The standard sign conjecture on algebraic cycles: the case of Shimura varieties

TL;DR

This paper links the standard sign conjecture for Shimura varieties to automorphic representations, showing how certain conjectures imply the sign conjecture and discussing current knowledge about these assumptions.

Contribution

It establishes a connection between the standard sign conjecture and automorphic representation conjectures for Shimura varieties, providing a new approach to the problem.

Findings

Standard sign conjecture can be deduced from automorphic representation conjectures.

Current knowledge about these automorphic conjectures is summarized.

The paper clarifies the relationship between algebraic cycles and automorphic forms.

Abstract

We show how to deduce the standard sign conjecture (a weakening of the K\"unneth standard conjecture) for Shimura varieties from some statements about discrete automorphic representations (Arthur's conjectures plus a bit more). We also indicate what is known (to us) about these statements.