Reciprocity laws and Galois representations: recent breakthroughs
Given a polynomial
with integer coefficients, a reciprocity law
is a rule which determines, for a prime
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.