Integral Points of Bounded Height:
Geometric and Analytic Background
Antoine Chambert-Loir
Institut de recherche mathématique de Rennes
Université de Rennes 1
Institut universitaire de France
Ein Gedi, January 3rd , 2011
Counting integral points of bounded height.
p. 1
Contents
1
Integral points: Three points of view
2
Height balls
3
Measures on analytic or adelic spaces
4
Volumes of height balls
Counting integral points of bounded height.
p. 2
Integral points: Three points of view
1
Integral points: Three points of view
Diophantine equations
Schemes
Analytic and adelic spaces
2
Height balls
3
Measures on analytic or adelic spaces
4
Volumes of height balls
Integral points: Three points of view.
p. 3
Diophantine equations
Maybe the older topic in mathematics:
We consider a family of polynomials f1 , . . . , fr ∈ Z[x1 , . . . , xn ] in n
variables with integral coefficients and we want to describe the
solutions of the corresponding system of equations.
Writing V for the “variety” defined by these equations, we want
to elucidate the set V(Z) (solutions in rational integers) within
the sets V(R) (solutions in real numbers) or V(Qp ) (solutions in
p-adic numbers) or within the adelic space V(AQ ).
The set V(Z) could be:
empty (x2 + 1 = 0; 3x3 + 4y3 + 5 = 0;...)
rare (Mordell conjecture);
Zariski dense (x2 − 2y2 = 1;...)
Restricting oneself to systems of homogeneous equations leads
to the topic of rational points.
Integral points: Three points of view.
Diophantine equations
p. 4
Schemes
As often in geometry, it may be useful or necessary to escape
the apparently simple context of equations and to reason in
more geometric terms.
This is already apparent for Diophantine equations in two
variables: as is know well-known, the complete picture involves
the genus of the corresponding plane curve (theorems of
Hasse–Minkowski, Siegel, Faltings...).
The “variety” from the preceding slide is the affine scheme:
V = Spec Z[x1 , . . . , xn ] (f1 , . . . , fr ).
For any ring A, one has
V(A) = {solutions in An of the system defining V}
= {morphisms of schemes Spec A → V}.
General Diophantine problems may involve general schemes,
and even stacks (theorems of Fontaine, Faltings).
Integral points: Three points of view.
Schemes
p. 5
Analytic spaces
If VQ is smooth and of finite type over Q, then V(R) is a finite
dimensional real analytic manifold.
The field of real numbers is the completion of Q for the usual
absolute value. For any prime p, let Qp be the completion of Q
for the p-adic absolute value. Let Zp be its subring of elements
of absolute value ≤ 1.
Again, if VQ is smooth and of finite type over Q, then V(Qp ) is a
finite dimensional p-adic analytic manifold and V(Zp ) is a
compact open subset of it.
With the exception of the trivial absolute value, the real and the
p-adic absolute values exhaust all absolute values on Q (up to
equivalence).
Recall the product formula: for any a ∈ Q∗ ,
Y
|a|∞
|a|p = 1.
p
We will set Q∞ = R.
Integral points: Three points of view.
Analytic and adelic spaces
p. 6
Adelic spaces
The (in)existence of integral points is often understood via the
geometry of the adelic space V(A) (Hasse principle, weak and
strong approximation, Brauer-Manin obstruction,...).
Recall:
the ring of adeles A is the restricted product
Y0
Q
Qp ; it is the subset of p≤∞ Qp consisting of families (xp )
p≤∞
such that xp ∈ Zp for all but finitely many primes p.
The adelic
space V(A) can also be defined as a restricted
Y0
Q
product
V(Qp ), subset of p≤∞ V(Qp ) consisting of
p≤∞
families (vp ) such that vp ∈ V(Zp ) for all be finitely many p.
It is a locally compact topological space.
V(Q) is embedded diagonally in V(A), with discrete image.
Integral points: Three points of view.
