Explicit Selmer groups for cyclic covers of $\mathbb {P}^1$

Michael Stoll ¹   ; Ronald van Luijk ²

¹ Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany
² Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA, Leiden, The Netherlands

Acta Arithmetica, Tome 159 (2013) no. 2, pp. 133-148

Résumé

For any abelian variety $J$ over a global field $k$ and an isogeny $\phi \colon J \to J$, the Selmer group $\mathop{\rm Sel}\nolimits^\phi(J,k)$ is a subgroup of the Galois cohomology group ${\rm H}^1(\mathop{\rm Gal}\nolimits({k^{\rm s}}/k), J[\phi])$, defined in terms of local data. When $J$ is the Jacobian of a cyclic cover of $\mathbb{P}^1$ 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.

Détail

DOI : 10.4064/aa159-2-4

Keywords: abelian variety nbsp global field nbsp isogeny phi colon selmer group nbsp mathop sel nolimits phi subgroup galois cohomology group mathop gal nolimits phi defined terms local jacobian cyclic cover nbsp mathbb prime degree nbsp selmer group has quotient subgroup order nbsp isomorphic fake selmer group whose definition amenable explicit computations paper define setting explicit selmer group which isomorphic selmer group itself just amenable explicit computations fake selmer group useful describing associated covering spaces explicitly may help developing methods second descents jacobians considered