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 $E/K$ be an elliptic curve defined over a number field, and let $P\in E\left(K\right)$ be a point of infinite order. It is natural to ask how many integers $n\ge 1$ fail to occur as the order of $P$ modulo a prime of $K$. For $K=\mathbb{Q}$, $E$ a quadratic twist of $y^2=x^3-x$, and $P\in E\left(\mathbb{Q}\right)$ as above, we show that there is at most one such $n\ge 3$.

We investigate possible orders of reductions of a point in the Mordell-Weil groups of certain abelian varieties and in direct products of the multiplicative group of a number field. We express the result obtained in terms of divisibility sequences.

We remark that Tate’s algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of $v\left(a_i\right)/i$. As an application, we deduce the behaviour of Kodaira types in tame extensions of local fields.

We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$\left\{\begin{array}{c}n=a_1+a_2+\cdots +a_{s-1},\hfill \\ a_1{a}_2\cdots a_{s-1}(a_1+a_2+\cdots +a_{s-1})=b^s\hfill \end{array}\right.$$ has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\ge 3.$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s=3$, but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s\ge 4$. As a corollary, for $s\ge 4$ and any positive integer $n$, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...