Analytic and adelic spaces
p. 7
Height balls
1
Integral points: Three points of view
2
Height balls
Heights on the projective space
Heights and line bundles
Heights and metrized line bundles
3
Measures on analytic or adelic spaces
4
Volumes of height balls
Height balls.
p. 8
Heights
Invented by Weil to generalize Fermat’s process of infinite
descent.
Basically: for a scheme V in the affine n-space Affn as above, a
point x ∈ V(Z) is a n-tuple (x1 , . . . , xn ) of integers; its height is
defined by
H(x) = max(1, |x1 | , . . . , |xn |).
If V lies in the projective space Pn , one may choose
homogeneous coordinates x = [x0 : . . . : xn ] ∈ Pn (Q) consisting of
coprime rational integers; its height is then
H(x) = max(|x0 | , |x1 | , . . . , |xn |)
Y
=
max(|x0 |p , . . . , |xn |p )
p≤∞
and the latter formula generalizes to arbitrary number fields.
Finiteness property (Northcott): for any B > 0, there are
only finitely many points P ∈ Pn (Q) such that H(P) ≤ B.
Height balls.
Heights on the projective space
p. 9
Heights and line bundles
More generally, heights are attached to line bundles—
“continuous families of lines parametrized by the scheme.”
The projective space Pn possesses a “tautological” line bundle
O(1): the line O(1)x over a point x ∈ Pn is the line corresponding
to x.
If φ : V → Pn is a morphism, one gets a line bundle φ∗ O(1) on V.
The height machine attaches a height function HL to any line
bundle L on a proper Q-variety V, well-defined up to
multiplication by a function whose log is bounded. It is
functorial; if L = φ∗ O(1), then HL = H ◦ φ.
Height balls.
Heights and line bundles
p. 10
Heights reflect geometry
Let V be a projective Q-scheme, L a line bundle on V. The height
function HL reflects the geometry of the line bundle:
Lower bounds:
For any global section f ∈ Γ(V, L), HL is bounded from below
outside of div(f);
If L is semiample (some power is generated by its global
sections), then HL is bounded from below ;
More generally, HL is bounded from below outside of the
intersection of the base loci of the powers of L.
Finiteness:
If L is ample, Northcott’s theorem holds: for any B > 0,
there are finitely many points x ∈ V(Q) such that HL (x) ≤ B.
If L is big, there exists a Zariski dense open set U ⊂ V such
that finiteness holds on U(Q);
More generally, finiteness holds on the largest open subset U
of V such that for any x ∈ U, there exists a decomposition
L = H + E with H ample, E effective, and x 6∈ E.
Height balls.
Heights and line bundles
p. 11
Heights and metrics on line bundles
Sections of line bundles don’t have a specific value at any point.
A real/p-adic metric on a line bundle—“continuous family of
norms”’— is a consistent way to measure them: it defines, for
any section f ∈ Γ(U, L), its norm kfkp , a continuous real function
on U(Qp ), satisfying
kfkp (x) ≥ 0
if a ∈ Γ(U, OV ), kafkp (x) = a(x)p kfkp (x) for any x ∈ U(Qp );
if f generates L around x, then kfkp (x) 6= 0.
Example: Global sections of O(1) on Pn correspond to
homogeneous polynomials of degree 1 in the
variables x0 , . . . , xn . The standard metric is so defined that
P(x)
p
kfP kp (x) =
,
x = [x0 : . . . : xn ].
max(|x0 |p , . . . , |xn |p )
An adelic metric on a line bundle is a family of p-adic metrics,
for all p ≤ ∞, subject to some uniformity assumption.
Height balls.
Heights and line bundles
p. 12
Heights and metrized line bundles
The notion of metrics allows to revisit the definition of the height:
Let us consider the particular case of the line bundle O(1) on Pn .
Let P be any homogeneous polynomial of degree 1, so that
P(x)
p
kfP kp (x) =
,
x = [x0 : . . . : xn ].
max(|x0 |p , . . . , |xn |p )
Observe that provided P(x) 6= 0,
Y Y
P(x)−1 = H(x)
kfP kp (x)−1 = H(x)
p
p≤∞
p≤∞
because of the product formula.
