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The purpose of this paper is to combine classical methods from transcendentalnumber theory with the technique of restriction to real scalars. We develop aconceptual approach relating transcendence properties of algebraic groups toresults about the existence of homomorphisms to group varieties over realfields. Our approach gives a new perspective on Mazur's conjecture on thetopology of rational points. We shall reformulate and generalize Mazur'sproblem in the light of transcendence theory and shall derive conclusions inthe direction of the conjecture. Next to these new theoretical insights, theaim of our application motivated Ansatz was to improve classical results oftranscendence, of algebraic independence in small transcendence degree and oflinear independence of algebraic logarithms. Thirty new corollaries, most ofwhich are generalizations of popular theorems, are stated in the seventhchapter. For example we shall prove:  Let a_1,a_2, a_3 be three linearly independent complex numbers, let \wp(z) bea Weierstrass function with algebraic invariants and let b be a non-zerocomplex numbers. If the four numbers satisfy certain hypotheses, then one amongthe 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 transcendentalnumber theory and arithmetic geometry.

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