数中的嵌入算法与高效代码实现。
We study the arithmetic properties of Weierstrass points on the modular curves $X_0^+(p)$ for primes $p$. In particular, we obtain a relationship between the Weierstrass points on $X_0^+(p)$ and the $j$-invariants of supersingular elliptic curves in characteristic $p$.
Gross lattices of supersingular elliptic curves
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。
Let $p>3$ be a fixed prime. For a supersingular elliptic curve $E$ over $\mathbb{F}_p$ with $j$-invariant $j(E)\in \mathbb{F}_p\backslash\{0, 1728\}$, it is well known that the Frobenius map $π=((x,y)\mapsto (x^p, y^p))\in \mathrm{End}(E)$ satisfies $π^2=-p$. A result of Ibukiyama tells us that $\mathrm{End}(E)$ is a maximal order in $\mathrm{End}(E)\otimes \mathbb{Q}$ associated to a (minimal) prime $q$ satisfying $q\equiv 3 \bmod 8$ and the quadratic residue $\bigl(\frac{p}{q}\bigr)=-1$ according to $\frac{1+π}{2}\notin \mathrm{End}(E)$ or $\frac{1+π}{2}\in \mathrm{End}(E)$. Let $q_j$ denote…
The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes
该研究在带符号Selmer群为余挠模的假设下，证明了$p$-超奇异椭圆曲线在虚二次域反循环$\u2124_p$-扩张子扩张上的Mordell-Weil秩有界，并给出其Tate-Shafarevich群$p$-部分增长的渐近公式。
Chevyrev and Galbraith recently devised an algorithm which inputs a maximal order of the quaternion algebra ramified at one prime and infinity and constructs a supersingular elliptic curve whose endomorphism ring is precisely this maximal order. They proved that their algorithm is correct whenever it halts, but did not show that it always terminates. They did however prove that the algorithm halts under a reasonable assumption which they conjectured to be true. It is the purpose of this paper to verify their conjecture and in turn prove that their algorithm always halts. More precisely, Chevyr…
Supersingular elliptic curves, quaternion algebras and applications to cryptography
本文综述了超奇异同源图及其密码学应用，介绍三个近期结果，并将讨论推广到超特殊同源图，同时证明了一个关于全实域上格生成的一般定理。
Let E/Q be an elliptic curve with good supersingular reduction at p with a_p(E)=0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the Zp-cyclotomic extension of a finite Galois extension of Q where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.
Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture
本文将 Kobayashi 的 ±-岩泽主猜想归约并证明为岩泽–Greenberg 主猜想，结合广义局部理论与 Beilinson–Flach 显式互反律，推出超奇异椭圆曲线结果及 BSD 公式 p-部分，并给出可证明完整 BSD 的无 CM 椭圆曲线无限族。
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.
Introduction to Dieudonné modules and supersingular abelian varieties revisited
本文介绍了Dieudonné模块和奇超阿贝尔簇，给出了奇超椭圆曲线乘积唯一性和Oort定理的简单证明。
A bound on the $μ$-invariants of supersingular elliptic curves
本文研究了超奇异椭圆曲线的 Iwasawa $μ$-不变量，证明了在固定 $λ$-不变量的条件下 $μ_p^\pm\leq 1$ 对几乎所有素数成立，从而为 Perrin-Riou 和 Pollack 的猜想提供了支持。
Let O be a maximal order in the quaternion algebra B_p over Q ramified at p and infinity. The paper is about the computational problem: Construct a supersingular elliptic curve E over F_p such that End(E) = O. We present an algorithm that solves this problem by taking gcds of the reductions modulo p of Hilbert class polynomials. New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice O^T of O, the order O is effectively characterized by the three successive minima and two other short vectors…
The SEA algorithm for endomorphisms of supersingular elliptic curves
针对超奇异椭圆曲线的自同态，提出一种推广的SEA算法，以O(n^4(log n)^2 + dLn^3)比特运算计算其迹，复杂度与SEA算法启发式复杂度匹配，且无需假设。
A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's $2$-variable $p$-adic $L$-functions) and algebraic objects (two "everywhere unramified'' Iwasawa modules) involving codimension two cycles in a $2$-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$, defined over $\…
Residual Supersingular Iwasawa Theory over quadratic imaginary fields
本文研究二次虚域中 p 完全分裂且椭圆曲线在 p 上超奇异时的 Z_p^2-扩张，定义精细双符号剩余 Selmer 群，并证明剩余同构曲线间带符号 μ-不变量零化的等价性及相关对偶无伪零子模。
An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines…
We study elliptic functions in quaternionic analysis, and prove some analogues of classical theorems from the complex case. The main result is a relation between the periods of closed differential 1-forms and 3-forms on H/L where L is a lattice in H. By applying to forms that are regular in the sense of quaternionic analysis, we obtain quaternionic analogues of Riemann's period relations for an abelian surface. We also obtain a quaternionic analogue of Legendre's relation, as an equality between $2π^2$ and a linear combination of the four quasi-periods of the quaternionic Weierstrass $ζ$-funct…
A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes
本文针对超奇异素数处带复乘的椭圆曲线，基于Burungale--Kobayashi--Ota的局部点结构研究突破，建立了反分圆扩张下余挠带符号Selmer群与精细Selmer群的特征理想关联，补充了Agboola--Howard的相关结果。
On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves
该研究针对孙智伟提出的截断雅可比符号行列式，通过统一方法结合超奇异椭圆曲线，证明了孙的多个相关猜想及强化形式，揭示了这类恒等式的通用机制。
Given an elliptic curve with supersingular reduction at an odd prime p, Iovita and Pollack have generalised results of Kobayashi to define even and odd Coleman maps at p over Lubin-Tate extensions given by a formal group of height 1. We generalise this construction to modular forms of higher weights.
If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$ and $\ell \geq 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.
Let $E$ be an elliptic curve over $\mathbb Q$ and let $p\geq5$ be a prime of good supersingular reduction for $E$. Let $K$ be an imaginary quadratic field satisfying a modified "Heegner hypothesis" in which $p$ splits, write $K_\infty$ for the anticyclotomic $\mathbb Z_p$-extension of $K$ and let $Λ$ denote the Iwasawa algebra of $K_\infty/K$. By extending to the supersingular case the $Λ$-adic Kolyvagin method originally developed by Bertolini in the ordinary setting, we prove that Kobayashi's plus/minus $p$-primary Selmer groups of $E$ over $K_\infty$ have corank $1$ over $Λ$. As an applicat…
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