Adele groups, p-adic groups, solenoidsgarrett/m/mfms/notes_2013-14/12_1_adeles.pdfPaul Garrett: Adele groups, p-adic groups, solenoids (April 29, 2014) with commuting triangles but

(April 29, 2014)

Adele groups, p-adic groups, solenoids

Paul Garrett [email protected] http://www.math.umn.edu/ garrett/

[This document ishttp://www.math.umn.edu/˜garrett/m/mfms/notes 2013-14/12 1 adeles.pdf]

1. Hensel’s lemma2. Metric definition of p-adic integers Zp and p-adic rationals Qp3. Elementary/clumsy definitions of adeles A and ideles J4. Uniqueness of objects characterized by mapping properties5. Existence of limits6. Zp and Z as limits7. Qp and A as colimits8. Abelian solenoids (R×Qp)/Z[1/p] and A/Q9. Non-abelian solenoids and SL2(Q)\SL2(A)

Although we will also give the more typical descriptions, we also show that the adeles A, p-adic integers Zp,p-adic rationals Qp, and related objects are already right under our noses, if only we can see them. Thesethings are not artificial or mere stylistic choices. It is unwise to ignore them.

Several important technical issues lead to such things. One is closer examination of Hecke operators, forexample on holomorphic modular forms:

TNf =∑γ∈XN

f |2kγ (f holomorphic weight 2k for Γ = SL2(Z))

where XN is integer-entry 2-by-2 matrices with determinant 1, with the weight-2k action

(f |2kγ

)(z) = f

(az + b

cz + d

)(cz + d)−2k (where γ =

(a bc d

))

An obstacle to a straightforward proof of the self-adjointness of these operators with respect to the Peterssoninner product

〈f, F 〉 =

∫Γ\H

f(z) F (z) y2k dx dy

y2

is that, although the sum over XN is a modular form of level one, the individual summands are typically notlevel one, but higher level, usually level N . That is, the most straightforward assertion we can rely upon isthat for γ ∈ XN the γth summand is a modular form only for

ΓN = {(a bc d

)=

(1 00 1

)mod N}

That is, the straightforward assertion is that(f |2kγ

)∣∣2kβ = f |2kγ (for β ∈ ΓN )

To see how the sufficient condition β ∈ ΓN arises, use the associativity of the action:

f |2k(γβ) = f |2k(γβγ−1 · γ) =(f |2kγβγ−1

)∣∣2kγ

Thus, f |2kγ is assured invariant under β when γβγ−1 ∈ Γ1, for f of level 1. For example,

γ

(a bc d

)γ−1 =

(a b/NcN d

)(for γ =

(1 00 N

))

1

Paul Garrett: Adele groups, p-adic groups, solenoids (April 29, 2014)

which is in Γ1 = SL2(Z) only for b = 0 mod N . Similarly,

γ

(a bc d

)γ−1 =

(a bNc/N d

)(for γ =

(N 00 1

))

is in Γ1 only for c = 0 mod N . Generally, with γ ∈ XN , similar computations show that β ∈ ΓN is thesimplest sufficient condition for (f |2kγ)|2kβ = f |2kγ.

To treat the summands in TNf directly, we might enlarge the space of modular forms to include those forΓN for all N . The full family of quotients ΓN\H, indexed by positive integers N ordered by divisibility, ispartly depicted in

. . . // Γ8\H // Γ4\H

##HHHHHHHHH

. . . // Γ12\H

::vvvvvvvvv// Γ6\H //

##HHHHHHHHHΓ2\H

##GGGGGGGG

. . . //

::vvvvvvvvvvv Γ9\H // Γ3\H // Γ1\H

. . .

::vvvvvvvvvvv // Γ10\H

DD// Γ5\H

;;wwwwwwww

. . . //

;;vvvvvvvvvvv Γ7\H

DD

(where ΓNz → ΓMz for M |N)

The poset [1] of positive integers ordered by divisibility is very interesting, but adds a further complication,so we might consider subfamilies corresponding to powers of a single prime, with their simpler partial orderstructures:

. . . // Γp3\H // Γp2\H // Γp\H // Γ1\H

In both cases, we want a single object on which all the functions on all spaces Γpn\H live, with quotient

maps to every Γpn\H, fitting together so that all triangles commute [2] in the diagram

???(())**++

. . . // Γp3\H // Γp2\H // Γp\H // Γ1\H

The corresponding diagram for the full family of ΓN\H is messier, but similar. The admittedly naturalcandidate H itself, with obvious quotient maps qN : H → ΓN\H fails in several ways. First, althoughfunctions f ∈ L2(ΓN\H) become functions on H by composing with qN , the function f ◦ qN is not in L2(H)with respect to the SL2(R)-invariant measure dx dy/y2. Second, the collection of all functions on H is fartoo large in the sense that most functions are not invariant under any ΓN . Third, certainly the true object??? fits into a diagram

. . . // Γp3\H // Γp2\H // Γp\H // Γ1\H

???

ddI I I I I

OO��

::vvvvv

55kkkkkkkkk

[1] Poset is the common abbreviation for partially ordered set, meaning a set S with an ordering ≤ with expected

properties: transitivity x ≤ y and y ≤ z implies x ≤ z, reflexivity x ≤ x, and x ≤ y and y ≤ x implies x = y, but

possibly lacking the dichotomy requirement that either x ≤ y or y ≤ x.

[2] For a diagram to commute means that the same outcomes are obtained no matter which route is followed throught

the diagram.

2


with commuting triangles but it turns out that there is no map ???→ H fitting into

H(())**++

. . . // Γp3\H // Γp2\H // Γp\H // Γ1\H

???

ddI I I I I

OO��

::vvvvv

55kkkkkkkkknon−existent

kk (fails ...)

We will see that the correct object ??? in the diagram is

SL2(Z[ 1p ])∖(

H× SL2(Qp)) ''((**,,

. . . // Γp3\H // Γp2\H // Γp\H // Γ1\H

where Qp is the field of p-adic numbers, the fraction field of the p-adic integers Zp, the latter often definedas the completion of Z with respect to the p-adic metric d(x, y) = |x− y|p with the p-adic norm

|p` ·m|p = p−` (for integers m prime to p)

and Z[ 1p ] is the ring of rational numbers with denominators only allowed to be powers of p. The action of

γ ∈ SL2(Z[ 1p ]) on H× SL2(Q2) is g(z × h) = gz × gh with linear fractional action on z ∈ H and left matrix

multiplication on h ∈ SL2(Qp). We will see that the diagonally-imbedded copy of SL2(Z[ 1p ]) is a discrete

subgroup of SL2(R)× SL2(Qp) much as SL2(Z) is a discrete subgroup of SL2(R).

We will see that this object does have the desired universal property, that for every other object X with acompatible family of maps (making triangles commute)

. . . // Γp3\H // Γp2\H // Γp\H // Γ1\H

X

ddII

II

I

OO��

::vv

vv

v

55jjjjjjjjjj

there is a unique map X → SL2(Z[ 1p ])\

(H× SL2(Q2)

)giving a commutative diagram

SL2(Z[ 1p ])\

(H× SL2(Qp)

) ''((**,,. . . // Γp3\H // Γp2\H // Γp\H // Γ1\H

X

ddII

II

I

OO��

;;vv

vv

v

55kkkkkkkkkkunique

ll

This situation also reveals a natural action of the group SL2(Qp) on the collection of quotients Γpn\H. Thisis very far from obvious!

Keeping in mind that H ≈ SL2(R)/SO(2,R), the full diagram using all ΓN ’s has a similar universal object,

3


with A the ring of rational adeles [3]

. . . // Γ2\H

##GGGGGGGG

SL2(Q)\SL2(A)/SO(2,R)

,,

))++++

,,YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY. . . // Γ6\H

;;wwwwwwww// Γ3\H // Γ1\H

. . . // Γ5\H

;;wwwwwwww

For many reasons we might want to get the group SO(2,R) out of the way, so that SL2(R) can act on theright as do the groups SL2(Qp), and we have a diagram

. . . // Γ2\SL2(R)

''OOOOOOOOOOO

SL2(Q)\SL2(A)

--

**++,,

,,ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ . . . // Γ6\SL2(R)

77ooooooooooo// Γ3\SL2(R) // Γ1\SL2(R)

. . . // Γ5\SL2(R)

77ooooooooooo

This reveals the natural action of SL2(A) on the right on the family of quotients ΓN\SL2(R), incorporatingan action of every group SL2(Qp) as well as SL2(R).

We will prove the universal mapping property that, for every space X and compatible family of mapsX → ΓN\SL2(R), there is a unique map X → SL2(Q)\SL2(A) fitting into a commutative diagram

. . . // Γ2\SL2(R)

''OOOOOOOOOOO

SL2(Q)\SL2(A)

--

**++,,

,,ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ . . . // Γ6\SL2(R)

77ooooooooooo// Γ3\SL2(R) // Γ1\SL2(R)

. . . // Γ5\SL2(R)

77ooooooooooo

X

unique

]] ;;xx

xx

xx

xx

xx

x

66mmmmmmmmmmmmmmmmm

22fffffffffffffff

::

_ ` a c d f g i jl

no

qs

u

33

��

��

}z

wt

ro

mk i h

Standard terminology for this property is that SL2(Q)\SL2(A) is the (projective) limit of the quotientsΓN\SL2(R).

To warm up to this situation, replace ΓN and SL2(R) by simpler objects, namely, replace ΓN by NZ and

[3] As discussed below, A contains a copy of every Qp as well as R, but is smaller than the product.

4


SL2(R) by R. In that case, we will find that Q\A is the universal object in analogous diagrams

. . . // 2Z\R

##FFFFFFFF

Q\A

,,

((****

,,XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX . . . // 6Z\R

;;wwwwwwwww// 3Z\R // Z\R

. . . // 5Z\R

;;xxxxxxxx

X

unique

XX =={{

{{

{{

{{

{{

88qqqqqqqqqqqqqq

33hhhhhhhhhhhhh

??

_ a b d f h j l np

su

x{

~

44

�

��

��

}z

vs

qn l j

Again, first with the simpler poset of powers pn of a fixed prime p, rather than the whole poset of positiveintegers ordered by divisibility: for every compatible family of maps X → pnZ\R, there is a unique mapX → Z[ 1

p ]\(R×Qp

)fitting into

Z[ 1p ]\(R×Qp

) ++ ** )) ((. . . // p3Z\R // p2Z\R // pZ\R // Z\R

X

OO��

::uu

uu

u

55jjjjjjjjjj

33ggggggggggggggggunique

jj

All the quotients NZ\R are circles, and the (projective) limit Z[ 1p ]\R is a solenoid.

As an even simpler examples of limits: the ring of p-adic integers Zp is the limit of the finite rings Z/pn: forevery compatible family of maps X → Z/pn, there is a unique X → Zp giving a commutative diagram

Zp** (( '' &&

. . . // Z/p4 // Z/p3 // Z/p2 // Z/p

X

OO��

;;ww

ww

w

55kkkkkkkkk

33ggggggggggggggunique

hh

This characterization of Zp is more useful than the more-elementary description as metric completion, below.

It is striking and profoundly important that these diagrammatic universal-mapping-property characteriza-tions determine the universal objects uniquely (up to unique isomorphism), as we will show. That is, forexample, any other object Y fitting into all the diagrams

Y** (( '' &&

. . . // Z/p4 // Z/p3 // Z/p2 // Z/p

X

OO��

;;ww

ww

w

55kkkkkkkkk

33ggggggggggggggunique

hh

is isomorphic to Zp, and there is a unique isomorphism Y → Zp compatible with all the maps to the Z/pn’s.Further, the uniqueness proof does not use internal details of either the objects or the maps, but only theshape of the diagrams! Thus, the proofs themselves are universal.

