Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields

2007 ◽  
Vol 50 (3) ◽  
pp. 409-417 ◽  
Author(s):  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Complex Multiplication ◽  
Number Fields ◽  
Quadratic Number ◽  
Endomorphism Rings ◽  
Quadratic Number Field ◽  
Curves Over Finite Fields

AbstractWe show that, for most of the elliptic curves E over a prime finite field p of p elements, the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over p is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over p.

scholarly journals Constructing Isogenies between Elliptic Curves Over Finite Fields

1999 ◽  
Vol 2 ◽  
pp. 118-138 ◽  
Author(s):  
Steven D. Galbraith
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Endomorphism Ring ◽  
Elliptic Curve Cryptography ◽  
Class Number ◽  
Probabilistic Algorithm ◽  
Worst Case ◽  
Curves Over Finite Fields ◽  
Exponential Complexity

AbstractLet E1 and E2 be ordinary elliptic curves over a finite field Fp such that #E1(Fp) = #E2(Fp). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp. The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny.The algorithm proposed in this paper has exponential complexity in the worst case. Nevertheless, it is efficient in certain situations (that is, when the class number of the endomorphism ring is small). The significance of these results to elliptic curve cryptography is discussed.

scholarly journals No Singular Modulus Is a Unit

2018 ◽  
Vol 2020 (24) ◽  
pp. 10005-10041 ◽  
Author(s):  
Yuri Bilu ◽  
Philipp Habegger ◽  
Lars Kühne
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Complex Multiplication ◽  
Computer Assisted ◽  
Algebraic Unit ◽  
Special Points ◽  
Effective Proof ◽  
Rule Out

Abstract A result of the 2nd-named author states that there are only finitely many complex multiplication (CM)-elliptic curves over $\mathbb{C}$ whose $j$-invariant is an algebraic unit. His proof depends on Duke’s equidistribution theorem and is hence noneffective. In this article, we give a completely effective proof of this result. To be precise, we show that every singular modulus that is an algebraic unit is associated with a CM-elliptic curve whose endomorphism ring has discriminant less than $10^{15}$. Through further refinements and computer-assisted arguments, we eventually rule out all remaining cases, showing that no singular modulus is an algebraic unit. This allows us to exhibit classes of subvarieties in ${\mathbb{C}}^n$ not containing any special points.

scholarly journals Higher genus theory

2020 ◽  
Author(s):  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Extension ◽  
Number Fields ◽  
Explicit Description ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Genus Theory ◽  
Narrow Class

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

scholarly journals THE AVERAGE NUMBER OF SUBGROUPS OF ELLIPTIC CURVES OVER FINITE FIELDS

2020 ◽  
Vol 71 (3) ◽  
pp. 781-822
Author(s):  
Corentin Perret-Gentil
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Short Interval ◽  
Divisor Function ◽  
Finite Abelian Groups ◽  
Cyclic Subgroups ◽  
Number Of Divisors ◽  
Curves Over Finite Fields ◽  
Euler Products

Abstract By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most $2$. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić–Zhai and Blomer. With the same tools, an asymptotic for the average of the number of divisors of the number of rational points could also be given.

scholarly journals Geometric-progression-free sets over quadratic number fields

2017 ◽  
Vol 147 (2) ◽  
pp. 245-262
Author(s):  
Andrew Best ◽  
Karen Huan ◽  
Nathan McNew ◽  
Steven J. Miller ◽  
Jasmine Powell ◽  
...  
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Geometric Progression ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Quadratic Number Fields ◽  
Geometric Progressions ◽  
Upper Density ◽  
Free Sets

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

scholarly journals RAY CLASS GROUPS OF QUADRATIC AND CYCLOTOMIC FIELDS

2010 ◽  
Vol 06 (05) ◽  
pp. 1169-1182
Author(s):  
JING LONG HOELSCHER
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Number ◽  
Class Groups ◽  
Quadratic Number Field ◽  
Cyclotomic Number ◽  
Class Field ◽  
Ray Class Field ◽  
Real Quadratic ◽  
Mod P

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Let [Formula: see text] be a real quadratic number field with a prime P lying above p in ℚ. If p splits in K/ℚ and p does not divide the big class number of K, then any pro-p extension of K ramified only at P is finite cyclic. If p is inert in K/ℚ, then there exist infinite extensions of K ramified only at P. Furthermore, for big enough integer k, the ray class field (mod Pk+1) is obtained from the ray class field (mod Pk) by adjoining ζpk+1. In the case of a regular cyclotomic number field K = ℚ(ζp), the explicit structure of ray class groups (mod Pk) is given for any positive integer k, where P is the unique prime in K above p.

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