The discriminant and the determinant of a hypersurface of even dimension

TL;DR

This paper establishes a relationship between the quadratic character of the Galois group and the square root of the discriminant for smooth even-dimensional hypersurfaces, linking algebraic geometry and number theory.

Contribution

It provides a formula connecting the Galois group's quadratic character to the hypersurface's defining polynomial discriminant, a novel insight in algebraic geometry.

Findings

Quadratic character computed via discriminant square root

Link between Galois group and polynomial discriminant

New method for analyzing hypersurface invariants

Abstract

For a smooth hypersurface of even dimension, the quadratic character of the absolute Galois group defined by the determinant of the l-adic cohomology of middle dimension is computed via the square root of the discriminant of a defining polynomial of the hypersurface.