5


1. Hensel’s lemma

Kurt Hensel’s 1897 conception of p-adic numbers can be illustrated in systematic solution of problems suchas x2 = −1 mod 5n for all powers 5n of 5. [4]

Starting with either x1 = 2 or x1 = 3 (whose squares are 4 and 9, which are −1 mod 5), one hopes toadjust the solution mod 5 to be a solution mod 52. Namely, one hopes that for some y the modified valuex2 = x1 + 5y will satisfy x2 = −1 mod 25. This condition simplifies usefully

(x1 + 5y)2 = −1 mod 52

x21 + 10x1y + 25y2 = −1 mod 52

(x21 + 1) + 10x1y = 0 mod 52

x21 + 1

5+ 2x1y = 0 mod 5

since x21 + 1 is divisible by 5. A critical point is that the y2 term disappears mod 52, leaving in any case a

linear problem in y. Since −2x1 = 1 mod 5 is invertible mod 5, we can solve for

y = (−2x1)−1 · x21 + 1

5= (−2 · 2)−1 · 22 + 1

5= 1 mod 5

and then

x2 = x1 + 5y = 2 + 1 · 5 = 7

satisfies

x22 = (x1 + 5y)2 = −1 mod 52

This process of improvement can be continued indefinitely, imitating the example just done, as follows. With

x2n = −1 mod 5n

try to find y mod 5 such that

(xn + 5ny)2 = −1 mod 5n+1

The same rearrangement gives

(xn + 5ny)2 = −1 mod 5n+1

x2n + 2 · 5nxny + 52ny2 = −1 mod 5n+1

(x2n + 1) + 2 · 5nxny = 0 mod 5n+1

x2n+15n + 2xny = 0 mod 5

[4] The abstracted notion of metric space did not exist until [Frechet 1906], so Hensel could not use such an idea in

1897. The notion of projective limit was not developed until after 1940, for example by S. Eilenberg and S. MacLane,

motivated by algebraic topology. Instead, Hensel attempted to make analogies to power series, but these analogies

did not quite work right.

6


using the fact that x2n + 1 is already divisible by 5n, and that the y2 term goes away. The last equation has

a unique solution

y = (−2xn)−1 · x2n + 1

5nmod 5

where the inverse need be taken only mod 5, not modulo any higher power of 5. The new solution is

xn+1 = xn + 5ny

and satisfiesxn+1 = xn mod 5n

Thus, x−1n = x−1

1 mod 5, so only a single multiplicative inverse need be computed, and the induction step is

y = (−2xn)−1 · x2n + 1

5nmod 5

This produces the sequence

2, 7, 57, 182, 1482, 13057, 25182, . . . ∼√−1

This procedure is an example of Hensel’s lemma. A little more generally:

[1.0.1] Claim: Let f(x) ∈ Z[x], p a prime, and x1 such that

f(x1) = 0 mod p and f ′(x1) 6= 0 mod p

Then the recursion [5]

xn+1 = xn − f(xn) · f ′(x1)−1 mod pn+1

(where f ′(x1)−1 is an inverse modulo p) determines a sequence of integers xn such that

f(xn) = 0 mod pn

with the compatibilityxn+1 = xn mod pn

Proof: Amusingly, we need a Taylor series expansion

f(x+ h) = f(x) + h · f ′(x) + (error term)

legitimate for purely algebraic reasons, for polynomials, of the form

f(x+ h) = f(x) + f ′(x) · h+ E · h2

where E is a polynomial in x and h with coefficients in Z. [6] Granting such an expression, letδ = −f ′(xn)−1 · f(xn), with inverse modulo pn, and evaluate

f(xn+1) = f(xn + δ) = f(xn) + f ′(xn) · δ + E · δ2

[5] This recursive formula is the Newton-Raphson formula, easily derived geometrically in the real-number case, by

sliding down the tangent, that is, by finding the intersection of the horizontal axis with the tangent line to the curve

y = f(x) at the point (xn, f(xn).

[6] The derivative of a polynomial can be defined without taking any limits, via the usual formula ddx (xn) = nxn−1,

and requiring that this map be linear over whatever commutative ring the polynomials’ coefficient lie in.

7


= f(xn)− f ′(xn) · f ′(xn)−1 · f(xn) + E · δ2 = f(xn)− f(xn) + E(xn) · δ2 = E · δ2

Since f(xn) = 0 mod pn, certainly xn+1 = xn mod pn, and f ′(xn) 6= 0 mod p, so has an inverse mod pn,since f and f ′ have coefficients in Z. And then δ = 0 mod pn, so δ2 = 0 mod p2n. Since E is a polynomialwith coefficients in Z, E · δ2 = 0 mod p2n. That is,

f(xn+1) = 0 mod p2n

For n ≥ 1, we have 2n ≥ n+ 1, meeting the requirement on the recursion.

In the expressionxn+1 = xn − f(xn) · f ′(xn)−1 mod pn+1

since f(xn) = 0 mod pn, we only need f ′(xn)−1 modulo p in order to know xn+1 mod pn+1. Thus, it sufficesto check that

f ′(x1)−1 = f ′(xn)−1 mod p

Indeed, since xn+1 = xn mod pn, xn = x1 mod p for all n. Since f ′ has coefficients in Z, we havef ′(xn) = f ′(x1) for all n. Since f ′(x1) 6= 0 mod p, the inverses mod p are all the same.

The factorials occurring in the usual form of the Taylor expansion appear to be a problem. However, anypolynomial P (x) =

∑i bix

i with coefficients in Z can be written in the form

P (x+ h) = c0 + c1 · h+ c2 · h2 + . . . (a finite expansion)

with ci polynomials in x by substituting x+ h in P and expanding in powers of h. Thus, the issue is to seethat, in this expansion

c1(x) = f ′(x)

The requisite expansion is linear [7] in the polynomial P , it suffices to consider P (x) = xn. By the BinomialTheorem

(x+ h)n = xn + nxn−1 · h+ E · h2

where E is a polynomial in x and h, with coefficients in Z. Since nxn−1 is the derivative of xn, we have thedesired sort of Taylor expansion, and Hensel’s procedure will succeed. ///

[1.0.2] Remark: No special properties of the ring Z were used above, so the same argument succeeds, andthis simple case of Hensel’s lemma applies, to prime ideals in arbitrary commutative rings with identity.

[1.0.3] Remark: The compatibility xn+1 = xn mod pn on a sequence of integers xn exactly says that

. . . // x3// x2

// x1

fits into the diagram

. . . // Z/p3 // Z/p2 // Z/p

[7] This linearity is that the expansion for the sum of two polynomials is the sum of the corresponding expansions.

8


2. Metric definition of p-adic integers Zp and p-adic rationals Qp

The p-adic norm or absolute value |n|p of an integer pem with m prime to p is

p-adic norm of m · pe = |m · pe|p = p−e

Additionally declare that |0|p = 0. The p-adic metric on Z is

p-adic distance m to n = |m− n|p

The p-adic norm on Q extends that on Z, namely

|pn · cd|p = p−n (integers c, d prime to p and n ∈ Z)

Perhaps counter-intuitively, p is small and 1/p is large:

|p|p =1

p|1p|p = p

In summary, high divisibility by p means p-adically small.

[2.0.1] Remark: In a context where the p-adic norm is the only norm used, we may suppress the subscript.

[2.0.2] Example: The sequences {xn} produced via Hensel’s lemma to achieve f(xn) = 0 mod pn (for given

f(x) ∈ Z[x]) are Cauchy sequences [8] in the p-adic metric, since xm+1 = xm mod pm implies that for allpairs of indices m ≤ n

|xm − xn|p ≤ |pm|p = p−m

And, the fact that f(xn) = 0 mod pn gives

limnf(xn) = 0 (in the p-adic metric)

We define p-adic things as completions [9] p-adic integers Zp = p-adic metric completion of Z

p-adic numbers Qp = p-adic metric completion of Q

but we should check that the p-adic metric really is a metric. The positivity and symmetry of the associatedp-adic metric are immediate, but the triangle inequality is not so immediate.

[2.0.3] Example: Partly for the visual effect, we note that

1 + 2 + 4 + 8 + 16 + . . . = −1 (in Z2)

[8] The p-adic sense of Cauchy sequence is completely analogous to that in R, namely, that a sequence {xn} is Cauchy

if for every ε > 0 there is N such that for all m,n ≥ N we have |xm − xn|p < ε.

[9] A metric space is complete if every Cauchy sequence converges. A completion of a metric space X is often defined

by a construction (given in the appendix below), but, as discussed shortly, the idea is that the completion is the

smallest complete metric space containing the given one.

9


This is genuinely valid in Z2. Superficially, thinking of the real numbers, this might be perceived as acorruption of the familiar identity

1 + r + r2 + r3 + . . . =1

1− r(for |r| < 1)

for real or complex r, and with the usual real or complex absolute value |r|. But it is not a flawed version atall, since it is literally correct 2-adically.

Regarding the triangle inequality, in fact, we have a strange stronger property: [10]

[2.0.4] Proposition: (ultrametric inequality) For x, y in Q,

|x+ y| ≤ max{|x|, |y|}

In fact, equality occurs in this last inequality, except possibly when |x| = |y|. Thus, in terms of the p-adicmetric d(x, y),

d(x, y) ≤ max{d(x, z), d(z, y)}

with equality except possibly when d(x, z) = d(z, y).

Proof: Let x = pm · a/b and y = pn · c/d with a, b, c, d prime to p, and positive or negative integers m,n.Without loss of generality, we can suppose that m ≤ n. Certainly

x+ y = pm · ab

+ pn · cd

=pmad+ pncb

bd= pm · ad+ pn−mcb

bd

Note that, by unique factorization, bd is still prime to p. For m < n, the numerator in the fraction in

x+ y = pm · ad+ pn−mcb

bd

is prime to p. Thus, for m < n, that is, for |x| > |y|,

|x+ y| = p−m = |x| = max{|x|, |y|}

When m = n, that is, when |x| = |y|, the numerator may be further divisible by p in some cases. Thus, for|x| = |y|,

|x+ y| ≤ p−m = max{|x|, |y|}

Then

d(x, y) = |x− y| = |(x− z) + (z − y)| ≤ max{|x− z|, |z − y|} = max{d(x, z), d(z, y)}

with equality unless possibly when d(x, z) = d(z, y). ///

An isometry f : X → Y of metric spaces X,Y is a set map from X to Y such that distances are preserved,namely,

dY (f(x), f(x′)) = dX(x, x′)

for all x, x′ in X. [11] A metric space is complete if every Cauchy sequence converges.

[10] It is traditional at this point to say the the ultrametric property proven in the proposition can be construed as

asserting that all p-adic triangles are isosceles.

[11] Beware that in some contexts an isometry is presumed to be a bijection to the target, in addition to preserving

distance. In our context we specifically do not assume that isometries are surjections to the target spaces.

10


[2.0.5] Definition: A completion of a metric space X is a complete metric space Y and an isometryi : X → Y such that, for every isometry j : X → Z to a complete metric space Z, there is a unique isometryJ : Y → Z giving a commutative diagram

Xi //

j @@@@@@@@ Y

J��

Z

As usual with mapping-property characterizations, there is at most one completion, up to unique isometricisomorphism. In the appendix below we give the usual construction, which proves existence.

