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.

We show how the size of the Galois groups of iterates of a quadratic polynomial f can be parametrized by certain rational points on the curves Cₙ: y² = fⁿ(x) and their quadratic twists (here fⁿ denotes the nth iterate of f). To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem...

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 $J$.

The variation of the rank of elliptic curves over $\mathbb{Q}$ in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over $\mathbb{Q}$ admits infinitely many quadratic twists of rank $\ge 1$. Most elliptic curves have even infinitely many twists of rank $\ge 2$ and examples of elliptic curves with infinitely many twists of rank $\ge 4$ are known. There are also certain density results. This paper studies the variation of the...

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...

In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.