منابع مشابه
Explicit isogeny descent on elliptic curves
In this note, we consider an `-isogeny descent on a pair of elliptic curves over Q. We assume that ` > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finitedimensional F`-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the `-part of the Birch and Swinnerton-Dyer...
متن کاملExploiting Isogeny Cordillera Structure to Obtain Cryptographically Good Elliptic Curves
The security of most elliptic curve cryptosystems is based on the intractability of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Such a problem turns out to be computationally unfeasible when elliptic curves are suitably chosen. This paper provides an algorithm to obtain cryptographically good elliptic curves from a given one. The core of such a procedure lies on the usage of successi...
متن کاملOn Elliptic Curves with an Isogeny of Degree
We show that if E is an elliptic curve over Q with a Q-rational isogeny of degree 7, then the image of the 7-adic Galois representation attached to E is as large as allowed by the isogeny, except for the curves with complex multiplication by Q( √ −7). The analogous result with 7 replaced by a prime p > 7 was proved by the first author in [8]. The present case p = 7 has additional interesting co...
متن کاملIsogeny cordillera algorithm to obtain cryptographically good elliptic curves
The security of most elliptic curve cryptosystems is based on the intractability of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Such a problem turns out to be computationally unfeasible when elliptic curves are suitably chosen. This paper provides an algorithm to obtain cryptographically good elliptic curves from a given one. The core of such a procedure lies on the usage of successi...
متن کاملGalois properties of elliptic curves with an isogeny
giving the action of GQ on Tp(E), the p-adic Tate module for E and for a prime p. If E doesn’t have complex multiplication, then a famous theorem of Serre [Ser2] asserts that the image of ρE,p has finite index in AutZp ( Tp(E) ) for all p and that the index is 1 for all but finitely many p. This paper concerns some of the exceptional cases where the index is not 1. If E has a cyclic isogeny of ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2018
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2018.04.017