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The Congruence subgroup problem

B.Sury

Indian Statistical Institute

Bangalore, India

[email protected]

17th May 2011

St.Petersburg Steklov Institute

Russia

B.Sury The Congruence subgroup problem

If one views Z simply as a discrete subgroup of R, many of its number-theoreticproperties lie hidden.

For instance, to ‘see’ prime numbers, one needs to use some other topology onZ. This is the so-called profinite (or artihmetic) topology. Here, arithmeticprogressions form a basis of open sets.

The Chinese remainder theorem essentially tells us that the arithmetic topologyis ‘built out of’ various p-adic topologies for all the primes p.

The congruence subgroup problem is a question which asks whether thisproperty generalizes to vastly more complicated groups. An affirmative answerwould again amount to the ‘topology given by subgroups of finite index beingbuilt out of the p-adic topologies’.


If one considers a matrix group with integral entries, say, SLn(Z), for somen ≥ 2, the study of its lattice of subgroups of finite index is a problem of grouptheory on the face of it.

But, one natural way of finding subgroups of finite index is number-theoretic;that is, to look at the natural ring homomorphism Z→ Z/kZ for a naturalnumber k and look at the kernel of the corresponding group homomorphismSLn(Z)→ SLn(Z/kZ). This is known as the principal congruence subgroup oflevel k.

Any subgroup of SLn(Z) which contains a principal congruence subgroup ofsome level is called a congruence subgroup.


Congruence subgroups show up in many situations in number theory.

For instance, the main step in Wiles’s proof of Fermat’s last theorem is toobtain a non-constant meromorphic map from the quotient of the upperhalf-plane by a suitable congruence subgroup into an elliptic curve defined overQ.

In fact, the classical theory of modular forms and Hecke operators which playssuch a central role in number theory is a theory which ‘lives’ on congruencesubgroups.

Interestingly, the proof of this fact requires the solution of the congruencesubgroup problem for the so-called Ihara modular group SL2(Z[1/p]), as shownby Serre and Thompson.


CSP - Naive form

The first question one may ask is whether all subgroups of finite index inSLn(Z) are congruence subgroups.

Note that the question is meaningful because of the existence of plenty ofcongruence subgroups in the sense that their intersection consists just of theidentity element.

Already in the late 19th century, Fricke and Klein showed that the answer tothis question is negative if n = 2.


Indeed, since the free group of rank 2 is the principal congruence subgroupΓ (2) of level 2, any 2-generated finite group is a quotient of this group.

It turns out that the finite, simple groups which can occur as quotients bycongruence subgroups are the groups PSL2(Fp) for primes p.

But there are many finite, simple 2-generated groups (like An (n > 5),PSL3(Fq) for odd prime q) which are not isomorphic to PSL2(Fp) (the latterhas abelian q-Sylow subgroups). So, the corresponding kernel cannot be acongruence subgroup of SL2(Z).


An explicit example of a noncongruence subgroup is the following.

Let k be a positive integer which is not a power of 2.

In the (free) group generated by the matrices X =

(1 20 1

)and Y =

(1 02 1

),

the group of all matrices which are words in X ,Y such that the exponents ofX ,Y add up to multiples of k, is a subgroup of finite index in SL2(Z) but isnot a congruence subgroup.

In fact, it can be shown that there are many more noncongruence subgroups offinite index in SL2(Z) than there are congruence subgroups.


It was only in 1962 that Bass-Lazard-Serre - and, independently, Mennicke -showed that the answer to the question is affirmative when n ≥ 3.

Later, in 1965, Bass-Milnor-Serre generalised this to the special linear and thesymplectic groups over number fields.

Mennicke showed later that subgroups of finite index in SL2(Z[1/p]) arecongruence subgroups and this is the principal reason behind the fact assertedabove that the classical Hecke operators for SL(2,Z) live only on congruencesubgroups.


Arithmetic, congruence groups in algebraic groups

How to pose the question for general (linear) algebraic groups over a numberfield k?

The algebraic groups we consider are all linear (therefore, affine) algebraicgroups; the formulation for abelian varieties is quite different.


Recall that an algebraic group defined over k (considered as a subfield of C) isa subgroup G of GL(N,C) which is also the set of common zeroes of a finiteset of polynomial functions P(gij , det(g)−1) in N2 + 1 variables withcoefficients from k.

The definition has to be slightly modified if k is a field of positive characteristic.

The group G(k) = G ∩ GL(N, k) turns out to be defined independent of thechoice of embedding G ⊂ GL(N,C) and, is called the group of k-points of G .


Here are some standard examples of algebraic groups defined over a subfield kof C.

(i) G = GL(n,C).

(ii) G = SL(n,C) = Ker(det : GL(n,C)→ C∗).

(iii) For any symmetric matrix M ∈ GL(n, k), the orthogonal group

O(M) = g ∈ GL(n,C) :t gMg = M.

(iv) For any skewsymmetric matrix Ω ∈ GL(2n, k), the symplectic group

Sp(Ω) = g ∈ GL(2n,C) :t gΩg = Ω.

(v) Let D be a division algebra with center k (its dimension as a k-vector spacemust be n2 for some n). Let vi ; 1 ≤ i ≤ n2 be a k-basis of D (then it is also aC-basis (as a vector space) of the algebra D ⊗k C).

The right multiplication by vi gives a linear transformation Rvi from D ⊗k C toitself, and thus, one has elements Rvi ∈ GL(n2,C) for i = 1, 2, · · · , n2.

G = g ∈ GL(n2,C) : gRvi = Rvi g ∀ i = 1, 2, · · · , n2.

This last group has its G(k) = the nonzero elements of D.


Here, k will denote an algebraic number field.

Let G ⊂ SLn be a k-embedding of a linear algebraic group. If Ok denotes thering of algebraic integers in k, we denote by G(Ok), the group G ∩ SLn(Ok).

For any non-zero ideal I of Ok , one has the normal subgroupG(I ) = Ker(G(Ok)→ SLn(Ok/I )) of finite index in G(Ok).

Note that it is of finite index because the ring Ok/I is finite.


Then, one could define a subgroup of G(Ok) to be a congruence subgroup if itcontains a subgroup of the form G(I ) for some non-zero ideal I .

The only problem is that, unlike G(k), the definitions of G(Ok) and G(I ) etc.depend on the k-embedding we started with.

However, it turns out that for a new k-embedding, the ‘new’ G(Ok) containsan ‘old’ G(I ) for some I 6= 0.

Thus, the following definition is independent of the choice of the k-embedding :

A subgroup Γ is a congruence subgroup if Γ ⊃ G(I ) for some I 6= 0.


The group G(Ok) can be realized as a discrete subgroup of a productG(R)r1 × G(C)r2 of Lie groups, where k has r1 real completions and r2

non-conjugate complex completions.

More generally, let S be any finite set of inequivalent valuations containing allthe archimedean valuations.

The non-archimedean valuations in S correspond to nonzero prime ideals of Ok .


