In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over Q. Several examples are included.

Given fields k ? L , our results concern one parameter L-parametric polynomials over and their relation to generic polynomials. The former are P ( T Y ) ? [ ] of group G which parametrize all Galois extensions via specialization in the latter those -parametric for every field ? . We show, example, that being with taken be single C V U is fact sufficient a polynomial generic. As corollary, we ob...

Based upon new global class field concepts leading to two-dimensional global Langlands correspondences, a modular representation of cusp forms is proposed in terms of global elliptic bisemimodules which are (truncated) Fourier series over R . As application, the conjectures of Shimura-Taniyama-Weil, Birch-Swinnerton-Dyer and Riemann are analyzed.

Given an L-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve L-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This c...

In this article, we study the Chow group of motive associated to a tempered global $L$-packet $\pi$ unitary groups even rank with respect CM extension, whose root number is $-1$. We show that, under some restrictions on ramification if central derivative $L'(1/2,\pi)$ nonvanishing, then $\pi$-nearly isotypic localization certain Shimura variety over its reflex field does not vanish. This proves...

The Weil conjectures constitute one of the central landmarks of 20th century algebraic geometry: not only was their proof a dramatic triumph, but they served as a driving force behind a striking number of fundamental advances in the field. The conjectures treat a very elementary problem: how to count the number of solutions to systems of polynomial equations over finite fields. While one might ...