We present a combinatorial mechanism for counting certain objects associated to a variety over a finite field. The basic example is that of counting conjugacy classes of the general linear group. We discuss how the method applies to counting these and also to counting unipotent matrices and pairs of commuting matrices.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

In this paper, we give asymptotic formulas for the number of cyclic quartic extensions of a number field.

For each transitive permutation group $G$ on $n$ letters with $n\le 4$, we give without proof results, conjectures, and numerical computations on discriminants of number fields $L$ of degree $n$ over $\mathbb{Q}$ such that the Galois group of the Galois closure of $L$ is isomorphic to $G$.

We give an asymptotic formula for the number of elliptic curves over ℚ with bounded Faltings height. Silverman (1986) showed that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in ℝ².

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.