One has the bigger ring OS of S-units of k; these are the elements of k whichadmit denominators only from primes in S .

Define G(OS) = G ∩ SLn(OS); then G(OS) is a discrete subgroup of theproduct

∏v∈S G(kv ) of real, complex and p-adic Lie groups, where kv denotes

the completion of k with respect to the valuation v .

One calls a subgroup Γ ⊂ G(OS) an S-congruence subgroup if and only if itcontains G(I ) (the elements of G(OS) which map to the identity element inG(OS/I )) for some nonzero ideal I of OS .

Of course, G(OS) itself depends on the k-embedding of G chosen.


Hence, the most general definition is the following :

A subgroup Γ ⊂ G(k) is an S-congruence subgroup if, for some (and,therefore, any) k-embedding of G , the group Γ contains G(I ) as a subgroup offinite index for some nonzero ideal I of OS .

One also defines :

A subgroup Γ ⊂ G(k) is an S-arithmetic subgroup if, for some (and, therefore,any) k-embedding, Γ and G(OS) are commensurable (i.e., Γ ∩ G(OS) hasfinite index in both groups).


The congruence kernel

Each of the two families (the S-arithmetic groups and the S-congruencegroups) defines a topology on G(k) as follows.

For any g ∈ G(k), one defines a fundamental system of neighbourhoods of gto be the cosets gG(I ) as I varies over the nonzero ideals of OS .

This topology Tc is called the S-congruence topology on G(k).

If one takes a fundamental system of neighbourhoods of g to be the cosets gΓas Γ varies over S-arithmetic subgroups of G(k), the corresponding topologyTa is called the S-arithmetic topology.

As S-congruence subgroups are S-arithmetic subgroups, Ta is finer. Note thatTa just gives the profinite topology on G(OS).


If the resulting completions of (uniform structures on) G(k) are denoted by Ga

and Gc , then there is a continuous surjective homomorphism π from Ga ontoGc .

The surjectivity of π is due to the fact that its image contains an opencompact subgroup (the closure of G(OS)) and a dense subgroup G(k).

Obviously, π is an isomorphism if all subgroups of finite index in G(OS) areS-congruence subgroups.In general, the kernel C(S ,G) := Ker(π), called the S-congruence kernel,measures the deviation.


A basic important property is that C(S ,G) is a profinite group.

Indeed, the closures Γa, Γc of G(OS) in Ga and Gc respectively, are profinite

groups and C(S ,G) ⊂ Γa.

The congruence subgroup problem (CSP) is the problem of determining thegroup C(S ,G) for any S ,G .


Margulis-Platonov conjecture

Let us return to a general algebraic group now.

There is a conjecture due to Serre which predicts precisely when the groupC(S ,G) is finite.

The finiteness of C(S ,G) has many interesting consequences as we shall see.

Therefore, in the modern liberal sense one says that the congruence subgroupproperty holds for G with respect to S if C(S ,G) is finite.

For any S , the association G 7→ C(S ,G) is a functor from the category ofk-algebraic groups to the category of profinite groups.


Given G , a general philosophy (which can be made very precise) is that thelarger S is, the ‘easier’ it is to compute C(S ,G).

In fact, when S consists of all the places v of k excepting the (finitely manypossible) nonarchimedean places T for which G is kv -anisotropic (equivalently,G(kv ) is compact for the topology induced from kv ), the computation ofC(S ,G) amounts to a conjecture of Margulis & Platonov which has beenproved in most cases.

In fact, in most cases T is empty which means that the triviality of C(S ,G)(for the S mentioned last) is equivalent to the simplicity of the abstract groupG(k)/center.


In my thesis, I was able to prove the case when the complement of S is finite(and this implied the Margulis-Platonov conjecture) for many types - this hasbeen given a completely uniform argument by Gopal Prasad and A.S.Rapinchuk.

One important case where the Margulis-Platonov conjecture has still not beenproved is that of the special unitary group of a division algebra with center kand with an involution of the second kind.


Necessary conditions for finiteness

First, we briefly indicate how the CSP for a general group reduces to CSP forsome special types of groups.

It reduces to connected algebraic groups G and, further, to the reductive groupG/Ru(G) using nothing more than the Chinese remainder theorem.

Here, Ru(G) is the unipotent radical of G ; it is the maximal, normal,connected, unipotent subgroup.


For tori T , the congruence kernel C(S ,T ) is the trivial group - this is anontrivial theorem due to Chevalley and is essentially a consequence ofChebotarev’s density theorem in global class field theory.

Hence, the problem reduces to that for semisimple groups.

For such G , the group G(OS) (for any embedding) can be identified with alattice in the group GS :=

∏v∈S G(kv ) under the diagonal embedding of G(k)

in GS .


Finally, it is necessary (as observed by Serre) for the finiteness of C(S ,G) thatG be simply-connected as an algebraic group - that is, there is no k-algebraicgroup G admitting a surjective k-map π : G → G with finite nontrivial kernel.

This is actually equivalent to G(C) being simply-connected in the usual sense.

Further, if one can compute C(S , G) for a simply-connected ‘cover’ G of G ,then one can compute C(S ,G).

Thus, the CSP reduces to the problem for semisimple, simply-connected groups.


If G is simply-connected and the group∏

v∈S G(kv ) is noncompact(equivalently, G(OS) is not finite), one has the strong approximation property.

This means that the closure Γc of G(OS) with respect to Tc can be identifiedwith

∏v 6∈S G(Ov ).

Here Ov is the local ring of integers in the p-adic field kv .

Note that for the additive group, the strong approximation is a consequence ofthe Chinese remainder theorem.

Thus, if C(S ,G) is trivial, the profinite completion of G(OS) is also equal to∏v 6∈S G(Ov ).

Thus, roughly speaking, when the congruence kernel is trivial the topologygiven by subgroups of finite index is built out of the p-adic topologies asasserted in the introduction.


For G(k) itself, strong approximation means that Gc can be identified with the‘S-adelic group’ G(AS), the restricted direct product of all G(kv ), v 6∈ S withrespect to the open compact subgroups G(Ov ).

The reason that G must be simply-connected in order that C(S ,G) be finite, isas follows.

Let there exist a k-map : G → G with kernel µ and G simply-connected.

Now, the S-congruence completion Gc can be identified with the S-adelicgroup G(AS). If π is the homomorphism from the S-arithmetic completion Ga

to Gc = G(AS), then it is easy to see that C(S ,G) contains the infinite groupπ−1(µ(AS))/µ(k).


As mentioned at the beginning, subgroups of finite index in SL(n,Z) arecongruence subgroups when n ≥ 3 while this is not so when n = 2.

What distinguishes these groups qualitatively is their rank.

If G ⊂ SLn is a k-embedding, one calls a k-torus T in G to be k-split, if thereis some g ∈ G(k) such that gTg−1 is a subgroup of the diagonals in SLn.

The maximum of the dimensions of the various k-split tori (if there exist k-splittori) is called the k-rank of G .

