On p-adic valuations of L(1) of elliptic curves with CM by √-3

Author： Qiu Derong  

Publisher： Royal Society of Edinburgh

ISSN： 1473-7124

Source： Proceedings Section A: Mathematics - Royal Society of Edinburgh, Vol.133, Iss.6, 2003-12, pp. : 1389-1407

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Abstract

For positive rational integers λ, we study the Hecke L-series attached to elliptic curves y2 = x3 - 2433Dλ over the quadratic field Q(√-3) and obtain various bounds of p(= 2, 3)-adic valuations of their values at s = 1 according to the cases of D and λ. In particular, for the case of even λ, we obtain a criterion of reaching the bounds of 3-adic valuations. From this, combining with the work of Coates and Wiles and Rubin, we obtain some results about the conjecture of Birch and Swinnerton-Dyer of these curves.

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