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Descent calculations for the elliptic curves of conductor 11. (English) Zbl 1052.11038
Elliptic curves of conductor 11 have been a test ground for the Iwasawa theory of elliptic curves since the work of B. Mazur [Invent. Math. 18, 183–266 (1972; Zbl 0245.14015)]. These elliptic curves form a single isogeny class J. Vélu [C. R. Acad. Sci., Paris, Sér. A 273, 73–75 (1971; Zbl 0225.14013)] and there are 3 explicit Weierstrass equations for them: \(A_1:=X_0(11) : y^2+y=x^3-x^2-10x-20\), \(A_2:=X_1(11): y^2+y=x^3-x^2\) and \(A_3 : y^2+y=x^3-x^2-7820x-263580\). One natural question in this context is about the variation of the Mordell-Weil rank as we pass up the tower of fields by adjoining 5-power division points.
Let \(A\) be any of the three previous elliptic curves. Mazur proved that rank\((A(\mathbb Q))=0\) and \(\text{ Ш}(A/\mathbb Q)(5)=0\). An extension of this result for \(\mathbb Q(\mu_{5^{\infty}})\), due to Greenberg, may be found in J. Coates and R. Sujatha [Galois cohomology of elliptic curves. Tata Institute of Fundamental Research. 88. New Delhi: Narosa Publishing House. Mumbai (2000; Zbl 0973.11059)]. In both cases the strategy is to use descent calculations to study the behavior of Selmer groups over the cyclotomic \(\mathbb Z_5\)-extension.
Elliptic curves of conductor 11 are the first modular elliptic curves that do not admit complex multiplication. The prime number 5 is used to make things simpler, for example there are degree 5 isogenies defined over \(\mathbb Q\) among the 3 previous elliptic curves. Moreover, the curves \(A_0\) and \(A_1\) have both a \(\mathbb Q\)-rational point of order 5, whereas \(A_2\) does not. By the properties of the Weil pairing it follows an isomorphism \(A_0[5]\cong\mu_5\oplus\mathbb Z/5\mathbb Z\) as Galois modules. Moreover, there are exact sequences \[ 0\to\mathbb Z/5\mathbb Z\to A_1\to A_0\to0,\qquad 0\to\mu_5\to A_2\to A_0\to0. \] The fields of 5-division points are \(k=\mathbb Q(\mu_5)\), \(K_1=\mathbb Q(A_1[5])\) and \(K_2=\mathbb Q(A_2[5])\). The fields \(K_1\) and \(K_2\) are non-abelian of degree 20 over \(\mathbb Q\), so the descent calculations are more involved. The author’s main results are:
rank\((A(K_1))=0\),
\(A_0(K_1)\cong(\mathbb Z/5\mathbb Z)^2\), \(\text{ Ш}(A_0/K_1)(5)\cong(\mathbb Z/5\mathbb Z)^8\),
\(A_1(K_1)\cong(\mathbb Z/5\mathbb Z)^2\), \(\text{ Ш}(A_1/K_1)(5)\cong(\mathbb Z/5\mathbb Z)^2\),
\(A_2(K_1)\cong\mathbb Z/5\mathbb Z\), \(\text{ Ш}(A_2/K_1)(5)\cong(\mathbb Z/5\mathbb Z)^4\oplus(\mathbb Z/25\mathbb Z)^8\),
rank\((A(K_2))=0\),
\(A_0(K_2)\cong(\mathbb Z/5\mathbb Z)^2\), \(\text{ Ш}(A_0/K_2)(5)\cong(\mathbb Z/5\mathbb Z)^8\),
\(A_1(K_2)\cong\mathbb Z/5\mathbb Z\), \(\text{ Ш}(A_1/K_2)(5)=0\),
\(A_2(K_2)\cong(\mathbb Z/5\mathbb Z)^2\), \(\text{ Ш}(A_2/K_2)(5)\cong(\mathbb Z/5\mathbb Z)^6\oplus(\mathbb Z/25\mathbb Z)^8\).

MSC:
11G05 Elliptic curves over global fields
11R23 Iwasawa theory
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