Assuming finiteness of the Tate–Shafarevich group, we prove that Birch–Swinnerton–Dyer conjecture correctly predicts parity rank semistable principally polarised abelian surfaces. If surface in question is Jacobian a curve, require curve has good ordinary reduction at 2-adic places.

Let p and q be two distinct primes and let Je denote the winding quotient at level pq. We give an explicit formula that expresses the special L-value of Je as a rational number, and interpret it in terms of the Birch-Swinnerton-Dyer conjecture.

We make several conjectures concerning the relations between the orders of the torsion subgroup, the arithmetic component groups, and the cuspidal subgroup of an optimal elliptic curve. These conjectures have implications for the second part of the Birch and Swinnerton-Dyer conjecture.

We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x + y = z in Q( √ d), conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.

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.

We survey the history of the Tate conjecture on algebraic cycles. The conjecture is closely related with other big problems in arithmetic and algebraic geometry, including the Hodge and Birch–Swinnerton-Dyer conjectures. We conclude by discussing the recent proof of the Tate conjecture for K3 surfaces over finite fields.

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.

We discuss a famous problem about right triangles with rational side lengths. This elementarysounding problem is still not completely solved; the last remaining step involves the Birch and Swinnerton-Dyer conjecture, which is one of the most important open problems in number theory (right up there with the Riemann hypothesis). 6.