Additive unit representations in global fields - A survey

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units. This has been investigated for several types of rings related to global fields, most importantly rings of algebraic integers. We also state some open problems and conjectures which we consider to be important in this field.Comment: 13 page

Betti maps, Pell equations in polynomials and almost-Belyi maps

We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation A(2) - DB2 = 1, with A, B, D is an element of C[t] and certain ramified covers P-1 -> P-1 arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of Andre, Corvaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials D that fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann existence theorem associates to the abovementioned covers certain permutation representations: We are able to characterize the representations corresponding to 'primitive' solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations when D has degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

LetSbe a smooth irreducible curve defined over a number fieldkand consider an abelianschemeAoverSand a curveCinsideA,bothdefinedoverk. In previous works, we provedthat, whenAis a fibred product of elliptic schemes, ifCis not contained in a proper subgroupscheme ofA, then it contains at most finitely many points that belong to a flat subgroupscheme of codimension at least 2. In this article, we continue our investigation and settle thecrucial case of powers of simple abelian schemes of relative dimensiong2. This, combinedwith the above-mentioned result and work by Habegger and Pila, gives the statement for generalabelian schemes which has applications in the study of solvability of almost-Pell equations inpolynomials

Additive unit representations in rings over global fields - A survey.

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. We focus on rings of integers in number fields and in function fields of one variable over perfect fields. The central problem is whether and how certain rings are (additively) generated by their units. In the final section we deal with matrix rings over quaternions and over Dedekind domains. Our point of view is number-theoretic whereas we do not discuss the general algebraic background

Unlikely intersections of curves with algebraic subgroups in semiabelian varieties

Let G be a semiabelian variety and C a curve in G that is not contained in a proper algebraic subgroup of G. In this situation, conjectures of Pink and Zilber imply that there are at most finitely many points contained in the so-called unlikely intersections of C with subgroups of codimension at least 2. In this note, we establish this assertion for general semiabelian varieties over Q¯. This extends results of Maurin and Bombieri, Habegger, Masser, and Zannier in the toric case as well as Habegger and Pila in the abelian case

Additive unit representations in rings over global fields - a survey

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. We focus on rings of integers in number fields and in function fields of one variable over perfect fields. The central problem is whether and how certain rings are (additively) generated by their units. In the final section we deal with matrix rings over quaternions and over Dedekind domains. Our point of view is number-theoretic whereas we do not discuss the general algebraic background

Multiplicative and Linear Dependence in Finite Fields and on Elliptic Curves Modulo Primes

For positive integers K and L, we introduce and study the notion of K-multiplicative dependence over the algebraic closure (F) over bar (p) of a finite prime field F-p, as well as L-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions phi(1), ... , phi(m), rho(1), ... , rho(n) is an element of Q(X) and an elliptic curve E defined over the rational numbers Q, for any sufficiently large prime p, for all but finitely many alpha is an element of (F) over bar (p), at most one of the following two can happen: phi(1)(alpha), ... , phi(m)(alpha) are K-multiplicatively dependent or the points (rho(1)(alpha), .), ... , (rho(n)(alpha), .) are L-linearly dependent on the reduction of E modulo p. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety G(m)(m) x E-n with the algebraic subgroups of codimension at least 2.As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases