Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

Iwasawa Theory for Elliptic Curves

The topics that we will discuss have their origin in Mazur’s synthesis of the theory of elliptic curves and Iwasawa’s theory of ZZp-extensions in the early 1970s. We first recall some results from Iwasawa’s theory. Suppose that F is a finite extension of Q and that F∞ is a Galois extension of F such that Gal(F∞/F ) ∼= ZZp, the additive group of p-adic integers, where p is any prime. Equivalentl...

Anticyclotomic Iwasawa Theory of Cm Elliptic Curves

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Zp-extension of the CM field, where p is a prime of good, ordinary reduction for E. When the complex L-function of E vanishes to even order, Rubin’s proof the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the p-power Selmer group over the anticyclotomic extension is a torsion Iwasawa ...

Iwasawa L-Functions of Elliptic Curves with Additive Reduction

9] Perrin-Riou, B.: Descente innnie et hauteur p-adique sur les courbes elliptiques a multiplication complexe.

Iwasawa Theory of Elliptic Curves with Complex Multiplication

Algorithms for the arithmetic of elliptic curves using Iwasawa theory

We explain how to use results from Iwasawa theory to obtain information about p-parts of Tate-Shafarevich groups of specific elliptic curves over Q. Our method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is g...