So, k-rank of SLn is n − 1 and k-rank Sp2n = n.

If F is a quadratic form over k, the group SO(F ) is a k-group whose k-rank isthe Witt index of F .


Serre’s conjecture

Consider a general semisimple, simply-connected k-group G .

Assume that G is absolutely almost simple (that is, G has no connected normalalgebraic subgroups).

For a finite S containing all the archimedean places, Serre formulated thecharacterization of the congruence subgroup property (that is, the finiteness ofC(S ,G)) conjecturally as follows.

First, it is easy to see that a necessary condition for finiteness of C(S ,G) isthat for any nonarchimedean place v in S , the group G has kv -rank > 0(equivalently, G(kv ) is not profinite by a theorem of Bruhat-Tits-Rousseau).Indeed, otherwise the whole of G(kv ) is a quotient of C(S ,G).


Serre’s Conjecture:

C(S ,G) is finite if, and only if, S-rank(G) :=∑

v∈S kv − rank(G) ≥ 2 andG(kv ) is noncompact for each nonarchimedean v ∈ S.

When C(S ,G) is finite, one says that the CSP is solved affirmatively or thatthe congruence subgroup property holds for the pair (G , S).

As mentioned in the introduction, for k = Q, S = ∞, p for some prime p andG = SL2, the CSP was solved affirmatively by Mennicke.

The finitness of C(S ,G) as against its being actually trivial, already has strongconsequences.


One of them - as pointed out by Serre - is super-rigidity; this means:

If C(S ,G) is finite, then any abstract homomorphism from G(OS) to GLn(C) isessentially algebraic.

In other words, there is a k-algebraic group homomorphism from G to GLn

which agrees with the homomorphism we started with, at least on a subgroupof finite index in G(OS).

In particular, Γ/[Γ, Γ ] is finite for every subgroup of finite index in G(OS).

This last fact has already been used to prove in some cases that C(S ,G) isNOT finite, by producing a Γ of finite index which surjects onto Z.


Relation with group cohomology

Consider the exact sequence defining C(S ,G). Recall that the strongapproximation theorem for G(k) means Gc can be identified with the ‘S-adelicgroup’ G(AS), the restricted direct product of all G(kv ), v 6∈ S with respect tothe open compact subgroups G(Ov ). Thus, we have the exact sequence

1→ C(S ,G)→ Ga → G(AS)→ 1

which defines C(S ,G).

By looking at continuous group cohomology H i with the universal coefficientsR/Z, one has the corresponding Hochschild-Serre spectral sequence whichgives the exact sequence

H1(G(AS))→ H1(Ga)→ H1(C(S ,G))G(AS ) → H2(G(AS)).


Now, the congruence sequence above splits over G(k), the last map actuallygoes into the kernel of the restriction map from H2(G(AS)) to H2(G(k)). So,if α denotes the first map, then we have an exact sequence

1→ Cokerα→ H1(C(S ,G))G(AS ) → Ker(H2(G(AS)→ H2(G(k))).

Here G(k) is considered with the discrete topology.

The last kernel is called the S-metaplectic kernel and it is a finite group.It has been computed in all cases by Gopal Prasad, Raghunathan andRapinchuk.

The cokernel of α is nothing but the S-arithmetic closure of [G(k),G(k)];hence it is also a finite group since [G(k),G(k)] has finite index in G(k).


In fact, one expects the cokernel to be trivial, and this is known in most cases.Therefore, the middle term H1(C(S ,G))G(AS ) is finite. But, this shows that thethe quotient C(S ,G)/[C(S ,G), Ga] is finite. In fact, we have :

C(S ,G) is finite if, and only if, it is contained in the center of Ga.

The centrality of C(S ,G) has been proved for all cases of S-rank at least 2other than the important case of groups of type An which have k-rank 0.


Several people have proved vanishing results for cohomology of lattices in Liegroups and the techniques of the CSP can often be applied.

For instance, the existence of cocompact arithmetic lattices ∆ in SO(n, 1)(n ≥ 5, n 6= 7) for which H1(∆,C) 6= 0 has been estabished in this manner.

It is relevant to recall a famous old conjecture of Thurston to the effect thatany compact (real) hyperbolic manifold admits a finite covering withnon-vanishing first Betti number. The various results mentioned here prove itexcept for 3-manifolds.


The connection with CSP arises as follows.

Embed the real hyperbolic manifold in a suitable complex hyperbolic manifold.That the latter admits a ‘congruence’ covering with non-vanishing first Bettinumber is a result of Kazhdan using his property T.

More precisely:

If H ⊂ G are Q-simple and H(R) = SO(n, 1) and G(R) = SU(n, 1), and Γ is acongruence subgroup of G(Q), then the restriction of H1(Γ,C) to the productof H1(gΓg−1 ∩ H,C) over all g ∈ G(Q), is injective.

The idea is to prove that the translates of C(S ,H) under the group (AutG)(Q)topologically generate a subgroup of finite index in C(S ,G).


There are other connections with cohomology in the form of Kazhdan’sproperty T, and Selberg’s property for minimal eigenvalue of Laplacian etc.

For instance, Thurston’s famous conjecture on the first Betti number is solvedfor some cases using these techniques.


Profinite group theory

Since C(S ,G) is a profinite group, it is conceivable that one can use generalresults on profinite groups profitably. This is indeed the case.

In fact, one can characterize the centrality by means of purelyprofinite-group-theoretic conditions. Platonov & Rapinchuk proved thatC(S ,G) is finite if, and only if, the profinite group Γa has bounded generation.

In other words, there are elements x1, · · · , xr (not necessarily distinct) in Γ so

that the set < x1 > · · ·< xr > coincides with the whole group Γa.


The fore-runner of this result (which was used by them crucially) is thetheorem of Lazard to the effect that among pro-p groups (for any prime p), thep-adic analytic groups are characterized as those admitting bounded generationas above.

After the solution of the restricted Burnside problem, one also has othercharacterizations for analyticity of pro-p groups like the number of subgroupsof any given index being a polynomial function of the index. These can also beadapted to characterize the finiteness of C(S ,G) in terms of polynomial index

growth for Γa.


Another characterization turns out to be true :

a (topologically) finitely generated profinite group ∆ which can be continuouslyembedded in the S-adelic group

∏p SL(n,Zp) for some n, has bounded

generation.

A special case of this was sufficient to prove Lubotzky’s conjecture that

C(S ,G) is finite if, and only if, G(OS) can be continuously embedded in theS-adelic group

∏p SL(n,Zp) for some n.


Aspects of subgroup growth have turned out to have deep relations with theCSP.

For instance, apart from revealing finer structural facts like the number ofnoncongruence subgroups being a higher order function (in a more preciselyknown form) than the number of congruence subgroups (when CSP does nothold), the methods allow an analogue of the CSP to be asked for lattices (evennonarithmetic ones ! ) in semisimple Lie groups, viz.,

If Γ is a lattice in a semisimple Lie group over a local field (of characteristic 0),is the subgroup growth type of Γ strictly less than nlog n ?


