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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
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##### References:
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