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Modular forms play an important role in number theory and geometry. These are well behaved, periodic functions on the upper half complex plane with suitable Fourier expansions. The Fourier coefficients of modular forms often encode important arithmetic information: for example, the number of ways of representing an integer as a sum of four squares or the number of points of elliptic curves over finite fields. An important arena of investigation today is the distribution patterns of these Fourier coefficients: in fact, the distribution laws that govern certain families of Fourier coefficients of some special modular forms are now explicitly understood. In this talk, we will survey classical and modern developments in the study of these distribution laws. Furthermore, we will attempt to answer the following questions:
1) How do the Fourier coefficients of these special modular forms fluctuate about their distribution measure?
2) What can we say about the spacings between these coefficients? Do their spacing patterns match those exhibited by iid random variables picked up from a uniform distribution?
The former is joint work with Neha Prabhu, while the latter is joint work with Baskar Balasubramanyam.
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