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I recently taught myself some algebraic number theory and am preparing to take a course in class field theory this fall. I understand the notion of a Haar measure on a locally compact topological group is important and was wondering if anyone could tell me specifically what I should know about it for algebraic number theory or class field theory.

I have a strong background in abstract algebra, but a comparatively weaker background in analysis and topology. So, I probably don't have time to learn the full theory of the Haar measure, only what will be necessary to understand the material of class field theory. The approach we are taking in the class is using both L-functions and cohomology.

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My guess is that you don't need to know too much, and that the following generally suffices:

  • The Haar measure, which is translation-invariant, exists and is unique on any locally compact topological group. (The uniqueness part might not be so important, but it's cool.)
  • It can be normalized for compact groups (so e.g. $\Bbb Z_p$ has measure $1$ wlog).
  • Integration can be performed over measure spaces, in particular topological groups.
  • What the measure looks like explicitly - specifically, (i) the measure of basic open sets for standard topoplogical bases, and (ii) how to decompose the kinds of regions of integration you'll be working with into basic open sets.
  • How to manipulate integrals. For instance, $\int\frac{dx}{|x|}$ can give us the multiplicative Haar measure of a set in terms of the additive Haar measure, and you'll need to be able to break apart and reparametrize integrals in a goal-oriented fashion.
  • Know how basic properties of the Fourier transform generalize to arbitrary LCA groups. (For instance, by Tate's thesis we know the functional equation of the Riemann zeta function is essentially a direct application of the Poisson summation formula over the adeles, IIRC.)

Of course you'll want to know the topology on local fields, adeles, ideles, and Galois groups. The above bullets are based on my exposure to number theory so far, so perhaps they could be better.

Much of the above is pretty much "have a working knowledge of calculus and Fourier theory, but as they apply to topological rings and groups that appear in number theory," which isn't too hard since we're usually familiar enough with usual calculus and Fourier theory, and familiar enough with the rings and groups in number theory, that we can combine the two knowledge bases without much trouble. As I understand it, the proof of existence of Haar measures isn't an easy or fun read (I haven't read it), but it won't be necessary.

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  • $\begingroup$ In the number theory case Haar measure is pretty easy to construct. E.g. on $Z_p$ it is just (the projective limit of) counting measure on the various finite groups $\mathbb Z/p^n$. On $\mathbb R$ is just Lebesgue measure. On the adeles, the product measure. So pretty straightforward, all in all. $\endgroup$ – user160609 Jul 18 '14 at 2:25

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