# Wach modules and Iwasawa theory for modular forms - Mathematics > Number Theory

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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 ofthe representation to the Iwasawa algebra, using the theory of Wach modules.Let f = suma n q^n be a normalized new modular eigenform and p an odd primeat which f is either good ordinary or supersingular. By applying our theory tothe p-adic representation associated to f, we define two Coleman maps withvalues in the Iwasawa algebra of Zp^* after extending scalars to someextension of Qp. Applying these maps to the Kato zeta elements gives adecomposition of the generally unbounded p-adic L-functions of f into linearcombinations of two power series of bounded coefficients, generalizing works ofPollack in the case a p=0 and Sprung when f corresponds to a supersingularelliptic curve. Using ideas of Kobayashi for elliptic curves which aresupersingular at p, we associate to each of these power series a cotorsionSelmer group. This allows us to formulate a -main conjecture-. Under sometechnical conditions, we prove one inclusion of the -main conjecture- and showthat the reverse inclusion is equivalent to Kato-s main conjecture.

Author: ** Antonio Lei, David Loeffler, Sarah Livia Zerbes**

Source: https://arxiv.org/