Some applications of potential theory to number theoretical problems on analytic curves

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Some applications of potential theory to number theoretical problems

on analytic curves

Antoine Chambert-Loir

Institut de recherche mathématique de Rennes, Université de Rennes 1 and

Institute for Advanced Study, Princeton

Athens/Atlanta Number Theory Seminar April 13^rd, 2010

. p. 1

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What is Potential Theory? — Classical Physics

Newton’s gravitation field is the gradient of a function (Lagrange), which Green and Gauß called apotential; also holds for electrostatics;

the equilibrium state is the solution of a Laplace equation(with boundary conditions);

the potential can be deduced by integrating the fundamental solution of the Laplacian (kernel):

Newtonian potential, 1/kx−ykⁿ⁻²in dimension n_¾3,

logarithmic potential, −logkx−yk in dimension n=2;

Robin’s problem: compute the mass distribution giving rise to the equilibrium potential.

Potential theory. From physics... p. 3

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Logarithmic potential theory and Complex function theory

We assume that the dimensionn=2.

IdentifyingR² withC, it appears thattwo-dimensional potential theoryis very closely connected to

one-dimensional complex analysis.

For example, the real part of a holomorphic function is harmonic, and conversely on a simply connected open set.

Applications (see, e.g., Ransford’s book):

uniform approximation (theorems of Bernstein–Walsh, Mergelyan...);

Schwarz’s lemma and its generalizations;

complex dynamics...

Generalizations to Riemann surfaces.

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Logarithmic potential theory

M= compact connected Riemann surface;o∈M, K⊂Mnon-polarcompact subset such thato6∈K. Green function: there exists a unique function g_K: M\{o}→Rwith the properties:

g_K is harmonic on M\({o}∪K);

g_K(p) +log|z(p)−z(o)|extends continuously ato (wherezis a local parameter aroundoon M);

gK(p)→0 whenp→∂K (up to a polar subset of ∂K).

Equilibrium measure: Laplacian ofg_K (in the sense of distributions) — probability measureμK supported by∂K. Robin constant(relative to the local parameterz):

−log capz(K) =_plim_→_ogK(p) +log|z(p)−z(o)|. Capacitary normon T^oM: defined by_∂z^∂=capz(K).

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Logarithmic potential theory and Classical number theory

In the 1920s, surprising connections with number theory were shown:

Define the Fekete–Szeg˝o set of a subsetA⊂Cas the set FS(A)ofalgebraic integersall of whose conjugates belong toA.

Theorem (Fekete–Szeg˝o)

Let K be a compact subset ofC, invariant under complex conjugation.

Ifcapz⁻¹(K)<1,FS(K)is finite;

Ifcapz⁻¹(K)¾1, then for any open neighborhood U of K,FS(U)is infinite.

Potential theory. ... to Number theory p. 6

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Logarithmic potential theory and more recent number theory

Ifais an algebraic number, let μ(a)be the Galois invariant probability measure onCsupported by the conjugates ofa: if the minimal polynomial of aisc^Q^d

j=1(X−a_j), then δ(a) = 1

d

X

j=1

δ_a_j (δ_z = Dirac mass atz).

Theorem (Serre, cf. Bilu, Rumely)

Let K be a compact subset ofC, invariant under complex conjugation such thatcapz⁻¹(K) =1.

The equilibrium measure μK of K is the unique probability measure on K which is a limit of measures of the

form δ(a), for algebraic integers a.

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Height of an algebraic number

Letabe an algebraic number, with minimal polynomial P=c^Q^d_j

=1(X−a_j).

height(a) = 1 d

Z ₂π 0

logP(e^iθ)dθ

2π Mahler measure

= 1

dlogc+1 d

d

X

j=1

log⁺(a_j) (Jensen’s formula)

= 1

dlogc+ Z

C

log⁺(|z|)dδ_a(z) and using all absolute values ofQ,

= ^X

p¶∞

1 d

d

X

j=1

log⁺(a_j_p).

Observe that forK={|z|¶1}, g_K =log⁺ andμ_K =dθ/2π.

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Generalizations

working on algebraic varieties usual Diophantine geometry setup Arakelov geometry defines the height of subvarieties (Faltings, Bost-Gillet-Soulé)

putting all places on equal foot: non-archimedean potential theory

Rumely’s theory (end of 1980s);

using Berkovich’s spaces (Baker/Rumely, Favre/Rivera-Letelier, Thuillier)

beyond potential theory

Néron functions in Diophantine geometry;

Green currents from Arakelov geometry;

function theory, Laplace operators, measures,... on Berkovich spaces.

