2-selmer Groups, 2-class Groups and Rational Points on Elliptic Curves of Conductor 4d
نویسنده
چکیده
Let E : y = F (x) be an elliptic curve over Q defined by a monic irreducible integral cubic polynomial F (x) with negative and square-free discriminant −D. We determine its 2-Selmer rank in terms of the 2-rank of the class group of the cubic field L = Q[x]/F (x). We then interpret this result as a mod 2 congruence between the Hasse-Weil L-function of E and a degree two Artin L-function associated to the cubic field L and provide several explicit examples. When L has odd class number and the root number of E is −1, the Birch and SwinnertonDyer conjecture predicts that E(Q) should have rank one. When E has conductor 4D, we construct a canonical point in E(Q) using Heegner points on a Shimura curve of level 4D. We naturally conjecture it to be of infinite order and we verify this conjecture explicitly for the case D = 11.
منابع مشابه
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تاریخ انتشار 2015