p-adic numbers and adeles - an introduction

"...we turn our attention to the most fundamental concept we've employed throughout our exploration - the notion of distance...We measured...distances [between real and rational numbers] using the absolute value on the real numbers, since that is the natural metric (distance measure) on R. This remark raises a basic question: How did the real numbers enter our analysis? We began by studying the rational numbers...The rationals were our footholds into the more mysterious realm of the irrational. But why were we looking at the rational numbers as a subset of the reals? The answer appears clear: because the rationals are a subset of the reals! Perhaps, however, the rational numbers are subsets of other interesting, mysterious realms.

...we [now] begin a journey into the basic idea of distance and discover new worlds of numbers that are as natural and as important as the reals but have a foreign feel and look. These new numbers, in fact, lead to a broader and deeper understanding...of number..."

E.B. Burger, Exploring the Number Jungle: A Journey into Diophantine Analysis (AMS, 2000) p.105

A map |.| from the rationals to the non-negative reals is called a norm (absolute value or valuation) if it satisfies the three following conditions:

(1) |x| = 0 if and only if x = 0
(2) For all rational x, y, we have |xy| = |x||y|
(3) For all rational x, y we have |x + y| < |x| + |y| (the triangle inequality)

The usual absolute value |.| clearly satisfies these properties, but what other kinds of norms can exist?

There's a trivial norm which works like this: |x| = 1 for all rationals x except 0, with |0| = 0. The non-trivial norms turn out to be very interesting indeed, and we shall consider them now.

A sequence {xn} is called a Cauchy sequence with respect to the norm |.| if it satisfies the following property:

Given any a > 0, there exists some N such that m,n > N implies |xm - xn| < a. Basically, a sequence is Cauchy if its terms become 'arbitrarily close' with respect to the norm |.|.

The definitions of norms and of Cauchy sequences can be easily generalised to other fields, but we shall restrict our attention here to the field of rational numbers Q.

Q is said to be Cauchy incomplete with respect to the usual absolute value |.| since there exist Cauchy sequences within Q which do not converge in Q. For example the sequence of rationals

{1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213...}

is clearly Cauchy, but its limit, the square root of 2, is not an element in Q.

The field of real numbers R is the result of completing the field of rationals Q with respect to the usual absolute value |.|. This completion works as follows:

First we define an equivalence relation on all Cauchy sequences in Q: {xn} and {yn} are said to be equivalent if |xn - yn| -> 0 as n goes to infinity. The field of real numbers can be defined in terms of the resulting equivalence classes. For example, sqrt(2) can be thought of as the equivalence class containing the sequence given above.

One doesn't generally think of this correspondence (involving equivalence classes of Cauchy sequences) when working with the real numbers, but this is how they are rigorously defined.

The general definition of completion goes like this:

A field K (with norm |.|K) is a completion of field k (with norm |.|) if
(1) K contains k
(2) |x|K = |x| for all x in k
(3) K is Cauchy complete with respect to |.|K
(4) k is dense in K with respect to the topology associated with |.|K

It turns out that by choosing different norms, we get different kinds of 'Cauchyness'. That is, the set of sequences of rational numbers which are Cauchy will generally change if we change to a different norm. We will also generally get a different set of equivalence classes when we change norms. This all means that different norms can lead to entirely different ways in which we can complete Q - different fields which contain Q, but which differ considerably from R and each other.

The trivial norm mentioned earlier (where |x| = 1 unless x = 0 when |x| = 0) leads to a 'trivial completion': Q itself. The only Cauchy sequences are those with constant tails, and the equivalence classes naturally correspond to individual rational numbers.

There is also an infinitude on non-obvious completions of the rationals. These are the p-adic fields Qp (where p is some fixed prime number) discovered by K. Hensel in 1902. Each p-adic field Qp is defined by completing Q with respect to the absolute value |.|p which is defined as follows:

Let x be a nonzero rational number. It can always be expressed as pk(m/n) where m and n are nonzero integers, neither divisible by the prime p, and k is an integer. We then have

|x|p = p-k

If we further define |0|p = 0, then it is not difficult to check that |.||x|p satisfies the necessary conditions above to be a norm on Q. It can be said to provide Q with an "arithmetical measure of distance". The set of equivalence classes of Cauchy (with respect to |.|p) sequences has a natural field structure. It is a completion of Q which we call the field of p-adic numbers. To see how it contains Q as a subfield, we simply put the rational number x in correspondence with the equivalence class containing the (obviously Cauchy) sequence {x, x, x, ...}.