An unexpected connection of the CSP with cohomology of finite simple groupsis revealed by some recent work.

For a finite group A, denote by h(A), the supremum of the number dimH2(A,M)dimM

over all primes p and all simple Fp[G ]-modules M.

For any finite simple group A, there is a bound h(A) = O(log |A|) and it wasconjectured that there is a constant c > 0 such that h(A) ≤ c for all finitesimple groups A.

Partial results can be obtained from the methods of the CSP. For instance, if Gis a fixed simple Chevalley scheme, then there is a constant c > 0 such thath(G(Fp)/center) ≤ c for all primes p.

The idea is to obtain the finite groups G(Fp) as quotients of the congruencecompletion of G(Z).


Status of finiteness of C (S ,G )

Having talked about the general problem, we return back to the past briefly.

For the classical case of SLn(Z) with n ≥ 3, centrality can be proved with thehelp of elementary matrices and reduction to SLn−1.

The key fact here is that any unimodular integral vector c which satisfiesc ≡ en mod r must be in the orbit of En(r), the normal subgroup generated bythose elementary matrices which are in the principal congruence subgroup Γ (r)of level r .


More precisely, look at G = SL3(Z) and the embedded SL2(Z) in its left handtop corner.

For any r , consider the normal subgroup E3(r) generated by U12(r) in G .

For any element g of Γ3(r) (for SL3), one may apply elementary row andcolumn operations and get x , y in E3(r), with xgy in the principal congruencesubgroup Γ2(r) of level r for the embedded SL2.

This implies easily that the action of G on Γ3(r)/E3(r) by conjugation is trivial.

Now, the first rows (a, b) of Γ2(r) for SL2(Z) give rise to corresponding

elements

a b 0∗ ∗ 00 0 1

of Γ3(r)/E3(r); these are the so-called Mennicke

symbols M(a, b) which will be shown to be trivial.


The Mennicke symbol has nice properties like:(i) M(a + tb, b) = M(a, b) for all t ∈ Z,

(ii) M(a, rta + b) = M(a, b) for all t ∈ Z,

(iii) M(a, b) is multiplicative in b,

(iv) M(a, b) = 1 if b ≡ ±1 mod a.

Using these, one can show that it is actually trivial.

Therefore, Γ3(r) = E3(r); so, a normal subgroup of finite index which containsE3(r) (as it must, for some r), also contains Γ3(r).


For general G whose k-rank is nonzero, there are unipotent elements in G(k)which play the role of the elementary matrices and which allow for similar(although much harder) proofs.

Briefly, we recall the case of K -rank at least 2 which was dealt with byRaghunathan.

For each parabolic k-subgroup Q with unipotent radical V , and each non-zeroideal a in OS , define V (a) = V ∩ G(a) and Q(a) = Q ∩ G(a).


Let EG(a) denote the subgroup of G(a) generated by V (a) for varyingparabolic K -subgroups Q.

This is a generalization to arbitrary G of the elementary subgroups in SLN(a).

Obviously, EG(a) is normal in G(a) (in fact in the whole group G(OS)).

Define similarly, the subgroups FG(a) as the ones generated by Q(a) as Qvaries.

The groups FG(a) are also normal in G(OS).


One innovation in Raghunathan’s proof is the introduction of theseintermediate groups EG(a) and FG(a) for the proof of centrality.

We have the inclusions EG(a) ⊂ FG(a) ⊂ G(a).

Furthermore, given a and an element g ∈ G(K), there exists a non-zero ideal a’such that gEG(a)g−1 ⊃ EG(a′).

Similarly for FG(a).

We then obtain new completions GF (resp. GE ) by requiring FG(a) (resp.EG(a)) to be open in G(K).

We also get the kernels CE/F and CF of the corresponding extensions GE of GF

and GF of Gc (which are also split over G(K)).


To prove the centrality of CF , one proceeds as follows.

Fix the highest k- root β, and set Uβ to be the corresponding root subgroup inG .

Set ∆ = Uβ(OS).

One then makes the key observation that for any non-zero ideal a, thecommutator subgroup [∆,G(a)] is contained in GF (a), as k-rank (G) ≥ 2.

The reason for this is essentially the following.

Since k-rank (G)≥ 2, for any two roots α and β with α 6= −β, the root groupsUα and Uβ lie in the unipotent radical of a parabolic subgroup of G definedover k.

Consequently, on the kernel CF , the action of ∆ is trivial, but the actionextends to all of G(K); the simplicity of G(K) modulo its center now impliesthat the action of G(K) is also trivial.


Similarly, centrality of CE/F is proved using simplicity of M(K) for the groupsM which are K -simple factors of the Levi part of Q.

One also uses the fact that there are only finitely many G(OS) conjugacyclasses of k-parabolic subgroups of G .

The final step that if CF and CE/F are centralized by G +(K), then so also isthe kernel C , is elementary but tedious.


For the groups of k-rank 1, the centrality was established by a rather differentmethod which also works in several cases of k-rank 0.

Note that for the groups of k-rank 0, there are no unipotents and one needs towork with concrete descriptions of them as unitary groups of hermitian formsetc.

The idea of this method goes back to Raghunathan in 1986 and Rapinchukwhich I was also able to carry out for a number of rank 0 groups.

These proofs have been shortened and simplified by Gopal Prasad andRapinchuk recently and we discuss them briefly now.

What we discussed earlier for number fields is also valid for global fields ofpositive characteristic; these are finite extensions of Fq(t). Note that in thepositive characteristic case, the valuations of k are non-archimedean.

One must remember though that some of the proofs need to be more carefullydone because lattices may not be finitely generated and the congruencesubgroup property can fail for unipotent groups.


The centrality of the congruence kernel (under the hypotheses of Serre’sconjecture) was reduced to the following general statement.

Let k be a global field and let G be an absolutely almost simple, simplyconencted k-group satisfying the Margulis-Platonov conjecture. Let S be afinite set of places containing the archimedean ones, S is disjoint from T andS-rank(G) ≥ 2. Suppose, for each v 6∈ S, we have subgroups Gv in the

S-arithmetic completion Ga, which pairwise commute and map onto G(kv )

under the map from Ga to Gc = G(AS). If the Gv (as v 6∈ S vary) generate a

dense subgroup of Ga, then C(S ,G) is central.


The proof of this reduction interestingly uses some abstract profinite grouptheory apart from some standard results on algebraic groups over local fields.

To use the above proposition, we need to locate such Gv ’s. We indicate insome detail how this is done when k-rank(G) ≥ 2.

One crucial fact one needs to prove in this approach is that the S-arithmeticand S-congruence topologies on G(k) induce the same topology on U(k)where U is the k-subgroup generated by Uα as α varies over positive roots (forany ordering). This fact is trivial for number fields but not so in positivecharacteristic.


Fix any ordering on the k-roots Φ (to be chosen later) and let U+ = UΦ+ andU− = UΦ− denote these unipotent k-subgroups normalized by thecorresponding maximal k-split torus.

