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Reciprocity laws and Galois representations: recent breakthroughs


Author: Jared Weinstein
Journal: Bull. Amer. Math. Soc. 53 (2016), 1-39
MSC (2010): Primary 11R37, 11R39, 11F80
DOI: https://doi.org/10.1090/bull/1515
Published electronically: August 25, 2015
MathSciNet review: 3403079
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Abstract: Given a polynomial $ f(x)$ with integer coefficients, a reciprocity law is a rule which determines, for a prime $ p$, whether $ f(x)$ modulo $ p$ is the product of distinct linear factors. We examine reciprocity laws through the ages, beginning with Fermat, Euler and Gauss, and continuing through the modern theory of modular forms and Galois representations. We conclude with an exposition of Peter Scholze's astonishing work on torsion classes in the cohomology of arithmetic manifolds.


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Additional Information

Jared Weinstein
Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts

DOI: https://doi.org/10.1090/bull/1515
Received by editor(s): May 18, 2015
Published electronically: August 25, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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