Recall motivation: CIT required for first-order axiomatisation of pseudo-exp,
specifically the Schanuel conjecture (over the kernel).
Let X (= G_m^N subvariety over \Qbar.
"Subgroup" of G_m^N means algebraic subgroup, i.e. of form
/\_i \Pi_j x_j^{n_ij} = 1
[and 'coset' means coset of connected such]
For H a subgroup (resp. coset) of G_m^N, a component S of H \cap X is a
_special subvariety_ (resp. _weakly special subvariety_) of X iff
\dim S > \dim X - \codim H
Conjecture [CIT]:
There are finitely many maximal special subvarieties.
Theorem [Weak CIT; Zilber / Bombieri-Masser-Zannier]:
The union Z (= X of the dim>0 weakly special subvarieties of X is Zariski
closed.
(Proof: Ax)
(Note: holds for X over \C)
Theorem [Habegger, BMZ]:
There are only finitely special points on X\Z.
[Refinements due to Rémond ("over parameters" versions, at least for abelian
varieties), Maurin (X^oa |-> X^ta for X a curve)]
Now let C (= G_m^N be a curve / \Qbar.
Say N=3 [for concreteness].
X := C^2: note Z=X!
[n] : G_m^3 -> G_m^3; (x_1,x_2,x_3) |-> (x_1^n,x_2^n,x_3^n)
If x \in C and some [n]x \in C, with n >= 2, then call x an "echoing point" of
C, and [n]x its "echo". Then
(x,[n]x)\in X \cap { (z_1,...,z_6) | z_4=z_1^n /\ z_5=z_2^n /\ z_6=z_3^n }
so (x,[n]x) is a special point of X.
Question: can C have infinitely many echoing points?
If N=2, then yes (consider dimensions).
If C is contained in a proper subgroup, C (= H < G_m^3, then again yes.
Else, CIT => no!
Remark: torsion points are echoing; MM => finitely many torsion points on C.
Let T (= G_m^3(\C) := { x | |x_1|=|x_2|=|x_3|=1 }.
Definition: evil locus := C(\C)\cap T
"Should" be finite: dim_R(C(\C)) = 2; dim_R(T) = 3; dim_R(G_m^3(C)) = 6...
but sometimes isn't...
[Mingxi Wang: z |-> i(z+1)/(z-1) is a bijection S^1 -> R\cup{\infty}]
Theorem [B-Habegger]:
Suppose C (= G_m^N (N>=2, C / \Qbar) is contained in no proper subgroup
and has finite evil locus. Then C has only finitely many echoing points.
Proof Part I : Ax and Pila-Wilkie
---------------------------------
WMA C is _free_ - contained in no proper *coset*
(if C is not free and has an echoing point, then C is contained in a
proper subgroup).
Work in the o-minimal structure <\R;+,*,exp|> where exp| is *complex* exp
restricted to
F := \R + i [0,2\pi) (= \C = \R+i\R ~= \R^2.
o-minimality is due to Wilkie.
Consider the definable family of definable sets
E_n := { (k,x) \in \R^3 * F^3 | ( exp|(x) \in C /\ exp|(nx-k2i\pi) \in C }
(so if k is integral, (k,x)\in E_n witnesses that exp|(x) is n-echoing)
and the projections to the k-coordinates
K_n := { k | \exists x. (k,x) \in E_n }.
If exp|(x) is n-echoing, then (k,x) \in E_n for some k \in {0,1,...,n}^3.
Pila-Wilkie counting theorem:
Let K_n^alg be the union of all positive-dimensional semi-algebraic
subsets of K_n, and let K_n^tr := K_n \ K_n^alg.
Then "rational points on K_n^tr are subpolynomially sparse":
for all \epsilon > 0, exists c, for all n,B:
| { q \in \Q^3 \cap K_n^tr | H(q) < B } |
<= c B^\epsilon
where H(q) = max_i H(q_i),
where H(a/b) = max{|a|,|b|} (assuming (a,b)=1)
(so for k\in\N, H(k) = k).
Claim: K_n^alg is empty for all n
Proof:
Suppose S (= K_n^alg is a semi-algebraic curve; let, in an elementary
extension, k\in S be generic over \R. So \trd(k/\R) = 1.
Let R' := dcl(R,k) >= \R, let C' := R'^2 >= \C.
Lemma [Ax]:
For x \in F(\Rfrak)^N
\trd(x,\exp|(x) / \C) > \ld_\Q(x / \C)
unless x \in \C^N.
Proof [after Wilkie]:
Define derivation \delta(f(k)) := f'(k)
for f a definable function / \R;
apply Ax's theorem.
