Iwasawa Theory for Modular Forms at Supersingular Primes

TL;DR

This paper extends Iwasawa theory to modular forms at supersingular primes, defining Coleman maps and relating p-adic L-functions to Selmer groups, and proves the main conjecture for CM modular forms.

Contribution

It generalizes previous work by defining Coleman maps for arbitrary weights and proves the main conjecture for CM modular forms.

Findings

Defined even and odd Coleman maps for modular forms of arbitrary weights.

Related Pollack's p-adic L-functions to Selmer groups.

Proved the main conjecture for CM modular forms.

Abstract

We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of even and odd Coleman maps for normalised new forms of arbitrary weights and relate Pollack's $p$-adic $L$-functions to the even and odd Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the "main conjecture" for CM modular forms.