More generally, a metrized line bundle L = (L, (k·kp )) gives rise to
a height function
Y
∗
kfk (x)−1
HL : V(Q) → R+
,
x 7→
p≤∞
where f is any non-vanishing local section of L.
Height balls.
Heights and metrized line bundles
p. 13
Height balls in adelic spaces
Let V be a projective Q-variety, L be an ample line bundle on V.
Let f ∈ Γ(V, L) be any non-zero section of L; set Vf = V \ div(f): it
is a Q-scheme of finite type.
For any fixed p, kfkp−1 defines a continuous exhaustion of Vf (Qp ).
Height function on the adelic space Vs (AQ ):
Y
kfkp (xp )−1 .
Hf ((xp )) =
p≤∞
It defines a continuous exhaustion Hs : Vf (AQ ) → R+ :
it is continuous and for any B, the set of points x ∈ Vf (AQ ) such
that Hf (x) ≤ B is compact.
Height balls.
Heights and metrized line bundles
p. 14
Measures on analytic or adelic spaces
1
Integral points: Three points of view
2
Height balls
3
Measures on analytic or adelic spaces
Differential forms, metrics, measures
Measures on adelic spaces
4
Volumes of height balls
Measures on analytic or adelic spaces.
p. 15
Differential forms and measures
We fix p ≤ ∞ as well as a Haar measure on Qp .
The measure attached to a differential form
ω = f (x1 , . . . , xn ) dx1 ∧ · · · ∧ dxn
on an open subset of Qnp is defined by
|ω| = f (x1 , . . . , xn ) dx1 . . . dxn .
By the formula about change of variables in multiple integrals, it
is invariant under analytic diffeomorphisms.
Consequently, any top-differential form ω on a real or p-adic
analytic manifold M defines a measure |ω| on M.
Example: If M is a Lie group, ω a left-invariant form, then |ω| is
a left-invariant Haar measure on M.
Measures on analytic or adelic spaces.
Differential forms, metrics, measures
p. 16
Gauge forms vs metrics
Some manifolds M admit natural gauge forms, i.e.,
non-vanishing top differential forms.
A more general point of view consists in considering metrics on
the canonical line bundle KM (whose sections are
top-differential forms). Then, one defines a measure by the local
formula
dμ(x) = d |ω| (x)/ kωk (x),
where ω is any local non-vanishing top differential form. It does
not depend on the choice of ω.
Measures on analytic or adelic spaces.
Differential forms, metrics, measures
p. 17
Metrics vs models
Assume that p is a prime number and M = V(Zp ), for some flat
Z-scheme V of finite type V, smooth over Q.
Then, KM has a canonical metric defined by the model (a
top-differential form has norm ≤ 1 if it extends to a section
of KV ).
Weil’s formula: If V is smooth over Z,
μ(M) = (μ(Zp )/ p)dim(M)#V(Fp ).
Measures on analytic or adelic spaces.
Differential forms, metrics, measures
p. 18
Measures on adelic spaces
Let V be a proper smooth geometrically connected Q-scheme.
Then V(Qp ) is a smooth compact p-adic manifold for any p ≤ ∞.
Endow KV with a metric; we then have a measure τV,p on each
V(Qp ).
Let U be an open subset of V. By restriction, we have a
measure τU,p on U(Qp ). How to define a measure on the adelic
space U(AQ )?
Q
The product of measures p≤∞ τU,p does not necessarily
converge to a Radon measure on U(AQ ) in general. One needs
convergence factors λp such that the infinite product
Y
(λp τU,p (U(Zp )))
p<∞
converges absolutely to a positive real number.
(NB: the individual U(Zp ) are not well-defined, but two definitions
agree for all but finitely many primes p.)
Measures on analytic or adelic spaces.
