A fundamental domain for the Fermat curves and their quotients.
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J. Guardia (2000)
Revista de la Real Academia de Ciencias Exactas Físicas y Naturales
R. Clement Fernández, J. M. Echarri Hernández, E. J. Gómez Ayala (2011)
Acta Arithmetica
Tomasz Jędrzejak (2016)
Banach Center Publications
This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) , and its Jacobians , where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of (resp. ). The main tools are computations of the zeta function of (resp. ) over for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp))...
David Grant (1996)
Acta Arithmetica
Jean-Marc Couveignes (1996)
Journal de théorie des nombres de Bordeaux
Le théorème de Belyi affirme que sur toute courbe algébrique lisse projective et géométriquement connexe, définie sur , il existe une fonction non ramifiée en dehors de . Nous montrons que cette fonction peut être choisie sans automorphismes, c’est-à-dire telle que pour tout automorphisme non trivial de , on ait . Nous en déduisons que si est une extension finie de , toute -classe d’isomorphisme de courbes algébriques lisses projectives géométriquement connexes peut être caractérisée...
Matthew Klassen, Pavlos Tzermias (1997)
Acta Arithmetica
Rafael von Känel (2014)
Journal de Théorie des Nombres de Bordeaux
Let be a hyperelliptic curve of genus over a number field with good reduction outside a finite set of places of . We prove that has a Weierstrass model over the ring of integers of with height effectively bounded only in terms of , and . In particular, we obtain that for any given number field , finite set of places of and integer one can in principle determine the set of -isomorphism classes of hyperelliptic curves over of genus with good reduction outside .
Shujuan Ji (1998)
Acta Arithmetica
We construct analogs of the classical Δ-function for quotients of the upper half plane 𝓗 by certain arithmetic triangle groups Γ coming from quaternion division algebras B. We also establish a relative integrality result concerning modular functions of the form Δ(αz)/Δ(z) for α in B⁺. We give two explicit examples at the end.
Jordi Guàrdia (2001)
Journal de théorie des nombres de Bordeaux
Arakelov invariants of arithmetic surfaces are well known for genus 1 and 2 ([4], [2]). In this note, we study the modular height and the Arakelov self-intersection for a family of curves of genus 3 with many automorphisms:Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of as a product of three elliptic curves. Using the corresponding...
Pavlos Tzermias (1998)
Acta Arithmetica
Ahmed Abbes, Emmanuel Ullmo (1997)
Journal für die reine und angewandte Mathematik
Francesc Bars, Aristides Kontogeorgis, Xavier Xarles (2013)
Acta Arithmetica
We determine all modular curves X(N) (with N ≥ 7) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of X(N) is PSL₂(ℤ/Nℤ), whence it coincides with the normalizer of Γ(N) in PSL₂(ℝ) modulo ±Γ(N).
Daeyeol Jeon, Chang Heon Kim (2004)
Acta Arithmetica
Dimitrios Poulakis (1998)
Acta Arithmetica
Dimitrios Poulakis (2003)
Acta Arithmetica
E. V. Flynn, N. P. Smart (1997)
Acta Arithmetica
Yukihiro Uchida (2011)
Acta Arithmetica
Tomasz Jędrzejak, Maciej Ulas (2010)
Acta Arithmetica
Tomasz Jędrzejak (2013)
Acta Arithmetica
Consider the families of curves and where A is a nonzero rational. Let and denote their respective Jacobian varieties. The torsion points of and are well known. We show that for any nonzero rational A the torsion subgroup of is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for (A ≠ 4) and . We also almost...
David Bourqui (2009)
Annales de l’institut Fourier
Nous établissons une version de la conjecture de Manin pour le plan projectif éclaté en trois points alignés, le corps de base étant un corps global de caractéristique positive.
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