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A note on the torsion of the Jacobians of superelliptic curves y q = x p + a

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) C q , p , a : y q = x p + a , and its Jacobians J q , p , a , where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of J 3 , 5 , a ( ) (resp. J q , p , a ( ) ). The main tools are computations of the zeta function of C 3 , 5 , a (resp. C q , p , a ) over l for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp))...

À propos du théorème de Belyi

Jean-Marc Couveignes (1996)

Journal de théorie des nombres de Bordeaux

Le théorème de Belyi affirme que sur toute courbe algébrique C lisse projective et géométriquement connexe, définie sur ¯ , il existe une fonction f non ramifiée en dehors de 0 , 1 , . Nous montrons que cette fonction peut être choisie sans automorphismes, c’est-à-dire telle que pour tout automorphisme non trivial a de C , on ait f 𝔞 f . Nous en déduisons que si 𝕂 est une extension finie de , toute 𝕂 -classe d’isomorphisme de courbes algébriques lisses projectives géométriquement connexes peut être caractérisée...

An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel (2014)

Journal de Théorie des Nombres de Bordeaux

Let C be a hyperelliptic curve of genus g 1 over a number field K with good reduction outside a finite set of places S of K . We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g , S and K . In particular, we obtain that for any given number field K , finite set of places S of K and integer g 1 one can in principle determine the set of K -isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside S .

Analogs of Δ(z) for triangular Shimura curves

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.

Arakelov computations in genus 3 curves

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: C n : Y 4 = X 4 - ( 4 n - 2 ) X 2 Z 2 + Z 4 . Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve C n in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of C n as a product of three elliptic curves. Using the corresponding...

Bielliptic and hyperelliptic modular curves X(N) and the group Aut(X(N))

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).

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak (2013)

Acta Arithmetica

Consider the families of curves C n , A : y ² = x + A x and C n , A : y ² = x + A where A is a nonzero rational. Let J n , A and J n , A denote their respective Jacobian varieties. The torsion points of C 3 , A ( ) and C 3 , A ( ) are well known. We show that for any nonzero rational A the torsion subgroup of J 7 , A ( ) is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to J 7 , A ( ) [ 2 ] (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 J 3 , A (A ≠ 4) and J 5 , A . We also almost...

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