Elliptic curves of rank 1 satisfying the 3-part of the Birch and Swinnerton–Dyer conjecture |
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Authors: | Dongho Byeon |
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Affiliation: | Department of Mathematics, Seoul National University, Seoul, Republic of Korea |
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Abstract: | Let E be an elliptic curve over Q of conductor N and K be an imaginary quadratic field, where all prime divisors of N split. If the analytic rank of E over K is equal to 1, then the Gross and Zagier formula for the value of the derivative of the L-function of E over K, when combined with the Birch and Swinnerton–Dyer conjecture, gives a conjectural formula for the order of the Shafarevich–Tate group of E over K. In this paper, we show that there are infinitely many elliptic curves E such that for a positive proportion of imaginary quadratic fields K, the 3-part of the conjectural formula is true. |
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Keywords: | Elliptic curves Quadratic fields Birch and Swinnerton&ndash Dyer conjecture |
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