padic 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 nonnegative 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 = xy
(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 nontrivial norms turn out to be very interesting indeed, and we shall consider them
now.
A sequence {x_{n}} 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 x_{m}  x_{n} < 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: {x_{n}}
and {y_{n}} are said to be equivalent if x_{n}  y_{n} > 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 nonobvious completions of the rationals. These are the padic
fields Q_{p} (where p is some fixed prime number) discovered by K. Hensel in 1902. Each
padic field Q_{p} 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 p^{k}(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 padic 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 'padically large' if it has a large power of p in its denominator
 a rational number is 'padically 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 padic norm equal to 1.
 all padic distances are powers (positive, negative or zero) of p
The sequence of rationals {1, p, p^{2}, p^{3},
p^{4}, p^{5}, ...} 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 p^{k}_{p} = p^{k}. Whereas
this sequence diverges rapidly according to the usual norm, it actually converges to 0
according to the padic 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 padically
close. Similarly, two rational numbers close together in 'real' terms could be far apart padically
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 Q_{p} must necessarily involve
the relinquishing of all 'common sense' Euclideantype 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
nonrational elements of Q_{p}? 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,...,p1}. You may be confused,
so here's an example (here p is some prime number bigger than 7)
...729464938675.542
would mean (reading righttoleft) 2p^{3} + 4p^{2} +
5p^{1} + 5 + 7p + 6p^{2} + 8p^{3}
+ ...
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
Q_{p}.
This notation allows us to easily describe the padic integers. These are the
padic numbers with nothing after the point. That is, sums of nonnegative powers of
p, or equivalently the limits of sequences of integers within Q_{p}.
Z_{p}, the ring of padic integers, is the maximal
subring of Q_{p}. Note that Z is not closed in Q_{p}.
Z_{p} is in fact its closure. The norm of a padic integer is always < 1
(consider the ultrametric inequality).
Note that a padic norm provides us with a notion of distance, hence a metric, a
notion of open discs, neighbourhoods, and hence a padic topology on the field Q_{p}
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 socalled 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,
pAdic 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 20^{th} 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
nonEuclidean and nonRiemannian geometry of space at small distances.
How can one construct a physical theory corresponding to a nonArchimedean
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 spacetime 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 nonperiodic 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 padic
norm for some fixed prime number p."
(from V.S. Vladimirov, I.V. Volovich, E.I. Zelenov,
pAdic 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
SpaceTime and Microphysics  Towards a Theory of Scale Relativity (World Scientific, 1993).)
Because padic 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 Q_{p} rather than
R as it traditionally has been. In fact there's a growing body of
"padic physics". The book by
Vladimirov, et.al. is a good starting point.
If we're going to use padic 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 padic norms (as well as the usual one which produces R)
simultaneously.
The adeles constitute a locally compact topological ring A_{Q},
individually taking the form
(a_{oo};a_{2}, a_{3},
a_{5}, a_{7}, a_{11},...)
a_{oo} is a real number (that subscript is meant to be 'infinity').
Each a_{p} is a padic number. We must require
that in all but a finite number of cases a_{p} is a padic integer.
a_{oo} is the archimedian entry. The rest are nonarchimedian.
Hence the semicolon separating them.
The notation a_{oo} conforms to the notion that whereas the field of
padic 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 "Q_{oo}".
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 padic integer for all but a finite number of primes p.
The ideles constitute the multiplicative group I_{Q} of
A_{Q}, that is those
(a_{oo};a_{2}, a_{3},
a_{5}, a_{7}, a_{11},...)
where a_{oo} is nonzero, a_{p}_{p} is
nonzero for all p and equals 1 for all but a finite number of p.
The ideles in I_{Q} turn out to be the units (multiplicatively invertible elements) within the ring
of adeles A_{Q}.
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 padic 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 padic numbers:
"2. Algebraic extensions of padic 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. pAdic 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 padic
numbers and are linearly independent. Any number in the extension can be written as a superposition
of roots with padic coefficients. Cyclic extensions determined by irreducible
polynomial P(x) = x^{n}1 are the simplest extensions.
3. The requirement that any padically real number (not all numbers of the extension)
allows square root leads to a 4dimensional extension for p > 2. The extension is
8dimensional for p = 2..."
"4. Canonical correspondence between the real and padic numbers
1. There are good motiviations for trying to find some kind of correspondence between
the real and padic numbers. The socalled canonical correspondence is
defined by the map
SUM(n)nx_{n}p^{n} > SUM(n) x_{n}p^{n}
of the padic numbers to real numbers. The image of a padic number is
always finite and the map is continuous. The inverse map is twovalued for the real numbers
with finite pinary digits...
2. The canonical correspondene makes it possible to associate to a padically
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 padic 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 padic 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, padic Numbers: An introduction (SpringerVerlag, 1993)
A.J. Baker, "An
Introduction to padic Numbers and padic Analysis" (2003 lecture notes)
M. Ram Murty, "Introduction
to pAdic Analytic Number Theory" (1999 lecture notes)
W.H. Schikhof, Ultrametric Calculus: An Introduction to pAdic 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,
pAdic numbers, padic analysis, and zetafunctions (Graduate Texts in Mathematics, Vol 58)
(SpringerVerlag, 1984)
N. Koblitz,
pAdic Analysis: A
Short Course on Recent Work (Cambridge University Press, 1980)
A.M. Robert, A
Course in padic Analysis, Graduate Texts in Mathematics
198 (SpringerVerlag, 2000).
K. Mahler, pAdic 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 padic numbers, and valued
fields", American Mathematical Monthly 108 (2001) 721728
A.A. Cuoco, "Visualizing the padic integers", American Mathematical
Monthly 98 (1991) 355364.
A. Robert, "Euclidean models of padic spaces, Lecture Notes in Pure
and Applied Mathematics 192 (1997) 95105
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