Abelian motives and Shimura varieties in nonzero characteristic

TL;DR

This paper proposes a new, weaker theory of motives that could underpin the study of Shimura varieties in nonzero characteristic, bypassing the currently inaccessible conjectures like Hodge and Tate.

Contribution

It introduces a feasible, alternative motive theory that retains key implications for Shimura varieties, challenging the reliance on traditional conjectures.

Findings

A new motive theory is proposed that is weaker but still meaningful.

The theory offers a pathway to progress in understanding Shimura varieties.

It suggests potential for proof where traditional conjectures are out of reach.

Abstract

Much of the work on Shimura varieties over the last thirty years has been devoted to constructing the theory that would follow from a good notion of motives, one incorporating the Hodge, Tate, and standard conjectures. These conjectures are believed to be beyond reach, and may not even be correct as stated. I argue in this article that there exists a theory of motives, accessible to proof, weaker than Grothendieck's, but with many of the same consequences.