×

zbMATH — the first resource for mathematics

A Riemann-Hurwitz formula for the Selmer group of an elliptic curve. (Une formule de Riemann-Hurwitz pour le groupe de Selmer d’une courbe elliptique.) (French) Zbl 0769.11027
Let \(E\) be an elliptic curve with complex multiplication, defined over a number field \(F\). Denote by \(p\) an odd prime. Adding certain \(p\)-torsion points of \(E\) to \(F\), we construct a \(\mathbb{Z}_ p\)-field \(F_ \infty\). We attach to \(F_ \infty\) a Selmer group which is closely related to the usual one for \(F\). For a Galois \(p\)-extension of \(F\), Wingberg established, under the usual arithmetic conjectures, a Riemann-Hurwitz formula for the codimension of the Selmer group at the top of the tower. We give a new proof of this result in the spirit of Chevalley and Weil. This points out the analogy with Kida’s formula for the classical lambda invariant, and with a well-known theorem of Deuring and Shafarevich on the Hasse invariant for Artin-Schreier curves. Next we obtain a generalization to a non-Galois case. In this context, we also have the analogue for Kani’s formulae on the quotient genus of algebraic curves.

MSC:
11G05 Elliptic curves over global fields
11R23 Iwasawa theory
11R32 Galois theory
11G15 Complex multiplication and moduli of abelian varieties
14H52 Elliptic curves
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] C. CHEVALLEY et A. WEIL, Über das verhalten der integrale erster gattung bei automorphismen des functionenkörpers, Hamb. Abh., 10 (1934), 358-361. · JFM 60.0098.01
[2] J. COATES, Infinite descent on elliptic curves, in arithmetic and geometry, papers dedicated to I. Shafarevitch, vol. 35, Progress in Math., Birkhäuser, 1983, pp. 107-137. · Zbl 0541.14026
[3] 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
[4] R. GOLD et M. MADAN, Kida’s theorem for a class of non-normal extensions, Proc. Am. Math. Soc., 104 (1988), 55-59. · Zbl 0678.12005
[5] R. GREENBERG, On the structure of certain Galois groups, Invent. Math., 47 (1978), 85-99. · Zbl 0403.12004
[6] K. IWASAWA, Riemann-Hurwitz formula and p-adic Galois representation for number fields, Tôhoku Math. J., 33 (1984), 263-288. · Zbl 0468.12004
[7] K. IWASAWA, On zp-extensions of algebraic number fields, Ann. of Math., 98 (1973), 243-326. · Zbl 0285.12008
[8] J.-F. JAULENT, Dualité dans LES corps surcirculaires, Sém. Th. Nbres Paris 1986-1987, 85 (1988), 183-220, Progress in Math., Birkhäuser. · Zbl 0679.12007
[9] J.-F. JAULENT, Genres des corps surcirculaires, Pub. Math. Fac. Sci. Besançon, 1985-1986 (1986). · Zbl 0614.12006
[10] J.-F. JAULENT et A. MICHEL, Classes des corps surcirculaires et des corps de fonctions, Sém. Th. Nbres Paris 1989-1990 (1991) (to appear). · Zbl 0751.11052
[11] E. KANI, Relations between the Hasse-Witt invariants of Galois covering of curves, Canad. Math. Bull., 28 (1985), 321-327. · Zbl 0557.14017
[12] Y. KIDA, L-extension of C.M. fields and Iwasawa invariants, J. Numb. Th., 12 (1980), 519-528. · Zbl 0455.12007
[13] M. MADAN et H. ZIMMER, Relations among Iwasawa invariants, J. Numb. Th., 25 (1987), 213-219. · Zbl 0608.12005
[14] A. MICHEL, Ubiquité de la formule de Riemann-Hurwitz, Thèse, Pub. Ec. Doc. Math. Univ. Bordeaux 1, 1992.
[15] B. MAZUR, Rational points of abelian varieties with values in towers of number fields, Invent. Math., 18 (1972), 183-266. · Zbl 0245.14015
[16] J. SILVERMAN, Arithmetic of elliptic curves, GTM 106, Springer-Verlag, New-York, 1986. · Zbl 0585.14026
[17] L. C. WASHINGTON, Introduction to cyclotomic field, GTM 83, Springer-Verlag, New-York, 1982. · Zbl 0484.12001
[18] K. WINGBERG, Galois groups of numbers fields generated by torsion points of elliptic curves, Nagoya Math. J., 104 (1986), 43-53. · Zbl 0621.12011
[19] K. WINGBERG, A Riemann-Hurwitz formula for the Selmer group of an elliptic curve with complex multiplication, Comment. Math. Helvetica, 63 (1988), 587-592. · Zbl 0682.12006
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.