Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes

TL;DR

This paper constructs and analyzes p-adic L-functions and Selmer groups for symmetric powers of CM modular forms at non-ordinary primes, confirming key conjectures in Iwasawa theory and extending the understanding of exceptional zero phenomena.

Contribution

It develops admissible and mixed plus/minus p-adic L-functions for symmetric powers of CM modular forms and proves the Main Conjecture and exceptional zero conjecture in this setting.

Findings

Construction of admissible p-adic L-functions for symmetric powers

Verification of the Main Conjecture relating Selmer groups and p-adic L-functions

Proof of the exceptional zero conjecture for these L-functions

Abstract

Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p-adic L-functions for the symmetric powers of f, thus verifying general conjectures of Dabrowski and Panchishkin in this special case. We also construct their "mixed" plus and minus counterparts and prove an analogue of Pollack's decomposition of the admissible p-adic L-functions into mixed plus and minus p-adic L-functions. On the arithmetic side, we define corresponding mixed plus and minus Selmer groups. We unite the arithmetic with the analytic by first formulating the Main Conjecture of Iwasawa Theory relating the plus and minus Selmer groups with the plus and minus p-adic L-functions, and then proving the exceptional zero conjecture for the admissible p-adic L-functions. The latter result takes advantage of recent work of Benois, while the former uses recent work of the second author, as well as the main conjecture of Mazur-Wiles.