Wach Modules and Iwasawa Theory for Modular Forms

Abstract

Abstract. We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.

Dieudonné crystals and Wach modules for p-divisible groups

“…For and , the Wach modules of Berger and Berger–Breuil provide a variant of the description of Kisin–Ren of stable ‐lattices in ‐representations whose restriction to is crystalline. As this variant figures prominently in applications (for example, , , , , , , , ), for the sake of completeness we now recall the relation between and .…”

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

“…The assumption on the eigenvalues of ϕ made in [6] are not necessary for our purposes here because the Perrin-Riou pairings can be defined by applying 1− ϕ to the (ϕ, G ∞ )-module of V * 2 (see [7] and Section 16.4 in [5]). We fix a non-zero elementω ∈ Fil −1 ‫ބ‬ cris (V * 2 (1)) and write…”

Signed Selmer groups over p-adic Lie extensions

“…We first recall our construction of the signed Selmer groups from [15]. Letν 1 ,ν 2 andn 1 ,n 2 be the bases of D cris (V (−1)) and N(V (−1)), respectively, as defined in [15, §3.2].…”

“…Let us first recall the description of the signed Selmer groups Sel i (E/Q(µ p ∞ )) in terms of p-adic Hodge theory as given in [15]. Let V = Q p ⊗ T where T = T p E is the p-adic Tate module of E, then V is a crystalline representation of G Qp , the absolute Galois group of Q p .…”

“…The details of this construction is given in Section 3.1. When E has good ordinary reduction at p, we have defined the Selmer groups Sel i (E/Q(µ p ∞ )) for i = 1, 2 in [15] in the same way as the good supersingular case. To justify the proposed definition of signed Selmer groups over K ∞ , we show in Section 3.2 that on extending our construction to the good ordinary case, Sel 2 (E/K ∞ ) again agrees with the usual Selmer group Sel(E/K ∞ ) for any finite extension K of Q.…”