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A talk on the p-adic numbers September 16, 2009

Posted by Akhil Mathew in General, algebraic number theory, number theory, math education.
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The start of the academic year has made it much more difficult for me to get in serious posts as of late, and the number theory series has slowed.  Things should clear up at least somewhat in a few more weeks.   In the meantime, I’ll do something that occurred to me a while back but I then forgot about: posting a talk.

I took an independent study course last semester on class field theory.  As is traditional, I gave a talk last May after the course on some aspects of the subject matter.  Several faculty members at the university and teachers in my school attended, along with some undergraduates there.  In the talk, I gave an elementary overview of the p-adic numbers, assuming no more than basic number theory and point-set topology.

Anyway, I am posting the (slightly corrected) presentation and the notes here.

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1. Qiaochu Yuan - September 16, 2009

It’s a shame combinatorialists and number theorists don’t talk more often. The construction of the ring R[[x]] of formal power series with coefficients in a commutative ring R is almost identical to the construction of the p-adic numbers (as you’ve discussed in some generality already) and combinatorialists owe all the nice convergence properties of formal power series to the properties of the x-adic topology, which is induced by essentially the same non-Archimedean metric as in the p-adic case. But I don’t think this fact gets recognized very often in combinatorics classes.

Akhil Mathew - September 16, 2009

It’s interesting that this p-adic material is useful in combinatorics. But in what combinatorial situations is it necessary to consider convergence, as opposed simply to formal series?

Steven Sam - September 16, 2009

There are formulas that vary on some parameter n, such that a term like q^n appears. One wants to say that in the limit n \to \infty, this term goes to 0.

Here’s a concrete example: let s_\lambda(x_1, x_2, \dots) denote the Schur function in infinitely many variables. One can show that the specialization s_\lambda(1, q, q^2, \dots, q^{n-1}, 0, 0, \dots) is equal to

q^{\sum_i (i-1) \lambda_i} \prod_{u \in \lambda} \frac{1-q^{n+c(u)}}{1-q^{h(u)}}

where u \in \lambda means a box in the Young diagram, and c and h (content and hook length) are some statistics that only depend on \lambda and not n. The point, using the notion of convergence, one can also make sense of the stable specialization

s_\lambda(1, q, q^2, \dots) = q^{\sum_i (i-1) \lambda_i} \prod_{u \in \lambda} (1-q^{h(u)})^{-1}.

Without the topology on the power series ring, one would have to waive hands and say that q is a number less than 1, but this is not convincing in my opinion.

Qiaochu Yuan - September 17, 2009

A basic situation is where we want to make sure that an infinite product or an infinite sum of series gives another series. Because of the non-Archimedean metric one has the easy conditions that an infinite sum converges if and only if its terms go to zero and an infinite product converges if and only if its terms go to one. Thus I can freely speak of generating functions such as \frac{1}{(1 - x)(1 - x^2)...} without worrying if their coefficients are well-defined, and I can also freely define composites f(g) with g(0) = 0, and so forth. I also want, for example, to show that the Catalan numbers are well-defined when given as a continued fraction. In other words I want to show that a recurrence converges to one of its fixed points, and although in that particular case the argument is direct there are probably examples where it’s easier to argue indirectly.

Akhil Mathew - September 17, 2009

Thanks for the explanations! I see your point about the utility- it’s because Q_p is nonarchimedean and complete.

2. soarerz - September 17, 2009

In the last section you put Kronecker-Weber under “Why does Q_p matter”. Is there any relation between them actually..?

3. soarerz - September 17, 2009

Oops I overlooked the local formulation. Please delete these two comments.


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