Curves and Their Reductions

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


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.


Singular Point Meromorphic Function Elliptic Curve Irreducible Component Abelian Variety 
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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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