On the generalised Tate conjecture for products of elliptic curves over finite fields

TL;DR

This paper proves the generalized Tate conjecture for the third cohomology of products of elliptic curves over finite fields, with full proof for up to three isogeny classes, and discusses complexities beyond that.

Contribution

It extends the proof of the generalized Tate conjecture to certain cases involving products of elliptic curves over finite fields, improving understanding of the conjecture's scope.

Findings

Proves the conjecture for H^3 of products of elliptic curves over finite fields.

Fully proves the conjecture when elliptic curves are in at most 3 isogeny classes.

Highlights increased complexity for H^4 and beyond with more than 3 isogeny classes.

Abstract

We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3 isogeny classes. We also show how things become more intricate from H^4 onwards, for more that 3 isogeny classes.