As expected, define p-adic integers Zp = p-adic metric completion of Z

p-adic numbers Qp = p-adic metric completion of Q

As usual, operations on the completions are defined as limits, and well-definedness must be proven. That is,for Cauchy sequences of rational numbers xn and yn with xn → a and yn → b p-adically, define

a+ b = limn

(xn + yn) a · b = limn

(xn · yn) |a| = limn|xn|

[2.0.6] Proposition: The p-adic norm is a continuous function on the completion. The p-adic norm ismultiplicative on the completions, that is,

|ab| = |a| · |b|

Addition, multiplication, and multiplicative inverse (away from 0) are continuous maps in the p-adic metric.Also,

Zp = {x ∈ Qp : |x| ≤ 1} = {x ∈ Qp : |x| < p}

In particular, Zp is both closed and open in Qp. The p-adic integers Zp form an integral domain, and Qp isits field of fractions. On Qp it is still true that the ultrametric inequality holds:

|x+ y| ≤ max{|x|, |y|} (with equality except possibly when |x| = |y|)

[2.0.7] Corollary: Polynomials with p-adic coefficients give continuous functions on Qp. ///

[2.0.8] Remark: The ultrametric property makes the set of x with |x| ≤ 1 a subring, since otherwise thisset would not be closed under addition. There is no analogous subring of R.

Proof: From the general theory of metric spaces the metric d(, ) on the completion is defined by takinglimits

d(a, b) = limnd(xn, yn)

where xn, yn are rational and xn → a and yn → b. Part of the general assertion is that this is well-defined,that is, is independent of the Cauchy sequences approaching a and b. In the present situation, the extensionof the p-adic norms to the completion is a special case, taking b = 0, so

|a| = d(a, 0) = limn|xn − 0| = lim

n|xn|

We have the expected

|a+ b| = |a− (−b)| = d(a, b) ≤ d(a, 0) + d(0, b) = |a|+ |b|

11


The p-adic continuity of the p-adic norm on Q is immediate from the continuity of the metric on thecompletion, which is a general fact about completions.

The multiplicativity |xy| = |x| · |y| follows for x, y ∈ Q from the fact that the ideal pZ is prime in Z. Thatis, writing x = pm · a/b and y = pn · c/d with a, b, c, d relatively prime to p,

xy = pm+n · (ac)/(bd)

and by the primality of p the products ac and bd are still prime to p. Then for xn → a and yn → b withxn, yn in Q,

|ab| = limn|xnyn| = lim

n(|xn| · |yn|) = lim

n|xn| · lim

n|yn| = |a| · |b|

by continuity of multiplication of real numbers, since |xn| → |a| and thus {|xn|} is Cauchy in R (as is {|yn|}).

Continuity of addition is easy, from

|(x+ y)− (x′ + y′)| ≤ |x− x′|+ |y − y′|

For multiplication,

|(xy)− (x′y′)| ≤ |x(y − y′)|+ |(x− x′)y′| ≤ |x(y − y′)|+ |(x− x′)(y′ − y)|+ |(x− x′)y|

= |x||y − y′|+ |x− x′||y′ − y|+ |x− x′||y|

Thus, given x, y in Qp and x′, y′ sufficiently close to them, the products are close.

For multiplicative inverses, let x 6= 0. From

1 = |1| = |x · x−1| = |x| · |x−1|

we have

|x−1| =1

|x|

and the continuity of inversion in R× gives the result.

Unsurprisingly, since Z is a subset of Q, there is the same containment relation between their completions.Since the p-adic absolute value of x ∈ Z is at most 1, we immediately have containment in one direction,namely

Zp ⊂ {x ∈ Qp : |x| ≤ 1}

On the other hand, for y ∈ Qp with |y| ≤ 1, since Qp is the completion of Q, there is r ∈ Q arbitrarily closeto y. For |y − r| ≤ 1,

|r| ≤ max{|r − y|, |y|} ≤ 1

so |r| ≤ 1 itself. Thus, it suffices to show that r itself can be approximated arbitrarily well by elements of Z.

Since |r| ≤ 1, r = pn · ab with a, b ∈ Z relatively prime to p and n ≥ 0. For b 6= 0 mod p, Hensel’s lemmagives a sequence of integers xi such that

b · xi = 1 mod pi

That is, xi is a Cauchy sequence of integers approaching a multiplicative inverse b−1 of b in Qp. By continuityof the norm,

|b−1| = limi|xi| = lim

n1 = 1

since p does not divide any of the integers xi. Thus,

limipn · a · xi = pn · a/b = r

12


That is, as i varies the integers pn · a · xi get close to r. This proves that Z is dense in {y ∈ Qp : |y| ≤ 1}, soZp is exactly the latter set, as claimed.

Since the possible values of the p-adic norm are only powers of p, the condition |x| < p implies |x| ≤ 1.

If ab = 0, then |ab| = 0, and by multiplicativity |a| · |b| = 0, and a or b is 0. Thus, Zp is an integral domain.

Since Q is a field, it follows fairly easily that its completion Qp is a field. To see that Qp has no propersubfield that contains Zp, recall that any element x ∈ Qp with |x| ≤ 1 lies in Zp. For |x| > 1, x cannot be 0,so has an inverse (since Qp is a field), and |x−1| < 1, so lies in Zp. This proves that Qp is the fraction fieldof Zp.

Finally, the ultrametric inequality persists, seen as follows. Given a, b ∈ Qp, let xn, yn be rational numberswith xn → a and yn → b. Given ε > 0, let n be large enough such that |xn − a| < ε and |yn − b| < ε. Thetriangle inequality gives

|xn + yn − (a+ b)| < |xn − a|+ |yn − b| < 2ε

and

|a+b| ≤ |(a−xn)+(b−yn)|+|xn+yn| ≤ 2ε+max{|xn|, |yn|} ≤ 3ε+max{|xn|−ε, |yn|−ε} < 3ε+max{|a|, |b|}

This is true for every ε > 0, giving the ultrametric inequality. ///

[2.0.9] Proposition: The units in Zp are

Z×p = {x ∈ Qp : |x| = 1}

The non-zero ideals in the ring Zp are pn ·Zp for 0 ≤ n ∈ Z. We have Z∩pnZp = pnZ, and the natural maps

Z/pnZ −→ Zp/pnZp

are isomorphisms.

Proof: For x, y ∈ Zp such that xy = 1, of course 1 = |1| = |xy| = |x| · |y|. Since |x| ≤ 1 and |y| ≤ 1, bothhave norm 1. Thus,

Z×p ⊂ {x ∈ Zp : |x| = 1}

On the other hand, take |x| = 1 and y ∈ Z such that |x− y| < 1. Then

|y| = |y − x+ x| = max{|y − x|, |x|} = |x| = 1

since the norms are unequal. Thus, y is an integer not divisible by p. An exercise using Hensel’s lemmashows y has an inverse in Zp. Then

x = x− y + y = y · (1 +x− yy

)

yields a convergent series expression for an inverse to x, namely

x−1 = y−1 ·

((1− x− y

y+

(x− yy

)2

−(x− yy

)3

+ . . .

)

which converges since |(x− y)/y| < 1. Thus, x−1 exists in Zp. The natural maps Z→ Zp and pnZ→ pnZpgive a map

Z/pnZ −→ Zp/pnZp

13


For x ∈ Z ∩ pnZp, we have |x| ≤ p−n, so pn divides x. That is, x ∈ pnZ, as claimed. Thus, these naturalmaps are injections. For y ∈ Zp, consider the coset y + pnZp. Using the density of Z in Zp, let x ∈ Z with|y − x| < p−n. Then y − x ∈ pnZp, so

y + pnZp = (y − x) + x+ pnZp = pnZp + x+ pnZp = x+ pnZp

proving the surjectivity of the natural map.

Given a non-zero ideal I in Zp, let M be the sup of the p-adic norms of elements of I. We claim that thesup is attained at some element of I, and that such a largest element generates I. Indeed, there are onlyfinitely-many values of the p-adic norm on Zp lying in any interval [δ, 1] for δ > 0, so the sup is attained.For x in I of maximum norm, for any y ∈ I, |y/x| ≤ 1, so y/x ∈ Zp. That is, y ∈ Zp · x. This shows thatI = Z·x. ///

[2.0.10] Remark: One should prove local compactness of Qp and Zp from this metric viewpoint as anexercise.

[2.0.11] Example: Let b be an integer relatively prime to a prime p. Let x1 be an inverse of b mod p. Thenthe recursion

xn+1 = xn − x−11 bxn

produces a sequence of integers xn such that

bxn = 1 mod pn

Indeed, letting f(x) = kx− 1, Hensel’s Lemma gives the recursion

xn+1 = xn − f(xn) · f ′(x1)−1 = xn − (bxn − 1) · b−1 = xn − (bxn − 1) · x1

using only the inverse mod p, not any higher power of p. For b 6= 0 mod p this gives a sequence of integersxn such that

xn+1 = xn mod pn and b · xn = 1 mod pn

3. Elementary/clumsy definitions of adeles A and ideles JWe give the quick-but-opaque common definitions of adeles A = AQ and ideles J = JQ of Q. The sense andmotivation and roles are discussed subsequently, in a slightly more sophisticated context.

For a finite set S of primes of Z, let

AS = R×∏p∈S

Qp ×∏p 6∈S

Zp

with the product topology. The full adele ring A is the ascending union, sometimes called a restricted product,

14


of the rings AS : [12]

A =⋃S

AS = {(x∞, x2, . . . , xp, . . .) ∈ R×∏p

Qp : all but finitely-many xp are in Zp}

A basis for a topology on A is given by the union of bases for the topologies of the subsets AS . That is,adeles are tuples of real numbers and p-adic numbers all-but-finitely-many of which are local integers.

Similarly, for a finite set S of primes of Z, let

JS = R× ×∏p∈S

Q×p ×∏p 6∈S

Z×p

with the product topology. The full idele group J is the ascending union, sometimes called a restrictedproduct, of the groups JS : [13]

J =⋃S

JS = {(x∞, x2, . . . , xp, . . .) ∈ R× ×∏p

Q×p : all but finitely-many xp are in Z×p }

A basis for a topology on J is given by the union of bases for the topologies of the subsets JS .

[3.0.1] Remark: At this point, one should check that, while J ⊂ A, the subspace topology on J from A isstrictly coarser than the topology on J.

4. Uniqueness of objects characterized by mapping properties

By object we mean an instance of a type of mathematical entity, such as group, ring, topological space, orsomething combining these or other structures. By map we mean a suitable map among the specified objects:for groups a map is a group homomorphism, for topological spaces a map is a continuous function.

We make some very mild assumptions: the notion of map includes closure under composition, associativityof the composition of maps, and an identity map idX of every suitable object X to itself, with the expectedrequirements that idX ◦ f = f for maps f : Y → X, and f ◦ idX = f for maps f : X → Y .

The collection of objects, together with the maps, is a category. [14]

[12] The term restricted product is only used in this clumsy definition of adeles and ideles. Since this notion does not

directly refer to anything else for comparison, it is useless except as a reminder that the adeles are not the whole

product of R and all the Qp’s. More unfortunately, that viewpoint fails to emphasize that A is a colimit of the rings

AS , meaning that for every collection of maps AS → R to another topological ring R, compatible with inclusions,

there is a unique map A→ R giving a commutative diagram

. . . // %%

((QQQQQQQQ AS // ''

!!CC

CC . . . A

~~R

The colimit characterization shows that there is no choice in the topology. We will take up this viewpoint below.

