The Hilbert-Polya strategy and height pairings

TL;DR

This paper explores how a cohomological framework related to the Hilbert-Polya strategy could imply positivity properties of height pairings, offering insights into the Riemann hypothesis for Hasse-Weil zeta functions.

Contribution

It proposes a cohomological approach to connect the Hilbert-Polya strategy with positivity of height pairings, advancing the theoretical understanding of the Riemann hypothesis.

Findings

Sketches how cohomological formalism implies positivity of height pairings

Provides a conjectural link between Hilbert-Polya strategy and height pairings

Suggests a new perspective on the Riemann hypothesis for Hasse-Weil zeta functions

Abstract

Previously we gave a conjectural cohomological argument for the validity of the Riemann hypotheses for Hasse-Weil zeta functions. In the present note we sketch how the same cohomological formalism would imply the conjectured positivity properties of the height pairings of homologically trivial cycles.