Applications of the p-adics

As I was preparing to spend a month studying p-adic analysis, I realized that I've never seen the theory of p-adic numbers applied in other branches of mathematics. I can certainly see that the field $\mathbb Q_p$ has many nice properties, e.g. $X\subset Q_p$ closed and bounded implies $X$ compact, $\sum\limits_{n=0}^\infty x_n$ exists iff $\lim\limits_{n\to\infty}x_n=0$, its ring of integers is compact, and any open set is a union of disjoint open balls. But how are properties of $\mathbb Q_p$ translated to applications to other mathematical problems? Are there any significant mathematical problems which have been resolved by applying the theory of p-adic numbers?

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–  Thomas Andrews Mar 20 '12 at 3:41
My master's thesis! Perhaps more importantly... they are used all over the place in number theory, as the answers have already hinted at. –  Alex Mar 20 '12 at 3:55
Take a look at mathoverflow.net/questions/84320/…. –  KCd Mar 20 '12 at 5:01

2 Answers

The Weil Conjectures are proved using $p$-adic numbers ($\ell$-adic cohomology); but the statement of these theorems does not make use of $p$-adic numbers. These are very important theorems in number theory and arithmetic geometry that explain the pattern you find when you count the number of points on an algebraic variety over a finite field and its extensions.

The generating function built from these point counts is a rational function, called a local zeta function. The Weil Conjectures give us an analogue of the Riemann Hypothesis: if we have the local zeta function of a curve, we know that all of its zeros lie on the critical line!

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This is almost like asking "what are real numbers good for". It would require several tomes of terse enumeration of key phrases to compile a comprehensive list.

Here is a random collection of terms for you to google (all these are mainly number theoretic, since that's where I come from, but there are plenty of applications in other areas; for each of these terms, it would require several months to several years to appreciate its depth): Hasse principle, local class field theory, $l$-adic Galois representations, Iwasawa theory and $p$-adic $L$-functions, rigid analytic geometry. I had better stop here.

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Except the real number are "obviously" good for something - certain kinds of real world measurements, comparison of lengths in Euclidean geometry, etc. The applications of $p$-adics are far from obvious, by comparison. –  Thomas Andrews Mar 20 '12 at 3:44
@Thomas From a number theorist's point of view, the reals are just another completion of $\mathbb{Q}$, no more real than the $p$-adics. How would you use real numbers to measure anything, anyway? –  Alex B. Mar 20 '12 at 3:47
Measuring the circumference of a circle by knowing its diameter, perhaps? –  Vhailor Mar 20 '12 at 3:55
Yeah, @AlexB, I'm aware, but my point is comparing the question to the same question about reals is pointless. The reason to ask about $p$-adics is because they don't represent anything intuitive or have any obvious applications. –  Thomas Andrews Mar 20 '12 at 3:56
@Vhailor And how precisely do you usually know the diameter of a circle? For all I care in terms of real life applications, $\pi$ might as well be rational, truncated after the first million decimal places. My point is that the only reason why you think that the reals are natural is because you have been taught about them at school. They are no more real than the $p$-adic or the imaginary numbers, as far as the platonic or the physical world are concerned. And who knows: perhaps I want to approximate the circumference of a circle $p$-adically, rather than in the Euclidean metric. –  Alex B. Mar 20 '12 at 4:16