Arithmetic properties of j-function

Das, Jishu (2018) Arithmetic properties of j-function. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Jishu Das (13MS109)) 13MS109.pdf - Submitted Version Restricted to Repository staff only Download (891kB)

Abstract

Modular forms play an significant role in number theory. They also appear in areas like algebraic topology and string theory. The j-function is a very special function in the theory of modular forms. In particular, modular forms of weight zero are polynomials in j-function which we prove in this thesis. The main goal of this thesis, however will be to explore some important properties of j-function, define modular functions for Γ₀ (m) and give a classification of it. Then we define the modular equation, explore some of its properties and use these properties to show that j(Ok)is an algebraic integer where Ok is the ring of integers of Q(√-n)for n being a square free natural number.

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