Explicit Selmer groups for cyclic covers of P^1

TL;DR

This paper introduces an explicit Selmer group for cyclic covers of P^1, which is computationally accessible and aids in explicit description of covering spaces, enhancing methods for second descents on Jacobians.

Contribution

It defines an explicit Selmer group for cyclic covers of P^1 that is isomorphic to the classical Selmer group and suitable for explicit computations.

Findings

The explicit Selmer group is isomorphic to the classical Selmer group.

It provides a more computationally accessible tool for studying Jacobians.

Potential applications include improved methods for second descents.

Abstract

For any abelian variety J over a global field k and an isogeny phi: J -> J, the Selmer group Sel^phi(J,k) is a subgroup of the Galois cohomology group H^1(Gal(ksep/k), J[phi]), defined in terms of local data. When J is the Jacobian of a cyclic cover of P^1 of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the `fake Selmer group', whose definition is more amenable to explicit computations. In this paper we define in the same setting the `explicit Selmer group', which is isomorphic to the Selmer group itself and just as amenable to explicit computations as the fake Selmer group. This is useful for describing the associated covering spaces explicitly and may thus help in developing methods for second descents on the Jacobians considered.