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Topological types of symmetries of elliptic-hyperelliptic Riemann surfaces and an application to moduli spaces.

José A. Bujalance, Antonio F. Costa, Ana M. Porto (2003)

RACSAM

Sea X una superficie de Riemann de género g. Diremos que la superficie X es elíptica-hiperelíptica si admite una involución conforme h de modo que X/〈h〉 tenga género uno. La involución h se llama entonces involución elíptica-hiperelíptica. Si g > 5 entonces la involución h es única, ver [1]. Llamamos simetría a toda involución anticonforme de X. Sea Aut±(X) el grupo de automorfismos conformes y anticonformes de X y σ, τ dos simetrías de X con puntos fijos y tales que {σ, hσ} y {τ, hτ} no...

Torelli theorem for stable curves

Lucia Caporaso, Filippo Viviani (2011)

Journal of the European Mathematical Society

We study the Torelli morphism from the moduli space of stable curves to the moduli space of principally polarized stable semi-abelic pairs. We give two characterizations of its fibers, describe its injectivity locus, and give a sharp upper bound on the cardinality of finite fibers. We also bound the dimension of infinite fibers.

Torsion des courbes elliptiques sur les corps cubiques

Pierre Parent (2000)

Annales de l'institut Fourier

On donne la liste (à un élément près) des nombres premiers qui sont l’ordre d’un point de torsion d’une courbe elliptique sur un corps de nombres de degré trois.

Torsion points in families of Drinfeld modules

Dragos Ghioca, Liang-Chung Hsia (2013)

Acta Arithmetica

Let Φ λ be an algebraic family of Drinfeld modules defined over a field K of characteristic p, and let a,b ∈ K[λ]. Assume that neither a(λ) nor b(λ) is a torsion point for Φ λ for all λ. If there exist infinitely many λ ∈ K̅ such that both a(λ) and b(λ) are torsion points for Φ λ , then we show that for each λ ∈ K̅, a(λ) is torsion for Φ λ if and only if b(λ) is torsion for Φ λ . In the case a,b ∈ K, we prove in addition that a and b must be ̅ p -linearly dependent.

Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

David Masser, Umberto Zannier (2015)

Journal of the European Mathematical Society

In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples...

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