Wach modules and Iwasawa theory for modular forms

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.

44 pages. To appear in the Asian Journal of Mathematics

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• Coleman maps and the p-adic regulator
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• Local epsilon isomorphisms
• Iwasawa theory for Rankin--Selberg products of $p$-non-ordinary eigenforms
• Factorisation of two-variable p-adic L-functions
• On pairs of p-adic L-functions for weight two modular forms
• Euler Characteristics and their Congruences for Multi-signed Selmer Groups
• Dieudonne crystals and Wach modules for p-divisible fgroups
• On fine Selmer groups and signed Selmer groups of elliptic modular forms
• Constructing Galois representations with large Iwasawa $\lambda$-Invariant
• On the cohomology of plus/minus Selmer groups of supersingular elliptic curves in weakly ramified base fields
• Iwasawa Theory for the Symmetric Square of a CM Modular Form at Inert Primes