(April 29, 2014) Adele groups, p-adic groups, solenoids Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/mfms/notes 2013-14/12 1 adeles.pdf] 1. Hensel's lemma 2. Metric definition of p-adic integers Zp and p-adic rationals Qp 3. Elementary/clumsy definitions of adeles A and ideles J 4. Uniqueness of objects characterized by mapping properties 5. Existence of limits 6. Zp and Zb as limits 7. Qp and A as colimits 8. Abelian solenoids (R × Qp)=Z[1=p] and A=Q 9. Non-abelian solenoids and SL2(Q)nSL2(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. These things 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, for example on holomorphic modular forms: X TN f = fj2kγ (f holomorphic weight 2k for Γ = SL2(Z)) γ2XN where XN is integer-entry 2-by-2 matrices with determinant 1, with the weight-2k action az + b a b fj γ(z) = f (cz + d)−2k (where γ = ) 2k cz + d c d An obstacle to a straightforward proof of the self-adjointness of these operators with respect to the Petersson inner product Z 2k dx dy hf; F i = f(z) F (z) y 2 ΓnH y is that, although the sum over XN is a modular form of level one, the individual summands are typically not level one, but higher level, usually level N. That is, the most straightforward assertion we can rely upon is th that for γ 2 XN the γ summand is a modular form only for a b 1 0 Γ = f = mod Ng N c d 0 1 That is, the straightforward assertion is that fj2kγ 2kβ = fj2kγ (for β 2 ΓN ) To see how the sufficient condition β 2 ΓN arises, use the associativity of the action: −1 −1 fj2k(γβ) = fj2k(γβγ · γ) = fj2kγβγ 2kγ −1 Thus, fj2kγ is assured invariant under β when γβγ 2 Γ1, for f of level 1. For example, a b a b=N 1 0 γ γ−1 = (for γ = ) c d cN d 0 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, a b a bN N 0 γ γ−1 = (for γ = ) c d c=N d 0 1 is in Γ1 only for c = 0 mod N. Generally, with γ 2 XN , similar computations show that β 2 ΓN is the simplest sufficient condition for (fj2kγ)j2kβ = fj2kγ. To treat the summands in TN f directly, we might enlarge the space of modular forms to include those for ΓN for all N. The full family of quotients ΓN nH, indexed by positive integers N ordered by divisibility, is partly depicted in ::: / Γ8nH / Γ4nH (where ΓN z ! ΓM z for MjN) v: HH vv HH vv HH vv HH vv H# ::: / Γ12nH / Γ6nH / Γ2nH : H D G vv HH © G vv HH © GG vv HH©© GG vv ©HH GG vv ©© # # :::v / Γ9nH © / Γ3nH / Γ1nH : ©© ; D vv © w vv © ww vv ©© ww vv © ww vv © w :::v / Γ10nH / Γ5nH ; vv vv vv vv vv :::v / Γ7nH 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 order structures: ::: / Γp3 nH / Γp2 nH / ΓpnH / Γ1nH In both cases, we want a single object on which all the functions on all spaces Γpn nH live, with quotient [2] maps to every Γpn nH, fitting together so that all triangles commute in the diagram + * ) ( ??? ::: / Γp3 nH / Γp2 nH / ΓpnH / Γ1nH The corresponding diagram for the full family of ΓN nH is messier, but similar. The admittedly natural candidate H itself, with obvious quotient maps qN : H ! ΓN nH fails in several ways. First, although 2 2 functions f 2 L (ΓN nH) become functions on H by composing with qN , the function f ◦ qN is not in L (H) 2 with respect to the SL2(R)-invariant measure dx dy=y . Second, the collection of all