Suppose E is an elliptic curve over Q and χ a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E,χ,1)=0. Via Birch Swinnerton-Dyer conjecture, this gives heuristic Mordell–Weil rank grows in abelian extensions Q. Using heuristic, we find large class infinite F where expect E(F) be finitely generated. Our work was inspired by ...

Let E/Q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of p-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton-Dyer conjecture when E(K) is infinite, and he proved that his conjecture is tru...

In a few earlier papers ([8], [9], [10]) attention was called to the striking parallel between the ideas surrounding the well-known conjecture of Birch and Swinnerton-Dyer for elliptic curves, and the mysterious section conjecture of Grothendieck [6] that concerns hyperbolic curves. We wish to explain here some preliminary ideas for ‘effective non-abelian descent’ on hyperbolic curves equipped ...

Given any integer N>1 prime to 3, we denote by C N the elliptic curve x 3 +y =N. We first study 3-adic valuation of algebraic part value Hasse–Weil L-function L(C ,s) over ℚ at s=1, and exhibit a relation between 3-part its Tate–Shafarevich group number distinct divisors which are inert in imaginary quadratic field K=ℚ(-3). In case where ,1)≠0 is product split primes K, show that order as predi...

We study the Galois groups of polynomials arising from a compatible family representations with big orthogonal monodromy. show that are usually as large possible given constraints imposed on them by functional equation and discriminant considerations. As an application, we consider Frobenius middle étale cohomology hypersurfaces in [Formula: see text] degree at least three. also text]-functions...

We review some of Kolyvagin’s results and conjectures about elliptic curves, then make a new conjecture that slightly refines Kolyvagin’s conjectures. We introduce a definition of finite index subgroups Wp ⊂ E(K), one for each prime p that is inert in a fixed imaginary quadratic field K. These subgroups generalize the group ZyK generated by the Heegner point yK ∈ E(K) in the case ran = 1. For a...