Motivic pieces of curves: $L$-functions and periods

TL;DR

This paper develops an algorithm to compute $L$-functions of motivic pieces of curves with group actions, enabling explicit factorization and verification of Deligne's Period Conjecture for these $L$-functions.

Contribution

It introduces a novel algorithm for explicit computation of motivic $L$-functions and demonstrates its application to curves with specific automorphism groups, advancing understanding of their arithmetic properties.

Findings

Successfully computed $L$-functions for curves with $C_3$, $C_4$, and $D_{10}$ actions.

Verified Deligne's Period Conjecture numerically for new classes of $L$-functions.

Provided a method to factor $L$-functions of curves with endomorphisms of Hecke type.

Abstract

Given a curve $C$ over a number field $K$ equipped with the action of a finite group $G$ by $K$-automorphisms, one obtains a factorisation of $L(C,s)$ into a product of $L$-functions of `motivic pieces of curves' associated to irreducible $G$-representations. We describe an algorithm for explicitly computing values of these $L$-functions, demonstrating implementations in the cases of certain curves with actions by $C_3$, $C_4$ and $D_{10}$. We explain how this algorithm can be used to factor $L$-functions of curves with endomorphisms of Hecke type. Towards applications, we explicitly formulate and numerically verify a version of Deligne's Period Conjecture for hitherto-uninvestigated $L$-functions arising from motivic pieces of superelliptic curves.