By the above remark, the map π : Ga → G(AS) maps U+(k) isomorphicallyonto U+(k) = U+(AS).

Let θ denote its inverse map.

For any root α, one may choose some ordering such that it is positive and, onecan then define θα as the restriction of θ to Uα(AS).

It can be checked that θα is independent of the ordering for which α > 0

because it is also the inverse of the isomorphism from Uα(k) to Uα(AS).

As Uα(kv ) is naturally contained in Uα(AS), the images θα(Uα(kv )) generate

(as α varies over roots) a subgroup Gv of Ga.


For k-rank(G) ≥ 2, the Kneser-Tits conjecture is valid for G over k as well asover all kv (note T is empty).

This means that G(k) is generated by Uα(k) and, for each v , G(kv ) isgenerated by Uα(kv ) as α varies over all roots.

This immediately implies that Gv maps onto G(kv ) for each v 6∈ S .

For each α, by strong approximation, Uα(kv )’s generate a dense subgroup ofUα(AS) as v 6∈ varies.

Thus, the closure in Ga of the subgroups generated by Gv ’s (as v varies)

contains each Uα(k) and thus, the closure is the whole of Ga.


The most interesting and difficult statement to prove is the one which isleft-over; viz., to show Gv and Gw commute in Ga for v 6= w .

For this, it suffices to prove that Uα,v := θα(Uα(kv )) and Uβ,w := θβ(Uβ(kw ))commute element-wise for any roots α, β and for v 6= w with v ,w 6∈ S .

If these two roots are independent or if one of them is a positive, integralmultiple of the other, there is an ordering of Φ such that both are positive. Inthis case, since Uα,v ,Uβ,w are contained in the images under θ of U+(kv ) andU+(kw ) respectively, we will be through.

Finally, we have the trickiest case when one of the roots is a negative integralmultiple of the other.

In this case, using the fact that for any root γ, the unipotent root subgroupU2γ ⊂ Uγ , the assertion reduces to the case when α is simple and β = −α.


For this, we proceed as follows. Fix an ordering so that α is simple and letβ = −α.

As k-rank(G) ≥ 2, there exists a simple root γ 6= α.

Look at the torus Tγ which is the connected component at the identity of theintersection of the kernels of all simple roots other than γ.

Let P+,P− be the parabolic k-subgroups generated by the reductive groupM := ZG (Tγ), the unipotent group U+ and, respectively, by ZG (Tγ),U−.

Note that M normalizes both Ru(P+) and Ru(P−), and contains the subgroupGα generated by Uα,U−α.


By a remark before beginning the proof, both the S-congruence andS-arithmetic topologies on G(k) induce the same topology on Ru(P+)(k) andsimilarly on Ru(P−)(k).

Thus, the restriction of π to Ru(P+)(k) as well as to Ru(P−)(k) are injective.

Since M(k) normalizes Ru(P+)(k) and Ru(P−)(k), we have

[M(k) ∩ C(S ,G), Ru(P+)(k)] ≤ C(S ,G) ∩ Ru(P+)(k) = 1

[M(k) ∩ C(S ,G), Ru(P−)(k)] ≤ C(S ,G) ∩ Ru(P−)(k) = 1

As Ru(P+)(k) and Ru(P−)(k) generate the whole of G(k), we have that

M(k) ∩ C(S ,G) commutes with G(k) and is, hence, contained in the center of

Ga.

In particular, the restriction of π to Gα(k) gives a central extension of Gα(AS)

which implies that the inverse images of Gα(kv ) and Gα(kw ) commute in Ga forv 6= w and v ,w 6∈ S .

Thus, further particularly, the inverse images of Uα(kv ) and of U−α(kw )

commute in Ga for v 6= w .


Let us indicate how the centrality of congruence kernel is proved forSL2(Z[1/p]).

That is, we are considering the case k = Q, S = ∞, p and G = SL2.

As before, the map π : Ga → G(AS) gives an isomorphism

U+(Q)→ U+(Q) = U+(AS)

andU−(Q)→ U−(Q) = U−(AS)

where U+,U− are the upper and lower triangular unipotent subgroups.

As Q is dense in AS by strong approximation, the maps

u+ : Q→ U+(Q) ; a 7→(

1 a0 1

)u− : Q→ U−(Q) ; b 7→

(1 0b 1

)give rise to continuous homomorphisms u± : AS → U±(Q) by composing theabove inverses.


For all q 6= p, Gq is defined to be the subgroup of Ga generated by u±(Qq).

As before, the only non-obvious part is to check that Gq’s commuteelement-wise.

The basic relation we need is:

[

(1 a0 1

)(1 0b 1

)] =

(1 a2b

ab−1

0 1

)(1

1−ab0

0 1− ab

)(1 0ab2

1−ab1

)


That is, writing h(t) = diag(t, t−1), we have

[u+(a), u−(b)] = u+( a2bab−1

)h( 11−ab

)u−( ab2

1−ab).

We need this while checking that [u+(Qq1 ), u−(Qq2 )] = 1 in Ga for any primesq1, q2 6= p.


Let us enumerate all the primes other than p as q1, q2, · · ·

We first observe that it suffices to get hold of one a0 ∈ Q∗q1and one b0 ∈ Q∗q2

such that [u+(a0), u−(b0)] = 1.

To see this, note that for any t ∈ Q∗, the inner conjugation on G(Q) by

diag(t, 1) also induces an automorphism σt of Ga.

Now, 1 = σt([u+(a0), u−(b0)]) = [u+(ta0), u−(t−1b0)].

As Q∗ is dense in Q∗q1×Q∗q2

, it follows that [u+(Qq1 ), u−(Qq2 )] = 1.


To get hold of a0, b0 we proceed as follows.

For any m ≥ 2, there exists n(m) ≡ 0 mod m! so that pn(m) ≡ 1 mod(q1q2 · · · qm)2m.

Therefore, we can get 1− pn(m) = ambm with (am, q1) = 1 = (bm, q2) and

am ≡ 0 mod (q2q3 · · · qm)m

bm ≡ 0 mod (q1q3 · · · qm)m.

There exists a subsequence mk such that

amk → a0 ∈ Q∗q1and

bmk → b0 ∈ Q∗q2.


Now [u+(am), u−(bm)] = u+(−a2mbmp−n(m))h(p−n(m))u−(amb2

mp−n(m)).

Since Qq’s commute in AS , we have that a2mbm, amb2

m → 0 in AS .

So, u+(−a2mbmp−n(m)), u−(amb2

mp−n(m))→ 1 in Ga.

Also, h(pn(m))→ 1 in Ga since m! divides n(m).

Thus, we have proved that [u+(a0), u−(b0)] = 1 and we are through.


This idea has also been exploited by Andrei Rapinchuk & Igor Rapinchuk toinvestigate the centrality of the congruence kernel for the elementary subgroupsover any commutative Noetherian ring.