functions on H is far too large in the sense that most functions are not invariant under any ΓN . Third, certainly the true object ??? fits into a diagram ::: / Γ 3 nH / Γ 2 nH / ΓpnH / Γ1nH p p 5 dI O v: k k I v k I v k k I v k I v k k ??? k [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 Paul Garrett: Adele groups, p-adic groups, solenoids (April 29, 2014) with commuting triangles but it turns out that there is no map ??? ! H fitting into + * ) ( H ::: / Γ 3 nH / Γ 2 nH / ΓpnH / Γ1nH (fails ...) k p p 5 dI O v: k k I v k I v k k non−existent I v k I v k k ??? k We will see that the correct object ??? in the diagram is / 1 , * ( ' SL2(Z[ p ]) H × SL2(Qp) ::: / Γp3 nH / Γp2 nH / ΓpnH / Γ1nH where Qp is the field of p-adic numbers, the fraction field of the p-adic integers Zp, the latter often defined as the completion of Z with respect to the p-adic metric d(x; y) = jx − yjp with the p-adic norm ` −` jp · mjp = p (for integers m prime to p) 1 and Z[ p ] is the ring of rational numbers with denominators only allowed to be powers of p. The action of 1 γ 2 SL2(Z[ p ]) on H × SL2(Q2) is g(z × h) = gz × gh with linear fractional action on z 2 H and left matrix 1 multiplication on h 2 SL2(Qp). We will see that the diagonally-imbedded copy of SL2(Z[ p ]) 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 a compatible family of maps (making triangles commute) ::: / Γp3 nH / Γp2 nH / ΓpnH / Γ1nH dI O : j5 I v j j I v j j I v j j I vj j X jv 1 there is a unique map X ! SL2(Z[ p ])n H × SL2(Q2) giving a commutative diagram 1 * ( ' SL2( [ ])n H × SL2( p) ::: /, Γ 3 nH / Γ 2 nH / ΓpnH / Γ1nH Z p Q p p 5 l dI O v; k k I v k k I v k unique I v k k I k k X kv This situation also reveals a natural action of the group SL2(Qp) on the collection of quotients Γpn nH. This is 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 Paul Garrett: Adele groups, p-adic groups, solenoids (April 29, 2014) [3] with A the ring of rational adeles ::: /, Γ2nH w; GG ww GG ww GG ww GG + w + ) # SL2(Q)nSL2(A)=SO(2; R) ::: / Γ6nH / Γ3nH / Γ1nH YYYY ; YYYYYY ww YYYYYY ww YYYYYY ww YYYYYY w YYYYYY w ::: /, Γ5nH For many reasons we might want to get the group SO(2; R) out of the way, so that SL2(R) can act on the right as do the groups SL2(Qp), and we have a diagram - ::: / Γ2nSL2(R) 7 O ooo OOO ooo OOO ooo OOO , oo + * O' SL2(Q)nSL2(A) ::: / Γ6nSL2(R) / Γ3nSL2(R) / Γ1nSL2(R) ZZZZ 7 ZZZZZZZ ooo ZZZZZZZ oo ZZZZZZ ooo ZZZZZZZ oo ZZZZZ, o ::: / Γ5nSL2(R) This reveals the natural action of SL2(A) on the right on the family of quotients ΓN nSL2(R), incorporating an 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 maps X ! ΓN nSL2(R), there is a unique map X ! SL2(Q)nSL2(A) fitting into a commutative diagram - ::: / Γ2nSL2(R) h3 o7 OO k i oo OO m ooo OOO o ooo OOO , r oo + * O' SL2(Q)nSL2(A) ::: /tΓ6nSL2(R) / Γ3nSL2(R) / Γ1nSL2(R) ] ZZZZ w ; 6 7 : ZZZZZZZ x m m oou ZZZZZZZz x m ooo ZZZZxZZZ m oo s } x ZZZZZmZZm oooq x m ZZZZZZ, o m::: / Γ5nSL2(Ro) unique £ x m m 2 n x m f f f l ¦ x m f f f j x m m f f i ¨ x m f f f g x mf f f d f X fm _ ` a c Standard terminology for this property is that SL2(Q)nSL2(A) is the (projective) limit of the quotients ΓN nSL2(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.