Luigi Cremona (1830 – 1903)

— I —

A Theorem

Main Goal

Theorem [C. - Junyi Xie].— Let Γ be a finite index subgroupof SLn(Z).

• If Γ embeds in Bir(PmC ) then m ≥ n − 1.

• If Γ embeds in Aut(AmC ) then m ≥ n.

Remark.— One can replace

• PmC by a complex variety V and get

dim(V ) ≥ n − 1.

• SLn(Z) by an arithmetic lattice in a simple linear algebraicgroup G (R) for which the congruence kernel is finite, and get

dim(V ) ≥ rankR(G (R)).

A more precise version

Theorem [C. - Junyi Xie].— Let Γ be a finite index subgroupof SLn(Z), n ≥ 3. Let V be a complex, irreducible projectivevariety. If Γ embeds in Bir(V ) then

dim(V ) ≥ n − 1

and in case of equality, V is rational and the action of Γ isconjugate to an action on Pm

C by regular automorphisms.

Finite groups

Problem.— If n > m + 1, and p is a large prime, show that

SLn(Z/pZ)

does not embed in Crm(C).

• this is known in dimension 2 and 3 (Dolgachev-Iskovskikh,Prokhorov).

• this is not known in dimension 4.

From complex to p-adic numbers

• Assume Γ ⊂ Aut(AmC ) is finitely generated:

S = {γ1, . . . , γs}

is a finite symmetric set of generators.

• Define

Coeff = {coefficients of the polynomial formulae defining the γi}

• Then

Lemma.— The ring Z[Coeff ] embedds into the ring of p-adic integers Zp (for infinitely many primes p).

From complex to p-adic numbers

• Zp = p-adic integers• If a ∈ Z and a = psq with q ∧ p = 1, then

|a|p = p−s .

• Qp = completion of Q for the norm∣∣∣ab

∣∣∣p

=|a|p|b|p

and

Zp = closure of Z

= {z ∈ Qp ; |z |p ≤ 1}

• Residue field = Zp/pZp = Z/pZ = Fp.

— II —

Local dynamics over C

Local dynamics: complex case

• Consider f (z) = z + εzd , with d ≥ 2 and ε > 0. Then,

One cannot find arbitrary small neighborhoods of 0 ∈ C whichare f -invariants.

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Local dynamics: Cremer Theorem

Theorem [Cremer].— Consider a polynomial transformationf : C→ C of type

f (z) = λz + a2z2 + . . .+ ad−1zd−1 + adzd

with d ≥ 2. Assume that |λ| = 1 and

∀ε > 0, ∃q ≥ 1, such that |λq − 1| ≤ εdq.

Then f has periodic points which are arbitrarily close tothe origin 0 ∈ C.

Remark.— |λq − 1| = |e2πiqθ − 1| ' 2π dist(qθ,Z).

Cremer Theorem: Proof

Sketch of Proof.–• Assume f is a monic polynomial (i.e. ad = 1).• Equation for periodic points

f q(z) = z

i.e.zdq

+ . . .+ (λq − 1)z = 0.

One root is z = 0, the other roots zi satisfy

dq−1∏i=1

zi = λq − 1.

If |λq − 1| < εdq−1, at least one root zi satisfies

|zi | < ε.

— III —

Local dynamics over Qp

p-adic analytic functions

• ‖ · ‖= Tate norm on Zp[x] = Zp[x1, . . . , xm]. If

f (x) =∑

J=(j1,...,jm)

aJxJ

then‖ f ‖= max

J|aJ |p.

• Tate Algebra = Zp〈x〉 = completion of Zp[x] for ‖ · ‖.• elements of Zp〈x〉 are power series

f (x) =∑

J=(j1,...,jm)

aJxJ

with |aJ |p → 0 as |J| = j1 + . . .+ jm goes to +∞.

Tate diffeomorphisms

• The polydisk of dimension m is U = Zmp .

• Analytic endomorphisms (or Tate endomorphisms) are maps

f : U → Ux 7→ (f1(x), . . . , fm(x))

where each fi is in Zp〈x〉.• Analytic diffeomorphisms (or Tate diffeomorphisms) form agroup

DiffTate(U).

• Write f (x) ≡ Id(x) (mod pc) if all the coefficients bJ in thepower series expansion of f (x)− Id(x) satisfy

bJ ∈ pcZp (i .e.|bJ |p ≤ p−c).

Jason Bell Theorem

Theorem [J. Bell, B. Poonen].— Let f be an analyticendomorphism of the polydisk U . Assume

f ≡ Id (mod pc) with c > 1/(p − 1).

There exists a Tate analytic map

Φ: Zp × U → U

such that

• Φ determines an analytic action of (Zp,+) on U .

