Explicit Selmer groups for cyclic covers of ℙ¹

Michael Stoll, and Ronald van Luijk. "Explicit Selmer groups for cyclic covers of ℙ¹." Acta Arithmetica 159.2 (2013): 133-148. <http://eudml.org/doc/286263>.

@article{MichaelStoll2013,
abstract = {For any abelian variety J over a global field k and an isogeny ϕ: J → J, the Selmer group $Sel^\{ϕ\}(J,k)$ is a subgroup of the Galois cohomology group $H¹(Gal(k^\{s\}/k),J[ϕ])$, defined in terms of local data. When J is the Jacobian of a cyclic cover of ℙ¹ 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.},
author = {Michael Stoll, Ronald van Luijk},
journal = {Acta Arithmetica},
keywords = {Selmer group; fake Selmer group; descent},
language = {eng},
number = {2},
pages = {133-148},
title = {Explicit Selmer groups for cyclic covers of ℙ¹},
url = {http://eudml.org/doc/286263},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Michael Stoll
AU - Ronald van Luijk
TI - Explicit Selmer groups for cyclic covers of ℙ¹
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 2
SP - 133
EP - 148
AB - For any abelian variety J over a global field k and an isogeny ϕ: J → J, the Selmer group $Sel^{ϕ}(J,k)$ is a subgroup of the Galois cohomology group $H¹(Gal(k^{s}/k),J[ϕ])$, defined in terms of local data. When J is the Jacobian of a cyclic cover of ℙ¹ 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.
LA - eng
KW - Selmer group; fake Selmer group; descent
UR - http://eudml.org/doc/286263
ER -