Consider the group ${SL}_2\left({\mathbf{O}}_K\right)$ over the ring of algebraic integers of a number field $K$. Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let ${SL}_2({\mathbf{O}}_K,t)$ be the number of matrices in ${SL}_2\left({\mathbf{O}}_K\right)$ with height bounded by $t$. We determine the asymptotic behaviour of ${SL}_2({\mathbf{O}}_K,t)$ as $t$ goes to infinity including an error term,$${SL}_2({\mathbf{O}}_K,t)=C{t}^{2n}+O\left(t^{2n-\eta }\right)$$with $n$ being the degree of $K$. The constant $C$ involves the discriminant of $K$, an integral depending only on the signature of $K$, and the value of the Dedekind zeta function...

In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: $$\pi \left(x\right)=\frac{q}{q-1}\frac{x}{{log}_{q}x}+\frac{q}{{(q-1)}^2}\frac{x}{{log}_q^{2}x}+O\left(\frac{x}{{log}_q^{3}x}\right),x=q^n\to \infty$$ where $\pi \left(x\right)$ denotes the number of monic irreducible polynomials in $F_q\left[t\right]$ with norm $\le x$.

Previous work on counting maximal independent sets for paths and certain 2-dimensional grids is extended in two directions: 3-dimensional grid graphs are included and, for some/any ℓ ∈ N, maximal distance-ℓ independent (or simply: maximal ℓ-independent) sets are counted for some grids. The transfer matrix method has been adapted and successfully applied

Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2(modp)$ is odd; it is $7/24$. Here we mimic successfully Hasse’s method to compute the density ${\delta }_q$ of monic irreducibles $P$ in ${𝔽}_q\left[X\right]$ for which the order of $X(modP)$ is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the ${\delta }_p$’s as $p$ varies through all rational primes.

We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large ; it is based on Shanks's baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic polynomial...

We find an asymptotic formula for the number of rational points near planar curves. More precisely, if f:ℝ → ℝ is a sufficiently smooth function defined on the interval [η,ξ], then the number of rational points with denominator no larger than Q that lie within a δ-neighborhood of the graph of f is shown to be asymptotically equivalent to (ξ-η)δQ².

We study the distribution of rational points on a certain exponential-algebraic surface and we prove, for this surface, a conjecture of A. J. Wilkie.

For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.