[13] Again, J is the colimit of the groups JS . Again, the colimit characterization of J shows that there is no choice of

correct topology on J: it is uniquely determined.

[14] We are doing naive category theory, as opposed to formal or axiomatic category theory, in the same way that

ordinary mathematics uses naive set theory, not formal or axiomatic set theory.

15


Given objects Xi with maps. . . // X1

// X0

let object X and maps X → Xi and object Y and maps Y → Xi fit into diagrams

X## $$

. . . // X1// X0 Y

## $$. . . // X1

// X0

such that, for all families of maps Z → Xi such that all triangles commute in

. . . // X1// X0

Z

==||

||

66mmmmmmmm

there are unique maps Z → X and Z → Y giving commutative diagrams

X## $$

. . . // X1// X0

Z

==||

||

66mmmmmmmm

`` Y## $$

. . . // X1// X0

Z

==||

||

66mmmmmmmm

``

Then

[4.0.1] Claim: There is a unique isomorphism Y → X giving a commutative diagram

X## $$

. . . // X1// X0

Y << ::

OO

. . . // X1// X0

Proof: First, we prove that the only map of a projective limit to itself compatible with all projections isthe identity map. That is, using X → Xi in the role of : Z → Xi, we find a unique map X → X making alltriangles commute in

X## $$

. . . // X1// X0

X

==||

||

66mmmmmmmm

``

The identity map idX : X → X fits the role, so by uniqueness the only such map is idX .

Now the main part of the proof. Let Y → Xi take the role of Z → Xi. Then there is a unique Y → X suchthat all triangles commute in

X## $$

. . . // X1// X0

Y

==||

||

66mmmmmmmm

``

16


To show that q is an isomorphism, reverse the roles of X and Y . Then there is a unique X → Y such thatall triangles commute in

Y## $$

. . . // X1// X0

X

==||

||

66mmmmmmmm

``

The composites Y → X → Y and X → Y → X are maps compatible with projections, so must be idY andidX , by the first part of this argument. That is, these are mutually inverse maps, so the map X → Y is anisomorphism. ///

[4.0.2] Remark: As usual in these categorical arguments, requirements on the maps are packaged (orhidden) in the quantification over all objects Z and all families of maps Z → Xi.

5. Existence of limits

Existence of limits of various types of objects can be proven by constructing the limits as subobjects ofproducts of those objects. Most often, products of concrete [15] objects are the cartesian products of theunderlying sets, with additional structure corresponding to the situation.

[5.1] Limits of sets As a warm-up, we consider sets with no further structures. We probably believethat the mapping-property product

∏iXi of a family {Xi : i ∈ I} of sets can be constructed as the usual

Cartesian product, namely, as∏i∈I

Xi = {functions f on I so that f(i) ∈ Xi}

This does indeed partly beg the question, by making it depend on a too-vague notion of function, but nevermind. The projections pj :

∏iXi → Xj are pj(f) = f(j). We first demonstrate that the Cartesian product of

sets does have the universal mapping property: given an arbitrary set Z and arbitrary set maps fj : Z → Xj ,there should be a unique map f : Z →

∏iXi such that fj = pj ◦ f : Z → Xj for all j. That is, there should

be a commutative diagram ∏iXi

""EEEEEEEEEE

((QQQQQQQQQQQQQQQ

**UUUUUUUUUUUUUUUUUUUUUUU

. . . Xj . . .

Z

OO��

>>||

||

66nnnnnnnn

exists unique

YY

Taking f(z) ∈∏iXi to be f(z)(j) = fj(z) gives existence of the map Z →

∏iXi. And for any other map

g : Z →∏iXi with fj = pj ◦ g, evaluating the latter condition at z ∈ Z gives

fj(z) = (pj ◦ g)(z) = pj(g(z)) = g(z)(j)

which agrees with f , proving that the Cartesian product of sets is the mapping-property product.

[15] An object is concrete if it is a set with additional structure, like a group, ring, vector space, topological space,

and so on. Thus, concrete objects really do have underlying sets.

17


Limits of sets can be constructed as subsets of products, and the construction can be discovered, rather thanmerely verified, as follows. For a limit with a simple indexing poset of the form

. . .p32 // X2

p21 // X1

given a compatible family of maps gj : Z → Xi, that is, giving a commutative diagram

. . .p32 // X2

p21 // X1

Z

g2==|

||

|g1

66mmmmmmmm

there is a unique map g : Z →∏iXi to the product, by the universal property of that product. The

compatibility property pj+1,j(gj+1(z)) = gj(z) on the maps Z → Xi make the image g(Z) ⊂∏iXi lie inside

the subsetX = {{x1, x2, . . .} : pj+1,j(xj+1) = xj for all j} ⊂

∏j

Xj

Unsurprisingly, on this subset the projections pj :∏iXi → Xj become compatible in the sense that

pj+1,j ◦ pj+1 = pj : X → Xj . That is, we have a commutative diagram

X

p2

((

p1

$$. . .

p32 // X2p21 // X1

Z

g2==|

||

|g1

66mmmmmmmmg

``

That is, X = limiXi can be constructed as compatible families, much as Hensel’s lemma creates compatiblesequences:

X = {{xj ∈ Xj : pj+1,j(xj+1) = xj for all j} = {. . .→ x2 → x1}

and pi : X → Xi by pi{xj} = xi. This all shows that limits of sets exist, by exhibiting a construction as asubset of the Cartesian product.

[5.2] Limits of concrete objects The previous argument shows that limits exist whenever products exist,and we showed that Cartesian products of sets are universal-mapping-property products. This correctlysuggests that when the objects are sets with additional structure, first, products exist by constructing themas the set-product with corresponding additional structure, and then limits exist by constructing them assubsets of the product with additional compatibility requirements.

Since we explicitly demonstrate below that otherwise-constructed objects are limits, we stop the generaldiscussion here.

6. Zp and Z as limits

[6.1] p-adic integers Zp as limit The metric-completion definition of the p-adic integers Zp really is thelimit of the quotients Z/pn:

[6.1.1] Theorem: The limit limn Z/pn is the p-adic metric completion Zp of Z, with projections

Zpquot // Zp/pnZp

isom // Z/pn

18


where the isomorphism Zp/pnZp ≈ Z/pn is the natural one exhibited earlier.

[6.1.2] Remark: We give two proofs, one emphasizing the limit viewpoint, the other emphasizing the metricviewpoint.

Proof: (First) The maps qn : Zp → Zp/pnZp ≈ Z/pn are a compatible family of (continuous!) maps to thelimitands in limn Z/pn, so induce a map of Zp to the limit. For each non-zero element x ∈ Zp, there is someexponent n such that the image of x in Zp/pnZp is non-zero, so Zp injects to the limit. Thus, we might

guess that Zp is the limit, and try to verify this. [16] Let fn : Z → Z/pn be a compatible family of maps

from another object For fixed z ∈ Z, for each n choose xn ∈ Z such that [17]

xn + pnZp = fn(z)

We claim that the sequence xn is a Cauchy sequence in Zp, so by completeness we could take a limit

f(z) = limnxn ∈ Zp

Cauchy-ness follows from the compatibility of the fn’s, and, then, from the necessary compatibility of theinteger representatives xn. This defines a map f : Z → Zp compatible with the fn’s and the projections.We need to show there is a unique such f to prove that Zp is the limit. For two maps f and g compatiblewith the projections and fn’s,

0 = pn (f(z)− g(z)) ∈ Z/pn ≈ Zp/pnZp

Taking the intersection over n gives the uniqueness f(z) = g(z), proving Zp is the limit. ///

Proof: (Second) We will prove that the limit is a completion of Z with respect to a metric which agrees onZ with the p-adic metric, and is complete, so (by the uniqueness of completions) naturally isomorphic to thecompletion Zp of Z. We need a bit of general discussion of products and limits of metric spaces:

[6.1.3] Claim: A countable product∏iXi of metric spaces Xi is metrizable. If every Xi is complete, then

the product is complete.

[6.1.4] Remark: We will use explicit but non-canonical expressions for the metric on the product. Forexample, letting di(, ) be the metric on Xi, the expression

d({xi}, {yi}) =∑n≥1

2−ndn(xn, yn)

1 + dn(xn, yn)

is a metric on the product. The oddness of this expression is put into context by several observations. First,the powers of 2 appearing can be replaced by any sequence of positive real numbers whose sum converges.Second, the expressions

di(xi, yi)

1 + di(xi, yi)

have the effect of specifying metrics giving the same topology on the Xi which are bounded by 1. The 1 inthe denominator can be replaced by any positive real number, still giving the same topology. For notationalease, let us replace each di(, ) by di(, )/(1 + di(, )), so that we effectively assume that the metric on each ofthe factors is already bounded by 1, to avoid carrying along the more complicated expressions.

[16] The argument so far applies as well to Z itself, which does indeed inject to the limit, but is a proper subobject.

[17] That there exists such xn is the content of the assertion that the natural map Z/pn → Zp/pnZp is an isomorphism.

19


[6.1.5] Remark: The extreme ambiguity of the constants reminds us that many different metrics can givethe same topology. Thus, a topology cannot possibly specify a canonical metric, in general. The uniquenessof the topology on a product has no canonical metric analogue.

Proof: (of claim) It is easy and not so interesting to verify that d(, ) gives a metric on the product. It is

more interesting to see that it gives the product topology. [18] The trick is that a condition

d({xi}, {yi}) < ε

gives no condition whatsoever on xi and yi for i large enough such that 2−i < ε/2, since

1

2i+

1

2i+1+

1

2i+2+

1

2i+3+ . . . =

2i

1− 12

= 2 · 2−i < ε

and since di(xi, yi)/(1 + di(xi, yi)) < 1 for all inputs. For fixed x = {xi} in the product, the collection ofy = {yi}’s within distance ε > 0 of x includes the whole Xi ×Xi+1 × . . . for some large-enough index i.

More precisely, given d(x, y) < ε, necessarily di(xi, yi) < 2i · ε for all i. Thus, given a collection of open

balls in the individual Xi’s, almost all [19] of them being the whole Xi, we can choose ε > 0 such that theresulting ball in the d(, ) metric is contained in the product of those balls.

Conversely, given x in the product and ε > 0, taking n large enough such that 2−n < ε/2, the product

(ε-ball at x1)× (ε-ball at x2)× . . . (ε-ball at xn)×Xn+1 ×Xn+2 × . . .

is contained in the ε-ball at x. Thus, the two topologies are the same.

The more serious part of the claim is the completeness. Given a Cauchy sequence {x(k)} in the productwith the funny metric, since the metric on the product dominates the (bounded-by-1) metrics on the factors,

for each index i the ith components {x(k)i } are Cauchy in Xi, so have a limit xi in Xi. It is irresistible to

suspect that the point x = {xi} in the product is the limit of the original Cauchy sequence. Indeed, givenε > 0, take n large enough such that 2−n < ε/2. Let N be large enough such that for k ≥ N for each of thefinitely-many i ≤ n

d(x(k)i , xi) < ε

Then it easily follows that for k ≥ N

d({x(k)i }, {xi}) <

ε

2+ε

4+ . . .+

ε

2n+

2−(n+1)

1− 12

<ε

2+ε

4+ . . .+

ε

2n+ε

2= ε

That is, the original Cauchy sequence does indeed have the anticipated limit. Thus, with this funny metric,the product is complete. ///

Keep in mind that projective limits are subsets of the corresponding products, and can be given the metricfrom the product by restriction. Since closed subsets of complete metric spaces are complete, [20] this provesthe completeness of the projective limit in this case.

