Last time we introduced the -adic integers and briefly described their topology. Today I’d like to provide some motivation for why people might care about -adic integers, although the motivation will spill into the next couple of posts.
Comparison of real and -adic topologies on the integers
Let’s begin this discussion with a question: if all one is interested in are integers, then why might one care about real numbers? This isn’t a frivolous question, as in practice rational (or algebraic) numbers are the only ones that we honestly come into contact with in our universe. We can never measure real quantities to enough accuracy to be sure we’ve nailed down a real number, like , and yet they are an incredibly useful class of numbers.
Here’s one very simple example to explain how real numbers can be used to study integers. The property we’ll use is that the integers are discrete in the real numbers. This means that each integer is isolated from every other integer in the usual real number line. So for example, if and are two integers, and we know that the distance between them satisfies , then in fact It's an absolutely obvious property of integers, and yet it comes up again and again in certain areas of mathematics.
Let's give a somewhat concrete example of how this can be used. It is often the case that certain discrete quantities have analytic expressions in terms of certain infinite series (I have in mind the examples of special values of -series). Imagine an equation of the form
discrete quantity, for example an integer infinite sum.
Mathematicians often encounter such equations and would like to compute the discrete quantity on the left, typically because it counts something of interest. Doing the counting directly is sometimes very difficult, as one might not have a precise description of what is being counted. For example, one might know that a certain polynomial equation has finitely many solutions, but might not know exactly how to find them efficiently. So it would be nice if one could use an analytic expression to compute the discrete quantity in an efficient way. It might seem impossible to use an infinite sum to compute a value exactly, as it would take an infinite number of additions. However, by the discreteness of the integers inside the real numbers, one only has to compute enough terms of the infinite sum to know that one is within of the full sum. Then, once we’ve computed that far out, we can stop. The discrete quantity in question must equal the nearest integer to the computed quantity, since the integers are spread out among the real numbers.
This strategy uses the discreteness of the integers inside the real numbers. The -adic integers have a topology all their own, and one of the rich properties of the -adic integers is that the integers do not sit discretely inside the -adic integers. If and are two integers, then the -adic distance between them (recall that we denoted it as ) was defined to be , where is the exact power of that divides the integer . As an example, the distance between the integers and is the very small quantity . So the two integers and are very close -adically, yet very far apart in the ordinary way that one measures distances between integers.
In contrast to the case of the usual distance for numbers, if for two integers, then all we can conclude is that divides . We can’t say like we could with the usual distance. This doesn’t mean that the -adic integers are less useful because they give less information than real numbers. Rather, they give us different information. It is best to use -adic numbers to study divisibility properties of integers. Thinking of integers as real numbers gives essentially no further knowledge of divisibility properties, but it can be useful in other ways, as was discussed above.
The point is that the integers have different topologies arising from the fact that they live inside many different topological spaces (in this discussion the real numbers and the -adic integers). These different topologies and spaces help to solve different problems.
I’d now like to turn to the topic of -adic interpolation, which is a rich and intriguing phenomenon. To explain why it’s such an interesting subject we’ll first discuss real interpolation. Let be a function on the integers. The graph of this function looks like a bunch of dots, one above each integer at height . The problem of real interpolation asks for a function on the entire real line which agrees with on all integers . The problem of interpolation stated in this form is very simple to solve: you just have to “connect the dots” in the graph of . In this case the discreteness of the integers in the real numbers makes the problem of interpolation trivial. To make the problem more interesting, one usually has to impose extra conditions on the interpolating function. For example, can one find a differentiable and convex function which interpolates the values?
To make this concrete and to motivate discussion of Morita’s -adic gamma function, let’s discuss the classical gamma function. First recall that the factorial function is a function on integers, denoted and read “-factorial” for each positive integer . It is defined as a product
This particular function pops up all over in mathematics, and so mathematicians have worked hard to find a particularly nice interpolating function. In a precise sense, the best interpolating function of the factorials is given by the gamma function defined by the integral
This is a convergent integral for all real numbers and satisfies the condition that for all integers . If you consider all smooth interpolating functions for the factorials, then is the unique one satisfying a certain convexity condition. It is thanks to this, and thanks to its ubiquity in mathematics, that mathematicians feel that the gamma function is the right real interpolating function for the factorials.
We can play the same game with the integers thought of as lying inside the -adic integers. Namely, given a function , can one find a continuous function which agrees with the original function on integers? This is a more difficult problem than the real interpolation problem, because the nondiscrete topology of the -adic integers intervenes. When we think of the integers as living inside the real numbers with the discrete topology, any function on them is continuous automatically. This is not so for the -adic topology inherited by the integers from the -adic integers. So we already see that our function to be inerpolated must satisfy certain continuity properties. For example, it must map and close together, because these two integers are close to one another -adically.
Last time we stated that the integers are dense in the -adic integers. This was just saying that if you chop off an infinite -adic number at any height, say
at , then you get honest integer, in this example . It turns out that if a function on the integers is continuous for the -adic topology on the integers, then since the integers are dense in the -adic integers, it interpolates automatically and uniquely to a continuous function on the entire set of -adic integers. This is analogous to the fact that a continuous function on the real line is determined uniquely by its values at the rational numbers. So the problem of -adic interpolation of a function comes down to checking the -adic continuity of alone. This is not true automatically as it was for real interpolation.
I’ve planned two more posts on this topic before I leave it. Here’s what is coming up:
- a discussion of the Skolem-Mahler-Lech theorem. This is a cool and easily stated theorem about linear recurrences (like the Fibonnaci sequence). Its statement doesn’t make any reference to -adic numbers, yet all known proofs of the theorem surprisingly use -adic interpolation. This should provide readers with some motivation for why one might care about -adic methods, and -adic interpolation in particular, even if one is only interested in elementary statements about integers!
- the definition of Morita’s -adic gamma function. This illustrates some of the delicacy of -adic interpolation. It’ll hopefully convince you that -adic interpolation is harder than real interpolation, and so the fact that many naturally occurring quantities can be interpolated -adically is quite striking.