Measures on adelic spaces
p. 19
Example of convergence factors
In the context of algebraic groups, the following convergence
factors are classical:
If U = Aff1 \ {0} is the multiplicative group, one takes
λp = (1 − 1/ p)−1 ;
More generally, if U is a torus, λp = Lp (1, X(U))−1 , the local
factor of Artin L-function of the group of characters of U
(Ono);
If U is a connected nilpotent algebraic group, or a connected
semi-simple algebraic group, one may take λp = 1;
Same if U = G/ H with G connected semi-simple, H without
characters (Borovoi-Rudnick);
If U = V, Peyre has shown that one may take λp = Lp (1, Pic(V)).
If U is a universal torsor over a proper toric variety, Salberger
has shown that one may take λp = 1.
Measures on analytic or adelic spaces.
Measures on adelic spaces
p. 20
Convergence factors
Geometric assumption: H1 (V, OV ) = H2 (V, OV ) = 0.
Then, we define two free Abelian groups with an action of
Gal(Q/ Q):
∗
invertible functions modulo constants: H0 (UQ , OV∗ )/ Q ;
Picard group modulo torsion: Pic(UQ )/ torsion.
Artin L-function:
∗
Lp (s, EP(U)) = Lp (s, H0 (UQ , OV∗ )/ Q )Lp (s, Pic(UQ )/ torsion)−1 .
Q
The Euler product
Lp (s, EP(U)) converges absolutely for
ℜ(s) > 1 ; it defines a meromorphic function on C with a pole at
s = 1 of order
rank H0 (U, OV∗ ) − rank Pic(U).
Theorem
One may take
λp = Lp (1, EP(U))−1 .
Measures on analytic or adelic spaces.
Measures on adelic spaces
p. 21
Volumes of height balls
1
Integral points: Three points of view
2
Height balls
3
Measures on analytic or adelic spaces
4
Volumes of height balls
Real semi-simple Lie groups
Adelic semi-simple algebraic groups
Volume estimates
Volumes of height balls.
p. 22
Real semi-simple Lie group
Let G be a semi-simple connected Lie group with trivial center,
μ a Haar measure on G.
Let ρ : G → GL(V) be a finite dimensional faithful representation
of G in a real vector space V. Let k·k be a norm on End(V).
For any T > 0, let BT = {g ∈ G ; ρ(g) ≤ T} — compact in G.
Theorem (Maucourant, 2004)
There exist d ∈ Q and e ∈ N, explicitly defined in terms of the
relative root system of G and the weights of ρ such that, when
T → ∞: one has the following volume estimate:
μ(BT ) ∼ cT d log(T)e−1 for some positive real number c. Moreover,
1 ≤ e ≤ rankR (G).
Volumes of height balls.
Real semi-simple Lie groups
p. 23
Adelic semi-simple algebraic groups
Let G be a connected semi-simple algebraic group over Q.
Let ρ : G → GL(V) be a faithful representation of G in a finite
dimensional Q-vector space V (with a unique highest weight).
For any p ∈ {prime numbers} ∪ {∞}, let k·kp be a p-adic norm
on End(V) ⊗ Qp (defined using a common matrix representation
for almost all p).
Q For any T > 0, let BT = {g = (gp ) ∈ G(A) ; p ρ(gp )p ≤ T}
— this is a compact set in G(A).
Volumes of height balls.
Adelic semi-simple algebraic groups
p. 24
Adelic semi-simple groups
BT = {g = (gp ) ∈ G(A) ;
Y
ρ(gp ) ≤ T}
p
p
Fix a Haar measure μ on G(A).
Theorem (Gorodnik, Maucourant, Oh, 2007)
There exist a positive real number c, a positive rational
number a and a non-negative integer b such that, when T → ∞,
one has the following volume estimate: μ(BT ) ∼ cT a log(T)b .
Again, a and b can be computed explicitly in terms of the
weights of ρ, the root system of G and the action of Gal(Q/ Q)
they possess.