[18] It is especially interesting to see that this metric gives the product topology if one still thinks of the product

topology as being disappointingly coarse. That is, it might seem unlikely that it could arise from a metric, but it

does.

[19] This standard usage of almost all means all but finitely-many.

[20] That topologically closed subsets of complete metric spaces are again complete is straightforward: a Cauchy

sequence in the subset does have a limit in the whole space by the completeness of the larger space, and the limit

point lies in the subset, by closedness.

20


Now we can complete the metric-space proof that the projective limit definition of Zp is the same as themetric one. The discussion just completed shows that limn Z/pn does have a structure of complete metricspace. What remains is to show that this is the same as that on Zp. To do so, we show that we can choose aparticular form of the metric on the limit such that the restrictions of the two metrics to Z are identical. Wealso must show that Z is dense in limn Z/pn, and then we’re done, by the uniqueness of metric completions.

Returning to the funny expressions for metrics on a countable product, give Z/pn the natural metric thatany discretely topologized set can be given, namely

dn(x, y) =

{1 (for x 6= y)0 (for x = y)

Being a little clever in choice of constants, we try

d({xn}, {yn}) =∑n≥1

p−n · d(xn, yn)

Unlike points in the whole product ΠnZ/pn, points in the limit have a compatibility condition. Thus, given{xn} 6= {yn} in the limit, there is a unique index N such that xn = yn for n ≤ N and xn 6= yn for n > N .With this N ,

d({xn}, {yn}) =∑

1≤n≤N

p−n · 0 +∑n>N

p−n · 1 =p−(N+1)

1− 1p

= p−N · 1

p− 1

When {xn} and {yn} come from integers x, y, the integer N is the maximal one such that pN divides x− y,so

d(x, y) = |x− y|p ·1

p− 1

That is, up to the easily reparable constant 1/(p− 1), this contrived metric agrees on Z with the p-adic one.

Finally, we should show that Z is dense in the limit. Denseness is an intrinsic topological property, notnecessarily metric, but since we’ve already contrived the metric we may as well use it again. Given acompatible sequence {xn} in the limit, and given ε > 0, let n be large enough such that p−(n+1) < ε. Let xbe an integer such that x = xn mod pn. Then the same sort of calculation gives

d(x, {xn}) ≤∑

1≤i≤n

p−i · 0 +∑

i > np−i · 1 =p−(n+1)

1− 1p

= p−n · 1

p− 1< 2ε

Thus, Z is dense in the limit, so the limit is its metric completion. This proves (a second time) that thelimit is the same as the metric completion definition of Zp. ///

[6.2] Z as limit The poset of positive integers ordered by divisibility requires more complicated diagramsthan powers of a single prime, the latter giving the much-simpler poset of non-negative integers ordered bysize.

A common definition is

Z =∏

p prime

Zp (product topology)

Also,

21


[6.2.1] Theorem: Z is the limit of the quotients Z/N with Z/N → Z/M for M |N by `+NZ→ `+MZ.That is, for all topological groups G and compatible families of continuous group homomorphisms G→ Z/N ,

there is a unique continuous group homomorphism G→ Z giving a commutative diagram

Z))

. . . // Z/N // . . .

G

__ =={{

{{

[6.2.2] Remark: This can be proven in imitation of the proof for Zp, but is also a corollary of a moregeneral fact, that projective limits commute with products. This general assertion holds in any category withproducts and projective limits, and so is proven diagrammatically. Indeed, projective limits and productsare examples of the same more-general construction. Then, since by Sun-Ze’s theorem the quotients Z/Nare products of Z/pe with prime powers pe appearing in the prime factorization of N ,

limN

Z/N = lime2,e3,e5,...

(Z/2e23e3 . . .) ≈∏p

(lime

Z/pe)≈∏p

Zp = Z

7. Qp and A as colimits

We compare characterization of the adeles A and the simpler case of Qp. The definition of Qp as p-adiccompletion of Q makes it clear (as earlier, above) that it is the field of fractions of Zp.

The ascending union Qp =⋃n p−nZp expresses Qp as a colimit of topological (additive) groups: that is, for

every compactible collection of continuous group homomorphisms p−nZp → X there is a unique continuousgroup homomorphism Qp → X giving a commutative diagram

Zp //

**VVVVVVVVVVVVV((1

pZp //

((RRRRRRRRR))1

p2Zp //

!!DD

DD

**. . . Qp

~~X

expresses Qp as a topological group, not as a ring. Nevertheless, the limitands in the colimit behavereasonably, in the sense that

p−m Zp · p−n Zp ⊂ p−(m+n) Zp

so that the ring structure on Zp (as limit) and the obvious multiplicative properties of powers of p give thecolimit a ring structure.

Similarly, the ring of finite adeles

Afin =⋃

finite S

Afin,S (with Afin,S =∏p∈S Qp ×

∏p 6∈S Zp)

is expressible as

Afin = =⋃n

1

nZ = colimn

1

nZ

22


indexing by positive integers ordered by divisibility. This gives a colimit expression for A as topologicalgroup. Similarly, the full adele ring

A = R× Afin

is similarly

Afin = =⋃n

1

nZ = colimn

1

nZ

indexing by positive integers ordered by divisibility. This gives a colimit expression for A as topologicalgroup.

[7.0.1] Remark: To talk about R at the same time as all the other completions Qp, often one talks asthough there were a prime ∞ whose corresponding completion is Q∞ = R. This has no content apart fromallowing a uniform language. It is more proper to speak of places rather than primes as a generalized notionthat includes both genuine primes and the standard metric that yields R, but insisting on using place ispointless. Thus, the infinite prime is just the index ∞ that allows us to talk about R in the same mannerwe talk about Qp.

8. Abelian solenoids (R×Qp)/Z[1p] and A/Q

Both p-adic numbers Qp and adeles A are discovered inside automorphism groups of families of ordinarycircles. These appearances are far more important than ad hoc definitions of Qp as a metric completion andA as restricted product. That is, p-adic numbers and the adeles appear inevitably in modestly complicatednatural structures, as parts of automorphism groups.

Projective limits of various families of circles are solenoids, since there is a wound-up copy of R inside.

This discussion of automorphisms of families of circles is a warm-up to the more complicated situation ofautomorphisms of families of objects acted upon by non-abelian groups.

An important theme is that when a group G acts transitively [21] on a set X, then X is in bijection withG/Gx, where Gx is the isotropy subgroup [22] in G of a chosen base point x in X, by gGx → gx. The pointis that such sets X are really quotients of the group G. Topological and other structures also correspond,under mild hypotheses. Isomorphisms X ≈ G/Gx are informative and useful, as seen subsequently.

[8.1] The p-solenoid limn R/pnZ As reported by MacLane in his autobiography, around 1942 Eilenbergtalked to him at a conference in Michigan about families of circles related to each other by repeated windings,and about understanding the limiting object. We make this precise. The point is that a surprisinglycomplicated physical object is made from families of circles.

As a model for the circle S1 take S1 = R/Z. Eilenberg and MacLane considered a family of circles and maps

. . .×p // R/Z

×p // R/Z×p // R/Z

with each circle mapped to the next by multiplying wrapping p times, more precisely, by multiplication byp on the quotients R/Z, namely

x+ Z −→ px+ Z (for x ∈ R)

[21] The spirit of action of group G on X is that G moves around elements of the set X: an action of G on a set X

is a map G×X → X such that 1G · x = x for all x ∈ X, and (gh)x = g(hx) for g, h ∈ G and x ∈ X. A group G acts

transitively on X when, for all x, y ∈ X, there is g in G such that gx = y.

[22] Recall that the isotropy subgroup Gx of a point x in a set X on which G acts is the subgroup of G fixing x, that

is, Gx is the subgroup of g ∈ G such that gx = x.

23


Since pZ ⊂ Z this is well-defined. That is, each circle is a p-fold cover of the circle to its immediate right inthe sequence. This sequence of circles with p-fold covering maps is the p-solenoid. We might ask what is thelimiting object

???&& && &&

. . .×p // R/Z

×p // R/Z×p // R/Z

A topologically equivalent model is a more convenient. Consider the sequence

. . .ϕ43 // R/p3Z

ϕ32 // R/p2Zϕ21 // R/pZ

ϕ10 // R/Z

where ϕn,n−1 : R/pnZ→ R/pn−1Z is induced from the identity map on R in the diagram

R

mod pn

��

id // R

mod pn−1

��R/pnZ

ϕn,n−1 //______ R/pn−1Z

That is,ϕn,n−1 : x+ pnZ −→ x+ pn−1Z

This second model has advantages: the maps ϕn,n−1 are locally distance-preserving on the circles, movingto the left in the sequence of circles the circles get larger, and there is the single copy of R mapping locallyisometrically to all the circles.

What is the limit of these circles?

One plausible, naive guess would be that since⋂n p

nZ = {0}, that perhaps the limit is R/{0} ≈ R. It isnot.

A concrete model of the limit X in a diagram

X

ϕ2##

ϕ1

$$

ϕ0

%%. . .

ϕ32 // X2ϕ21 // X1

ϕ10 // X0

is the collection of all sequences xo, x1, x2, . . . such that the transition maps ϕn,n−1 map them to each other,that is,

ϕn,n−1(xn) = xn−1 (for all indices n)

We may write a compatible family of elements as

. . . −→ x3 −→ x2 −→ x1 −→ xo

The projection to the ith limitand R/piZ maps the compatible family to xi ∈ R/piZ.

For the abelian solenoid limnR/pnZ, xo is in the circle R/Z. Take a representative xo ∈ R. There are exactlyp choices of x1 ∈ R/pZ, with representatives xo, xo + 1, xo + 2, . . . , x+ p− 1. Similarly, for each such choicex1 = xo + `1, there are p compatible choices for x2, expressible as x2 = xo + `2 with `2 = `1 mod p. Thesequence of integers ì with compatibilities ì+1 = ì mod pi is an element ` of Zp.

Thus, we are tempted to say that the limit is R/Z×Zp. However, the way that ` ∈ Zp determines an elementof the limit depends somewhat on xo ∈ R/Z, and this is a slightly misleading picture, and can be improved

as follows. [23]

[23] What we literally have at this point is that the p-solenoid is a fiber bundle over R/Z with fibers Zp. We can do

better.

24


[8.2] Automorphisms of solenoids

Make sense of automorphisms of a limit X by looking at automorphisms of the diagram. With a large-enoughgroup G of automorphisms to act transitively on X, X is a quotient of G:

X ≈ G/Gx = {Gx-cosets in G} = {gGx : g ∈ G} (with isotropy subgroup Gx ⊂ G of x ∈ X)

This is an isomorphism of G-spaces, meaning (topological) spaces A,B on which G acts continuously. Asexpected, a map of G-spaces is a set map ψ : A→ B such that

ψ(g · a) = g · ψ(a)

for a ∈ A, g ∈ G, where on the left the action is of G on A, and on the right it is the action on B.

The solenoid is itself a group, being a limit of groups, but is also a quotient of more familiar and simplerobjects.

One kind of automorphism f of the p-solenoid is a collection of maps fn : R/pnZ → R/pnZ such that allsquares commute in

. . .ϕ43 // R/p3Z

ϕ32 //

f3

��

R/p2Zϕ21 //

f2

��

R/pZ

f1

��

ϕ10 // R/Z

f0

��. . .

