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...

Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over $\mathbb{Z}$. We will present the method and some of the surprising results that have been proved using it. For instance, the probability that a plane curve over ${𝔽}_2$ is smooth is asymptotically $21/64$ as its degree tends to infinity. Much of this paper is an exposition...

We examine an arithmetical function defined by recursion relations on the sequence ${\left\{f\left(p^k\right)\right\}}_{k\in \mathbb{N}}$ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.

Let $H\left(x\right)={\sum }_{n\le x}\frac{\phi \left(n\right)}{n}-\frac{6}{{\pi }^2}x$. Motivated by a conjecture of Erdös, Lau developed a new method and proved that $\#\{n\le T:H(n\left)H\right(n+1)\<0\}\gg T.$ We consider arithmetical functions $f\left(n\right)={\sum }_{d\mid n}\frac{b_d}{d}$ whose summation can be expressed as ${\sum }_{n\le x}f\left(n\right)=\alpha x+P(log\left(x\right))+E\left(x\right)$, where $P\left(x\right)$ is a polynomial, $E\left(x\right)=-{\sum }_{n\le y\left(x\right)}\frac{b_n}{n}\psi \left(\frac{x}{n}\right)+o\left(1\right)$ and $\psi \left(x\right)=x-\lfloor x\rfloor -1/2$. We generalize Lau’s method and prove results about the number of sign changes for these error terms.

We propose a definition of sign of imaginary quadratic fields. We give an example of such functions, and use it to define new invariants that are roots of the classical Ramachandra invariants. Also we introduce signed ordinary distributions and compute their signed cohomology by using Anderson's theory of double complex.

Si $K$ est une extension abélienne de $\mathbf{Q}$ de degré impair, l’étude du 2-groupe des classes (au sens ordinaire) de $K$ (et même celle de la parité du nombre de classes $h$ de $K$) est non triviale, et les algorithmes connus ne dépassent guère le cas $[K:\mathbf{Q}]=3$.L’expression analytique de $h$ s’interprète à l’aide d’indices convenables de groupes d’unités cyclotomiques (Hasse et Leopoldt) ; ce dernier point de vue permet une caractérisation de la parité de $h$, en fonction de l’existence d’unités cyclotomiques totalement...