Generalized Jacobians and explicit descents

TL;DR

This paper generalizes the cohomological description of explicit descents on curves using generalized Jacobians, linking them to abelian coverings and providing tools for computing Mordell-Weil groups.

Contribution

It introduces a new cohomological framework for explicit descents via generalized Jacobians, extending known results for hyperelliptic curves and connecting to class field theory.

Findings

Describes multiplication by n as an isogeny on generalized Jacobians.

Relates n-coverings to abelian unramified coverings of the curve.

Provides methods for computing Mordell-Weil groups of Jacobians.

Abstract

We develop a cohomological description of various explicit descents in terms of generalized Jacobians, generalizing the known description for hyperelliptic curves. Specifically, given an integer $n$ dividing the degree of some reduced effective divisor $\mathfrak{m}$ on a curve $C$, we show that multiplication by $n$ on the generalized Jacobian $J_\frak{m}$ factors through an isogeny $\varphi:A_{\mathfrak{m}} \rightarrow J_{\mathfrak{m}}$ whose kernel is naturally the dual of the Galois module $(\operatorname{Pic}(C_{\overline{k}})/\mathfrak{m})[n]$. By geometric class field theory, this corresponds to an abelian covering of $C_{\overline{k}} := C \times_{\operatorname{Spec}{k}} \operatorname{Spec}(\overline{k})$ of exponent $n$ unramified outside $\mathfrak{m}$. The $n$-coverings of $C$ parameterized by explicit descents are the maximal unramified subcoverings of the $k$-forms of this ramified covering. We present applications of this to the computation of Mordell-Weil groups of Jacobians.