Katz \(p\)-adic \(L\)-functions, congruence modules and deformation of Galois representations. (English) Zbl 0739.11022

Hida, H.; Tilouine, J.

\(L\)-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 \(\mathbb{Q}_ p\).

Reviewer: S.Kamienny (Los Angeles)

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

Keywords:

CM field; anticyclotomic main conjecture; divisibility relations; Katz \(p\)-adic \(L\)-function; power series; deformations of Galois representations