Um estudo sobre a transcendência de períodos de funções elípticas

Abstract

This work is based on the book ”Transcendental Numbers”by M. Ram Murty and Puru- sottam Rath and aims to present results that intertwine studies on elliptic functions and the theory of transcendental numbers, proving that it is possible to obtain transcendental numbers from the periods of elliptic functions. Initially, we review some preliminary results from complex analysis, with emphasis on properties of holomorphic functions, power series, and the residue theorem, as well as fundamentals of algebraic number theory, including algebraic numbers, symmetric polynomials, and Siegel’s lemma. Subsequently, we address the emergence of the theory of transcendental numbers, highlighting historical milestones such as Liouville’s Theorem, and later results (such as Hermite’s and Lindemann’s proofs of the transcendence of the number e and π, respectively). Finally, we explore the deep connection between the theory of transcendental numbers and the theory of elliptic functions, evidenced by the study of the periods of these functions. With this, the work seeks to illustrate how objects of an analytical nature, such as elliptic functions and their periods, contribute significantly to advancing the understanding of the transcendence of certain numbers, unifying ideas from different areas of mathematics.