Hijikata-Pizer-Shemanske曲线上的Heegner点以及Birch和Swinnerton-Dyer猜想

Pub Date : 2018-07-01 DOI:10.5565/PUBLMAT6221803

M. Longo, V. Rotger, Carlos de Vera-Piquero

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引用次数: 3

摘要

我们研究了椭圆曲线上的Heegner点，或者更一般的模阿贝尔变体，这些点来自于一种相当一般的四元数阶的Shimura曲线的均匀化。我们在这个大背景下解决了Birch和Swinnerton-Dyer (BSD)猜想引起的几个问题。特别地，在温和的技术条件下，我们证明了椭圆曲线上非扭转Heegner点在BSD猜想预测其存在的所有情况下的存在性。

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Heegner points on Hijikata-Pizer-Shemanske curves and the Birch and Swinnerton-Dyer conjecture

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rathergeneral type of quaternionic orders. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.

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