The Tate conjecture for surfaces of geometric genus one -- embracing singularities

TL;DR

This paper advances the proof of the Tate conjecture for certain algebraic surfaces with singularities and applies these results to verify the Birch--Swinnerton-Dyer conjecture for specific elliptic curves over function fields.

Contribution

It introduces new techniques for analyzing singular models of surfaces of geometric genus one, completing a major step in the Tate conjecture program.

Findings

Proves the Tate conjecture for a broad class of singular surfaces.

Shows elliptic curves of height one over certain function fields satisfy BSD.

Develops methods to handle singularities in algebraic surface analysis.

Abstract

In this article, we aim to largely complete the program of proving the Tate conjecture for surfaces of geometric genus one, by introducing techniques to analyze those surfaces whose "natural models" are singular. As an application, we show that every elliptic curve of height one over a global function field of genus one and characteristic $p \ge 11$ satisfies the Birch--Swinnerton-Dyer conjecture.