On The Birch and Swinnerton-Dyer Conjecture for CM Elliptic Curves over $\BQ$

TL;DR

This paper proves that for certain CM elliptic curves over rationals with analytic rank one, the p-part of their Shafarevich-Tate group matches the BSD conjecture's prediction for all suitable primes.

Contribution

It establishes the BSD conjecture's prediction for the p-part of the Shafarevich-Tate group in the CM case under specific conditions.

Findings

p-part of Shafarevich-Tate group matches BSD prediction

Results hold for all potential good ordinary primes not dividing roots of unity

Advances understanding of BSD conjecture for CM elliptic curves

Abstract

For CM elliptic curve over rational field with analytic rank one, for any potential good ordinary prime p, not dividing the number of roots of unity in the complex multiplication field, we show the p-part of its Shafarevich-Tate group has order predicted by the Birch and Swinnerton-Dyer conjecture.