Arithmetic geometry of moduli spaces and applications

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Number fields are obtained when one wishes to define solutions to polynomial equations. They are the primary object of study of number theory. Abelian varieties are a generalization of elliptic curves and are one of the primary objects of study of algebraic geometry. The spaces that parameterize abelian varieties are a generalization of modular curves, called Shimura varieties. Shimura varieties play a central role in modern number theory. They are intimately related to Galois representations (projections of the absolute Galois group, whose study, arguably, is the chief purpose of algebraic number theory), explicit class field theory (that attempts to give an explicit handle on number fields), modular forms (examples of which are theta functions which encode the number of times a quadratic form represents an integer, and have many applications to physics and combinatorics), nonabelian harmonic analysis and more.