A note on Tate's conjectures for abelian varieties

TL;DR

This paper provides an exposition and proof of Tate's two conjectures for algebraic cycles on specific products of elliptic curves and abelian surfaces over number fields.

Contribution

It offers a clear explanation and proof of Tate's conjectures for particular classes of abelian varieties, expanding understanding in this area.

Findings

Proof of Tate's conjectures for certain products of elliptic curves

Proof of Tate's conjectures for certain abelian surfaces

Clarification of the conjectures' implications for algebraic cycles

Abstract

In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.