Abstract

We consider the problem of explicitly computing Beilinson–Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes, between the height of certain limit mixed Hodge structures and certain Beilinson–Bloch heights obtained from odd-dimensional hypersurfaces with a node. This congruence suggests a new method to compute Beilinson–Bloch heights. Here we explain how to compute the relevant limit mixed Hodge structures in practice, then apply our computational method to a nodal quartic curve and a nodal cubic threefold. In both cases we explain the nature of the primes occurring in the congruence.

摘要

我们考虑在数域上定义的变体上显式计算同源平​​凡循环的 Beilinson–Bloch 高度的问题。最近的结果已经在一定极限混合霍奇结构的高度与从具有节点的奇数维超曲面获得的某些 Beilinson-Bloch 高度之间建立了一致，直到素数对数的有理跨度。这种一致性提出了一种计算 Beilinson-Bloch 高度的新方法。在这里，我们解释了如何在实践中计算相关极限混合霍奇结构，然后将我们的计算方法应用于节点四次曲线和节点三次三重曲线。在这两种情况下，我们都解释了同余中出现的素数的性质。