On the Mordell-Weil and Shafarevich-Tate Groups for Weil Elliptic Curves

Abstract

A second-order many-body method developed from the many-body scheme introduced by Schneider et a1 is applied to elastic electron-helium scattering. Three models derived from the method are discussed and results for differential, integrated and total cross sections are presented at energies ranging from 30 to 200 eV and compared with absolute experimental data.

Values of L-functions and p-adic Cohomology

“…4, it is proved in [37], [60] that (E5) In [37], improving upon previous works [35], [36], Kolyvagin proves that, if y ∈ E(K) is of infinite order, then…”

“…(E5) E : "Heegner points" on Jacobians of modular (or Shimura) curves ( [6], [9], [23], [24], [25], [35], [36], [37], [38], [39], [40], [41]) defined over ring class fields of an imaginary quadratic field…”

Realising the cup product of local Tate duality

“…It plays a crucial role in Kolyvagin's work in [3] and [4], where he applies Euler systems to elliptic curves and thereby provides evidence for Birch and Swinnerton-Dyer Conjecture.…”

Indivisibility of class numbers of imaginary quadratic fields

“…The Heegner hypotheses are a set of conditions about how the rational primes of bad reduction of an elliptic curve split in an imaginary quadratic field. The work of Kolyvagin on the Birch and Swinnerton-Dyer Conjecture (see [19,20]) is based on the existence of suitable quadratic twists of elliptic curves in which the twisting discriminant satisfies prescribed Heegner hypotheses. Combining his work with an important theorem of Gross and Zagier, who showed that the height of the Heegner point is a multiple of the derivative of the L-series of the elliptic curve at 1, it follows that the Birch and Swinnerton-Dyer Conjecture holds when the analytic rank is at most 1.…”