ϕ43 // R/p3Zϕ32 // R/p2Z

ϕ21 // R/pZϕ10 // R/Z

There are obvious families of maps fn. For example, since all circles are quotients of R in a compatiblefashion, a simple sort of family of maps fn is obtained by letting r ∈ R act, by

fn(xn + pnZ) = xn + r + pnZ

Orbits of this action are highly-wound-up copies of R, earning the name solenoid.

Another simple family of maps is obtained by sequences of integers yn and maps

fn(xn + pnZ) = xn + yn + pnZ

requiring that the sequence yn be chosen so that the squares in the diagram commute. That is, we musthave

(xn + yn + pnZ) + pn−1Z = xn−1 + yn−1 + pn−1Z

Since already(xn + pnZ) + pn−1Z = xn−1 + pn−1Z

it is necessary and sufficient that

(yn + pnZ) + pn−1Z = yn−1 + pn−1Z

That is, the compatible sequence of integers

. . . −→ yn −→ . . . −→ y2 −→ y1

gives an element in the limit Zp.

Still without worrying about the topology,

[8.2.1] Claim: The product group R× Zp acts transitively on the p-solenoid. The point . . . → 0 → 0 → 0in the solenoid has isotropy group which is the diagonally imbedded copy of the integers

Z∆ = {(`,−`) ∈ (Z× Z) ⊂ R× Zp : ` ∈ Z}

25


Proof: Given a compatible family

. . .→ x3 + p3Z→ x2 + p2Z→ x1 + pZ→ x0 + Z

of elements xn + pnZ ∈ R/pnZ, act by r ∈ R as above such that x0 + r = 0 ∈ R/Z. Since the xn’s arecompatible, it must be that (r+x1) mod 1 = (x0+r) = 0, (x2+r) mod p = (x1+r), (x3+r) mod p2 = x2+r,and so on. That is, every xn + r ∈ Z, and the sequence yn = xn + r gives a compatible family

. . .→ y3 + p3Z→ y2 + p2Z→ y1 + pZ→ y0 + Z

which gives an element in Zp. That is, the further action by −yn on the solenoid will send every element to

(xn + r)− yn = (xn + r)− (xn + r) = 0

proving transitivity.

To determine the isotropy group of a point, let r be a real number and yn an integer modulo pn, such thatthe 0-element

. . . 0→ 0→ 0→ 0

is mapped to itself. That is, require that

0 + r + yn ∈ 0 + pnZ

for all n. First, this implies that r ∈ Z. Then yn, which is only determined modulo 2n anyway, is completelydetermined modulo pn by

yn + pnZ = −r + pnZ

That is, yn = −r mod pn. These conditions are visibly sufficient, as well, to fix the 0. Thus, the isotropygroup truly is the diagonal copy of Z. ///

Again, for a group G acting transitively on a set X, for fixed x ∈ X, there is a bijection

X ←→ G/Gx = {gGx : g ∈ G} by gx←→ gGx

Indeed, the map from G to X by g → gx is a surjection, since G is transitive. This map factors throughG/Gx and is injective, since gx = hx if and only if h−1gx = x, if and only if h−1g ∈ Gx, if and only ifgGx = hGx as desired. The topological features are discussed in a supplemental note. Thus, we have thecorrected expression for the limit in terms of R× Zp:

[8.2.2] Corollary: The p-solenoid is

limn

R/pnZ ≈ (R× Zp)/Z∆

where Z∆ is the diagonal copy.

[8.3] Bigger diagrams and more automorphisms Bigger diagrams with the same limit make visiblemore automorphisms of the common limit object. For the p-solenoid, we will find a copy of the p-adic rationalnumbers Qp acting on the p-solenoid, rather than merely the p-adic integers Zp. This is a big change: Qp isnon-compact, Zp is compact. We had already seen that

p-solenoid = limn

R/pnZ ≈ (R× Zp)/Z∆

with Z∆ the diagonal copy of Z. Having found a larger group of automorphisms, we will have

p-solenoid ≈ (R×Qp)/Z[1/p]∆

26


The diagonal copy Z[1/p]∆ of

Z[1/p] = Z +1

p· Z +

1

p2· Z +

1

p3· Z + . . .

(rational numbers with denominators restricted to be powers of p) is discrete in R×Qp.

Different (related, of course) diagrams can give the same limit object. We find a larger diagram with nobottom object, but giving the same limit. So far, the p-solenoid X is a limit fitting into a diagram

X++ ** ))

. . . // R/p2Z // R/pZ // R/Z

Given a point x on the solenoid, let xn be its projection to R/pnZ, and think of x as the a compatible familyof points on the respective circles, written

x = (. . .→ x2 → x1 → x0)

We already found that Zp = limn Z/pn acts on the p-solenoid. As earlier, given a point x on X, an elementr ∈ R on all circles simultaneously, to make 0th projection 0 ∈ R/Z. That is, the new values

x = (. . .→ x2 → x1 → x0 = 0)

are in Z, and form a compatible family inside

. . .mod p3// Z/p3 mod p2// Z/p2 mod p // Z/p mod 1 // Z/1

In the diagram defining the p-solenoid there is no compulsion to stop at the circle R/Z. We can continue tothe right with ever-shrinking circles, as in

. . . // R/p2Z // R/pZ // R/Z // R/ 1pZ // R/ 1

p2Z // R/ 1p3Z // . . .

[8.3.1] Claim: The limit of this larger diagram is naturally isomorphic to the limit of the original diagram.

Proof: Let X be a projective limit fitting into a commutative diagram

X)) ((

. . . // X1// X0

and consider also an enlarged diagram with projective limit Y :

Y)) (( **

. . . // X1// X0

// . . . // X−n // . . .

We claim that there is a natural isomorphism X → Y , induced from the commutative diagram

X)) ((

. . . // X1// X0

Y 55 66 44. . . // X1// X0

// . . . // X−n // . . .

A map from Y to the projective limit X is determined uniquely by a compatible family of maps from Y tothe Xn with n ≥ 0, provided by the projections of Y to the Xn with n ≥ 0. To get a map from X to Y is togive a compatible family of maps from X to all the Xn, now with n ∈ Z. For n ≥ 0 the projections of X to

27


Xn work. For −n < 0, there are many possibilities. For example, map X to X0 and then map to X−n bythe transition maps in the diagram for Y .

Thus, we obtain unique maps f : Y → X and g : X → Y compatible with all the projections and equalities.Then f ◦ g : Y → Y is a self-map of Y preserving all the projections, so, by the uniqueness of the projectivelimit, is the identity map. Similarly, g ◦ f is the identity on X. Thus, X ≈ Y . ///

The larger diagram for the same object makes more automorphisms visible, as follows.

Given a point

x = (. . .→ x1 → x0 → x−1 → . . .)

in the larger diagram, since there is no bottom circle to normalize, we have the further auxiliary choice of aninteger n, and let R act to rotate xn ∈ R/pnZ to 0. Take Z 3 −n ≤ 0, and let R act by xi → xi + r for allindices i, with r chosen to rotate x−n to 0 in R/p−nZ. Thus, since the arrows are group homomorphisms,

. . .→ x1 → x0 → x−1 → . . .→ x−n = 0→ 0→ 0→ . . .

That is, at and after the −nth place, all the xi have become 0.

Thus, x−n = 0 ∈ p−nZ/p−nZ, and there are exactly p choices for x−n+1, namely p−nZ mod p−n+1Z. Foreach of these p choices, there are p choices of x−n+2, and so on. The choice x−n = 0 on the nth circleR/p−nZ means that x−n+i is in p−nZ modulo p−n+iZ. The collection of all such compatible families for afixed choice of −n fits together as

. . .→ p−nZ/p2 → p−nZ/p→ p−nZ/1→ p−nZ/1

pZ→ p−nZ/

1

p2Z→ . . .→ p−nZ/p−nZ = {0}

Let X(n) be the projective limit of this,

X(n)++ ++ ++ ++

. . . // p−nZ/2 // p−nZ/1 // . . . // p−nZ/p−n+1 // p−nZ/p−n

The family of these diagrams fits together, giving an ascending chain of larger-and-larger limits. Indeed,

[8.3.2] Claim: The diagram

X(n)** ++

. . . // p−nZ/p−n+1 //

inc

��

p−nZ/p−n

inc

��X(n+1)

44 44 33. . . // p−(n+1)Z/p−n+1 // p−(n+1)Z/p−n // p−(n+1)Z/p−(n+1)

induces a unique injective map X(n) → X(n+1) compatible with all the projections (where the vertical mapsare the obvious inclusions).

Proof: A map to a projective limit is a compatible family of maps to the limitands. By composition withthe inclusions, we obtain the dashed arrows

X(n)** ++

**TTTTTTTTT

,,YYYYYYYYYYYYYYYYYY . . . // p−nZ/p−n+1 //

inc

��

p−nZ/p−n

inc

��X(n+1)

44 44 33. . . // p−(n+1)Z/p−n+1 // p−(n+1)Z/p−n // p−(n+1)Z/p−(n+1)

28


Since the initial diagram commutes, there is a map

p−nZp = X(n) → p−(n+1)Z/p−(n+1)

given by composition with any choice of inclusion map from the top row to the bottom. Thus, there is aunique induced dotted arrow in the commuting diagram

X(n)** ++

**TTTTTTTTT

,,YYYYYYYYYYYYYYYYYY

--[[[[[[[[[[[[[[[[[[[[[[[[[[[

��

. . . // p−nZ/p−n+1 //

inc

��

p−nZ/p−n

inc

��X(n+1)

44 4444. . . // p−(n+1)Z/p−n+1 // p−(n+1)Z/p−n // p−(n+1)Z/p−(n+1)

The induced map is injective, because an element in a projective limit of groups or topological groups is 0if and only if all its projections are 0.

Given non-zero x ∈ X(n), at least one projected image xi ∈ p−nZ/p−n+i is non-zero. The inclusion top−(n+1)Z/p−n+i is still non-zero, so the image under the induced map to X(n+1) cannot be 0. Thus, the(abelian) group homomorphism X(n) → X(n+1) has trivial kernel, so is injective. ///

Thus, we have a family of inclusions

X(0) inc // X(1) inc // X(2) inc // . . .

Zp 1p · Zp

1p2 · Zp

of groups acting on the p-solenoid. The action of X(n+1) matches that of X(n) when restricted to X(n),giving an action on the p-solenoid of the colimit

Qp =

∞⋃n=0

p−nZp =

∞⋃n=0

X(n) = colimnX(n)

Now determine the isotropy subgroup of the point 0 in the p-solenoid, under the action of R × Qp. Recallthat r ∈ R acts by

r · (. . .→ xi + piZ→ . . .) = . . .→ r + xi + piZ→ . . .

Similarly, each y ∈ Qp is of the form (for some n, depending on y)

. . . −→ yi + piZ −→ . . . −→ y−n+1 + p−n+1Z −→ y−n = 0 + p−nZ −→ 0→ 0→ . . .

with y−n+i lying in p−nZ/p−n+iZ. The action on x in the p-solenoid is

y · x =(. . . −→ yi + xi + piZ −→ . . .

