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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
##### Keywords:
elliptic curves; descent calculations
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