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Metrics vs. Néron functions

Xprojective variety overQ, Deffective Cartier divisor on X Lthe line bundleOX(D),s_D the canonical section ofL

Néron function forD:functionλD onX(Cp)\ |D|such that λ_D,p(x) +log|f|p(x)

extends continuously for any local equationf ofD. Green function for D: function on X(C)\ |D|such that

dd^cg_D+δ_D=ω_D, a smooth (1,1)-form onX(C). Continuous/smooth Metric forL: consistent way of defining the norm of local non-vanishing sections ofLas continuous/smooth positive functions.

Equivalent concepts: use the formulae

λ_D,p(x) =−logks_Dkp(x), g_D(x) =−logks_Dk_∞(x). ω_D is the Chern formof the metrized line bundleL.

Potential theory. ... and Arakelov geometry p. 10

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Metrized line bundles and local heights — archimedean picture

Xprojective complex variety purely of dimensionm Db_j= (D_j, g_D_j), 0¶j_¶m, divisors and Green functions, intersecting properly.

Multiplyingm of the Chern formsω_D_j furnishes a

differential form of type(m, m)onX(C), hence a signed measure.

Inductive definition of thelocal height pairing:

(D^b₀· · ·D^b_m|X)∞= (D^b₀· · ·D^b_m−1|D_m)∞+ Z

X(C)

g_D_mω_D₀. . . ω_D_m₋₁

Remarks:

the integral converges;

multilinear;

independent on the order of theD^b_j.

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Metrized line bundles and local heights — non-archimedean picture

Xprojective variety purely of dimensionmover ap-adic field

Db_j= (D_j, g_D_j), 0¶j_¶m, divisors and “smooth” Néron functions, intersecting properly.

Local height pairing defined usingarithmetic intersection theoryonZp-schemes.

There are no Chern forms anymore, but one maydefine measures on the Berkovich spaceXp so that the inductive formula holds:

(D^b₀· · ·D^b_m|X)p= (D^b₀· · ·D^b_m₋₁|D_m) + Z

Xp

g_D_mω_D₀. . . ω_D_m₋₁

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Metrized line bundle and heights — admissible metrics

Both in the archimedean and non-archimedean cases, it is necessary to enlarge the settings of smooth metrized line bundles.

One good notion is that of anadmissible metrized line bundle(Zhang, 1995).

Semipositive metrics: uniform limits of smooth metrics with positive first Chern forms, resp. given by a

numerically effective model.

Admissible metric: quotient of two semipositive metrics.

For those metrics, one may definelocal heightsand measures(ACL, 2006) by approximation.

The inductive formula still holds (“Mahler formula”, ACL &

Thuillier, 2009).

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Metrized line bundle and heights — global picture

Xprojective variety overQ, purely of dimensionm

Db_j= (D_j,(g_D_j_,p)p), 0¶j_¶m, divisors and associated Green functions at all places, intersecting properly.

“Adelic condition”: for almost allp, the g_D_j_,pare defined using a fixedZ-scheme.

Global height pairing(in)finite sum over all primes:

(D^b₀. . .Db_m|X) = ^X

p¶∞

(D^b₀. . .Db_m|X)p.

Properties:

multilinear;

independent on the order of theD^b_j;

vanishes forD^b₀=div(^d f) = (div(f),(log|f|⁻¹_p )): metrized line bundles define global heights.

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Example: algebraic curves and potential theory

Xalgebraic curve over C. o∈X, za local parameter ato.

K compact subset ofXsuch thato6∈K. bo= (o, g_K).

Robin constants are local heights:

((ob−div(^d z))·ob|X)∞= (ob−div(^d z)|o)∞+ Z

X_∞

g_Kμ_K

=_qlim_→_o(bo−div(^d z)|q)∞

=_q→olimg_K(q) +log|z(q)|

=−log capz(K).

Similar computation at all places: the self-intersection (bo·ob) is the opposite of Rumely’s adelic logarithmic capacity.

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Bounding heights from below

Xprojective variety overQ, purely of dimensionm. Lline bundle onX plus metrics at all places.

Successive minima: e₀_¾· · ·¾e_m, with e_j(L) = sup

S,codim(S)>jinf

x6∈S(bc₁(L)|x).