They prove:

Let G be a universal Chevalley-Demazure group scheme corresponding to areduced, irreducible root system of rank ≥ 2. Let R be a commutativeNoetherian ring where 2 is assumed to be a unit if the root system is of typeCn(n ≥ 2) or G2. Consider the corresponding elementary subgroup E(R).Then, the congruence kernel of E(R) is central.

Note that in particular, the theorem applies to SLn(Z[t1, t2, · · · , tr ]) for n ≥ 3.


We indicate the main steps in the proof of centrality.

Note that unlike the classical situation where G(k) has an action on thecongruence kernel, there is no such k here.

As usual, let E(R) denote the subgroup of G(R) generated by Uα(R) := eα(R)as α varies in Φ.

One defines the arithmetic and congruence topologies on any subgroup ofG(R) - we are interested in Γ := E(R).

We have the corresponding exact sequence

1→ C(Γ )→ Γπ→ Γ → 1

Note that the congruence topology on E(R) is induced by that on G(R) butthe arithmetic one may not be for all we know.


Step I.

The profinite completion R of the ring R can be naturally identifiedtopologically with

∏m Rm where m runs through maximal ideals of finite index

and Rm is the m-adic completion of R.

Proof uses only Chinese remainder theorem.

Step II.

For some N > 0, every element of G(R) is a product of at the most N

elements of the set S = eα(t) : t ∈ R, α ∈ Φ.

The proof of this reduces to local rings by step I.

For such rings, one reduces to the big cell and to G over the residue field.


Step III.

The congruence completions G(R) and E(R) are both equal to G(R).

Further, there is N > 0 so that each element of this congruence closure is aproduct of at the most N elements of the set S which is the closure incongruence topology of eα(t) : α ∈ Φ, t ∈ R.

This just follows from step II.


Step IV.

The profinite and congruence topologies on E(R) induce the same topology oneach Uα(R).

This means that for each normal subgroup N of finite index in E(R), there isan ideal of finite index in R such that eα(I ) ⊂ N ∩ Uα(R) for each root α.

The proof involves explicit computations with commutation relations and is,therefore, case by case.


The proof of the theorem proceeds similarly to the number theoretical caseabove.

Since π gives isomorphisms from Uα(R) to Uα(R), and

since we have a continuous homomorphism eα : R → G(R) = G(R),

one has isomorphisms eα : R → Uα(R) for each α of topological groups.

Define, for each maximal ideal m of finite index in R, the subgroup Γm of Γwhich is generated by the image of Rm under eα for varying α.

Then,

(i) [eα(s), eβ(t)] =∏

eiα+jβ(Nαβijsi t j) for α 6= −β and s, t ∈ R;

(ii) Γm and Γ ′m commute element-wise in Γ .


Combining this with an old result of M.Stein asserting that the kernel of πrestricted to Γm is in its center for any maximal ideal m of finite index, thecentrality of Ker(π) in Γ can be deduced using some observations abouttopological groups.

We remark that in the number-theoretic situation there is an action of G(k) onC which makes it easier; here, we needed Stein’s result as well as the boundedgeneration property of E(R) mentioned in step III.


One may pose an analogue of the congruence subgroup problem for the groupsAut(Fn), where Fn is the free group of rank n.

Note that this group does not have faithful finite-dimensional representations ifn ≥ 3 but does possess such representations when n = 2.

M.Asada, K-U.Bux, M.Ershov and A.Rapinchuk have proved also an analogueof the congruence subgroup property for the group Aut(F2).The problem is still open for n ≥ 3.


For any finitely generated, residually finite group G , one may pose the problemfor Aut(G) as follows.

Note that Aut(G) ia also finitely generated and residually finite.

Note also that G injects into its profinite completion G and Aut(G) is finitelygenerated as a topological group.

For any normal subgroup N of finite index in G , consider the subgroup

ΓN := σ ∈ Aut(G) : σ(N) = N, σ|G/N = I

of finite index in Aut(G).Note that if N is taken to be characteristic, then this is a normal subgroup.


One may ask if every subgroup of finite index in Aut(G) contains Γ (N) forsome N as above.

As before, one may define the topologies Ta,Tc and get a continuous

homomorphism π from the completion Aut(G) to Aut(G).

We note that Aut(G) = Aut(G) where the latter is the group of continuousautomorphisms.

Thus the problem is :

Is π : Aut(G)→ Aut(G) is injective?


Look at G = Fn for any n ≥ 2.It is known that Aut(Fn) is not finitely generated as a topological group; henceπ cannot be surjective unlike the classical situation!

M.Asada had shown that any normal subgroup of finite index in Aut(F2) whichcontains Int(F2), contains some ‘congruence’ subgroup Γ (N).

Later, the other 3 authors re-interpreted and simplified Asada’s proof andshowed that the injectivity of π holds for Aut(F2) without any assumption.


The proof involves explicit calculations in the free groups.

We merely mention the fact that the proof of the congruence subgroupproperty for Aut(F2) uses strongly the fact that Out(F2) = GL2(Z) is virtuallyfree and, very interestingly, this fact is what prevents the latter group fromhaving the congruence subgroup property!


Coming to the classical cases of Dedekind rings, the centrality/finiteness ofC(S ,G) (for the cases where it is conjectured to be finite) is still open for twovery important cases where new ideas may be required but these cases have alot of relevance to the theory of automorphic forms.

These two cases are:

(i) G = SL(1,D) for a division algebra with center k, and

(ii) G = SU(1,D) where D is a division algebra with a center K which is aquadratic extension of k and D has a K/k-involution.


Structure of infinite C (S ,G )

Let us turn to the case when Serre’s conjecture has a negative answer; that is,when C(S ,G) is infinite.

One theme which has been exploited in proving finiteness of C(S ,G) is : IfC(S ,G) is not finite, it has to be really huge.

This can be used to show that if C(S ,G) is infinite, it is infinitely generated asa profinite group. Melnikov used results on profinite groups to prove that, forSL2(Z), one has C(∞, SL2) ∼= Fω, the free profinite group of countablyinfinite rank. This proof uses the existence of a central element of order 2 inSL2(Z).


In joint work with Alec Mason, Alexander Premet and Pavel Zalesskii(appeared in Crelle’s Journal), we completely determine the structure ofC(S ,G) when G(OS) can be identified with a lattice in G(kv ) for somenonarchimedean completion kv .

We use group actions on trees and profinite analogues as well as a detailedanalysis of unipotent radicals in every rank 1 group over a local field of positivecharacteristic, to deduce the following structure theorem :


Theorem

Let G , k, S be as above, with S-rank (G) = 1. Then,

(i) if G(OS) is cocompact in G(kv )(in particular, if char. k = 0), thenC(S ,G) ∼= Fω, and

(ii) if G(OS) is nonuniform (therefore, necessarily char. k = p > 0), thenC(S ,G) ∼= Fω t T , a free profinite product of a free profinite group ofcountably infinite rank and of the torsion factor T which is a free profiniteproduct of groups, each of which is isomorphic to the direct product of 2ℵ0

copies of Z/pZ.


