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...

Let E be an optimal elliptic curve of conductor N , such that the L-function of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K also vanishes to order one at s = 1. In view of the Gross-Zagier theorem, the second part of the Birch and Swinnerton-Dyer conjecture says that the index in E(K) of...

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 ...

Let p be a prime number and E denote the elliptic curve y 2 = x 3 + . It is known that for which congruent to 1 , 9 modulo 16, rank of over Q equal 0 Under condition Birch Swinnerton-Dyer conjecture true, we give necessary sufficient in terms constant term some polynomial defined by recurrence formula.

Introduction 2 1. The Group Law on Pell Conics and Elliptic Curves 2 1.1. Group Law on Conics 2 1.2. Group Law on Elliptic curves 3 2. The Group Structure 3 2.1. Finite Fields 3 2.2. p-adic Numbers 3 2.3. Integral and Rational Points 4 3. Applications 4 3.1. Primality Tests 4 3.2. Factorization Methods 5 4. 2-Descent 5 4.1. Selmer and Tate-Shafarevich Group 5 4.2. Heights 6 5. Analytic Methods ...

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...