Theorem (Zhang)

If L is ample and L is semipositive, then

e₀(L)¾

(bc₁(L)^m⁺¹|X) (m+1)(c₁(L)^m|X)^¾

1 m+1

e₀(L) +· · ·+em(L). The first inequality is a direct application of the analogue of Hilbert-Samuel theorem in Arakelov geometry.

The second inequality corresponds to the analogue of the Nakai-Moishezon criterion for ampleness.

Applications. Fekete–Szeg˝o p. 17

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Application to the theorem of Fekete–Szeg˝ o

The first minimume₀(L)is the threshold at which Zariski dense sets of points of small heights exist: the set of pointsx∈X(Q)such that(bc₁(L)|x)< α is:

not Zariski-dense ifα < e₀(L);

Zariski-dense ifα > e₀(L).

Hence, the inequalitye₀(L)¾. . . is a “negative

Fekete–Szeg˝o”: if(bc₁(L)^m⁺¹|X)>0, then only finitely many points have small height.

When moreovere_m(L)¾0, the second inequality implies a

“positive Fekete–Szeg˝o”: if(bc₁(L)^m⁺¹|X)¶0, then

e₀(L)¶0, so both vanish — there exist points of arbitrary small positive height.

Applications. Fekete–Szeg˝o p. 18

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Algebraic dynamics

LetF be a number field andφ∈F(T)be a rational function of degreed_¾2.

Dynamical height: defined by a metrized line bundleL_φ characterized byφ^∗L_φ 'L^d

φ.

(This metric is defined through a limit process, in the spirit of Tate’s construction of the Néron–Tate height, and

requires the formalism of admissible metrics.) Functional equation:

(bc₁(L_φ)|φ(x)) =d(bc₁(L_φ)|x); (bc₁(L_φ)²|P¹_F) =0. In that case,e₀(L_φ) =e₁(L_φ) =0.

Vanishing: (bc₁(L_φ)|x) =0⇔xpreperiodic

Applications. Algebraic dynamics p. 19

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Algebraic dynamics (followed)

Petsche/Szpiro/Tucker’sdynamical pairing:

〈φ, ψ〉= (bc₁(L_φ)bc₁(L_ψ)|P¹_F).

Controls the essential minimum of(bc₁(L_φ)|x) + (bc₁(L_ψ)|x).

Is positive unless strong coincidences:

(bc₁(L_φ)bc₁(L_ψ)|P¹_F) =−1 2

(bc₁(L_φ)−bc₁(L_ψ))²|P¹_F

=−1 2

X

v

Z

P¹

Fv

u_vdd^cu_v

=1 2

X

v

ku_vk²_Dir,

whereuv=log(kskφ,v/kskψ,v)is the difference of the metrics ofL_φ andL_ψ at placev, and k·k_Dir is the Dirichlet semi-norm.

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Algebraic dynamics (followed)

Dynamical pairing: 〈φ, ψ〉= ¹₂^Pvku_vk²_Dir.

Nonnegative; vanishes if and (almost) only ifu_v≡0 for allv.

This condition implies,e.g., that all Julia sets coincide, that φandψ have the same preperiodic points, etc. (PhD

Thesis ofArman Mimar, 1997)

Positivity is one aspect of the Hodge index theorem in Arakelov geometry, the other being the positivity of Néron–Tate height.

Other approach by Petsche/Szpiro/Tucker

Generalizations to any field by Baker/DeMarco, and in any dimension by Yuan/Zhang (2009).

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Equidistribution

Theorem (Szpiro/Ullmo/Zhang,..., Yuan)

Assume that(bc₁(L)^m⁺¹|X) =0 and that there exists a sequence(x_n)of points of X(Q)such that(bc₁(L)|x_n)→0. Then for any place p, the measures δ(xn)converge to the measure c₁(L)^m on X_p.

These measures are products of Chern forms ifp=∞, live on Berkovich spaces otherwise.

The proof relies on a variational argument: apply Zhang’s inequality to small perturbations ofL, at least ifLis ample.

For the general case, one needs to apply an estimate of arithmetic volumes due to Yuan (arithmetic analogue of a holomorphic Morse inequality proved by Demailly and Siu).

Applications. Equidistribution p. 22

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Theorem of Jentzsch–Szeg˝ o

Letf =^P^∞_j₌₀a_jz^j∈C[[z]] be a power series in one variable.

Radius of convergenceR∈(0,∞).

