The Birch and Swinnerton-Dyer conjecture states that the rank of the Mordell-Weil group of an elliptic curve E equals the order of vanishing at the central point of the associated L-function L(s, E). Previous investigations have focused on bounding how far we must go above the central point to be assured of finding a zero, bounding the rank of a fixed curve or on bounding the average rank in a ...

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.

This article explores the Birch and Swinnerton-Dyer Conjecture, one of the famous Millennium Prize Problems. In addition to providing the basic theoretic understanding necessary to understand the simplest form of the conjecture, some of the original numerical evidence used to formulate the conjecture is recreated. Recent results and current problems related to the conjecture are given at the en...

In [Tei], Teitelbaum formulates a conjecture relating first derivatives of the Mazur– Swinnerton-Dyer p-adic L-functions attached to a modular forms of even weight k ≥ 2 to certain L-invariants arising from Shimura curve parametrisations. This article formulates an analogue of Teitelbaum’s conjecture in which the cyclotomic Zp extension of Q is replaced by the anticyclotomic Zp-extension of an ...

We prove the Rankin-Selberg convolution of two cuspidal automorphic representations are automorphic whenever one of them arises from an irreducible representation of an abelian-by-nilpotent Galois extension, which extends the previous result of Arthur-Clozel. Moreover, if one of such representations is of dimension at most 3 and another representation arises from a nearly nilpotent extension or...

We produce explicit elliptic curves over Fp(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptoti...