This paper continues the study of beta-expansions of 1 where β is a Pisot or Salem number. Special attention is given to regular Pisot numbers, and the Salem numbers that approach these Pisot numbers.

Let k \mathbf {k} be a global function field, and let A \mathbf {A} be the elements of k \mathbf {k} regular outside a fixed place ∞ \infty . Let ϕ : A → K { τ } \phi :\mathbf {A}\to K\{\tau \} be a Drinfeld module of generic characteristic and rank  n n . For a prime ℘ \wp of K K of good reduction, let F ℘ \mathbb {F}_\wp be the residue field at ℘ \wp , and let χ A ( ϕ ( F ℘ ) ) \chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big ) be the Euler-Poincaré characteristic of F ℘ \mathbb {F}_\wp viewed as an A \mathbf {A} -module. We determine the normal order of the number of distinct prime ideals of A \mathbf {A} dividing χ A ( ϕ ( F ℘ ) ) \chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big ) , denoted by ω A ( χ A ( ϕ ( F ℘ ) ) ) \omega _\mathbf {A}\Big (\chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big )\Big ) , as ℘ \wp runs over primes of K K of degree x x with a specified splitting behaviour. Furthermore, let a ∈ K a\in K be non-torsion for ϕ \phi , and let f a ( ℘ ) f_a(\wp ) be the Euler-Poincaré characteristic of the submodule of ϕ ( F ℘ ) \phi (\mathbb {F}_\wp ) generated by a a modulo ℘ \wp . We also consider the problem of determining the distribution of ω A ( f a ( ℘ ) ) \omega _\mathbf {A}\big (f_a(\wp ) \big ) as ℘ \wp runs over primes of K K of degree x x with a specified splitting behaviour. Note that we do not make restrictions on A \mathbf {A} , ϕ \phi , or its endomorphism ring E n d K {sep} ( ϕ ) \rm {End}_{K^\textrm {{sep}}}(\phi ) .

The ring A = $\mathbb{F}$r[T] and its fraction field k, where r is a power of a prime p, are considered as analogues of the integers and rational numbers respectively. Let K/k be a finite extension and let φ be a Drinfeld A-module over K of rank d and Γ ⊂ K be a finitely generated free A-submodule of K, the A-module structure coming from the action of φ. We consider the problem of determining the number of primes ℘ of K for which the reduction of Γ modulo ℘ is equal to $\mathbb{F}$℘ (the residue field of the prime ℘). We can show that there is a natural density of primes ℘ for which Γ mod ℘ is equal to $\mathbb{F}$℘. In certain cases, this density can be seen to be positive.

No abstract

Let X be a smooth, projective curve defined over a finite field Fr, and ∞ a (closed) point of X. Let κ be the function field of X and A the elements of κ regular everywhere except possibly at ∞. Let φ : A → K{τ } be a Drinfeld module defined over K, where ι : A → K is an injective map that identifies K as a finite extension of κ. Suppose that the rank of φ is n 2.For a place ℘ of good reduction for φ, reducing the coefficients of φ modulo ℘ equips the residue field F℘ with an A-module structure. We establish that the number of places ℘ of K of good reduction for φ, of degree x, and such that φ(F℘) is a cyclic A-module, has a natural density. Furthermore, this density is positive if and only if there are infinitely many primes ℘ such that φ(F℘) is cyclic as an A-module.We do not make further restrictions on either A or the ring of endomorphisms EndKsep (φ).