Let f be a newform of weight 2 on Γ0(N), and let Af be the corresponding optimal abelian variety quotient of J0(N). We describe an algorithm to compute the order of the component group of Af at primes p that exactly divide N . We give a table of orders of component groups for all f of level N ≤ 127 and five examples in which the component group is very large, as predicted by the Birch and Swinn...

Let E be an elliptic curve over Q and let be an odd, irreducible twodimensional Artin representation. This article proves the Birch and Swinnerton-Dyer conjecture in analytic rank zero for the Hasse-WeilArtin L-series L(E, , s), namely, the implication L(E, , 1) = 0 ⇒ (E(H)⊗ ) = 0, where H is the finite extension of Q cut out by . The proof relies on padic families of global Galois cohomology c...

In this paper, we introduce arithmetic Heilbronn characters that generalize the notion of the classical Heilbronn characters, and discuss several properties of these characters. This formalism has several arithmetic applications. For instance, we obtain the holomorphy of suitable quotients of L-functions attached to elliptic curves, which is predicted by the Birch–Swinnerton–Dyer conjecture, an...

To a large extent, the investigations to be brought up today arise from a curious inadequacy having to do with the arrow on the left. On the one hand, it is widely acknowledged that the theory of motives finds a strong source of inspiration in Diophantine geometry, inasmuch so many of the structures, conjectures, and results therein have as model the conjecture of Birch and Swinnerton-Dyer, whe...

We will discuss similarities of L-functions between in the number theory and in the geometry. In particular the Hesse-Weil congruent zeta function of a smooth curve defined over a finite field will be compared to the zeta function associated to a disrete dynamical system. Moreover a geometric analog of the Birch and Swinnerton-Dyre conjecture and of the Iwasawa Main Conjecture will be discussed. 1

We refine the techniques of our previous paper [KM1] to prove that the average order of vanishing of L-functions of primitive automorphic forms of weight 2 and prime level q satisfies ∑ f∈S2(q)∗ ords= 1 2 L(f, s) ≤ C|S2(q) ∗| with C = 6.5, for all q large enough. On the Birch and Swinnerton-Dyer conjecture, this implies rank J0(q) ≤ C dim J0(q) for q prime large enough.

In this note, we consider an `-isogeny descent on a pair of elliptic curves over Q. We assume that ` > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finitedimensional F`-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the `-part of the Birch and Swinnerton-Dyer...

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 ...

This proposal falls broadly in the area of number theory and more specifically in arithmetic geometry. It is concerned with a part of the Birch and Swinnerton-Dyer (BSD) conjecture on elliptic curves and abelian varieties. A fundamental problem of number theory is: given a set of polynomial equations with rational coefficients, find all of its rational solutions and investigate their structure....

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...