Note following facts:

  • a rational number is 'p-adically large' if it has a large power of p in its denominator
  • a rational number is 'p-adically small' if it has a large power of p in its numerator
  • a rational number without factors of p in either numerator or denominator (that is the vast majority of rational numbers for any given p) will have p-adic norm equal to 1.
  • all p-adic distances are powers (positive, negative or zero) of p

The sequence of rationals {1, p, p2, p3, p4, p5, ...} is a straightforward example of a sequence which is not convergent (and hence not Cauchy) in Q according to the usual absolute value |.|, but which is Cauchy in Q according to |.|p. For we know that |pk|p = p-k. Whereas this sequence diverges rapidly according to the usual norm, it actually converges to 0 according to the p-adic norm.

Two rational numbers can be a huge distance apart in 'real' terms, but if the numerator of their difference (in reduced terms) happens to be divisible by a high power of p, then they can be p-adically close. Similarly, two rational numbers close together in 'real' terms could be far apart p-adically as a result of the denominator of their difference (in reduced terms) being divisible by a high power of p.

Any attempt to 'picture' the field Qp must necessarily involve the relinquishing of all 'common sense' Euclidean-type concepts of space and distance, something which is not easy to achieve.

The mathematical existence and validity of such norms and their accompanying fields (completions of Q) is clear, but how can we actually refer to or work with the non-rational elements of Qp? It turns out that there is a very usable notation, a sort of "base p" (some people call it "pinary") notation which has a kind of "reverse decimal" quality. The digits available are {0,1,2,...,p-1}. You may be confused, so here's an example (here p is some prime number bigger than 7)


would mean (reading right-to-left) 2p-3 + 4p-2 + 5p-1 + 5 + 7p + 6p2 + 8p3 + ...

As with familiar decimal notation, the digits to the left of the point deal with (higher, the further left we go) positive powers, and digits to the right of the point deal with (higher the further right we go) negative powers. But the situation has been turned on its head: the negative powers always terminate, while the positive ones can continue indefinitely. In the 'sensible' world of the usual absolute value |.|, numbers like this could not be finite. They are described by what amounts to a divergent infinite series. But if we are using |.|p then they can be convergent and thereby elements of Qp.

This notation allows us to easily describe the p-adic integers. These are the p-adic numbers with nothing after the point. That is, sums of nonnegative powers of p, or equivalently the limits of sequences of integers within Qp. Zp, the ring of p-adic integers, is the maximal subring of Qp. Note that Z is not closed in Qp. Zp is in fact its closure. The norm of a p-adic integer is always < 1 (consider the ultrametric inequality).

Note that a p-adic norm provides us with a notion of distance, hence a metric, a notion of open discs, neighbourhoods, and hence a p-adic topology on the field Qp and its subfield Q.

There is a stronger inequality for an absolute value |.| than the triangle inequality which is known as the ultrametric inequality or strong triangle inequality:

|x + y| < max{|x|,|y|}

Any norm |.| satisfying this is called nonarchimedean (or ultrametric). A norm which does not satisfy it is called archimedean. The usual norm on the real line is clearly archimedian - in fact there is an archimedean axiom:

"Let us turn our attention to axioms of Euclidean geometry. In a list of axioms there exists the so-called Archimedean axiom, which was at first pointed out and analyzed by Veroneze and Hilbert. According to the Archimedean axiom any given large segment on a straight line can be surpassed by successive addition of small segments along the same line. Really, this is a physical axiom which concerns the process of measurement. Two different scales are compared by this axiom. It means that we can measure distances as small as we want."

(from V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, 1994))

A more formal statement of the axiom woud be that if 0 < |x| < |y| then there is some positive integer multiple nx of x such that |nx| > |y|. As the excerpt above indicates, this property corresponds to common sense experiences of measurement. If one distance is smaller than another, then by simply taking enough copies of the smaller distance and stacking them end to end, we can eventually produce something which exceeds the bigger distance.

