Shafarevich-Tate groups of elliptic curves upon quadratic extension and several applications

Details Shafarevich-Tate groups of elliptic curves upon quadratic extension and several applications Shafarevich-Tate groups of elliptic curves upon quadratic extension and several applications

2010-03-23

arxiv.org/abs/1003.4393v2

Mathematics

Number Theory

29 pages; a remark is added on page 5; a reference [Dok] is added.

Scientific paper

Let $ E $ be an elliptic curve over a number field $ F $ and $ K = F (\sqrt{D}) $ be a quadratic extension of $ F. $ In this paper, for $ E $ and its quadratic $ D-$twist $ E_{D}, $ by calculating the cohomology groups, we obtain an explicit formula relating the orders of the Shafarevich-Tate groups $\amalg(E/F),\amalg(E_{D}/F),\amalg(E/K)$ and the ranks of the groups of $ F-$rational points of E and $E_{D}$. Then, assuming the finiteness of Shafarevich-Tate groups, we prove by a simple way different from the recent paper [Dok] that each square free positive integer $n\equiv5, 6\ \text{or} \ 7 (\text{mod} 8) $ is a congruent number; and for several families of elliptic curves $ E_{n}: y^{2} = x^{3} - n^{2} x, $ we prove that the orders of $ \amalg\amalg(E_{n}/ \Q(\sqrt{n})) (\cong \amalg\amalg(E_{1}/ \Q(\sqrt{n}))) $ are equal to the squares of a product of $ 2-$th powers with the $ n (\text{or} \ n / 2)-$th Fourier coefficients of some modular forms of weight $ 3/2. $ In particular, unconditionally, we obtain the values of $ \amalg\amalg(E / \Q(\sqrt{D})) $ for some elliptic curves $ E $ and integers $ D, $ e.g., we show that all $ \amalg(E/\Q(\sqrt{D})) $ are trivial for the elliptic curve $ E $ of conductor $ 37 $ and the 23 integers $ D $ in Kolyvagin's papers.

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