Iwasawa Theory for Modular Forms at Supersingular Primes

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.

Has been expanded and revised incorporating comments made by the referees. To appear in Compositio Mathematica

Cited by in corpus (11)

• Coleman maps and the p-adic regulator
• On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions
• Iwasawa theory for Rankin--Selberg products of $p$-non-ordinary eigenforms
• On fine Selmer groups and signed Selmer groups of elliptic modular forms
• On Selmer groups in the supersingular reduction case
• Non-commutative p-adic L-functions for supersingular primes
• Constructing Galois representations with large Iwasawa $\lambda$-Invariant
• Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes
• Iwasawa Theory for the Symmetric Square of a CM Modular Form at Inert Primes
• On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic ${\mathbf{Z}}_p$-towers
• On the weak Leopoldt conjecture and coranks of Selmer groups of supersingular abelian varieties in $p$-adic Lie extensions