We study a problem in discrete tomography on $\mathbf{Z}^{n}$, and show that there is an intimate connection between the problem and the study of the Hodge cycles on abelian varieties of CM-type. This connection enables us to apply our results in tomography to obtain several infinite families of abelian varieties for which the Hodge conjecture holds.

In this paper, we prove some congruences for higher-order Euler numbers.

This is the first part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. The main result of this part is a constructibility result for the analytic Gröbner fan of a parametric ideal in the ring of analytic differential operators. In this part, the main tool is the notion of reduced generic standard basis.

This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in the case of one function. We also make an extensive study of an example for which we give an expression of a generic (and under some conditions, a relative) Bernstein-Sato polynomial.

In this paper we show a $2$-dimensional subgroup in $\mathit{PU}(2,1)$ which contains elliptics is discrete if and only if all its subgroups generated by two elliptics are discrete. This generalize the well-known discreteness criterion first established by T. Jørgensen.

In this paper, we will study the discreteness criterion for non-elementary subgroups in $\mathit{PU}(1,n;C)$. Several discreteness criteria are obtained. As an application, the convergence theorem of discrete subgroups in $\mathit{PU}(1,n;C)$ is discussed.

We calculate the $q$-Eulerian distribution $W(t,q)$ of the elliptic Weyl group of type $A_1^{(1,1)}$, which is a formal power series in $\mathbf{Z}[[t,q]]$, and classically defined for any Coxeter system $(W,S)$.

We estimate the bounds for the difference between the ordinary height and the canonical height on elliptic curves over number fields. Our result is an improvement of the recent result of Cremona, Prickett, and Siksek (J. Number Theory 116 (2006), 42--68). Our bounds are usually sharper than the other known bounds.