by the mordell-weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎there is no known algorithm for finding the rank of this group‎. ‎this paper computes the rank of the family $ e_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

For modular elliptic curves over number fields of narrow class one, and with multiplicative reduction at a collection p-adic primes, we define new invariants. Inspired by Nekovář Scholl's plectic conjectures, believe these invariants control the Mordell–Weil group higher rank support our expectations numerical experiments.

We give a counterexample to conjecture by Miasnikov, Ventura and Weil, stating that an extension of free groups is algebraic if only the corresponding morphism their core graphs onto, for every basis ambient group. In course proof we present partition set homomorphisms between which independent interest.

In this paper, a super-optimal pairing based on the Weil pairing is proposed with great efficiency. It is the first approach to reduce the Miller iteration loop when computing the variants of the Weil pairing. The super-optimal pairing based on the Weil pairing is computed rather fast, while it is slightly slower than the previous fastest pairing on the corresponding elliptic curves.

We propose an improved implementation of modified Weil pairings. By reduction of operations in the extension field to those in the base field, we can save some operations in the extension field when computing a modified Weil pairing. In particular, computing e`(P, φ(P )) is the same as computing the Tate pairing without the final powering. So we can save about 50% of time for computing e`(P, φ(...

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the SteinWatkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.

In the previous article (Found Phys. Lett. 16 325-341), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a finite group.

We describe in terms of the j-invariant all elliptic surfaces π : X → C with a section, such that h(X) = rankNS(X) and the Mordell-Weil group of π is finite. We use this to give a complete solution to infinitesimal Torelli for elliptic surfaces over P with a section.