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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
Received by editor(s):
May 18, 2015
Published electronically:
August 25, 2015
Article copyright:
© Copyright 2015
American Mathematical Society