A surprising consequence of the theorem is that C(S ,G) depends only on thecharacteristic of k when G(OS) is a lattice in a rank 1 group over anonarchimedean local field.

As a by-product of the above result, we also obtain the following surprisingresult which is of independent interest :


Theorem

Let U be the unipotent radical of a minimal kv -parabolic subgroup of G whereG is as in (ii) of the above theorem. Then either U is abelian or isautomatically defined over k.


We note that although the profinite completion of a free group of finite rank ris the free profinite group of rank r , the profinite completion of a free group ofcountably infinite rank is not the group Fω appearing here.

The profinite completion is much larger and, the group Fω can be constructedfrom a countably infinite set X by looking at the free group F (X ) and at onlythose normal subgroups of finite index in F (X ) which contain all but finitelymany elements of X .

One further distinction between the case of free profinite groups and those of(discrete) free groups is that closed subgroups of free profinite groups may notbe free profinite.

For instance, Zp is a closed subgroup of Z.


We recall the two open problems which may be termed outstanding here. Thefirst is the finiteness of the C(S ,G) in Serre’s conjecture is still very much openfor the groups of type An with k-rank 0. Solving them would lead toapplications to the theory of automorphic forms. These G , S are described asfollows :


(i) Let D be a division algebra with center k and the k-group G is SL(1,D).For S with S-rank(G) ≥ 2, is C(S ,G) finite?

(ii) Let D be a division algebra with center K which is a quadratic extension ofk and suppose D admits an involution of the 2nd kind whose fixed field is k.Then the k-group G is SU(1,D). For S so that S-rank(G) ≥ 2, is C(S ,G)finite?


The second question is when k is a number field, S consists only of thearchimedean places, and G is a k-group such that S-rank of G is 1, then whatis the structure of C(S ,G) ? Recently, it has been proved by Lubotzky that

C(S ,G) is finitely generated as a normal subgroup of G(OS). However, thestructure is unknown as yet. This is unknown excepting the case of SL(2,Z);our theorem above deals with all the cases where S contains a nonarchimedeanplace.


Clopen subgroups

I don’t give any more details of the proof of the above results. However, Idiscuss a recent result which leads to a simplification in the proof of one of theprincipal results there.

Specifically Zel’manov’s solution of the restricted Burnside problem fortopological groups is no longer required there.

The result we will prove is already stated as a lemma by Raghunathan in 1986but he merely gave a very brief sketch of the proof and it was not completelyclear that it works.

We work out a detailed proof especially for the benefit of group-theorists whoare not experts in algebraic groups.

By applying the lemma to S-arithmetic lattices in G of K -rank one, wherechar(K) 6= 0 and |S | = 1, we provide a lower estimate for the number ofsubgroups of a given index in such a lattice which are not S-congruence.


Raghunathan’s Lemma:

Let K be a global field and let S be a finite nonempty set of places of Kcontaining all those of archimedean type. Suppose that G is connected,simply-connected and K-simple with strictly positive S-rank. Let Γ be anS-arithmetic subgroup of GK , the K-rational points of G. If N is anynoncentral subgroup of GK which is normalized by Γ then the closure of N inthe S-congruence topology is also open.


Applying this to the classical case of an S-arithmetic group which occurs as alattice Λ in GKv , where char(K) 6= 0, G has K -rank one and S = v, we provea result on the ubiquity of finite index subgroups of Λ which are notS-congruence.

To describe it, recall that associated with GKv is its Bruhat-Tits tree T .

Bass-Serre theory shows how a presentation for Λ can be inferred from itsaction on T , via the structure of the quotient graph Λ\T .


It is known that the first Betti number of Λ\T , b1(Λ\T ), is finite. We have:

Theorem. Let Fr be the free group on r generators, where r = b1(Λ\T ) and letf (r , n) denote the number of index n subgroups of Fr . Let nc(Λ, n) be thenumber of S-noncongruence subgroups of index n in Λ. Then there exists aconstant n0 = n0(Λ) such that, if n > n0, then

nc(Λ, n) ≥ f (r , n).

Moreover, if r ≥ 1, then for all n > n0, there exists at least one normal,S-noncongruence subgroup of index n in Λ.


In some important special cases, it is possible prove an explicit version ofRaghunathan’s lemma in an elementary way. Here K has any characteristic andS is arbitrary.

Let H be a subgroup of SL2(O(S)).

Recall that the order of H, o(H), is the O(S)-ideal generated by allh12, h21, h11 − h22, where (hij) ∈ H.

For each O(S)-ideal q, define

ψ(q) =

12q , char(K) = 0q4 , char(K) 6= 0

Lemma.Let N be a noncentral normal subgroup of SL2(O(S)), with o(N) = n( 6= 0).Let q be any nonzero O(S)-ideal. If n′ = n + q, then

SL2(ψ(n′)) ≤ N.SL2(q).


Theorem.

Let N be a noncentral normal subgroup of SL2(O(S)). Then

N =⋂

q6=0

N.SL2(q) = N.SL2(ψ(n)),

where n = o(N).

Under further restrictions, this can be improved. For example, from the resultsof Mason it follows that, if o(N) is prime to 6, then

N =⋂

q 6=0

N.SL2(q) = N.SL2(n).

In particular, if o(N) = O(S), then N = SL2(O(S)).


We indicate now some pieces of the proofs of our results on the structure ofthe congruence kernel.

We recall what a free profinite product is and also some important facts aboutprofinite groups which we need.

The free profinite product G1 t · · · t Gn of profinite groups G1, · · · ,Gn can bedefined by an obvious universal property but it can also be constructed as thecompletion of the free product G1 ∗ · · · ∗ Gn with respect to the topology givenby the collection of all normal subgroups N of finite index in the free productsuch that N ∩ Gi is open in Gi for each i .


One has the profinite counterpart of Schreier’s formula :

Let H be an open subnormal subgroup of a free profinite group Fr of rank r .Then, H is a free profinite group of rank 1 + (r − 1)[Fr : H].

In general, one says that a profinite group ∆ satisfies Schreier’s formula if thefollowing holds. Let d(∆) denote the smallest cardinality of a set of generatorsof ∆ converging to 1. Here, a subset X of a profinite group ∆ is said to be aset of generators converging to 1 if every open subgroup of ∆ contains all butfinitely many elements in X and < X > is dense in ∆. Then, ∆ is said tosatisfy Schreier’s formula if, for every open normal subgroup H, one has

d(H) = 1 + (d(∆)− 1)[∆ : H].

Free profinite groups satisfy Schreier’s formula.


One also uses the following results :

Let ∆ be a finitely generated pro-p group, for some prime p. Then, ∆ is freepro-p if, and only if, it satisfies Schreier’s formula.

On the other hand, a free pronilpotent group ∆ with d(∆) ≥ 2 never satisfiesSchreier’s formula. Note that this is not true if d(∆) = 1. In fact, any directproduct ∆ = ∆1 ×∆2 with d(∆) ≥ 2 does not satisfy Schreier’s formula. Thisis one of the key (although simple to prove) facts we use in our proof.


