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Curves and Their Reductions

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Part of the Progress in Mathematics book series (PM, volume 218)

Abstract

Let k be a complete non-archimedean valued field and qk with \( 0 < \left| q \right| < 1 \). We will use both the multiplicative group Gm,k over k and its analytification \( G_{m,k}^{an} \). One writes < q > for the subgroup of k* generated by q. The elements in < q > are seen as automorphisms of \( G_{m,k}^{an} \). The Tate curve is the object \( \mathcal{T}: = G_{m,k}^{an} /\left\langle q \right\rangle \) (we keep this somewhat heavy notation in order to avoid confusions which might arise from the notation k* / < q >). In the sequel we will explain the rigid analytic structure of \( \mathcal{T} \), compute the field of meromorphic functions on it and show that \( \mathcal{T} \) is the analytification of an elliptic curve over k of a special type.

Keywords

Singular Point Meromorphic Function Elliptic Curve Irreducible Component Abelian Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  1. 1.Théorie des nombres et Algorithmique arithmétique (A2X) UMR 5465Université Bordeaux 1TalenceFrance
  2. 2.Department of MathematicsUniversity of GroningenGroningenThe Netherlands

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