# A Weil-etale version of the Birch and Swinnerton-Dyer formula over function fields

## TL;DR

This paper reformulates the Birch and Swinnerton-Dyer conjecture over function fields using Weil-etale cohomology and demonstrates its validity assuming the Tate-Shafarevich group is finite.

## Contribution

It introduces a Weil-etale cohomology framework for the BSD conjecture over function fields and proves its validity under certain finiteness assumptions.

## Key findings

- Reformulation of BSD conjecture via Weil-etale cohomology
- Proof of conjecture validity assuming Tate-Shafarevich group finiteness
- New cohomological perspective on BSD over function fields

## Abstract

We give a reformulation of the Birch and Swinnerton-Dyer conjecture over global function fields in terms of Weil-etale cohomology of the curve with coefficients in the Neron model, and show that it holds under the assumption of finiteness of the Tate-Shafarevich group.