Let Fr be the free profinite group of finite rank r ≥ 2. Let H be a closednormal subgroup of Fr of infinite index. If Fr/H does not satisfy Schreier’sformula, then H is a free profinite group of countably infinite rank.

The analogue of Kurosh subgroup theorem holds for open subgroups of freeprofinite products but not for closed subgroups.

For closed normal subgroups of free profinite products, one has a similarstatement which involves projective profinite groups in place of free profinitegroups.


Projective profinite groups are closed subgroups of free profinite groups. Weuse the following result :

Let F be a free profinite group of infinite rank and let P be a projectiveprofinite group with d(P) ≤ d(F ). Then the free profinite product F t P ∼= F .

Note that the above result also shows that free factors of a free profinite groupneed not be free profinite groups.

Another fact we use is :

Let F be a free profinite group of rank r (any cardinal number) ≥ 2. Then, aclosed normal subgroup H of infinite index with d(F/H) < r must be a freeprofinite group and it has rank max(r ,N0).

Some of these facts that we use are proved using the notion of a profinitegraph of profinite groups and a notion of the fundamental group of such aprofinite graph of groups.


Crucial for our purposes is the following surprising result. It ensures that thestructure of any C(U) can deduced from a detailed description of one particularU. (This result in fact holds for any global field.)

If U is nonabelian, then U is defined over k.


Proof.

Let P = P(K) and U = U(K). Obviously, P is a parabolic subgroup of G andU = Ru(P), the unipotent radical of P. Choose a maximal torus T of Gcontained in P and let Φ denote the root system of G relative to T . Let ∆ bea basis of simple roots in Φ.

Adopt Bourbaki’s numbering of simple roots and denote by α the highest rootof Φ with respect to ∆.Let α∨ denote the coroot corresponding to α ∈ Φ, an element inX∗(T ) ⊂ X∗(G).

Recall α∨(K×) is a 1-dimensional torus in T . As usual, we letUα = xα(t) | t ∈ K denote the root subgroup of G corresponding to α. Givenx ∈ G we denote by ZG(x) the centraliser of x in G.


We discuss here one main case; viz., we suppose that G is not of type C3; thenG is of type A2, A3, D4 or D5.

A quick look at the Tits indices reveals that P is G-conjugate to the normaliserin G of the 1-parameter unipotent subgroup Uα.

From this it follows that in our present case the derived subgroup of U hasdimension 1 as an algebraic group and coincides with the centre Z of U .

Moreover, Z is G-conjugate to Uα.


By our assumption, the derived subgroup [U,U] contains an element u 6= 1.

Thenu ∈ [U,U] ⊂ [U(kv ),U(kv )] ⊂ [U ,U ] = Z.

Since the subgroup Z is T -invariant, the preceding remark implies that there isa long root β ∈ Φ such that U = Uβ .

Then u = xβ(a) for some nonzero a ∈ K.

We claim that the centraliser ZG(u) is defined over k. To prove this claim itsuffices to verify that the orbit morphism g 7−→ gug−1 of G is separable.

The latter amounts to showing that the Lie algebra of ZG(u) coincides withgu := X ∈ g | (Ad u)(X ) = X.


After adjusting ∆, possibly, we can assume that β = α.

For each α ∈ Φ we choose a nonzero vector Xα in gα = LieUα and let t denotethe Lie algebra of T .

Denote by g′ the K-span of all Xγ with γ 6∈ ±α and setg(α) := g−α ⊕ t⊕ gα.

Clearly, g = g′ ⊕ g(α).


It is easy to observe that both g′ and g(α) are (Ad u)-stable and one canchoose Xα such that

(Ad u)(Xγ) = Xγ + a[Xα,Xγ ] (∀ Xγ ∈ g′).

Since α is long, standard properties of root systems and Chevalley bases implythat if γ ∈ Φ is such that γ 6= −α and γ + α ∈ Φ, then [Xα,Xγ ] = λγXα+γ forsome nonzero λγ ∈ K.

From this it follows that gu ∩ g′ coincides with the K-span of all Xγ such thatγ 6∈ ±α and α + γ 6∈ Φ.

On the other hand, the commutator relations imply that each such Xγ belongsto LieZG(u). Therefore, gu ∩ g′ ⊂ LieZG(u).


The differential dα is a linear function on t.

Since G is simply connected, we have that dα 6= 0 and [Xα,X−α] 6= 0 (it iswell-known that dα = 0 if and only if p = 2 and G is of type A1 or Cn).

As(Ad u)(h) = h − a(dα)(h)Xα (∀ h ∈ t),

this implies that gu ∩ g(α) = gα ⊕ ker dα.

But then gu ∩ g(α) ⊂ LieZG(u), forcing gu ⊆ LieZG(u).

Since LieZG(x) ⊆ gx for all x ∈ G, we now conclude that the group ZG(u) isdefined over k.

Hence the connected component ZG(u) is defined over k, too.


Let C denote the connected component of the centraliser ZG(Uα).

The argument above shows that Lie C = LieZG(u).

Since C ⊆ ZG(u), we must have the equality ZG(u) = C.

Then C is a k-group, hence contains a maximal torus defined over k, say T ′.

As ker α ⊂ ZG(u), the torus T ′ has dimension l − 1, where l = rkG.

Let H denote the centraliser of T ′ in G.

By construction, H is a connected reductive k-group of semisimple rank 1containing Uα.


Since G is simply connected, so is the derived subgroup of H.

As Uα is unipotent, it lies in [H,H].

Since 1 6= u ∈ G(k) ∩ [H,H], the group [H,H] is k-isotropic.

The classification of simply connected k-groups of type A1 now shows that[H,H] ∼= SL2(K) as algebraic k-groups.

As a consequence, u belongs to a k-defined Borel subgroup of [H,H]; call it B.

Since u commutes with Uα, it must be that Uα = Ru(B).


Let S be a k-defined maximal torus of B.

Since [H,H] is k-isomorphic to SL2(K), there exists a k-defined cocharacterµ : K× → [H,H] such that

S = µ(K×), µ(t)xα(t′)µ(t)−1 = xα(t2t′) (∀ t, t′ ∈ K).

Then S is k-split in G, and hence it is a maximal kv -split torus of G (recall thatG has kv -rank 1).


Since S normalises Uα, it lies in P.

As P is defined over kv , there exists a kv -defined cocharacter ν : K× → P suchthat P = P(ν).

Since µ(K×) and ν(K×) are maximal kv -split tori in P, they are conjugate byan element of U .

In conjunction with the earlier remarks this yields that rν = Int x sµ for somex ∈ U and some positive integers r and s.

But thenP(µ) = P(sµ) = P(Int x sµ) = P(rν) = P.

Since µ is defined over k, so are P and U.


the congruence subgroup problem - indian statistical institutesury/csp170511.pdf · b.sury the...

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