Iwasawa Theory for the Symmetric Square of a CM Modular Form at Inert Primes

TL;DR

This paper constructs and compares two p-adic L-functions for the symmetric square of a CM modular form at inert primes, introducing plus and minus variants and relating them to Selmer groups through a main conjecture.

Contribution

It introduces a new framework for plus and minus p-adic L-functions and Selmer groups for symmetric squares of CM modular forms at inert primes, extending existing theories.

Findings

Two distinct p-adic L-functions constructed with different Frobenius eigenvalues.

Definition of plus and minus p-adic L-functions à la Pollack.

Proposal of a main conjecture relating these L-functions and Selmer groups.

Abstract

Let f be a CM modular form and p an odd prime which is inert in the CM field. We construct two p-adic L-functions for the symmetric square of f, one of which has the same interpolating properties as the one constructed by Delbourgo-Dabrowski, whereas the second one has a similar interpolating properties but corresponds to a different eigenvalue of the Frobenius. The symmetry between these two p-adic L-functions allows us to define the plus and minus p-adic L-functions \`a la Pollack. We also define the plus and minus p-Selmer groups analogous to Kobayashi's Selmer groups. We explain how to relate these two sets of objects via a main conjecture.