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

Harry Spencer

TL;DR

The paper develops and tests a framework in which the $L$-function of a curve $C$ with a finite group action $G$ factors into motivic pieces $L(C^ au,s)$ attached to irreducible $G$-representations $ au$. A key contribution is Algorithm 3 (The algorithm), which reduces the computation of Euler factors for $L(C^ au,s)$ to counting points on auxiliary curves, enabling explicit numerical verifications in cases with $G=C_3$, $C_4$, and $D_{10}$, and allowing factorisations of Jacobians of GL$_2$-type via Hecke-type endomorphisms. The work further connects these motivic pieces to Deligne’s Period Conjecture, providing a practical method to compute Deligne periods for motivic pieces arising from superelliptic curves and giving substantial numerical evidence for the conjecture in exponent-three and exponent-four settings. Overall, this framework links explicit point-counting, modularity phenomena, and period conjectures to provide both computational tools and numerical validation for conjectures in arithmetic geometry related to motivic L-functions.

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.

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

TL;DR

The paper develops and tests a framework in which the -function of a curve with a finite group action factors into motivic pieces attached to irreducible -representations . A key contribution is Algorithm 3 (The algorithm), which reduces the computation of Euler factors for to counting points on auxiliary curves, enabling explicit numerical verifications in cases with , , and , and allowing factorisations of Jacobians of GL-type via Hecke-type endomorphisms. The work further connects these motivic pieces to Deligne’s Period Conjecture, providing a practical method to compute Deligne periods for motivic pieces arising from superelliptic curves and giving substantial numerical evidence for the conjecture in exponent-three and exponent-four settings. Overall, this framework links explicit point-counting, modularity phenomena, and period conjectures to provide both computational tools and numerical validation for conjectures in arithmetic geometry related to motivic L-functions.

Abstract

Given a curve over a number field equipped with the action of a finite group by -automorphisms, one obtains a factorisation of into a product of -functions of `motivic pieces of curves' associated to irreducible -representations. We describe an algorithm for explicitly computing values of these -functions, demonstrating implementations in the cases of certain curves with actions by , and . We explain how this algorithm can be used to factor -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 -functions arising from motivic pieces of superelliptic curves.
Paper Structure (14 sections, 13 theorems, 45 equations, 1 table, 1 algorithm)

This paper contains 14 sections, 13 theorems, 45 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivic pieces of curves
  3. The algorithm
  4. Endomorphisms of Jacobians
  5. Endomorphisms of Hecke type
  6. $\mathop{\mathrm{GL}}\nolimits_2$-type and modularity
  7. An example
  8. Periods
  9. Motives
  10. Deligne's Period Conjecture
  11. Superelliptic curves
  12. Computations
  13. Exponent three
  14. Exponent four

Key Result

Theorem 1.4

For the $14$ genus $3$ superelliptic curves with $L(C,1)\ne0$ considered in §sec:examples, there exist $a,b,c\in\mathbb{Z}$ with $|a|,|b|\le 16$ and $|c|\le160$ such that where $\tau$ is a non-$\mathbb{Q}$-rational character of $C_3$ or $C_4$, $L^{\text{approx}}(C^\tau,1)$ is our numerical value for $L(C^\tau,1)$, $\Omega^\text{approx}_{C^\tau}$ is that for the Deligne period as in §sec:super_per

Theorems & Definitions (63)

  • Conjecture 1.2
  • Theorem 1.4
  • Remark 1.6
  • Remark 1.7
  • Example 1.8: = Example \ref{['ex:pic1']}
  • Remark 1.11
  • Proposition 1.12: = Proposition \ref{['prop:mod_form']}
  • Definition 2.1: = DGKM2
  • Definition 2.2
  • Remark 2.3
  • ...and 53 more