In fact, thanks to Tauberian theorems, the proof is a
straightforward consequence of the analytic behaviour of some
Mellin transform, which was proved by Shalika, Tschinkel,
Takloo-Bighash.
Volumes of height balls.
Adelic semi-simple algebraic groups
p. 25
Motivation/consequence of these
estimates
These volume estimates are one step in understanding the
number of
points in Γ ∩ BT , where Γ is a lattice of the Lie group G —
lattice points in balls;
points in G(Q) ∩ BT — rational points of “bounded
height”.
When T → ∞, and for adequate representations ρ, the obtained
estimates are
#(Γ ∩ BT ) ∼ V(T)/ μ(G/ Γ);
#(G(Q) ∩ BT ) ∼ V(T)/ μ(G(A)/ G(Q)) — with a deliberately
ignored twist caused by automorphic characters.
Volumes of height balls.
Adelic semi-simple algebraic groups
p. 26
Previously on “N(T) ∼ V(T)”
There are many results of this sort in the litterature.
Lattice points in homothetic (convex) bodies;
Circle method;
Some homegeneous spaces G/ H:
Duke, Rudnick, Sarnak;
Eskin, McMullen;
Eskin, Mozes, Shah;
Borovoi, Rudnick;
Partial equivariant compactifications of vector groups or tori
(Moroz,...) ;
etc.
Volumes of height balls.
Adelic semi-simple algebraic groups
p. 27
Volume estimates for quasi-projective
varieties
Goal: Generalize these results when the algebraic group is
replaced by any quasi-projective variety U.
We assume that U = V \ D, complement of a divisor with strict
normal crossings in a smooth proper Q-scheme.
We metrize OV (D) and introduce a new measure
dτ(V,D) (x) = kfD k (x)−1 dτV (x)
on U(Qp ).
Let L be a metrized line bundle on V and f be a global section
of L vanishing on D.
We are interested in the height ball defined in U(Qp ) by the
inequality kfk (x) ≥ 1/ B, and in the asymptotic behaviour, when
B → ∞, of its volume:
Z
V(B) =
dτ(V,D) (x).
kfk(x)≥1/ B
Volumes of height balls.
Volume estimates
p. 28
Proving general volume estimates over
local fields
Z
V(B) =
dτ(V,D) (x).
kfk(x)≥1/ B
To establish an asymptotic estimate for V(B), when B → ∞, our
main idea relies in the consideration of the Mellin-Stieltjes
transform
Z∞
Z
B−s dV(B) =
Z(s) =
0
U(Qp )
kfk (x)s dτ(V,D) (x).
It turns out that the latter integral is a kind of Igusa zeta
function, and can be studied using classical methods in that
field. We are able to understand analytic properties of Z
(convergence, meromorphic continuation, rightmost poles).
A Tauberian theorem allows to conclude.
Volumes of height balls.
Volume estimates
p. 29
Geometric Igusa integrals
To elucidate the integral
Z
Z(s) =
V ( Qp )
kfk (x)s dτ(V,D) (x),
we express it in local coordinates around points of V.
P
Let Dα , for α
P∈ A, be the components of D; write D = dα Dα and
div(f) = E + λα Dα , where E is minimal and effective.
Let q ∈ V(Qp ) and let A = Aq be the set of α such that q ∈ Dα .
We use the assumption that D has strict normal crossings by
saying in a neighbourhood Ωq of q, there are local coordinates
defining the divisors Dα , for α ∈ A. Then, the integral defining
Z(s) can be written locally as
Z
Y
|xα |λα s−dα φ(s, x) dx.
Ωq α∈A
We observe absolute convergence for ℜ(s) > max((dα − 1)/ λα ) as
α∈A
well as a pole of order the number of α achieving equality.
Volumes of height balls.
Volume estimates
p. 30
Example: Compactifications of algebraic
groups
Assume here that V is an equivariant compactification of an
algebraic group G. In other words, we require that the
multiplication on G extends to an action of G on V.
