L-functions. A TCC course, Jan-Mar 2018.

Coordinates

Thursdays 1600-1800, room 6M42, Huxley Building (i.e. maths department), Imperial College, starts Thursday 18th Jan, last lecture 8th March.

The syllabus.

John Tate's 1950 PhD thesis explained an absolutely wonderful new way to think about the proof of the meromorphic continuation and functional equation for the Riemann zeta function; Tate's method involved algebraic arguments as well as analytic ones, and was to a certain extent "local" (it treats all primes equally, including 'the infinite prime'). Tate's argument explained in a beautifully coherent way where the mysterious "fudge factors" in the functional equation came from, and, crucially, the beautiful arguments were amenable to vast vast generalisation (they apply to a wide class of automorphic forms).

The prerequisites for reading Tate's thesis itself are a basic understanding of p-adic numbers and some basic facts about Fourier analysis. I will start a little "further back" however. The prerequisites for my course are first courses in group and ring theory, and algebraic number theory (up to and including uniqueness of factorization of ideals into prime ideals). I will go over the more analytic arguments carefully (as I'll be less familiar with them myself). For example I will be going over the basic definitions of the Fourier transform rather than assuming everyone knows it.

I intend to cover

* Introduction. Definition of the Riemann zeta function; basic properties. Proof of meromorphic continuation and functional equation.

* The p-adic numbers; completions of number fields at prime ideals; general local fields.

* Topological groups and the Haar measure.

* The Fourier transform and Poisson summation.

* Tate's Thesis.

Note that the first topic is analytic number theory, the second is algebraic number theory, the third is algebra and the fourth analysis. Finally, Tate's Thesis gives a fundamental application of all of the above topics which is still being used today.

It is very hard to think of a reference for Tate's thesis that is better-written than Tate's thesis itself, although the thesis might not have enough background for the typical student of this course and perhaps they might prefer one of the other expositions of the thesis which have appeared in books more recently.

References: [1] is Tate's thesis (available as a chapter in a book), [2] and [3] are books containing expositions of the thesis plus more background material, and [4] is a letter from Iwasawa to Dieudonne dated 1952 where Iwasawa seems to independently have the same idea as Tate but is constrained by space and doesn't develop the idea as far, but makes other independent observations.

[1] "Fourier Analysis in Number Fields and Hecke's Zeta-Functions", by J. Tate (MR0217026), published as chapter 15 of "Algebraic Number Theory" by J.W.S. Cassels and A. Froehlich (Academic Press 1967), (MR0215665).

[2] "Fourier Analysis on Number Fields", by D. Ramakrishnan and R. J. Valenza, Springer Graduate Texts in Mathematics 186, Springer 1999 (MR1680912).

[3] "Automorphic forms and representations", by D. Bump (Cambridge Studies in Advanced Mathematics 55, CUP 1997) (MR1431508)

[4] "Letter to J. Dieudonne", by K. Iwasawa (MR1210798), published in "Zeta functions in geometry", Advanced Studies in Pure Mathematics 21, 1992 (MR1210779).

[5] More recently (after I had learnt the material) Andreas Holmstrom pointed out this link to his Masters Thesis on the subject, which people might well find useful.

What happened last time.

I taught this course once before, nearly ten years ago, and my treatment is unlikely to change substantially. I reserve the right to tinker with both the course and the example sheets, however you will certainly not be wasting your time if you take a look at the old slides and the example sheets: Sheet 1, Sheet 2, Sheet 3, and Sheet 4.