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There is now a whole world of conjectures on the geometry and arithmetic of Shimura varieties. It led to the so-called André-Oort conjecture, which says that subvarieties of Shimura varieties having lots of CM points must be Shimura varieties themselves. Since then, they have played a key role in the Langlands programme, providing a precious source of Galois and automorphic representations.Ī question going back to Robert Coleman was, up to isomorphism, how many algebraic curves of genus g have the property that their Jacobian (an abelian variety) has complex multiplication (CM)? This is really a question about the moduli space of principally polarised abelian varieties of dimension g, which is an example of a Shimura variety. In view of this we decided to go beyond the prescribed syllabus and try to understand more advanced topics. German mathematician Carl Friedrich Gauss (17771855) said, 'Mathematics is the queen of the sciencesand number theory is the queen of mathematics. The students in my class had some background in group theory and complex analysis, and more importantly were extremely enthusiastic. Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. Goro Shimura introduced many examples of Shimura varieties as he sought to generalise the fact that every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients, otherwise known as the Kronecker-Weber theorem. In Fall 2018 I taught Basic Number Theory at IIT Bombay. Although arising as geometric objects, they possess rich algebraic and arithmetic structure. Shimura varieties are playgrounds for many kinds of mathematics. It covers the basic background material that an IMO student should be familiar with.
The geometry and arithmetic of Shimura varieties (Dr Chris Daw) Number Theory Naoki Sato <> 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. Number Theory Author Product Type Digital Learning Options Friendly Introduction to Number Theory, A (Classic Version), 4th Edition Elementary Number.