Curves and Their Reductions

Abstract

Let k be a complete non-archimedean valued field and q ∈ k with $0 < \left| q \right| < 1$. We will use both the multiplicative group G_m,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.