scholarly journals On the Iwasawa invariants of elliptic curves

2000 ◽  
Vol 142 (1) ◽  
pp. 17-63 ◽  
Author(s):  
Ralph Greenberg ◽  
Vinayak Vatsal
Keyword(s):  
Elliptic Curves ◽  
Iwasawa Invariants

ON THE 2-ADIC IWASAWA INVARIANTS OF ORDINARY ELLIPTIC CURVES

2008 ◽  
Vol 04 (03) ◽  
pp. 403-422
Author(s):  
KAZUO MATSUNO
Keyword(s):  
Elliptic Curves ◽  
Explicit Formula ◽  
Selmer Groups ◽  
Quadratic Twists ◽  
Iwasawa Invariants

In this paper, we give an explicit formula describing the variation of the 2-adic Iwasawa λ-invariants attached to the Selmer groups of elliptic curves under quadratic twists. To prove this formula, we extend some results known for odd primes p, an analogue of Kida's formula proved by Hachimori and the author and a formula given by Greenberg and Vatsal, to the case where p = 2.

scholarly journals On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms Over Cyclotomic Extensions

2017 ◽  
Vol 69 (4) ◽  
pp. 826-850 ◽  
Author(s):  
Antonio Lei ◽  
David Loeffler ◽  
Zerbes Sarah Livia
Keyword(s):  
Asymptotic Behaviour ◽  
Modular Form ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Upper Bounds ◽  
Selmer Groups ◽  
Asymptotic Growth ◽  
Supersingular Elliptic Curves ◽  
Cyclotomic Extensions ◽  
Iwasawa Invariants

AbstractWe study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic ℤp-extension of ℚ under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

Elliptic Curves

1997 ◽  
Author(s):  
Henry McKean ◽  
Victor Moll
Keyword(s):  
Elliptic Curves

The Joint Universality for L-Functions of Elliptic Curves

2004 ◽  
Vol 9 (4) ◽  
pp. 331-348
Author(s):  
V. Garbaliauskienė
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

2019 ◽  
Author(s):  
Anna ILYENKO ◽  
Sergii ILYENKO ◽  
Yana MASUR
Keyword(s):  
Information Systems ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Computational Efficiency ◽  
Digital Signature ◽  
Discrete Logarithm ◽  
Algebraic Groups ◽  
Schnorr Signature ◽  
Stability Of Algorithms

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

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