Reciprocity laws and Galois representations: recent breakthroughs

Weinstein, Jared

doi:10.1090/bull/1515

Reciprocity laws and Galois representations: recent breakthroughs
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Bull. Amer. Math. Soc. 53 (2016), 1-39 Request permission

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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