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Explicit Selmer groups for cyclic covers of P¹

Title data

Stoll, Michael ; van Luijk, Ronald:
Explicit Selmer groups for cyclic covers of P¹.
In: Acta Arithmetica. Vol. 159 (2013) . - pp. 133-148.
ISSN 0065-1036
DOI: https://doi.org/10.4064/aa159-2-4

Official URL: Volltext

Abstract in another language

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.

Further data

Item Type: Article in a journal
Refereed: Yes
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II - Univ.-Prof. Dr. Michael Stoll
Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Result of work at the UBT: Yes
DDC Subjects: 500 Science > 510 Mathematics
Date Deposited: 28 Nov 2014 08:09
Last Modified: 28 Nov 2014 08:09
URI: https://eref.uni-bayreuth.de/id/eprint/4181