Book Review: Iwasawa theory of elliptic curves with complex multiplication

Iwasawa Theory of Elliptic Curves with Complex Multiplication

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...

Recent Results in the GL2 Iwasawa Theory of Elliptic Curves without Complex Multiplication

Recently new results have been obtained in the GL2 Iwasawa theory of elliptic curves without complex multiplication. This article is a survey of some of those results. Mathematics Subject Classification: 11G05, 11R23

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 ...

The main conjecture of Iwasawa theory for elliptic curves with complex multiplication over abelian extensions at supersingular primes

We develop the plus/minus p-Selmer group theory and plus/minus padic L-function theory for an elliptic curve E with complex multiplication over an abelian extension F of the imaginary quadratic field K given by the complex multiplication of E when p is a prime inert over K/Q (i.e. supersingular). As a result, we prove that the characteristic ideal of the Pontryagin dual of the plus/minus p-Selm...