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On the \(\mu\)-invariant of \(p\)-adic \(L\)-functions attached to elliptic curves with complex multiplication. (English) Zbl 0615.12018

The aim of this paper is to prove the vanishing of \(\mu\) (the Iwasawa invariant) in the ordinary elliptic case; let \(K\) be a principal imaginary quadratic field and \(p\) a prime \(\ne 2,3\) which splits as \(p=\mathfrak pp^-_{\infty}\) be obtained by adjoining to \(K\) all the \({\mathfrak p}^ n\)-division points on \(E\) \((n=1,\ldots)\) and \(M_{\infty}\) the maximal abelian \(p\)-extension of \(F_{\infty}\) unramified outside \({\mathfrak p}\); the result means that \(\mathrm{Gal}(M_{\infty}/F_{\infty})\) has no \({\mathbb{Z}}_ p\)-torsion. Its proof uses a result of algebraic independence for the elliptic curve. The method is similar (in this much more difficult case) to the one introduced by W. Sinnott [Invent. Math. 75, 273–283 (1984; Zbl 0531.12004)].
The result has been generalized by the reviewer [J. Reine Angew. Math. 358, 76–91 (1985; Zbl 0551.12011)].

MSC:

11R23 Iwasawa theory
11S40 Zeta functions and \(L\)-functions
11R18 Cyclotomic extensions
14H52 Elliptic curves
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References:

[1] Bernardi, D.; Goldstein, C.; Stephens, N., Notes \(p\)-adiques sur les courbes elliptiques, J. Reine Angew. Math., 351, 129-170 (1984) · Zbl 0529.14018
[2] Coates, J.; Goldstein, C., Some remarks on the main conjecture for elliptic curves with complex multiplication, Amer. J. Math., 103, 411-435 (1983)
[3] Coates, J.; Wiles, A., On the conjecture of Birch and Swinnerton-Dyer, Invent. Math., 39, 223-251 (1977) · Zbl 0359.14009
[4] Coates, J.; Wiles, A., On \(p\)-adic \(L\)-functions and elliptic units, J. Austral. Math. Soc. Ser. A, 26, 1-25 (1978) · Zbl 0442.12007
[5] R. Gillard; R. Gillard · Zbl 0615.12019
[6] Goldstein, C.; Schappacher, N., Séries d’Eisenstein et fonctions \(L\) de courbes elliptiques à multiplication complexe, J. Reine Angew. Math., 327, 184-218 (1981) · Zbl 0456.12007
[7] Greenberg, R., On the structure of certain Galois groups, Invent. Math., 47, 85-99 (1978) · Zbl 0403.12004
[8] Lubin, J., One parameter formal Lie groups over \(p\)-adic integer rings, Ann. of Math., 80, 464-484 (1964) · Zbl 0135.07003
[9] Sinnott, W., On the μ-invariant of a rational function, Invent. Math., 75, 273-283 (1984) · Zbl 0531.12004
[10] Gillard, R., Unités elliptiques et fonctions \(Lp\)-adiques, Compositio Fascicule, 1, 57-88 (1980) · Zbl 0446.12012
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