)Already R × Zp was transitive, so certainly R × Qp is transitive. The isotropy group of the point x = 0 inthe p-solenoid is the collection of r ∈ R and y ∈ Qp such that

r + yi ∈ piZ (for all i ∈ Z)

where for each y ∈ Qp there is an integer n ≥ 0 such that all yi lie in p−nZ. For fixed y with associated n,taking i = −n, since y−n ∈ p−nZ,

r ∈ −y−n + p−nZ = p−nZ

29


For all indices 0 ≥ i ∈ Z, by the isotropy condition,

y−n+i = −r (in p−nZ/p−n+iZ)

For all n ≥ 0, we have the diagonal copy of p−nZ imbedded in X(n) = p−nZp induced from

p−nZp = X(−n)++ ++

. . . // p−nZ/p−n+1 // p−nZ/p−n = 0

p−nZ

ffN N N N N N

88rrrrrrrrrr

33gggggggggggggggggggggggg

That is, for all n ≥ 0, inside the isotropy group(p−nZ

)∆= {(δ,−δ) : δ ∈ p−nZ} ⊂ R×Qp

Taking the ascending union, the diagonal copy of Z[1/p] is the isotropy group. Thus, as R×Qp -spaces,

p-solenoid = limn

R/pnZ ≈ (R×Qp)/Z[1/p]∆

To see the discreteness of the diagonal copy of Z[1/p] in the product R × Qp, recall that Zp is open in Qp,because

Zp = {x ∈ Qp : |x|p < p}

Thus, R× Zp is open in R×Qp, and

(R× Zp) ∩ Z[1/p]∆ = Z∆

The projection of Z∆ to R is discrete, so Z[1/p]∆ is discrete in R×Qp. ///

[8.4] Projections (R×Qp)/Z[1/p] −→ R/piZ Limits are not genuinely described without giving theprojection maps.

[8.4.1] Claim: The quotient(R×Qp

)/Z[1/p]∆ −→

(R×Qp

)/(Z[1/p]∆ + piZp

)is naturally isomorphic to R/piZ.

Proof: First, Z[1/p] is dense in Qp: given x ∈ p−nZp, approximate pnx well by y ∈ Z, depending on n.Then p−ny is close to x.

Thus, given r×x in R×Qp, adjust by Z[1/p] to put x into piZp, and then act by piZp to give a representativeof the form r × 1. The only allow further possible adjustments are exactly by by

Z[1/p] ∩ piZp = piZ

That is, (R × Qp)/(Z[1/p] + piZp) has representatives in R × {1}, and two representatives give the sameimage exactly when they differ by piZ. ///

[8.5] The big solenoid limN R/NZ ≈ A/Q Replacing {pn : 1 ≤ n ∈ Z} by integers divisible only by aspecified set of primes, possibly all primes, an analogous discussion gives an analogous result. In particular,allowing divisibility by all primes gives the biggest solenoid of this type,

big solenoid = limN

R/NZ ≈ A/Q∆

30


because

A =⋃N

(R× 1

NZ) = colimN (R× 1

NZ)

Further, the diagonal copy of Q in A is discrete: Z is open in the finite adeles, so R × Z is open in A, and(R× Z) ∩Q∆ = Z∆, which is discrete when projected to R.

The projections areA/Q −→ A/(Q +N Z) ≈ R/NZ

Indeed, Q is dense in Afin, so, given r+x in A = R×Afin, adjust by Q to put x into N Z, and then act by N Zto make the finite-prime component trivial. To avoid any further adjustments disturbing this normalization,the only allow further possible adjustments are exactly by

Q ∩N Z = NZ

That is, A/(Q + N Z) has representatives in R × {1}, and two representatives give the same image exactlywhen they differ by NZ. Thus,

A/Q −→ A/(Q +N Z) = (R +N Z) / (Q +N Z) ≈ R/NZ

gives the projections.

9. Non-abelian solenoids and SL2(Q)\SL2(A)

The non-abelian solenoid of interest is the projective limit limN ΓN\H, with transition maps ΓN ·z −→ ΓM ·zfor M |N . It is better to use H ≈ SL2(R)/SO(2,R), and consider over-lying objects Γ(pn)\SL2(R) whosequotient on the right by SO(2,R) is Γ(pn)\H, as this allows SL2(R) to act on the right on SL2(Z)\SL2(R).Write

G∞ = SL2(R)

Thus, consider the non-abelian solenoid limN ΓN\G∞, with transition maps ΓN · g −→ ΓM · g for M |N .

As with the abelian solenoids, to simplify the indexing poset, we carry out the discussion for the somewhatsimpler solenoid limn Γ(pn)\G∞ with prime p.

Model elements of this projective limit by sequences

. . . −→ g2 −→ g1 −→ go

meeting the compatibility condition Γ(pn−1) · gn = Γ(pn−1) · gn−1 for all n. That is, gn = γn · go for asequence gn ∈ Γ(1) with Γ(pn−1) · γn = Γ(pn−1) · γn−1.

[9.0.1] Claim:limn

Γ(1)/Γ(pn) ≈ limnSL2(Z/pn) ≈ SL2(Zp)

and this group has a natural action on limn Γ(pn)\G∞.

Proof: First, because Γ(pn) is normal in Γ(1) = SL2(Z), the quotient group Γ(1)/Γ(pn) acts on Γ(pn)\G∞by

γ ·(Γ(pn) · g

)= γΓ(pn)γ−1 · γg = Γ(pn) · γg

For γ ∈ Γ(pn), the action is trivial since γ is absorbed: γ · Γ(pn) = Γ(pn). Thus, group elements

. . .→ γ2 → γ1 → γ0 (γn ∈ Γ(1)/Γ(pn))

31


with the compatibilityγn+1 · Γ(pn) = γn · Γ(pn)

give an automorphism of the limit by

(. . .→ γ2 → γ1 → γ0) · (. . .→ g2 → g1 → g0) = (. . .→ γ2g2 → γ1g1 → γ0g0)

It remains to check thatlimn

Γ(1)/Γ(pn) ≈ limnSL2(Z/pn) ≈ SL2(Zp)

Surjectivity of Γ(1)→ SL2(Z/pn) is an exercise.

The kernel of this homomorphism is Γ(pn). The elements of the projective limit are compatible families

. . .mod p3 //

[a3 b3c3 d3

]mod p2 //

[a2 b2c2 d2

]mod p //

[a1 b1c1 d1

]

Thus, that each of the four sequences of entries is a compatible family of elements in the projective limitZp = limn Z/pn. That is, limn SL2(Z/pn) ≈ SL2(Zp). ///

Similarly as for the abelian solenoids, for each go ∈ G∞, an element γ ∈ SL2(Zp) given by a compatiblesequence . . .→ γ1 → γo, specifies an element in limn Γ(pn)\G∞ by . . .→ γ1go → γogo. However, as occurredalready with the abelian solenoids, this does not present the limit as nicely as is possible.

[9.1] Automorphisms of the solenoid In the same fashion as Zp = limn Z/pn appeared in theautomorphisms of the p-solenoid, the limit SL2(Zp) = limn Γ(pn)\Γ(1) gives some automorphisms of thelimit, by

(. . . γ1 → γo) · (. . .→ x1 → xo) = (. . . γ1x1 → γoxo) (for (. . .→ x1 → xo) ∈ limn Γ(pn)\G∞)

As R acted on the abelian solenoid, g ∈ G∞ acts by

g · (. . . −→ x1 −→ xo) = (. . . −→ x1g−1 −→ xog

−1) (for (. . .→ x1 → xo) ∈ limn Γ(pn)\G∞)

with inverse so that this is an associative action written on the left. Thus, G∞×SL2(Zp) acts on this solenoid.Given . . . → x1 → xo in the projective limit, act by g = xo to put this into the form . . . → x2 → x1 → 1.The compatibility implies that xi ∈ Γ(1) and Γ(pi)xi+1 = Γ(pi)xi. Thus, . . . → x2 → x1 → 1 gives anelement γ ∈ SL2(Zp). This shows that G∞ × SL2(Zp) acts transitively.

The isotropy group of . . . → 1 → 1 → 1 consists of (g, γ) with γ = (. . . → γ2 → γ1 → γo) in SL2(Zp) withγi ∈ Γ(1), and g ∈ G∞, such that

(. . .→ γ2g−1 → γ1g

−1 → γog−1) = (. . .→ γ2 → γ1 → γo) · g · (. . .→ 1→ 1→ 1) = ·g · (. . .→ 1→ 1

Thus, γig−1 ∈ Γ(pi) for all i. In particular, g ∈ Γ(1). Then γi ∈ Γ(pi)g. Since γi is really in Γ(1)/Γ(pi), we

can say γi = g for all i. Thus, the isotropy subgroup is a diagonal copy SL2(Z)∆ of SL2(Z) in G∞×SL2(Zp),and

limn

Γ(pn)\G∞ ≈ (G∞ × SL2(Zp)) / SL2(Z)∆

Or, purely notationally, using right actions rather than left,

limn

Γ(pn)\G∞ ≈ SL2(Z)∆\(G∞ × SL2(Zp))

This is more convenient when looking at the induced translation action on functions on the solenoid.

32


[9.2] Larger diagram, more automorphisms Next, we find a larger cofinal diagram to see moreautomorphisms. We allow movement outside the group SL2(Z), although not far.

A p-power congruence subgroup is a subgroup Γ of SL2(Q) which contains some Γ(pn) with finite index.That is, for some 0 ≤ n ∈ Z

Γ ⊃ Γ(pn) [Γ : Γ(pn)] <∞

[9.2.1] Claim: For g ∈ SL2(Z[ 1p ]), the action Γ→ gΓg−1 stabilizes the set of p-power congruence subgroups

of SL2(Q).

Proof: Let Γ be a p-power congruence subgroup. Given g ∈ SL2(Z[ 1p ]), we must show that gΓg−1 contains

some Γ(p`) with finite index. That is, we want to show that Γ contains some g−1Γ(p`)g with finite index.Since the subgroups Γ(p`) are of finite index in each other, it suffices to verify the finite-index property forsufficiently small Γ(p`).

Let m be large enough such that there are integral matrices A,B such that

g = 12 + p−mA g−1 = 12 + p−mB

For γ = 12 + pnN in Γ(pn), where N is an integral matrix,

g−1γg = (1 + p−mB) (1 + pnN) (1 + p−mA)

= 1 + p−mB + p−mA+ p−2mBA+ pnN + pn−mBN + pn−mNA+ pn−2mBNA

The first four summands sum to 1, since g g−1 = 1, so this is

1 + pnN + pn−mNA+ pn−mBN + pn−2mBNA

For n > 2m, we have g−1γg ∈ Γ(pn−2m), so

g−1 Γ(pn) g ⊂ Γ(pn−2m) ⊂ Γ (n large enough such that Γ(pn−2m) ⊂ Γ)

That is, for large enough n such that Γ(pn−2m) ⊂ Γ, we have the desired containment

Γ(pn) ⊂ gΓ(pn−2m)g−1 ⊂ gΓg−1

For the finite-index condition,

[gΓg−1 : Γ(pn)] = [Γ : g−1Γ(pn)g] = [Γ : Γ(pn−2m)] · [Γ(pn−2m) : g−1Γ(pn)g]

since the indices are unaltered by conjugation.

