Wach modules and Iwasawa theory for modular forms

TL;DR

This paper develops Coleman maps from Wach modules for crystalline p-adic Galois representations associated with modular forms, leading to a decomposition of p-adic L-functions and the formulation of a main conjecture relating Selmer groups and L-functions.

Contribution

It introduces a new family of Coleman maps for modular forms using Wach modules, generalizing previous results and connecting to Kato's main conjecture.

Findings

Decomposition of p-adic L-functions into bounded power series.

Construction of associated cotorsion Selmer groups.

Partial proof of the main conjecture under certain conditions.

Abstract

For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(a_n q^n) be a normalized new modular 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 two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). 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 a_p=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 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.