Truncationsf_n=^Pⁿ_j₌₀a_jz^j. Theorem (Jentzsch)

Any point of the circle C_R: {|z|=R}is a limit point of zeroes of truncations f_n.

Probability measureν_n= ¹_nf^∗

n δ₀given by the zeroes off_n. Theorem (Szeg˝o)

In a subsequence such that|a_n|¹^{/ n}→1/ R, ν_n converges to the invariant probability measure on the circle C_R.

Applications. Zeroes of polynomials p. 23

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A general theorem

LetMbe an analytic curve,o∈M, local parameterzat o. LetK⊂Mbe a compact subset such thato6∈K.

Let(f_n)be a sequence of regular functions on M\o, k_n=−ordo(f_n), order of the pole.

Leading coefficient: lc^z(f_n) =_plim_→_o|f_n|(p)|z(p)−z(o)|^kⁿ. Measures: νn= _k¹_nf_n^∗δ₀.

Theorem

Make the three assumptions:

lim supn 1

knlogkf_nkK¶0;

lim infⁿ_k¹_nlog lc^z(f_n)−log capz(K)¾0; for any compact subset E⊂K, ν˚ n(E)→0. Then, νn→μK.

Statement and proof inspired by Andrieveskii/Blatt’s treatment of the Jentzsch–Szeg˝o theorem.

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Irreducibility of truncations

LetF be ap-adic field

f =^Pja_jz^j∈F[[z]] a power series with radius of convergenceR∈(0,∞);

f_n the truncation off in degree n

Classical examples (Schur,...) overQwhere all f_nare irreducible, for examplef =e^z.

Corollary

Let d be a positive integer. In a sequence such that

|a_n|¹^{/ n}→1/ R, the number of irreducible factors of f_n whose degree is_¶d iso(n).

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Irreducibility of truncations — Sketch of proof

Theorem

Let d be a positive integer. In a sequence such that

|a_n|¹^{/ n}→1/ R, the number of irreducible factors of f_n with degree_¶d iso(n).

Sketch of proof:

The general Jentzsch–Szeg˝o applies and shows that the measuresν_nof zeroes of f_n converge to the equilibrium measure of the diskD_R.

That measure is the Dirac measure at the Berkovich point of the affine line corresponding to the Gauß norm of the diskD_R.

Let F_d be thefinite extension ofFgenerated by numbers of degree¶d.

If irreducible factors off_n with degree¶d weren’t negligible, the measures ν_nwould give some mass on the compact subset P¹(F_d)ofP¹.

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Borel, Dwork, Pólya, Bertrandias

LetF be a number field.

For any placevofF, let K_v be a bounded subset ofCv. Letf ∈F[[z⁻¹]].

Theorem Assume that

there exists a finite set S of places such that the coefficients of f belong too_F,S;

for any place v, f defines a meromorphic function on Cv\Kv;

If, moreover,−^P_vlog cap(K_v)>0, then f is the expansion of a rational function.

History: Borel, 1896 ; Pólya, 1925 ; Dwork, 1955 ; Bertrandias, 1960s ; Cantor, Rumely; Harbater, 1980s.

Applications. Rationality of formal functions p. 27

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Formal functions on an algebraic curve

LetMbe an algebraic curve over a number field F, o∈M(F),za local parameter at o.

For any placevofF, let K_v be a compact/affinoid subspace ofM(Cv)such thato6∈K_v, let U_v=M(Cv)\K_v.

Letf ∈O^bM,o be a formal function onMat o. Theorem (with J.-B. Bost)

Assume that

there exists a finite set S of places such that f ∈oF,S[[z]];

for any place v, f defines a meromorphic function on Ωv.

If, moreover,deg TÕ ^oM=−^P_vlog capz(K_v)>0, then f is the expansion of a rational function on M.

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Formal functions on an algebraic curve

Proof in two steps:

1 Algebraicity of f: techniques from Diophantine approximation.

2 Rationality of f: application of the Hodge index theorem in Arakelov geometry.

The theorem of Borel–Dwork–Pólya–Bertrandias then appears as an analogue of the following theorem in algebraic geometry:

Theorem (Hartshorne; Hironaka/Matsumura)

Let X be a smooth projective connected complex surface.

Let D be a divisor on X.

Let f be a (formal) meromorphic function on a neighborhood of D.

If the self-intersection(D, D) =degNDX >0, then f extends uniquely to a rational function on X.

Some applications of potential theory to number theoretical problems on analytic curves