Supersingular elliptic curve
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Analogues of Vélu’s formulas for isogenies on alternate models of elliptic curves Open
Isogenies are the morphisms between elliptic curves and are, accordingly, a topic of interest in the subject. As such, they have been well studied, and have been used in several cryptographic applications. Vélu's formulas show how to expli…
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Elliptic stable envelopes Open
We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of Maulik and Okounkov [Astérisque 408 (2019), ix+209]. We apply them to the…
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Computing endomorphism rings of supersingular elliptic curves and connections to path-finding in isogeny graphs Open
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.We give a n…
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3d Mirror Symmetry and Elliptic Stable Envelopes Open
We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We give formulas for the elliptic stable envelopes on both side…
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Arithmetic Considerations for Isogeny-Based Cryptography Open
In this paper we investigate various arithmetic techniques which can be used to potentially enhance the performance in the supersingular isogeny Diffie-Hellman (SIDH) key-exchange protocol which is one of the more recent contenders in the …
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Arithmetic properties of signed Selmer groups at non-ordinary primes Open
We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over -cyclotomic extensions. First, …
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Bounds for Serre’s open image theorem for elliptic curves over number fields Open
For an elliptic curve [math] without potential complex multiplication we bound the index of the image of Gal [math] in GL [math] , the representation being given by the action on the Tate modules of [math] at the various primes. The bound …
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Efficient Isogeny Computations on Twisted Edwards Curves Open
The isogeny-based cryptosystem is the most recent category in the field of postquantum cryptography. However, it is widely studied due to short key sizes and compatibility with the current elliptic curve primitives. The main building block…
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A criterion to rule out torsion groups for elliptic curves over number fields Open
We present a criterion for proving that certain groups of the form $\mathbb {Z}/m\mathbb {Z}\oplus \mathbb {Z}/n\mathbb {Z}$ do not occur as the torsion subgroup of any elliptic curve over suitable (families of) number fields. We apply thi…
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Tropical mirror symmetry for elliptic curves Open
Mirror symmetry relates Gromov–Witten invariants of an elliptic curve with certain integrals over Feynman graphs [10]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gr…
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Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks Open
Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$ -Selmer groups, have given new upper bounds on …
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Composite images of Galois for elliptic curves over $\mathbf {Q}$ and entanglement fields Open
Let $E$ be an elliptic curve defined over $\mathbf {Q}$ without complex multiplication. For each prime $\ell$, there is a representation $\rho _{E,\ell }\colon \operatorname {Gal}(\overline {\mathbf {Q}}/\mathbf {Q}) \rightarrow \operatorn…
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Computing Supersingular Isogenies on Kummer Surfaces Open
We apply Scholten's construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over $$\mathbb {F}_{p^2}$$ and corresponding abelian surfaces over $$\mathbb {F}_{p}$$…
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Elliptic Springer theory Open
We introduce an elliptic version of the Grothendieck–Springer sheaf and establish elliptic analogues of the basic results of Springer theory. From a geometric perspective, our constructions specialize geometric Eisenstein series to the res…
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The Order of Edwards and Montgomery Curves Open
The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA) [2]. It is well known that the problem of discrete logarithm is NP-hard on group on elliptic curve (EC) [5]. The…
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A classification of isogeny‐torsion graphs of Q‐isogeny classes of elliptic curves Open
Let $\\mathcal{E}$ be a $\\mathbb{Q}$-isogeny class of elliptic curves defined\nover $\\mathbb{Q}$. The isogeny graph associated to $\\mathcal{E}$ is a graph\nwhich has a vertex for each elliptic curve in the $\\mathbb{Q}$-isogeny class\n$…
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CODIMENSION TWO CYCLES IN IWASAWA THEORY AND ELLIPTIC CURVES WITH SUPERSINGULAR REDUCTION Open
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 analyt…
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Iwasawa theory for Rankin-Selberg products ofp-nonordinary eigenforms Open
Let f and g be two modular forms which are non-ordinary at p. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution f⊗g, one for each choice of p-stabilisations of …
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Integral Tate modules and splitting of primes in torsion fields of elliptic curves Open
Let [Formula: see text] be an elliptic curve over a finite field [Formula: see text], and [Formula: see text] a prime number different from the characteristic of [Formula: see text]. In this paper, we consider the problem of finding the st…
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The Iwasawa Main Conjecture for elliptic curves at odd supersingular primes Open
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.
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Torsion points on elliptic curves over number fields of small degree. Open
Let d be an integer and let K be a number field of degree d over Q. By the Mordell- Weil theorem we know that if E is an elliptic curve over K then there exist unique integers m, n > 0 and r \geq 0 such that E(K)_{tors} = Z/mZ x Z/mnZ x Zr…
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Deuring for the people: Supersingular elliptic curves with prescribed endomorphism ring in general characteristic Open
Constructing a supersingular elliptic curve whose endomorphism ring is isomorphic to a given quaternion maximal order (one direction of the Deuring correspondence) is known to be polynomial-time assuming the generalized Riemann hypothesis …
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A heuristic for boundedness of ranks of elliptic curves Open
We present a heuristic that suggests that ranks of elliptic curves E over \mathbb Q are bounded. In fact, it suggests that there are only finitely many E of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich…
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On the torsion of rational elliptic curves over quartic fields Open
Let E be an elliptic curve defined over Q and let G = E(Q)_tors be the\nassociated torsion subgroup. We study, for a given G, which possible groups G\n<= H could appear such that H=E(K)_tors, for [K:Q]=4 and H is one of the\npossible torsi…
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Automorphisms of Elliptic K3 surfaces and Salem numbers of maximal degree Open
Using elliptic structures, we show that any supersingular K3 surface of Artin invariant 1 in characteristic p = 5, 7, 13 has an automorphism the entropy of which is the natural logarithm of a Salem number of degree 22.
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Supersingular Curves With Small Non-integer Endomorphisms Open
We introduce a special class of supersingular curves over $\mathbb{F}_{p^2}$, characterized by the existence of non-integer endomorphisms of small degree. A number of properties of this set is proved. Most notably, we show that when this s…
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Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves Open
We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_ε(|{\rm Disc}(K)|^{1/2+ε})$ by Brauer--Siegel). This yields cor…
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Elliptic and K-theoretic stable envelopes and Newton polytopes Open
In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes co…
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Elliptic Hypergeometric Functions Associated with Root Systems Open
We give a survey of elliptic hypergeometric functions associated with root systems, comprised of three main parts. The first two in essence form an annotated table of the main evaluation and transformation formulas for elliptic hypergeomet…
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Attacking the linear congruential generator on elliptic curves via lattice techniques Open
In this paper we study the linear congruential generator on elliptic curves from the cryptographic point of view. We show that if sufficiently many of the most significant bits of the composer and of three consecutive values of the sequenc…