Luscious Langlands

There are very few papers that have remained in my possession, after years of travelling and moving about. One of these is Stephen Gelbart's An Elementary Introduction to the Langlands Program (Bull. Amer. Math. Soc. 10, 2 (1984) 177-215). These days there is no need to canvas mathematicians with a dim hope that they will see some connection of this to physics. I still understand very little of the Langlands program, but no longer for want of articles on the subject. Actually, I met Langlands once, back in the mid 90s. I had no idea who he was, much to the shock of the mathematicians I was socialising with at the time. Really cool guy - looks 20 years younger than he is.

Anyway, Gelbart's paper begins with a quote by Browder: that the possible number fields of degree n are restricted by the irreducible infinite dimensional representations of GL(n) was the visionary conjecture of R. P. Langlands. The basic concept needed is that of an L-function associated to a representation $\pi$. We can start with the Artin version, which depends on a morphism $\sigma: G \rightarrow GL_{n}(C)$ for a certain Galois group $G$, and takes the form

$L (s, \sigma) = \prod_p (\textrm{det} (I_n - \sigma (F_p) p^{-s}))^{-1}$

where $I_n$ is an identity and $F_p$ is a Frobenius map. Oh yes, this is supposed to look a bit like the Riemann zeta function. Mahndisa will be pleased to know that the p-adics soon show up, what with primes floating about all over the place.

Langlands set about finding a correspondence $\sigma \rightarrow \pi (\sigma)$. The $n = 2$ case ends up being about automorphic forms. Remember that pretty fundamental domain, related to the modular group, that came up when we were talking about the moduli space for the punctured torus? One considers nice functions

$f(z) = \sum_{1}^{\infty} a_n e^{2 \pi i n z}$

with respect to such domains, so that when the set of $a_n$ is multiplicative, $f$ is an eigenfunction for Hecke operators.