Title:

Iwasawa theory for modular forms at supersingular primes

Let f=\sum a_nq n be a normalised eigennewform of weight k\ge2 and p an odd prime which does not divide the level of f. We study a reformulation of Kato's main conjecture for f over the Zpcyclotomic extension of Q. In particular, we generalise Kobayashi's main conjecture on psupersingular elliptic curves over Q with a_p=0, which asserts that Pollack's padic Lfunctions generate the characteristic ideals of some \pmSelmer groups which are cotorsion over the Iwasawa algebra \Lambda=Zp[[Zp]]. We begin by studying the padic Hodge theory for the padic representation associated to f in the case when a_p=0. It allows us to give analogous definitions of Kobayashi's \pmColeman maps and \pmSelmer groups. The Coleman maps are used to show that the Pontryagin duals of these new Selmer groups are torsion over \Lambda as in the elliptic curve case. As a consequence, we formulate a main conjecture stating that Pollack's padic Lfunctions generate their characteristic ideals. Similar to Kobayashi's works, we prove one inclusion of the main conjecture using an Euler system constructed by Kato. We then prove the other inclusion of the main conjecture for CM modular forms, generalising works of Pollack and Rubin on CM elliptic curves. As a key step of the proof, we generalise the reciprocity law of CoatesWiles and Rubin. Next, we study Wach modules associated to positive crystalline padic representations in general and generalise the construction of the Coleman maps. By applying this to modular forms with much more general a_p, we define two Coleman maps and decompose the classical padic L functions of f into linear combinations of two power series of bounded coefficients generalising works of Pollack (in the case a_p=0) and Sprung (when f corresponds to an elliptic curve over Q with a_p\ne0). Once again, this leads to a reformulation of Kato's main conjecture involving cotorsion Selmer groups and padic Lfunctions of bounded coefficients. One inclusion of this new main conjecture is proved in the same way as the a_p=0 case. Finally, we explain how the \pmColeman maps can be extended to LubinTate extensions of height 1 in place of the Zpcyclotomic extension. This generalises works of Iovita and Pollack for elliptic curves over Q.
