Generalized Jacobians and explicit descents

Abstract:

We develop a cohomological description of 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, and base point free divisor $\frak {m}$ on a curve $C$, we show that multiplication by $n$ on the generalized Jacobian $J_\frak {m}$ factors through an isogeny $\varphi :A_\frak {m} \to J_\frak {m}$ whose kernel is dual to the Galois module of divisor classes $D$ such that $nD$ is linearly equivalent to some multiple of $\frak {m}$. By geometric class field theory, this corresponds to an abelian covering of $C_{\overline {k}} := C \times _{\mathrm {Spec}{k}} \mathrm {Spec}(\overline {k})$ of exponent $n$ unramified outside $\frak {m}$. We show that 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 to the computation of Mordell–Weil ranks of nonhyperelliptic curves.