Arithmetic on elliptic threefolds
Arithmetic on elliptic threefolds
复制标题
椭圆三重的算术
作者：
R. Wazir
在最近的一篇论文中，罗森和西尔弗曼表明，泰特的猜想代数周期意味着一个公式的长尾，这使排名的椭圆表面的加权平均纤维Frobenius迹值。在本文中，我们将他们的结果推广到椭圆三重的情况。我们的论点的主要成分是一个Shioda-Tate-like椭圆三倍公式，以及奇异纤维上的有理点的“平均”数量与这些纤维上的伽罗瓦作用之间的关系。
In a recent paper, Rosen and Silverman showed that Tate's conjecture on algebraic cycles implies a formula of Nagao, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. In this article, we extend their result to the case of elliptic threefolds. The main ingredients of our argument are a Shioda–Tate-like formula for elliptic threefolds, and a relation between the ‘average’ number of rational points on singular fibers and the Galois action on those fibers.