It is very difficult to imagine a situation where this axiom does not hold, but in fact the very space and time we inhabit have both been shown by 20th century science to be unequivocally nonarchimedian: The archimedean axiom breaks down at the Planck scale, that is for distances less than 1.6 x 10-33 metres and durations less than 5.4 x 10-44 seconds. Despite our entrenched belief that space and time are continuous, homogeneous, infinitely divisible quantities, we are now confronted with the fact that below this scale, distances and durations cannot scaled up in order to produce macroscopic distances and durations. Equivalently, we cannot meaningfully measure distances or durations below this scale.

"...So a suggestion emerges to abandon the Archimedean axiom at very small distances. This leads to a non-Euclidean and non-Riemannian geometry of space at small distances.

How can one construct a physical theory corresponding to a non-Archimedean geometry? As it is well known there is an analytical description of geometry. One uses coordinates to describe a geometrical picture.

There are two equivalent approaches

geometry <---> number system

The usual Euclidean geometry is described by means of real numbers. If we want to abandon the standard geometry for description of small distance in physical space-time we have to abandon real numbers. What should be used instead of real numbers?

In computations in everyday life, in scientific experiments and on computers we are dealing with integers and fractions, that is with rational numbers and we never have dealings with irrational numbers - infinite non-periodic decimals. Results of any practical action we can express only in terms of rational numbers which are considered to have been given to us by God. Certainly, there exists generally accepted confidence that if we carry out measurements more and more precisely, then in principle we can get any large number of decimal digits and interpret a result as a real number. However, this is an idealization and as it follows from the previous discussion we should be careful with such statements. Thus, let us take as our starting point the field Q of rational numbers....

...What norms do exist on Q? There is a remarkable Ostrowski theorem describing all norms on Q. According to this theorem any nontrivial norm on Q is equivalent to either ordinary absolute value or p-adic norm for some fixed prime number p."

(from V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, 1994))

Incidentally, French physicist Laurent Nottale has introduced a concept of scale relativity where the Planck length and time play comparable roles to the role played by the speed of light in the theory of special relativity, that is, invariant under a fundamental set of transformations (in this case scale transformations). Einstein took as his starting point the observed fact that the speed of light is invariant under changes of inertial frame of reference (acceleration, basically), and followed this counterintuitive fact to its logical conclusion, which was that spacetime must be curved (i.e. conforms to a hyperbolic geometry). Notalle takes as his starting point the invariance of the Planck length with respect to scale transformations, and follows this counterintuitive idea to its astonishing logical conclusion - that spacetime has an intrinsically fractal quality. (See L. Nottale, Fractal Space-Time and Microphysics - Towards a Theory of Scale Relativity (World Scientific, 1993).)

Because p-adic fields are nonarchimedian, and space and time have revealed themselves as nonarchimedian, it's natural to consider whether or not physics might not better be formulated in terms of Qp rather than R as it traditionally has been. In fact there's a growing body of "p-adic physics". The book by Vladimirov, et.al. is a good starting point.

If we're going to use p-adic fields to describe physical phenomena, there arises the obvious question "which p do we use?". It turns out that there is an approach which involves ALL p-adic norms (as well as the usual one which produces R) simultaneously.

The adeles constitute a locally compact topological ring AQ, individually taking the form

(aoo;a2, a3, a5, a7, a11,...)

aoo is a real number (that subscript is meant to be 'infinity').

Each ap is a p-adic number. We must require that in all but a finite number of cases ap is a p-adic integer.

aoo is the archimedian entry. The rest are non-archimedian. Hence the semicolon separating them.

The notation aoo conforms to the notion that whereas the field of p-adic numbers relates to the finite prime number p, the field of real numbers relates to the prime at infinity. Some people call it the real prime (as opposed to the finite primes 2, 3, 5, 7, ...). The real prime is something of a mysterious entity. There is a recent book by Haran which explores it in some depth, called The Mysteries of the Real Prime (OUP, 2001).

There is a meaningful sense in which we can think of the usual absolute value |.| as "|.|oo" and of R as "Qoo".

