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

Volume 159 / 2013

Michael Stoll, Ronald van Luijk Acta Arithmetica 159 (2013), 133-148 MSC: Primary 11G30; Secondary 11G10, 14G25, 14H30, 14H40. DOI: 10.4064/aa159-2-4

Abstract

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.

Authors

  • Michael StollMathematisches Institut
    Universität Bayreuth
    95440 Bayreuth, Germany
    e-mail
  • Ronald van LuijkMathematisch Instituut
    Universiteit Leiden
    Postbus 9512
    2300 RA, Leiden, The Netherlands
    e-mail

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