Hypergeometric decomposition of symmetric K3 quartic pencils

Area of Science:

• Algebraic Geometry
• Number Theory
• Motivic Theory

Background:

• Delsarte K3 quartic hypersurfaces are fundamental objects in algebraic geometry.
• Understanding their associated hypergeometric functions is crucial for deeper insights into their structure.
• One-parameter deformations offer a way to study families of these hypersurfaces.

Purpose of the Study:

• To compute Picard-Fuchs differential equations for five one-parameter deformations of Delsarte K3 quartic hypersurfaces.
• To count points on these hypersurfaces using Gauss sums and finite-field hypergeometric sums.
• To connect these computations to the zeta function and global L-functions, providing a description of their motives.

Main Methods:

• Computation of Picard-Fuchs differential equations.
• Point counting over finite fields using Gauss sums.
• Rewriting results in terms of finite-field hypergeometric sums.
• Matching differential equations to factors of the zeta function.

Main Results:

• Explicit computation of all Picard-Fuchs differential equations for the studied hypersurfaces.
• A complete description of the motives for these pencils in terms of hypergeometric motives.
• The zeta function factors are expressed in terms of global L-functions.

Conclusions:

• The study provides a comprehensive and explicit description of the motives associated with these K3 hypersurface families.
• The methods used successfully link differential equations, point counting, and L-functions.
• This work advances the understanding of hypergeometric motives in algebraic geometry.