Fourier coefficients of modular forms and arithmetic embedding problems
by Michael Volpato
Abstract: We start by reviewing the history of some low rank algebraic structures and embeddings into certain non-associative structures, outlining how, from this, one could have predicted the existence of certain classes of exotic modular forms. We then illustrate how, through the optic of representation theory, one can construct such modular forms, whose associated Fourier coefficients count these arithmetic embeddings. If time permits, we will describe how these Fourier coefficients relate M. Bhargava's 'higher composition laws' to Euler product expansions of certain automorphic L-functions.
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