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Tamagawa number divisibility of central \(L\)-values of twists of the Fermat elliptic curve. (English. French summary) Zbl 1496.11081

Summary: Given any integer \(N>1\) prime to 3, we denote by \(C_N\) the elliptic curve \(x^3+y^3=N\). We first study the 3-adic valuation of the algebraic part of the value of the Hasse-Weil \(L\)-function \(L(C_N, s)\) of \(C_N\) over \(\mathbb{Q}\) at \(s=1\), and we exhibit a relation between the 3-part of its Tate-Shafarevich group and the number of distinct prime divisors of \(N\) which are inert in the imaginary quadratic field \(K=\mathbb{Q}(\sqrt{-3})\). In the case where \(L(C_N, 1)\ne 0\) and \(N\) is a product of split primes in \(K\), we show that the order of the Tate-Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.

MSC:

11G05 Elliptic curves over global fields
14H52 Elliptic curves
11R23 Iwasawa theory

Software:

SageMath; Magma
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Full Text: DOI arXiv

References:

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