It occured to me that I haven’t defined a -adic number on this blog yet. I’ve been reading about Morita’s -adic gamma function and the Gross-Koblitz formula recently, so I’m going to take this opportunity to introduce -adic integers, discuss how one uses interpolation to define important -adic functions, and give the example of the -adic gamma function. At some point I may come back and discuss the Gross-Koblitz formula, to help motivate why one might care about the existence of a -adic gamma function (although, up to this point in my blog, I’ve been pretty consistant in not coming back and doing follow up posts. I’ll pretend that this is a consequence of the forward momentum of my work, and not of my laziness…)
Below I plan to:
- introduce -adic integers in as down-to-earth a manner as possible
- introduce the topology on -adic integers. The discussion will become more technical here, but this is necessary to discuss interpolation
After this I’ll pause for the day and come back later in the week to discuss -adic interpolation and give examples, in particular Morita’s -adic gamma function.
Last week I met with my advisor and mentioned a way that I could speed up a computation I’ve been working on, essentially just by switching from rational arithmetic to -adic arithmetic. He responded as though doing anything else would be absurd; indeed, he remarked that “-adic numbers are much simpler than rational numbers, just as real numbers are simpler than rational numbers”. It was a simple observation that every number theory student is told when they first encounter -adic numbers, but one doesn’t become fully conscious of it until one has spent considerable time working with -adic numbers in a concrete manner. The same technical machinery is used to construct the -adic numbers from the rational numbers as is used to construct the real numbers from the rational numbers, but our everyday familiarity with decimal expansions seems to make this technical baggage more palatable when constructing the real numbers.
I won’t give the definition of -adic numbers in terms of completions, which would stress the analogy between -adic numbers and real numbers. Instead I’ll give a simpler but more ad hoc definition of -adic integers, which provides a concrete description of what -adic integers look like, and how one does arithmetic with them in practice.
First we must fix a prime number . The set of all -adic integers will be denoted . It consists of all formal infinite sums
where each is an integer in the range . If you’ve seen calculus, this should remind you of a Taylor series in the “variable” , or alternatively, as an infinite expansion of an integer in “base “. Such an infinite sum doesn’t make sense if you interpret the terms as rational numbers, as it would blow up to infinity in this interpretation. If this makes you queasy, just take this definition of a -adic integer as a formal construction and imagine that the plus signs in the expression above are simply placeholders used to separate terms. Note, though, that there is a precise way to make sense of this infinite sum; we’ll give a few details in the section on topology.
A crucial property is that -adic integers can be added, subtracted and multiplied. They are an example of a mathematical gadget known as a ring. Two -adic integers are added by the elementary school rule of “addition with carries”. That is, add the leftmost two terms. If the result is bigger than , then you’ll need to carry to the next term to the right and continue. Here is an example: we’ll consider the two -adic integers and . Their sum is easy to compute:
Continuing this process indefinitely shows that all the terms vanish as they are carried higher and higher, so that the result is zero (Note: if you’re concerned about such infinite processes, there is a way to make this precise using limits). This example shows that the series acts as an additive inverse to in the -adic integers. This justifies writing as elements of , as it is this property of acting as an additive inverse for which characterizes the integer .
We won’t write out the precise details, but one can define multiplication of -adic numbers analogously by extending the elementary school rule for multiplying integers using carries, possibly an infinite number of them. Then one can check that multiplying by allows one to subtract -adic numbers. That is, we have explained how to multiply and add -adic numbers, so to work out , one can instead compute .
By writing each positive integer in terms of a base expansion, one can think of every positive integer as a -adic integer. For example, the base expansion of is given by , which allows us to think of as a -adic integer. If is a negative integer, then we can write the positive integer as a -adic integerr as above and multiply by to obtain a -adic expression for . Note that this -adic expression for a negative integer will be in terms of an infinite number of nonzero terms, while for a positive integer it will be given simply by the (finite) base expansion. This shows that each integer defines a -adic integer (we have not argued that all the -adic expansions of usual integers are distinct, but this is the case. This is clear for positive integers and only slightly less so for negative ones). In spite of this, the collection of -adic integers is much larger than the integers alone. Indeed, a Cantor-style diagonal argument shows that the set of -adic integers is not countable.
The -adic topology
The -adic topology on comes from an absolute value defined in the following way: let be a -adic integer. If then we define the -adic absolute value to be . Otherwise we may let be the smallest positive integer such that . In this case we define . This definition means that if a -adic integer is “highly divisible by “, then it is very small. The sequence of integers does not converge in the ordinary topology on real numbers, but it converges to zero in the -adic topology.
We’ll illustrate this with an example. Consider two -adic numbers and , where we suppose so that we don’t have to carry to make sense of either expression. Then of course is equal to , which means that by definition. Thus and are somewhat close -adically. If we think of them as integers, though, then the difference is still , which is quite large in the ordinary way that we measure distances between numbers.
The important property of this topology that we’ll need is that the integers are dense in the -adic integers. This property means that there are integers as close as we like to any given -adic integer. In fact, truncating an infinite expression at some high power of gives a positive integer which approximates the -adic integer to a high precision. This is the -adic analogue of the fact that there is a rational number between any two distinct real numbers.
Anyways, I’m visiting Jenna now and so don’t want to spend too much time at the keyboard. I’ll post more in some spare time later in the week.
Added: Here is a link to part 2.