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Katz $$p$$-adic $$L$$-functions, congruence modules and deformation of Galois representations. (English) Zbl 0739.11022
$$L$$-functions and arithmetic, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 153, 271-293 (1991).
[For the entire collection see Zbl 0718.00005.]
The authors generalize results of Mazur and Tilouine giving a proof of the anticyclotomic main conjecture. More precisely, they prove a generalization, to the $$CM$$ case, of certain divisibility relations (predicted by the main conjecture) between a Katz $$p$$-adic $$L$$-function and certain characteristic power series. They also re-prove some of the divisibility results of Mazur-Tilouine without the additional hypotheses needed in the original proof. The proof uses the theory of deformations of Galois representations over finite extensions of $$\mathbb{Q}_ p$$.

##### MSC:
 11F85 $$p$$-adic theory, local fields 11R23 Iwasawa theory 11S40 Zeta functions and $$L$$-functions 11F80 Galois representations 11G15 Complex multiplication and moduli of abelian varieties 11R32 Galois theory