Reciprocity laws and Galois representations: recent breakthroughs
Author:
- 25 August 2015
- journal article
- Published by American Mathematical Society (AMS) in Bulletin of the American Mathematical Society
- Vol. 53 (1) , 1-39
Abstract
Given a polynomial f ( x ) f(x) with integer coefficients, a reciprocity law is a rule which determines, for a prime p p , whether f ( x ) f(x) modulo p 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.Keywords
Funding Information
- National Science Foundation (DMS-1303312)
- Alfred P. Sloan Foundation (DMS-1303312)
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