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

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