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

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.

💡 Research Summary

The paper studies curves C over a number field K that admit a finite group G of K‑automorphisms. The presence of such a symmetry forces the ℓ‑adic Tate module of the Jacobian, TℓJac(C), to decompose as a direct sum of irreducible G‑representations. For each irreducible representation τ of G the authors define a “motivic piece” Cτ by taking the τ‑isotypic component of the first ℓ‑adic cohomology H¹ℓ(C). The associated L‑function L(Cτ,s) is then defined in the usual way via Frobenius traces on this τ‑piece. This yields a formal factorisation \