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University of Connecticut Department of Mathematics Colloquium |
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| Modular forms are certain holomorphic functions on the complex upper half-plane that possess an infinity of symmetry. Number theorists are interested in the systems of eigenvalues obtained from the action of Hecke operators on modular forms. These (a priori complex) numbers are algebraic integers which are often of arithmetic significance. One is interested in studying congruences modulo (powers of) a prime p between these eigenvalues. This is most efficiently done through a systematic study of p-adic analytic variation of modular forms. In my talk I will survey the progress in this area from the conception of the notion of a p-adic analytic family of modular forms to Coleman-Mazur’s construction of the eigencurve which is in some sense the “universal” such family. |
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