Now let \gamma := exp(k2i\pi)\in ([n]C - C).
Then by Ax:
\ld_\Q(k2i\pi / \C) <= \trd(k / \C) + \trd(\gamma / \C) - 1
<= 1 + 2 - 1
<= 2 < N .
C hence ([n]C - C) is free, so \gamma isn't generic, so
\trd(\gamma / \C) = 1 hence \ld_\Q(k / \C) = 1.
So say \pi : \G_m^N -->> \G_m^{N-1} such that \pi(\gamma) \in \C.
By definition of K_n, say \alpha in C such that [n]\alpha-\gamma \in C.
If \alpha \in \C then [n]\alpha-\gamma is generic in C but
\pi([n]\alpha-\gamma)\in\C, contradicting freeness of C
Hence \alpha is generic in C and \pi(\alpha) is generic in \pi(C).
But then [n]\pi(\alpha)-\pi(\gamma)\in \pi(C) implies
[n]\pi(C)-\pi(\gamma)=\pi(C), contradicting freeness of C.
Now for (n,k) integral, there are <\aleph_0 (k,x) \in E_n since there are
<\aleph_0 n-echoing points; so by uniform finiteness there is a
bound uniform in (n,k) on the number of such x. So Pila-Wilkie => for all
\epsilon > 0, exists c, for all n
|{n-echoing points}| <= c n^\epsilon (*)
Proof Part II: degree lower bounds
----------------------------------
Say C is over F.
Lemma: exists c such that for n large enough, if x is non-torsion and
n-echoing, then
\deg(x/F) := [F(x):F] >= c n^{1/11}.
But the conjugates of x over F are also n-echoing, so (picking \epsilon = 1/12
say) we contradict (*) if there are infinitely many non-torsion echoing
points.
By Manin-Mumford, there are only finitely many torsion points on C, so we're
done.
Sketch Proof of Lemma:
Echoing points are algebraic.
h : G_m^N -> \R
; (x_1,...,x_N) |-> \Sigma_i h(x_i)
where
h : \Qbar -> \R
; x |-> \Sigma_{v normalised abs value} max(0, log |x|_v)
Fact [corollary of Rémond's toric Vojta inequality]:
Echoing points have bounded height.
[but not, a priori, their echos!]
Now points of bounded height can't have all conjugates "close together",
so by finite evil,
any echoing point has some conjugate bounded away from 1 in modulus.
"Tropical geometry" to study behaviour "near 0 and \infty" :
Claim:
Exist A,B and finitely many dim 1 cosets H_1+b_1,...,H_s+b_s such that
for all x \in C(\C), if
L := || \log|x| || > A
then for some i
log|| (x / H_i) - (b_i / H_i) || <= -BL
Proof:
Else, by compactness, in an elementary extension R' of the real
field \R there exists a point x\in X(C') (where C' is the complex
field of R', C' = R'+iR' \supset \R+i\R = \C) such that
(i) for some i, x_i > \R or x_i^{-1} > \R
(ii) for all dim 1 subtori H, and all b \in \C,
|| x/H - b/H || > |x_i|^q for all i and all rationals q\neq 0.
We work with the standard part map st : C' \maps \C and the
corresponding valuation v.
Recall the transcendental valuation inequality (TVI):
for all y \in G_m^N(C')
trd(y) >= trd(st(y)) + ld_\Q(v(y))
By (i) and TVI, \ld_\Q(v(x))=1. So say v(x/H)=0, H a dim 1 torus, and
let \alpha := st(x/H); by TVI again, \alpha \in \Qalg^{N-1}.
Let \beta := x/H - \alpha.
By (ii), v(\beta_i) < qv(x_j) for all i,j and q != 0. (*)
But \trd(\beta, x) = \trd(x) = 1,
so by TVI, \ld_\Q(v(\beta), v(x)) = \ld_\Q(v(x)) = 1,
so v(\beta) \in \spanof{v(x)}_\Q.
So by (*), v(\beta_i) <= 0 for all i.
But this contradicts st(\beta)=0.
Now let x be n-echoing.
Baker-Wüstholz effective "linear forms in logarithms" =>
log||([n]x / H_i - b_i / H_i)|| is (if defined) bounded below
proportionally to -deg(x)^11 log(n) and h(x), h(b_i).
[also have to deal with case [n]x / H_i = b_i / H_i... but let's ignore
that]
Latter are bounded, so by claim, || log|[n]x| || is bounded below
proportionally to -deg(x)^11 log(n).
So by lower bound (after replacing x with a conjugate) on || log|x| ||,
deg(x) >= c n^{1/11}