One of the best things that pops regularly into my pigeonhole is the European Mathematical Society Newsletter. The current issue, number 104, is particularly rich in interesting articles.
For example, there is the transcript of a talk by Christian Krattenthaler (without the musical interludes, sad to say) on “Mathematics AND Music?”. He avoids many of the usual clichés about the relationship between these two disciplines, and defends the thesis
Both Mathematics AND Music are food for the soul AND the brain.
It may be difficult to explain to the non-mathematician that there is soul in mathematics; Christian refers them to the moving moment in the BBC documentary on Fermat’s Last Theorem when Andrew Wiles describes how he finally overcame the difficulties in the proof.
This brings me to the article I really want to talk about: Henri Darmon on “Andrew Wiles’ Marvellous Proof”, in which he explains the relationship between what Wiles did and the Langlands programme, which he describes as “the imposing, ambitious edifice of results and conjectures that has come to dominate the number theorist’s view of the world”. He gives a “beginner’s tour” of the Langlands programme, first excusing his own shortcomings. I want to say a bit about this, but his shortcomings are insignificant compared to mine! I shall also be much briefer, and refer you to the article.
The context is the solution of Diophantine equations, solutions of a polynomial equation in several variables over the integers (or maybe the rational numbers). But rather than tackle this head-on, we first count solutions over the finite field of order pr. (These are the Galois fields, constructed and proved unique by Galois.) For fixed p and varying r, we get an infinite sequence of numbers, which can be represented by a zeta function. Dwork and Deligne proved two remarkable facts about such zeta-functions: first, they are rational functions whose numerators and denominators are independent of the prime p (if a few bad primes are avoided), and second, the reciprocals of their roots are powers of √p. The last is the Riemann hypothesis over function fields, part of the work for which Deligne won the Abel prize.
The new developments involve allowing p to vary. For degree 2 equations in a single variable, the relationship between different primes is precisely described by Gauss’ quadratic reciprocity law. For higher degrees, things get a bit more complicated. We must combine the local zeta functions into a single global zeta function, and show that it is a modular form, in the sense that it is transformed in a very simple manner by linear fractional transformations of its argument.
This brings us to the Shimura–Taniyama conjecture, stating (more or less) that the zeta function of an elliptic curve is a modular form of weight 2. This is what Wiles proved (in the semistable case) and which led, by earlier work of Frey and Ribet, to a proof of Fermat’s Last Theorem. (The semistability assumption was later removed.)
So what does this have to do with Langlands? We have to look at Galois representations, linear representations of the Galois group of the algebraic numbers over the rationals (more precisely, of quotients corresponding to extensions of the rationals unramified outside a finite set S of primes). One can define a zeta function of such a representation; work of Weil, Grothendieck and others shows that, if the diophantine equation has good reduction outside S, then its zeta function is the quotient of the zeta functions of two such Galois representations. Now representations can be decomposed into irreducible representations, and the corresponding zeta functions multiply; so we can look at irreducible representations. Now there are notions of “modular” and “geometric” for Galois representations (the latter corresponding to realisation in an étale cohomology group, as the representations involved in zeta functions of diophantine equations do); the “main conjecture of the Langlands programme” states:
All geometric Galois representations are modular.
One of the main ingredients of Wiles’ work is a lifting theorem allowing the proof of this under suitable (local-to-global) hypotheses.
One detail I have not mentioned is the connection of the Dedekind eta-function with the generating function for the partition numbers, which featured in the work of Ramanujan; Darmon says it “plays a starring role alongside Jeremy Irons and Dev Patel in a recent film about the life of Srinivasa Ramanujan”.
Which brings me back to Krattenthaler’s article. In explaining how mathematics, like music, can contain humour, he outlines the proof of “Ramanujan’s most beautiful theorem”, the statement that the number of partitions of 5n+4 is always divisible by 5. For this, a certain amount of detail about q-series and Jacobi’s triple product formula is required before we get to the punchline of the joke!
The message from all this is that there is are deep-level correspondences between some superficially very different parts of mathematics!
I do urge you to read these articles yourself. Better, why not join the European Mathematical Society and get the Newsletter?