The twisting representation of the $L$-function of a curve

TL;DR

This paper explores how twisting a curve affects its L-function by relating the associated l-adic representations through Artin representations, providing explicit relations for genus 2 curves with specific automorphism groups.

Contribution

It establishes explicit relations between the local factors of L-functions of a curve and its twists using Artin representations, especially for genus 2 curves with automorphism groups D_8 or D_{12}.

Findings

Explicit relations between local L-factors of curves and their twists.

Determination of local factors for genus 2 curves with specific automorphism groups.

Application to a broad class of twists of selected curves.

Abstract

Let C be a smooth projective curve defined over a number field and let C' be a twist of C. In this article we relate the l-adic representations attached to the l-adic Tate modules of the Jacobians of C and C' through an Artin representation. This representation induces global relations between the local factors of the respective Hasse-Weil L-functions. We make these relations explicit in a particularly illustrating situation. For every Qbar-isomorphism class of genus 2 curves defined over Q with automorphism group isomorphic to D_8 or D_{12}, except for a finite number, we choose a representative curve C/Q such that, for every twist C' of C satisfying some mild condition, we are able to determine either the local factor L_p(C'/Q,T) or the product L_p(C'/Q,T)L_p(C'/Q,-T) from the local factor L_p(C/Q,T).