We set D = V \ G Then, KV is linearly equivalent to a divisor
supported on D and τ(V,D) is a Haar measure on V.
For a line bundle, we take L = −(KV + D) is the log-anticanonical
line bundle of the pair (V, D).
We assume that U is a G-stable open subset with G ⊂ U ⊂ V;
more precisely,
we assume that U is of the form
S
U = V \ α∈AU Dα , for AU ⊂ A.
We see that the pole of ZU (s) is at s = 1, and its order is equal to
the maximal number of components Dα , for α ∈ AU , whose
intersection has a Qp -point.
Volumes of height balls.
Volume estimates
p. 31
Clemens complexes
In the context U = V \ D, where D has strict normal crossings
an (V, D) as follows—we call
in V, we define a simplicial complex CQ
p
it the p-adic analytic Clemens complex:
Its points are the components of D which have Qp -points;
Its edges are the intersections of two components when they
have Qp -points;
Generally, its faces of dimension k are the intersections of
k + 1 distinct components of D which have Qp -points.
We have seen how such complexes are related to the asymptotic
behaviour of volumes.
They can be explicitly described for toric varieties, or for the
wonderful compactifications of semisimple groups. We then
recover the results of Maucourant and Gorodnik-Maucourant-Oh.
Volumes of height balls.
Volume estimates
p. 32
Towards a general description
Generalize these results when the algebraic group is replaced by
any quasi-projective variety U.
We assume that U = V \ D, complement of a divisor with strict
normal crossings in a smooth proper Q-scheme.
To have a notion of integral points, we also fix a flat Z-scheme of
finite type U extending U.
We consider the log-anticanonical line bundle −(KV + D); assume
that it is ample on V, or at least big. We give ourselves a metric
on it. Let NU (B) be the number of integral points of U of
height ≤ B.
Volumes of height balls.
Volume estimates
p. 33
Counting integral points — expected
behaviour
Goal: find as many interesting examples as possible where
NU (B) ∼ cB(log B)e ,
an
e = ords=1 L(s, EP(U)) + dim C∞
(V, D),
for some positive real number c > 0.
Relate c to the asymptotic behaviour of the volume of the adelic
height balls, and other geometric invariants of the pair (V, D).
As for rational points:
It is necessary to assume that the set U(Z) of integral points
is Zariski dense in U;
It may be necessary to restrict U, or to pass to a larger
number field, in order to get the desired asymptotic;
There may exist counterexamples.
Volumes of height balls.
Volume estimates
p. 34
Equivariant compactifications of additive
groups
We consider here a projective equivariant compactification V of
a vector group G = Gna over Q. We assume that V \ G is a divisor
with strict normal crossings. We fix a G-stable open subset U
such that D = V \ U is a divisor.
Let U be a model of U over Z.
Fix a metric on the log-anticanonical line bundle −(KV + D) and
consider the associated height function H. This also gives a Haar
measure on G(A).
The two quantities in the game are:
Number of integral points of bounded height:
NU (B) = #{x ∈ G(Q) ∩ U(Z) ; H(x) ≤ B};
Haar volume of the integral locus in the adele group:
Y
V(B) = volG {x ∈ G(A) ∩ G(R)
U(Zp ) ; H(x) ≤ B}.
p<∞
Volumes of height balls.
Volume estimates
p. 35
Equivariant compactifications of vector
groups
Theorem
When B → ∞,
NU (B) ∼ V(B) ∼ cB(log B)b ,
where c is a positive real number and
b = rank(Pic(U)) + dim CRan (V, D).
We also have a precise formula for the constant c, as a product
of rational numbers, and of two volumes:
Volume for the measure τ(V,D) of the integral locus
Q
p<∞ U(Zp ) in the finite adelic space;
Integrals for some “residue measures” over the strata of
minimal dimension of D(R).
Volumes of height balls.
Volume estimates
p. 36