Computing Periods of Hypersurfaces

TL;DR

This paper presents an algorithm that efficiently computes the periods of smooth projective hypersurfaces of any dimension, extending previous methods limited to plane curves, and enables high-precision arithmetic analysis.

Contribution

It introduces a novel algorithm that reduces period computation to solving Picard-Fuchs differential equations, applicable to hypersurfaces of any dimension.

Findings

Algorithm computes periods with high precision.

Extends period computation from plane curves to higher-dimensional hypersurfaces.

Provides exact initial conditions for cohomology pairings on Fermat hypersurfaces.

Abstract

We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard-Fuchs type. In this way, the periods can be computed to extreme-precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis.