Interdisciplinary Information Sciences
Online ISSN : 1347-6157
Print ISSN : 1340-9050
ISSN-L : 1340-9050
Special Section: Japan-Korea Joint Seminar on Number Theory and Related Topics 2008
Congruence Relations Connecting Tate-Shafarevich Groups with Hurwitz Numbers
Yoshihiro ÔNISHI
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2010 年 16 巻 1 号 p. 71-86

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Let p > 3 be a rational prime congruent to 3 modulo 4, and h(−p) be the class number of the imaginary quadratic field Q(\\sqrt{-p}). Then h(−p)≡−2B\\frac{p+1}{2}modp, where Bn is the n-th Bernoulli number. This is a quite classical congruence. Under the full BSD conjecture, we provide an easy method to obtain the natural explicit generalization of this, which is a congruence between the conjectural order of the Tate-Shafarevich group for certain elliptic curve with Mordell-Weil rank 0 and a coefficient of power series expansion of an elliptic function associating the elliptic curve.

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© 2010 by the Graduate School of Information Sciences (GSIS), Tohoku University
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