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

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.