On the anticyclotomic main conjecture. (Sur la conjecture principale anticyclotomique.) (French) Zbl 0707.11079

Tilouine, J.

Duke Math. J. 59, No. 3, 629-673 (1989).

The purpose of the author is to relate a congruence module to \(p\)-adic series associated with \(p\)-adic \(L\)-functions introduced by Katz and Yager. This allows the author to finish the proof of the so-called “anticyclotomic main conjecture” in the Iwasawa theory of abelian extensions of quadratic imaginary fields: preceding work of the author did half of the job by relating the congruence module to characteristic functions in Iwasawa theory [see the author, Compos. Math. 65, No. 3, 265–320 (1988; Zbl 0663.12008) et B. Mazur and the author, Publ. Math., Inst. Hautes Étud. Sci. 71, 65–103 (1990; Zbl 0744.11053)].
Here the work is mainly \(p\)-adic analysis. Classically, the Rankin product is used to compute a Dirichlet series in terms of a Petersson product. Its \(p\)-adic analog is studied to give a \(p\)-adic series in 3 variables which is computed in terms of the Katz-Yager series. The characteristic series of the congruence module appears in the final formula. A careful study of this one gives the hoped divisibility.
Finally, note that this theory can be adapted in the case of CM fields (see a forthcoming paper by Hida and the author). This paper is quite technical and finishes the proof of a very nice result.

Reviewer: Roland Gillard (Saint-Martin-d’Hères)

MSC:

11R23	Iwasawa theory
11R20	Other abelian and metabelian extensions
11S80	Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)

Keywords:

congruence module; p-adic series; p-adic L-functions; Iwasawa theory

Citations:

Zbl 0663.12008; Zbl 0744.11053

References:

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