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 $J_{3,5,a}\left(\mathbb{Q}\right)$ (resp. $J_{q,p,a}\left(\mathbb{Q}\right)$). 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))...

Le théorème de Belyi affirme que sur toute courbe algébrique $C$ lisse projective et géométriquement connexe, définie sur $\overline{\mathbb{Q}}$, il existe une fonction $f$ non ramifiée en dehors de $\left\{0,1,\infty \right\}$. 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 $f\circ 𝔞\ne f$. Nous en déduisons que si $𝕂\subset ℂ$ est une extension finie de $\mathbb{Q}$, toute $𝕂$-classe d’isomorphisme de courbes algébriques lisses projectives géométriquement connexes peut être caractérisée...

Let $C$ be a hyperelliptic curve of genus $g\ge 1$ 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 $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.

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 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:$$C_n:Y^4=X^4-(4n-2)X^2{Z}^2+Z^4.$$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...