Hirzebruch-Zagier classes and rational elliptic curves over quintic fields
Number Theory
2024-02-19 v4
Abstract
Conditionally on a conjecture on the \'etale cohomology of Hilbert modular surfaces and some minor technical assumptions, we establish new instances of the equivariant BSD-conjecture in rank with applications to the arithmetic of rational elliptic curves over quintic fields. The key ingredients are a refinement of twisted triple product -adic -functions, the construction of a compatible collection of Hirzebruch-Zagier cycles and an explicit reciprocity law relating the two.
Cite
@article{arxiv.2005.02520,
title = {Hirzebruch-Zagier classes and rational elliptic curves over quintic fields},
author = {Michele Fornea and Zhaorong Jin},
journal= {arXiv preprint arXiv:2005.02520},
year = {2024}
}
Comments
Revised version. Corrected several minor mistakes. Note that Sangiovanni-Skinner announced a proof of (a generalization of) our conjecture on the etale cohomology of Hilbert modular surfaces