Hodge and Tate conjectures for hypergeometric sheaves

TL;DR

This paper investigates the Hodge and Tate conjectures for hypergeometric sheaves associated with Gel'fand Zelevinski functions, proving them up to Fermat motifs using cohomological Mellin transform techniques.

Contribution

It formulates the Hodge and Tate conjectures for hypergeometric sheaves and proves these conjectures up to Fermat motifs, advancing understanding in algebraic geometry.

Findings

Proved Hodge and Tate conjectures for hypergeometric sheaves up to Fermat motifs.

Utilized cohomological Mellin transform to establish main results.

Formulated conjectures in terms of variation of Hodge structure and l-adic sheaves.

Abstract

A constructible sheaf corresponding to Gel'fand Zelevinski hypergeometric functions on a torus is called hypergeometric sheaf. We consider Hodge and Tate conjectrue for hypergeomtric sheaves. Hodge conjecture is formulated in terms of variation of Hodge strucure and Tate conjecture is done for l-adic sheaves on an open set of torus. We prove Hodge and Tate conjecture up to Hodge and Tate cycle of Fermat motifes. We use cohomological Mellin transform to get the main theorem. This is the final revision for preprint.