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Quelques aspects de la descente sur une courbe elliptique dans le cas de réduction supersingulière. (Some aspect of the descent on an elliptic curve in the case of supersingular reduction). (French) Zbl 0604.14017
Let E be an elliptic curve over a number field F, with complex multiplication by a quadratic imaginary field K, and assume that K is contained in F. Let N be any field containing F and contained in the \({\mathbb{Z}}^ 2_ p\)-extension \(F_{\infty}\) of F which is itself contained in the field generated over F by the coordinates of the p-power division points of E. The author studies the torsion submodule of the Pontryagin dual S(N) of the Selmer group S(N) \((for\quad p)\) of \(E_{/N}\) when p is a prime of good supersingular reduction. Assuming Leopoldt’s conjecture for all fields N as above, the author shows that for \(N=F_{\infty}\) there is a pseudo-isomorphism between the torsion submodule of \(S(F_{\infty})\) and the Pontryagin dual of the module \(\Sigma (F_{\infty})=Ker(S(F_{\infty})\to \prod_{v| p}H^ 1(F_{\infty,v},E_{p^{\infty}})).\)
Now, let \(\gamma\) and \(\gamma\) ’ denote two generators of the Iwasawa algebra \(\Lambda\) that correspond to the elements T and T’ of \({\mathbb{Z}}_ p[[T,T']]\) under the usual isomorphism sending T to (1-\(\gamma)\) and T’ to (1-\(\gamma\) ’). Assume that (\(\gamma\)-1) is prime to the characteristic power series of \(S(F_{\infty})_{tors}\), and let \(\Lambda_ 2=\Lambda /(\gamma -1)\Lambda\). If \(L_{\infty}\) denotes the \({\mathbb{Z}}_ p\)-extension fixed by \(\gamma\) the author shows that the characteristic power series \(g_{\Lambda_ 2}(T)\) of \(\Sigma (L_{\infty})\) is, up to a unit, the characteristic power series \(f_{\Lambda_ 2}(\frac{1}{1+T}-1)\) of \(S(L_{\infty})_{tors}\). The author also gives several applications, one of which gives conditions under which \(S(L_{\infty})\) is trivial.
Reviewer: S.Kammienny

14H25 Arithmetic ground fields for curves
14H52 Elliptic curves
11R18 Cyclotomic extensions
14H45 Special algebraic curves and curves of low genus
14K22 Complex multiplication and abelian varieties
Full Text: Numdam EuDML
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