• Φ(n, x) = f n(x) for all n ∈ Z

In particular, f is a diffeomorphism of U , withf −1(x) = Φ(−1, x).

Proof of Bell Theorem after Poonen

• Consider the operator

∆f : h(x) 7→ h ◦ f (x)− h(x).

Since f (x) ≡ Id(x) (mod pc), one gets

‖ ∆jf (h) ‖≤ p−jc ‖ h ‖

• The series

Φ(t, x) =∑j≥0

(tj

)∆j

f (Id(x))

=∑j≥0

t · (t − 1) · · · (t − j + 1)

1 · 2 · · · j∆j

f (Id(x))

converges because

p-adic valuation(j!) = [j

p] + [

j

p2] + ... ≤ j

p − 1

Proof of Bell Theorem after Poonen

• Moreover, for n ∈ Z+,

Φ(n, x) =

n∑j=0

(nj

)∆j

f

(Id(x))

= (Id + ∆f )n(Id(x))

= f n(x)

• From this relation one gets

Φ(n,Φ(n′, x)) = Φ(n + n′, x)

for all n and n′ in Zp.

Bell Theorem, and vector fields

• Bell theorem says :

• Start with an analytic diffeomorphism f ∈ DiffTate(U) withf ≡ Id (mod p)

• The action of (Z,+) generated by f extends to an analyticaction of

Zp = pro-p completion of Z

= projective limit of the Z/p`Z.

= a one dimensional p-adic analytic group

• The vector field

v(x) =∂Φ(t, x)

∂t |t=0

is an analytic vector field whose flow is given by Φ(t, x).

— IV—

Bell Theorem for SLn(Z)

The congruence subgroup property

• Principal congruence subgroups of SLn(Z) = kernels of

SLn(Z)→ SLn(Z/qZ)

(q a positive integer).• Γ < SLn(Z) is a congruence subgroup if it contains aprincipal congruence subgroup.

Theorem [Bass, Milnor, Serre ; Mennicke].— If n ≥ 3,the group SLn(Z) satisfies the congruence subgroup prop-erty: every finite index subgroup of SLn(Z) is a congruencesubgroup.

• Consequence: up to finite index

pro-p completion of Γ = open subgroup of SLn(Zp)

A non-abelian version of Bell Theorem

Theorem [C., J. Xie].— Fix a prime p ≥ 3.Let Γ be a finite index subgroup of SLn(Z). Assume Γ actsby analytic diffeomorphisms on U = Zm

p and

∀f ∈ Γ, f (x) ≡ Id(x) (mod p).

Then the action of Γ on U extends to an analytic action ofthe p-adic group Γ ⊂ SLn(Zp) on U .

Here,Γ = open subgroup of SLn(Zp).

and one getsΦ: Γ× U → U .

Conclusion I

• Setting

• Γ is a finite index subgroup in SLn(Z).

• Γ acts faithfully on AmC

• Γ acts faithfully on AmQp

by polynomial automorphisms (which

are defined over Zp), for some prime p ≥ 3.

• Changing Γ in a finite index subgroup,

f (x) = A0 + A1(x) +∑j≥2

Aj(x)

withA0 = 0 (mod p2) and A1 = Id (mod p).

Then

p−1f (px) = p−1A0 + A1(x) +∑j≥2

pj−1Aj(x)

≡ Id (mod p)

Conclusion II

• Thus, an open subgroup of SLn(Zp) acts analytically onU = Zm

p .

• Taking derivative :The Lie algebra of SLn(Qp) embeds into the Lie algebra of Tateanalytic vector fields on U .

• Take z0 a generic point of U : the subset of vector fieldsvanishing at z0 forms a Lie algebra of codimension ≤ m in sln(Qp).

• Lie theory : n − 1 ≤ m (classification of maximal subalgebrasin sln)

• Jacobian determinant = 1 implies n ≤ m

— V—

Comments

Conclusion II

• Alternative proof in dimension 2

• blow-up all points of P2 and take the limit of Neron-Severigroups.

• classes with self-intersection 1 = an infinite dimensionalsymmetric space = H∞, the infinite dimensional hyperbolicspace.

• an action of SLn(Z) on such a space has a fixed point if n ≥ 3(Kazhdan; Faraut and Harzallah)

• a fixed point gives an invariant polarization.

Conclusion II

• Alternative proof for regular automorphisms of complexprojective varieties

• Γ acts on the cohomology of V .

• for n ≥ 3, Margulis super-rigidity implies that the actionextends to an action of SLn(R).

• Hodge index theorem and representation theory implydim(V ) ≥ n if the action on cohomology is not trivial.