Note that we can easily embed Q in the ring of adeles. Any rational x can be associated with the adele (x;x, x, x,...). Note that x can only have a finite collection of primes appearing in the factorisation of its denominator, so x must be a p-adic integer for all but a finite number of primes p.

The ideles constitute the multiplicative group IQ of AQ, that is those

(aoo;a2, a3, a5, a7, a11,...)

where aoo is nonzero, |ap|p is nonzero for all p and equals 1 for all but a finite number of p.

The ideles in IQ turn out to be the units (multiplicatively invertible elements) within the ring of adeles AQ.

It is possible to generalise the notions of adeles and ideles to fields other than Q (we can define subrings of integers and prime ideals. The prime ideals can be used to construct norms (analogous to p-adic norms) on the field. Some of these turn out to be archimedean and others nonarchimedean.

The term global is sometimes applied in the adelic setting, as opposed to the use of the term local in connection with the valuations |.| or |.|p.

Here are two useful excerpts from M. Pitkänen's online notes on p-adic numbers:

"2. Algebraic extensions of p-adic numbers

1. Real numbers allow only complex numbers as their algebraic extension. The extension is obtained by requiring that each number allows square root.

2. p-Adic numbers allow infinite number of algebraic extensions with all possible dimensions. One class of extensions of dimension n can be defined using irreducible polynomial of degree n. Irreducibility implies that the roots are not p-adic numbers and are linearly independent. Any number in the extension can be written as a superposition of roots with p-adic coefficients. Cyclic extensions determined by irreducible polynomial P(x) = xn-1 are the simplest extensions.

3. The requirement that any p-adically real number (not all numbers of the extension) allows square root leads to a 4-dimensional extension for p > 2. The extension is 8-dimensional for p = 2..."

"4. Canonical correspondence between the real and p-adic numbers

1. There are good motiviations for trying to find some kind of correspondence between the real and p-adic numbers. The so-called canonical correspondence is defined by the map

SUM(n)nxnpn --> SUM(n) xnp-n

of the p-adic numbers to real numbers. The image of a p-adic number is always finite and the map is continuous. The inverse map is two-valued for the real numbers with finite pinary digits...

2. The canonical correspondene makes it possible to associate to a p-adically analytic function a real function and these functions have fractal like appearance. Also higher dimensional fractal can be defined using algebraic extensions."

3. the canonical correspondence makes it possible to define p-adic definite integral and the definition makes it possible to formulate variational principles with desired properties (in particular, total divergence reduces to a surface integral)..."


additional resources

Mathworld entry

Wikipedia article on p-adic numbers

Wikipedia article on the ring of adeles

J. Baez, This Week's Finds in Mathematical Physics week 218 (part of a discussion relevant to noncommutative geometry and related topics).

F.Q. Gouvea, p-adic Numbers: An introduction (Springer-Verlag, 1993)

A.J. Baker, "An Introduction to p-adic Numbers and p-adic Analysis" (2003 lecture notes)

M. Ram Murty, "Introduction to p-Adic Analytic Number Theory" (1999 lecture notes)

W.H. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis (C.U.P. 1984)

E. Lapid, "Notes on the adeles" (July 2002)

S.D. Miller, "Adeles, Automorphic Forms and Representations" (2002 course notes)

N. Koblitz, p-Adic numbers, p-adic analysis, and zeta-functions (Graduate Texts in Mathematics, Vol 58) (Springer-Verlag, 1984)

N. Koblitz, p-Adic Analysis: A Short Course on Recent Work (Cambridge University Press, 1980)

A.M. Robert, A Course in p-adic Analysis, Graduate Texts in Mathematics 198 (Springer-Verlag, 2000).

K. Mahler, p-Adic numbers and their functions, Cambridge Tracts in Mathematics 76 (C.U.P., 1980)

A. Weil, Adeles and Algebraic Geometry, Progress in Mathematics 23 (Birkhauser, 1982)

J.E. Holly, "Pictures of ultrametric spaces, the p-adic numbers, and valued fields", American Mathematical Monthly 108 (2001) 721-728

A.A. Cuoco, "Visualizing the p-adic integers", American Mathematical Monthly 98 (1991) 355-364.

A. Robert, "Euclidean models of p-adic spaces, Lecture Notes in Pure and Applied Mathematics 192 (1997) 95-105

more books and links

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