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

Abstract

Abstract. Let f be a modular form with complex multiplication (CM) and p an odd prime that 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 and Dabrowski (A. Dabrowski and D. Delbourgo, S-adic L-functions attached to the symmetric square of a newform, Proc. Lond. Math. Soc. 74(3) (1997), 559-611), whereas the other 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-functionsà la Pollack (R. Pollack, on the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118 (3) (2003), 523-558). We also define the plus and minus p-Selmer groups analogous to the ones defined by Kobayashi (S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152(1) (2003), 1-36). We explain how to relate these two sets of objects via a main conjecture.

Iwasawa theory for Rankin-Selberg products of p-nonordinary eigenforms

“…Remark 3.1.3. We note that this decomposition was exploited in [Lei12] in the CM case. In fact, this decomposition holds as G Q -representations (not just G Qp -representations) when f is of CM type.…”

“…Our running hypothesis a p (f ) = 0 yields a G Qp -equivariant decomposition T = R * 1,χ ⊕ R * 2,χ . Exploiting this decomposition, we define signed Coleman maps as in [Lei12]. More precisely, we define Λ O (Γ)-morphisms…”

“…Exploiting this decomposition, we define signed Coleman maps as in [Lei12]. More precisely, we define Λ O (Γ)-morphisms…”

Iwasawa theory for symmetric squares of non-$p$-ordinary eigenforms

“…Remark 4.1.3. We note that this decomposition was exploited in [Lei12] in the CM case. In fact, this decomposition holds as G Q -representations (not just G Qp -representations) when f is of CM type.…”