[Γ(pn−2m) : g−1Γ(pn)g] = [gΓ(pn−2m)g−1 : Γ(pn)] ≤ [gΓ(pn−4m)g−1 : Γ(pn)] < ∞

by the same computation as above. This proves that conjugation by elements of SL2(Z[ 1p ]) stabilizes the set

of p-power congruence subgroups. ///

Consider the larger family of limitands Γ\G∞ where Γ is a p-power congruence subgroup, with the naturalmaps

Γ\G∞ −→ Γ′\G∞ (for Γ ⊂ Γ′)

[9.2.2] Claim: The family Γ\G∞ of quotients with p-power congruence subgroup Γ has cofinal subfamilyof the quotients Γ(pn)\G∞ by principal congruence subgroups Γ(pn), giving a natural isomorphism

limp -power Γ

Γ\G∞ ≈ limn

Γ(pn)\G∞

33


Proof: Each such Γ contains some Γ(pn), so there is a surjection

Γ(pn)\G∞ −→ Γ\G∞

That is, the collection of quotients by principal congruence subgroups is cofinal. Cofinal limits are naturallyisomorphic. ///

[9.2.3] Corollary: An element g ∈ SL2(Z[ 1p ]) acts on the limit of the p-power congruence quotients

limΓ Γ\G∞ by an action induced from the compatible family of isomorphisms

g : Γ\G∞ −→ gΓg−1\G∞ by g · (Γ · x) = gΓg−1 · (g · x) (for x ∈ G∞)

Proof: A map of a limit X = limi∈I Xi to itself is given by a compatible family of maps X → Xi to thelimitands. One way to give such a family is as follows. Let pj : X → Xj be the projection to the jth

limitand. Let σ be a an order-preserving permutation of the index set I, and suppose we are given a familyof isomorphisms fi : Xσ(i) −→ Xi compatible in the sense that for i > j there is a commutative diagram

Xi// Xj

Xσ(i)

fi

OO

// Xσ(j)

fj

OO

Define a family of maps Fi : X → Xi byFi = fi ◦ pσ(i)

giving a commutative diagramX

%% ''Xi

// Xj

X77 77

Fi

>>}}

}}

}}

}}

}

Fj

88ppppppppppppp

F

OO

Xσ(i)

OO

// Xσ(j)

OO

with uniquely induced map F : X → X. This idea applies to the p-power congruence subgroups with thenatural isomorphisms

Γ\G∞ −→ gΓg−1\G∞ by Γ · x −→ gΓg−1 · gx

Thus, SL2(Z[ 1p ]) acts on the projective limit. ///

The natural actions of G∞×SL2(Zp) and of SL2(Z[ 1p ]) on limn Γ(pn)\G∞, the latter also expressible as the

limit over p-power congruence subgroups, can be combined.

[9.2.4] Claim: SL2(Z[1/p]) is dense in SL2(Qp).

Proof: First, Z[1/p] is dense in Qp: given y ∈ Qp, some pny is in Zp. Using the density of Z in Zp,approximate pny well by xo ∈ Z, with closeness of approximation depending on n, of course. Then pnxo

approximates y well. Approximate the entries in given h =

(a bc d

)∈ SL2(Qp) closely by ho ∈ SL2(Z[1/p])

with respective entries ao, bo, co, do in Z[1/p], modifyied to keep the determinant-one condition. ///

Thus, an element x of limΓ Γ\G∞ (limit over p-congruence subgroups) is a compatible family {xΓ} indexedby Γ, rather than by non-negative integers as for the sub-family Γ = Γ(pn). Given h ∈ SL2(Qp), for each Γ

34


let hΓ ∈ SL2(Z[1/p]) approximate h well enough so that h = hΓ · kΓ with kΓ ∈ SL2(Zp) close enough to 1so that kΓ acts trivially on Γ\G∞. Then

h · (. . . −→ ΓxΓ −→ . . .) = (. . . −→ (hΓkΓ) · ΓxΓ −→ . . .) = (. . . −→ hΓ · ΓxΓ −→ . . .)

= (. . . −→ hΓΓh−1Γ · hΓxΓ −→ . . .)

Since G∞×SL2(Zp) was already transitive on the limit, certainly G∞×SL2(Qp) is transitive. The isotropysubgroup of the basepoint (. . .→ Γ · 1→ . . .) is the collection of g × h ∈ G∞ × SL2(Qp) such that

(. . . −→ Γ · 1 −→ . . .) = (g × h) · (. . . −→ Γ · 1 −→ . . .) = (. . . −→ hΓΓh−1Γ · hΓg

−1 −→ . . .)

That is, for every p-congruence Γ, we require hΓg−1 ∈ hΓΓh−1

Γ . That is, hΓ ∈ g · Γ for all Γ. Since hΓ isambiguous by Γ, we may as well take hΓ = g for all Γ. In particular, g ∈ SL2(Z[1/p]), and the isotropygroup is the diagonal copy SL2(Z[1/p])∆ of SL2(Z[1/p]) in G∞×SL2(Qp). The sided-ness of the action canbe reversed, if desired. Thus, we have proven

[9.2.5] Theorem:

limn

Γ(pn)\G∞ = limp−congruence Γ

Γ\G∞ ≈ SL2(Z[1/p])∆ \(G∞ × SL2(Qp)

)

[9.3] Projections SL2(Z[1/p]) \ (G∞×SL2(Qp)) −→ Γ(pn)\G∞ These complete the description ofthe limit. There is no necessity of giving projections to the more general p-congruence subgroups’ quotients,although one can do so. Let

K(pn) = {(a bc d

)∈ SL2(Zp) :

(a bc d

)=

(1 00 1

)mod pn}

We claim that the quotient on the right by K(pn)

SL2(Z[1/p]) \ (G∞ × SL2(Qp)) −→ SL2(Z[1/p]) \ (G∞ × SL2(Qp)) /K(pn)

produces an object isomorphic to Γ(pn)\G∞. Indeed, we can adjust given g×h in G∞SL2(Qp) to move h intoK(pn), since SL2(Z[1/p]) is dense in SL2(Qp). Further, right multiplication by K(pn) gives a representativeof the form g × 1. Two such representatives g × 1 and g′ × 1 are equivalent left-modulo SL2(Z[1/p]) andright modulo K(pn) exactly when there is γ ∈ SL2(Z[1/p]) and k ∈ K(pn) such that

γ · (g × 1) · k = g′ × 1

Projecting γ to SL2(Qp), we have γ · k = 1. Thus, γ is in the intersection SL2(Z[1/p]) ∩K(pn) = Γ(pn).///

[9.4] Bigger solenoid and adele-group action The projective limit over all ΓN = Γ(N) rather than just

Γ(pn) for fixed prime p is treated analogously. First, we would note that limN SL2(Z/N) = SL2(Z) acts,and

limN

ΓN\G∞ ≈ SL2(Z) \ (G∞ × SL2(Z))

being aware by this point that it is natural to take the diagonal copy of SL2(Z) rather than just the copyinside G∞.

Looking at the larger diagram in which Γ ranges over all congruence subgroups of SL2(Q), that is, all

subgroups containing some ΓN with finite index, and noting that Afin = Q · Z, we see an action of SL2(Afin).

35


Since A = R × Afin and SL2(A) = SL2(R) × SL2(Afin), and realizing that the diagonal copy of SL2(Q) isnatural,

limΓ

Γ\G∞ ≈ SL2(Q) \ (G∞ × SL2(Afin)) = SL2(Q) \SL2(A)

To describe the projection to ΓN\G∞ for principal congruence ΓN , let

K(N) = {(a bc d

)∈ SL2(Z) :

(a bc d

)=

(1 00 1

)mod N}

There is the natural quotient map

SL2(Q) \SL2(A) −→ SL2(Q) \SL2(A) /K(N)

and we claim thatSL2(Q) \SL2(A) /K(N) ≈ ΓN\G∞

Indeed, SL2(Q) is dense (see below) in SL2(Afin), so the finite-prime part of an element of SL2(A) can bemoved to K(N), and then made trivial by right adjustment by K(N). For g×1 and g′×1 in G∞×SL2(Afin),if for γ ∈ SL2(Q) and k ∈ K(N) we have

γ · (g × 1) · k = g′ × 1

then the projection of γ to SL2(Afin) must be k−1. Thus, projected to SL2(Afin), we have γ ∈SL2(Q) ∩K(N) = ΓN .

[9.5] Density of Q in Afin Although Q is discrete in A, dropping the factor R makes the copy of Q dense,demonstrated as follows.

Given finite adele x, the set of primes p so that the p-component xp of x is not in Zp is finite. Since Z[1/p]is dense in Qp, and since 1/p is in Zq for primes q 6= p, we can adjust x by a finite sum of elements from

various copies of Z[1/p] to make x lie in Zp for all p. That is, x ∈ Z.

The diagrammatic characterization of Z already shows that the natural image of Z is dense, since its closurein Z has all the properties of a projective limit of the quotients Z/N .

This density result is the larger part of the argument that SL2(Q) is dense in SL2(Afin).

Bibliographic notes:

[Chevalley 1939] introduced ideles to formulate classfield theory for infinite extensions. [Matchett 1946],[Iwasawa 1950], [Tate 1950], and [Iwasawa 1952] treated Hecke’s L-functions (compare [Hecke 1918]) inadelic/idelic formulations. [Tamagawa 1960] and [Tamagawa 1963] were the earliest examples of treatmentof automorphic forms on adele groups, giving evidence of the advantages of this viewpoint. [GGPS 1969],a translation of an earlier Russian edition, systematically rewrote the basic theory of automorphic forms inthe context of adele groups and their representations. [Jacquet-Langlands 1971] redid parts of [GGPS 1969],and added its own emphasis. [Jacquet 1972] rewrote [Rankin 1939] and [Selberg 1940] in adelic terms.

[Chevalley 1936] C. Chevalley, Generalization de la theorie de classes pour les extensions infinities. J. deMath. Pures et App. 15 (1936), 359-371.

[GGPS 1969] I. Gelfand, M. Graev, I. Piatetski-Shapiro, Representation Theory and Automorphic Functions,W.B. Saunders Co., Philadelphia, 1969.

[Hecke 1918] E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen der Verteilung derPrimzahlen, Math. Z. 1 no. 4 (1918), 357-376; 6 no. 1-2 (1920), 11-51.

36


[Hensel 1897] K. Hensel, Uber eine neue Gegrundung der Theorie der algebraischen Zahlen, JahresberichtDeutsche Math. Vereinigung 6, 83-88.

[Iwasawa 1950/1952] K. Iwasawa, [brief announcement], in Proceedings of the 1950 International Congressof Mathematicians, Vol. 1, Cambridge, MA, 1950, 322, Amer. Math. Soc., Providence, RI, 1952.

[Iwasawa 1952/1992] K. Iwasawa, Letter to J. Dieudonne, dated April 8, 1952, in Zeta Functions in Geometry,editors N. Kurokawa and T. Sunada, Advanced Studies in Pure Mathematics 21 (1992), 445-450.

[Jacquet-Langlands 1971] H. Jacquet and R. P. Langlands, Automorphic forms on GL2, Lecture Notes inMathematics 114, Springer-Verlag, Berlin and New York, 1971.

[Jacquet 1972] H. Jacquet, Automorphic forms on GL2, part II, SLN 278, Springer-Verlag, 1972.

[Matchett 1946] M. Matchett, On the Zeta Function for Ideles, thesis, Indiana University, 1946.

[Rankin 1939] R. Rankin, Contributions to the theory of Ramanujan’s function τ(n) and similar arithmeticfunctions, I, Proc. Cam. Phil. Soc. 35 (1939), 351-372.

[Selberg 1940] A. Selberg, Bemerkungen uber eine Dirichletsche Reihe, die mit der Theorie der Modulformennahe verbunden ist, Arch. Math. Naturvid 43 (1940), pp. 47-50.

[Tamagawa 1960] T. Tamagawa, On Selberg’s trace formula, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960),363-386.

[Tamagawa 1963] T. Tamagawa, On the ζ-functions of a division algebra, Ann. of Math. (2) 77 (1963),387-405.

[Tate 1950/1967] J. Tate, Fourier analysis in number fields and Hecke’s zeta functions, thesis, Princeton(1950), in Algebraic Number Theory, J. Cassels and J. Frolich, editors, Thompson Book Co., 1967.

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