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 of a finite prime field Fp, 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 φ1,…,φm,ϱ1,…,ϱn ∈ Q(X) and an elliptic curve E defined over the integers Z, for any sufficiently large prime p, for all but finitely many α in the algebraic closure of F_p, at most one of the following two can happen: φ1(α),…,φm(α) are K-multiplicatively dependent or the points (ϱ1(α),⋅),…,(ϱn(α),⋅) 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^k_m×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

Algorithms in algebraic number theory

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers.Comment: 34 page

On the discrepancy of some generalized Kakutani's sequences of partitions

In this paper we study a class of generalized Kakutani’s sequences of partitions of [0,1], constructed by using the technique of successive refinements. Our main focus is to derive bounds for the discrepancy of these sequences. The approach that we use is based on a tree representation of the sequence of partitions which is precisely the parsing tree generated by Khodak’s coding algorithm. With the help of this technique we derive (partly up to a logarithmic factor) optimal upper bound in the so-called rational case. The upper bounds in the irrational case that we obtain are weaker, since they heavily depend on Diophantine approximation properties of a certain irrational number. Finally, we present an application of these results to a class of fractals

Analytic Number Theory

Analytic number theory is a subject which is central to modern mathematics. There are many important unsolved problems which have stimulated a large amount of activity by many talented researchers. At least two of the Millennium Problems can be considered to be in this area. Moreover in recent years there has been very substantial progress on a number of these questions

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes effectiveness in rational points on curves and especially on modular curves, modularity, L-functions, and many other topics

On elements with index of the form 2a3b in a parametric family of biquadratic fields

In this paper we give some results about primitive integral elements α in the family of bicyclic biquadratic fields Lc= Q ( ((c-2) c)1/2, ((c+4) c)1/2) which have index of the form μ( α) =2a3b and coprime coordinates in given integral bases. Precisely, we show that if c≥11 and α is an element with index μ( α) =2a3b≤ c+1, then α is an element with minimal index μ( α) =μ( Lc) =12. We also show that for every integer C0≥3 we can find effectively computable constants M0( C0) and N0( C0) such that if c≤ C0, than there are no elements α with index of the form μ( α) =2a3b, where a>M( C0) or b>N( C0)

A Generalised abc Conjecture and Quantitative Diophantine Approximation

The abc Conjecture and its number field variant have huge implications across a wide range of mathematics. While the conjecture is still unproven, there are a number of partial results, both for the integer and the number field setting. Notably, Stewart and Yu have exponential abc bounds for integers, using tools from linear forms in logarithms, while Győry has exponential abc bounds in the number field case, using methods from S-unit equations [20]. In this thesis, we aim to combine these methods to give improved results in the number field case. These results are then applied to the effective Skolem-Mahler-Lech problem, and to the smooth abc conjecture. The smooth abc conjecture concerns counting the number of solutions to a+b = c with restrictions on the values of a, b and c. this leads us to more general methods of counting solutions to Diophantine problems. Many of these results are asymptotic in nature due to use of tools such as Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory". We make these lemmas effective rather than asymptotic other than on a set of size δ > 0, where δ is arbitrary. From there, we apply these tools to give an effective Schmidt’s Theorem, a quantitative Koukoulopoulos-Maynard Theorem (also referred to as the Duffin- Schaeffer Theorem), and to give effective results on inhomogeneous Diophantine Approximation on M0-sets, normal numbers and give an effective Strong Law of Large Numbers. We conclude this thesis by giving general versions of Lemmas 1.4 and 1.5 of Harman's "Metric Number Theory"