Displaying similar documents to “The class group of a one-dimensional affinoid space”

Etale coverings of a Mumford curve

Marius Van Der Put (1983)

Annales de l'institut Fourier

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Let the field K be complete w.r.t. a non-archimedean valuation. Let X / K be a Mumford curve, i.e. the irreducible components of the stable reduction of X have genus 0. The abelian etale coverings of X are constructed using the analytic uniformization Ω → X and the theta-functions on X . For a local field K one rediscovers G . Frey’s description of the maximal abelian unramified extension of the field of rational functions of X .

The Brauer–Manin obstruction for curves having split Jacobians

Samir Siksek (2004)

Journal de Théorie des Nombres de Bordeaux

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Let X → 𝒜 be a non-constant morphism from a curve X to an abelian variety 𝒜 , all defined over a number field k . Suppose that X is a counterexample to the Hasse principle. We give sufficient conditions for the failure of the Hasse principle on X to be accounted for by the Brauer–Manin obstruction. These sufficiency conditions are slightly stronger than assuming that 𝒜 ( k ) and Ш ( 𝒜 / k ) are finite.

A group law on smooth real quartics having at least 3 real branches

Johan Huisman (2002)

Journal de théorie des nombres de Bordeaux

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Let C be a smooth real quartic curve in ℙ 2 . Suppose that C has at least 3 real branches B 1 , B 2 , B 3 . Let B = B 1 × B 2 × B 3 and let O ∈ B . Let τ O be the map from B into the neutral component Jac ( C ) ( ℝ ) 0 of the set of real points of the jacobian of C , defined by letting τ O ( P ) be the divisor class of the divisor ∑ P i - O i . Then, τ O is a bijection. We show that this allows an explicit geometric description of the group law on Jac ( C ) ( ℝ ) 0 . It generalizes the classical geometric description of the group law on the neutral component of the set of real...

On the variety of linear series on a singular curve

E. Ballico, C. Fontanari (2002)

Bollettino dell'Unione Matematica Italiana

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Let Y be an integral projective curve with g := p a ⁢ Y ≥ 2 . For all positive integers d , r let W d r ⁢ Y ⁢ * A * be the set of all L ∈ Pic d ⁢ Y with h 0 ⁢ Y , L ≥ r + 1 and L spanned. Here we prove that if d ≤ g - 2 , then dim ⁡ W d r ⁢ Y ⁢ * A * ≤ d - 3 ⁢ r except in a few cases (essentially if Y is a double covering).