Density of rational points on commutative group varieties and small transcendence degree (long version)

TL;DR

This paper integrates transcendental number theory with real scalar restriction techniques to explore rational points on algebraic groups, leading to new results on algebraic independence and a reformulation of Mazur's conjecture.

Contribution

It introduces a novel approach linking transcendence properties of algebraic groups to real field homomorphisms, generalizing Mazur's conjecture and improving classical transcendence results.

Findings

Proved that certain algebraic numbers involving Weierstrass functions and exponential functions are transcendental under specific conditions.

Derived 30 new corollaries generalizing well-known theorems in transcendence and algebraic independence.

Reformulated Mazur's conjecture in the context of transcendence theory and algebraic groups.

Abstract

The purpose of this paper is to combine classical methods from transcendental number theory with the technique of restriction to real scalars. We develop a conceptual approach relating transcendence properties of algebraic groups to results about the existence of homomorphisms to group varieties over real fields. Our approach gives a new perspective on Mazur's conjecture on the topology of rational points. We shall reformulate and generalize Mazur's problem in the light of transcendence theory and shall derive conclusions in the direction of the conjecture. Next to these new theoretical insights, the aim of our application motivated Ansatz was to improve classical results of transcendence, of algebraic independence in small transcendence degree and of linear independence of algebraic logarithms. Thirty new corollaries, most of which are generalizations of popular theorems, are stated in the seventh chapter. For example we shall prove: Let a_1,a_2, a_3 be three linearly independent complex numbers, let \wp(z) be a Weierstrass function with algebraic invariants and let b be a non-zero complex numbers. If the four numbers satisfy certain hypotheses, then one among the six numbers \wp(a_j), e^{ba_j} is transcendental. This version of our paper is very elaborated and essentially self-contained. It should be admissible for master students specializing in transcendental number theory and arithmetic geometry.