It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field $k$ and any finite set of places $S$ of $k$, one can effectively compute the set of isomorphism classes of hyperelliptic curves over $k$ with good reduction outside $S$. We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus $g$ would imply an effective version of Siegel’s theorem for integral points on...

We give some easy necessary and sufficient criteria for twists of abelian varieties by Artin representations to be simple.

Ce texte est un survey concernant la question du rang d’une variété abélienne $A$ sur un corps de fonctions $K$ en une variable sur un corps de base $k$. Il s’agit non seulement de discuter une borne supérieure pour ce rang, mais aussi d’étudier le comportement de cette borne si on prend une extension abélienne finie $L$ de $K$. On se demande aussi : que se passe-t-il quand on enlève cette dernière hypothèse ? Dans un cas particulier, on discute de la validité d’un analogue du théorème de Lang-Néron. Pour...

We study the family of curves $F_m\left(p\right):x^p+y^p=m$, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves $F_m\left(p\right)$. As a corollary we conclude that the jacobians of the curves $F_m\left(5\right)$ with even analytic rank and those with odd analytic rank are equally distributed.

Let $\ell \ge 7$ be a prime, $C$ be the non-singular projective curve defined over $\mathbb{Q}$ by the affine model $y(1-y)=x^{\ell }$, $\infty$ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$, and $\phi :C\to J$ the albanese embedding with $\infty$ as base point. Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$. Taking care of a case not covered in [12], we show that $\phi \left(C\right)\cap J_{tors}\left(\overline{\mathbb{Q}}\right)$ consists only of the image under $\phi$ of the Weierstrass points of $C$ and the points $(x,y)=(0,0)$ and $(0,1)$, where $J_{tors}$ denotes the torsion points of $J$.

We consider an irreducible curve $𝒞$ in $E^n$, where $E$ is an elliptic curve and $𝒞$ and $E$ are both defined over $\overline{\mathbb{Q}}$. Assuming that $𝒞$ is not contained in any translate of a proper algebraic subgroup of $E^n$, we show that the points of the union $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, where $A$ ranges over all proper algebraic subgroups of $E^n$, form a set of bounded canonical height. Furthermore, if $E$ has Complex Multiplication then the set $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, for $A$ ranging over all algebraic subgroups of $E^n$ of codimension at least $2$, is finite. If $E$ has no Complex Multiplication...

In this expository note, we describe an arithmetic pairing associated to an isogeny between Abelian varieties over a finite field. We show that it generalises the Frey–Rück pairing, thereby giving a short proof of the perfectness of the latter.

Let $K$ be a number field, and let $A/K$ be an abelian variety. Let $c$ denote the product of the Tamagawa numbers of $A/K$, and let $A{\left(K\right)}_{\mathrm{tors}}$ denote the finite torsion subgroup of $A\left(K\right)$. The quotient ${c/|A\left(K\right)}_{\mathrm{tors}}|$ is a factor appearing in the leading term of the $L$-function of $A/K$ in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over $\mathbb{Q}$ or quadratic extensions $K/\mathbb{Q}$, and for abelian surfaces $A/\mathbb{Q}$. The smallest possible ratio...

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...