The relationship between arithmetic objects (such as global fields, or varieties over global fields) and the analytic properties of their Lfunctions poses many deep and difficult questions. The theme of this paper is the Birch and Swinnerton Dyer conjecture, and certain refinements that were proposed by Mazur and Tate. We will formulate analogues of these conjectures over imaginary quadratic fi...

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 develop a theory of ‘non-abelian higher special elements’ in the non-commutative exterior powers Galois cohomology p-adic representations. explore their relation to organising matrices and thus module structure Selmer modules. In concrete applications, we relate our general formulation refined conjectures Birch Swinnerton-Dyer type Tate–Shafarevich groups abelian varieties.

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.

In this paper we present a method for explicitly computing rational points on elliptic curves using Heegner points. This method was crucial to the proof of Gross-Zagier, which proves the rank one case of the Birch and Swinnerton-Dyer Conjecture. Although this use of Heegner points can be found in many books and articles, we strive here to present it in a more concrete and complete form, and usi...