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On $$p$$-adic $$L$$-functions and the Riemann-Hurwitz genus formula. (English) Zbl 0545.12011
Let $$F$$ be a CM-field; in its cyclotomic tower $$\cup_{n}F_ n$$, Iwasawa’s formula for the $$p$$-relative class number of $$F_ n$$ is characterized by the classical $$\mu^-_ F$$, $$\lambda^-_ F$$ invariants. If $$E/F$$ is a Galois extension of CM-fields, of $$p$$-power degree, the author proves, in full generality, that $$\mu^-_ E=0\Leftrightarrow \mu^-_ F=0,$$ and that, when $$\mu^-_ F=0$$, $$\lambda^-_ E$$ is given explicitly, from $$\lambda^-_ F$$, via Y. Kida’s formula [J. Number Theory 12, 519–528 (1980; Zbl 0455.12007)].
The proof of the author involves only the theory of $$p$$-adic $$L$$-functions of totally real fields, and uses the properties of the corresponding $$p$$-adic pseudomeasures of Deligne-Ribet; the main argument is that if $$\chi$$,$$\psi$$ are even characters, $$\psi$$ of $$p$$-power order, the congruence “ $$\chi \psi \equiv \chi mod \pi$$ ” (for an evident $$\pi \mid p$$) gives a congruence between $$L^*_ p(\chi \psi)$$ and the product of $$L^*_ p(\chi)$$ with suitable Euler factors (this has been observed also by K. A. Ribet [Sémin. Delange-Pisot-Poitou, 19e Année 1977/78, Théor. des Nombres, Fasc. 1, Exp. 9 (1978; Zbl 0394.12007)]), where the $$L^*_ p$$ are suitable series giving $$L_ p$$-functions of characters of the cyclotomic tower; this gives a relation between analytic $$\lambda$$-invariants $$\lambda$$ ($$\chi \psi)$$, $$\lambda$$ ($$\chi)$$, and then Kida’s formula comes from the classical analytic class number formula involving $$L$$-functions at $$s=0.$$ As it is explained by the author, the case of abelian extensions $$F,E$$ over $$\mathbb Q$$ was given by the reviewer, and the Galois representation aspects of Kida’s theory, by K. Iwasawa [Tôhoku Math. J., II. Ser. 33, 263–288 (1981; Zbl 0468.12004)].

##### MSC:
 11S40 Zeta functions and $$L$$-functions 11R23 Iwasawa theory 11R18 Cyclotomic extensions
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##### References:
 [1] P. Cassou-Noguès : Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques . Inv. Math. 51 (1979) 29-59. · Zbl 0408.12015 · doi:10.1007/BF01389911 · eudml:142621 [2] J. Coates : p-adic L-functions and Iwasawa theory . In: Algebraic Number Fields , ed. by A. Fröhlich, Academic Press, New York (1977) pp. 269-353. · Zbl 0393.12027 [3] C. Chevalley and A. Weil : Über das Verhalten der Integrale erster Gattung bei Automorphismen des Funktionenkörpers . Hamb. Abh. 10 (1934) 358-361. · Zbl 0009.16001 · doi:10.1007/BF02940687 [4] P. Deligne and K. Ribet : Values of abelian L-functions at negative integers over totally real fields . Inv. Math. 59 (1980) 227-286. · Zbl 0434.12009 · doi:10.1007/BF01453237 · eudml:142740 [5] Leslie Jane Federer : Ph.D. thesis , Princeton Univ., Princeton, 1982. [6] G. Gras , Sur la construction des fonctions L p-adiques abéliennes , Seminaire Delange-Pisot-Poitou (Théorie des nombres), 1978/79, n^\circ 22. · Zbl 0427.12014 · numdam:SDPP_1978-1979__20_2_A1_0 · eudml:111036 [7] G. Gras , Sur les invariants lambda d’Iwasawa des corps abéliens . Pub. Math. de la Fac. des Sci. de Besançon (1978/79). · Zbl 0472.12009 [8] K. Iwasawa : On r-extensions of algebraic number fields . Bull. Amer. Math. Soc. 65 (1959) 183-226. · Zbl 0089.02402 · doi:10.1090/S0002-9904-1959-10317-7 [9] K. Iwasawa : Lectures on p-adic L-functions . Ann. Math. Studies 74, Princeton University Press, Princeton (1972). · Zbl 0236.12001 · doi:10.1515/9781400881703 [10] K. Iwasawa : Riemann-Hurwitz formula and p-adic Galois representations for number fields . Tôhoku Math. J. (Second Series) 33(2) (1981) 263-288. · Zbl 0468.12004 · doi:10.2748/tmj/1178229453 [11] Y. Kida : l-extensions of CM-fields and cyclotomic invariants . J. Number Theory 12 (1980) 519-528. · Zbl 0455.12007 · doi:10.1016/0022-314X(80)90042-6 [12] K. Ribet : Report on p-adic L-functions over totally real fields . Soc. Math. de France, Astérisque 61 (1979) 177-192. · Zbl 0408.12016 [13] J.-P. Serre : Sur le résidu de la fonction zêta p-adique d’un corps de nombres . C.R. Acad. Sc. Paris 287 (1978) 183-188. · Zbl 0393.12026
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