The study of reciprocity laws is a venerable tradition in number theory, going back to the quadratic reciprocity law studied by Euler and Legendre and proved by Gauss. From a modern view-point,
reciprocity laws can be understood as being certain precise
relationships between representations of Galois groups (Galois
representations, for short) and automorphic forms. My goal
in these lectures is to explain this view-point, and to describe
some recent developments in the study of reciprocity laws.
From the modern view-point, quadratic reciprocity is a special
case of abelian reciprocity (because it is related to quadratic,
and hence abelian, extensions of the rational numbers, and also
to automorphic forms on the abelian group $GL_1$). The general
abelian reciprocity law was established by Emil Artin (building
on the work of many others), and is the fundamental statement of
class field theory.
The study of non-abelian reciprocity laws (or, what is the same
thing, non-abelian class field theory) is an ongoing area of
investigation. The nature of such laws are the subject of very
general conjectures of Langlands, and Fontaine and Mazur. There
has been much recent progress in the direction of these conjectures
in the first non-abelian context, namely reciprocity between
two-dimensional Galois representations and automorphic forms on
the group $GL_2$, and describing this progress will be the main focus
of my lectures.
The first lecture is intended to provide an introduction to the
topic for non-experts. We will introduce the key ideas in the
theory, and state some of the recent results in the direction
of the Fontaine-Mazur-Langlands conjecture for two-dimensional
Galois representations.
The second and third lectures will then explain some of the methods
of proof of these results. We will place particularly emphasis
on two tools: the $p$-adic local Langlands correspondence for the
group $GL_2$, and so-called local-global compatibility for the
$p$-adically completed cohomology of modular curves, which is a
powerful reciprocity law relating the $p$-adic local Langlands
correspondence to the theory of automorphic forms on $GL_2$.