×

zbMATH — the first resource for mathematics

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.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. Coates and A. Wiles, On \(p\)-adic \(L\)-functions and elliptic units , J. Austral. Math. Soc. Ser. A 26 (1978), no. 1, 1-25. · Zbl 0442.12007
[2] J. Coates, \(p\)-adic \(L\)-functions and Iwasawa’s theory , Algebraic number fields: \(L\)-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) ed. A. Fröhlich, Academic Press, London, 1977, pp. 269-353. · Zbl 0393.12027
[3] J. Coates and A. Wiles, Kummer’s criterion for Hurwitz numbers , Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976) ed. Iyanaga, Japan Soc. Promotion Sci., Tokyo, 1977, pp. 9-23. · Zbl 0369.12009
[4] R. Gillard, Fonctions \(L\) \(p\)-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes , J. Reine Angew. Math. 358 (1985), 76-91. · Zbl 0551.12011
[5] R. Greenberg, On the structure of certain Galois groups , Invent. Math. 47 (1978), no. 1, 85-99. · Zbl 0403.12004
[6] R. Greenberg, Iwasawa’s theory and \(p\)-adic \(L\)-functions for imaginary quadratic fields , Number Theory Related to Fermat’s Last Theorem ed. N. Koblitz, Progress in Math., vol. 26, Birkhauser-Verlag, 1982, pp. 275-286. · Zbl 0505.12006
[7] B. Gross, Arithmetic on elliptic curves with complex multiplication , Lecture Notes in Mathematics, vol. 776, Springer, Berlin, 1980. · Zbl 0433.14032
[8] H. Hida, Congruence of cusp forms and special values of their zeta functions , Invent. Math. 63 (1981), no. 2, 225-261. · Zbl 0459.10018
[9] H. Hida, On congruence divisors of cusp forms as factors of the special values of their zeta functions , Invent. Math. 64 (1981), no. 2, 221-262. · Zbl 0472.10028
[10] H. Hida, A \(p\)-adic measure attached to the zeta functions associated with two elliptic modular forms I , Invent. Math. 79 (1985), no. 1, 159-195. · Zbl 0573.10020
[11] H. Hida, Iwasawa modules attached to congruences of cusp forms , Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 231-273. · Zbl 0607.10022
[12] H. Hida, Galois representations into \(\mathrm GL_ 2(\mathbbZ_ p[[X]])\) attached to ordinary cusp forms , Invent. Math. 85 (1986), no. 3, 545-613. · Zbl 0612.10021
[13] H. Hida, Hecke algebras for \(\mathrm GL_ 1\) and \(\mathrm GL_ 2\) , Séminaire de Théorie des Nombres, Paris 1984-85, Progress in Math., vol. 63, Birkhauser, Boston, MA, 1986, pp. 131-163. · Zbl 0648.10020
[14] H. Hida, A \(p\)-adic measure attached to the zeta functions associated with two elliptic modular forms II , à paraître aux Ann. Inst. Fourier 38 (1988) disponible comme Hokkaido University Preprint in Math. · Zbl 0645.10028
[15] H. Hida, Modules of congruence of Hecke algebras and \(L\) functions associated with cusp forms , Amer. J. Math. 110 (1988), no. 2, 323-382. JSTOR: · Zbl 0645.10029
[16] N. Katz, \(p\)-adic interpolation of real analytic Eisenstein series , Ann. of Math. (2) 104 (1976), no. 3, 459-571. JSTOR: · Zbl 0354.14007
[17] B. Mazur and J. Tilouine, Représentations galoisiennes, différentielles de Kähler et conjectures principales , à paraître. · Zbl 0744.11053
[18] B. Perrin-Riou, Arithmétique des Courbes Elliptiques et Théorie d’Iwasawa , Mém. Soc. Math. France (N.S.) 17 (1984), 130. · Zbl 0599.14020
[19] G. Robert, Unités Elliptiques , vol. 36, Mém. Soc. Math. France, Paris, 1973. · Zbl 0314.12006
[20] K. Rubin, On the main conjecture of Iwasawa theory for imaginary quadratic fields , Invent. Math. 93 (1988), no. 3, 701-713. · Zbl 0673.12004
[21] E. de Shalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication , Perspectives in Math., vol. 3, Academic Press, Boston, MA, 1987. · Zbl 0674.12004
[22] G. Shimura, On the zeta function of an abelian variety with complex multiplication , Ann. of Math. (2) 94 (1971), 504-533. JSTOR: · Zbl 0242.14009
[23] G. Shimura, On elliptic curves with complex multiplication as factors of the jacobians of modular function fields , Nagoya Math. J. 43 (1971), 199-208. · Zbl 0225.14015
[24] G. Shimura, Introduction to the arithmetic theory of automorphic functions , Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971. · Zbl 0221.10029
[25] J. Tate, \(p-divisible\) \(groups.\) , Proc. Conf. Local Fields (Driebergen, 1966), Springer, Berlin, 1967, pp. 158-183. · Zbl 0157.27601
[26] J. Tilouine, Fonctions \(L\) \(p\)-adiques à deux variables et \( \mathbbZ^ 2_ p\)-extensions , Bull. Soc. Math. France 114 (1986), no. 1, 3-66. · Zbl 0606.14024
[27] J. Tilouine, Un sous-groupe \(p\)-divisible de la jacobienne de \(X_ 1(Np^r)\) comme module sur l’algèbre de Hecke , Bull. Soc. Math. France 115 (1987), no. 3, 329-360. · Zbl 0677.14006
[28] J. Tilouine, Kummer’s criterion over \(\Lambda\) , Technical Report Series in Math. 4, Hokkaïdo University, August 1987. · Zbl 0663.12007
[29] J. Tilouine, Une conséquence de la conjecture principale dans la théorie d’Iwasawa d’un corps quadratique imaginaire , C. R. Acad. Sci. Paris sér. I Math. 306 (1988), no. 5, 217-221. · Zbl 0674.12002
[30] J. Tilouine, Théorie d’Iwasawa classique et de l’algèbre de Hecke ordinaire , Compositio Math. 65 (1988), no. 3, 265-320. · Zbl 0663.12008
[31] L. Washington, Introduction to Cyclotomic Fields , Graduate Texts in Math., vol. 83, Springer-Verlag, New York, 1982. · Zbl 0484.12001
[32] A. Weil, On a certain type of characters of the idele-class group of an algebraic number field , Collected Papers, vol. 2, Springer-Verlag, 1979, pp. 255-261.
[33] R. Yager, On two variable \(p\)-adic \(L\)-functions , Ann. of Math. (2) 115 (1982), no. 2, 411-449. JSTOR: · Zbl 0496.12010
[34] R. Gillard, Transformation de Mellin-Leopoldt des fonctions elliptiques , J. Number Theory 25 (1987), no. 3, 379-393. · Zbl 0615.12019
[35] B. Gross, On the factorisation of \(p\)-adic \(L\)-series , Invent. Math. 57 (1980), no. 1, 83-95. · Zbl 0472.12011
[36] B. Ferrero and L. Washington, The Iwasawa invariant \(\mu _p\) vanishes for abelian number fields , Ann. of Math. (2) 109 (1979), no. 2, 377-395. JSTOR: · Zbl 0443.12001
[37] L. Schneps, On the \(\mu\)-invariant of \(p\)-adic \(L\)-functions attached to elliptic curves with complex multiplication , J. Number Theory 25 (1987), no. 1, 20-33. · Zbl 0615.12018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.