Formal functions on arithmetic surfaces A generalization of the criterion of Borel-Dwork-Pólya-Bertrandias

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Formal functions on arithmetic surfaces

A generalization of the criterion of Borel-Dwork-Pólya-Bertrandias

Antoine Chambert-Loir

Institut de recherche mathématique de Rennes, Université de Rennes 1 Joint work with Jean-Benoît Bost

Département de mathématiques, Université de Paris-Sud Orsay

Workshop on Arithmetic Geometry :

Diophantine approximation and Arakelov theory Toronto, October 20–24, 2008

. p. 1

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Rationality of formal power series

Let a formal power seriesf =

∞

P

n=0

anxⁿbe given.

How to decide if it is the Taylor expansion of a rational function ?

We search for a criterion of an arithmetic nature :

1 we assume that the coefficientsanlie in a number field ;

2 we impose arithmetic conditions on the coefficients (e.g., that they are algebraic integers) ;

3 geometric conditions on the analytic behaviour (radius of convergence, of meromorphy), possibly at all places of the number field.

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The theorem of Borel-Dwork-Pólya-Bertrandias

Theorem

Let F be a number field, let f ∈F[[x⁻¹]].

Assumptions :

1 the coefficients of f are S-integral for a finite set S of finite primes in F ;

2 for any place v ∈S, f defines a meromorphic function on the complement of a bounded subset Kv ofCv.

3 the product of thetransfinite diametersof the K_v is smaller than1.

Then f is a rational function.

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Transfinite diameter

Thetransfinite diameterof a bounded metric space(X,d)is the limit tdiam(X)of the decreasing sequence(d_n)defined by

d_n =



 sup

(x₁,...,xn)∈Xⁿ

Y

i6=j

d(x_i,x_j)





1/n(n−1)

.

Examples:

diskin a valued field : tdiam=radius ;

lemniscate: {|P(z)|61}in a valued field,P =a_dz^d+. . . : tdiam=|a_d|⁻¹^/d;

segment in C: tdiam=length/4;

compact subset in C: tdiam=capacity with respect to the point at infinity.

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The theorem of BDPB

History and applications

1 É. Borel, 1894,Sur une application d’un théorème de M. Hadamard. f ∈Z[[1/x]],K_v =D(0,r),r <1.

2 G. Pólya, 1928,Über notwendige Determinantenkriterien für die Fortsetzbarkeit einer Potenzreihe.

3 B. Dwork, 1960,On the rationality of the zeta function of an algebraic variety.

⇒IfV is an algebraic variety over a finite fieldFq, then its zeta functionζV(t) =exp



 X

n>1

#V(F_qⁿ)tⁿ n



is a rational function.

4 F. Bertrandias, 1963, see Y. Amice’s bookLes nombres p-adiques.

5 D. Cantor, 1980, R. Rumely, 1989.

6 J-P. Bézivin, P. Robba, 1989 : New proof of the theorem of Lindemann-Weierstrass.

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The theorem of BDPW

Remarks

1 Iff is holomorphic, this is easy. For example consider f ∈Z[[x]], holomorphic onD(0,r), withr >1.

Writef =^Pa_nxⁿ.

By Cauchy inequalities,|a_n|6M/rⁿgoes to 0.

Sincea_n∈Z,a_n =0 forn0 : f is a polynomial.

2 In some cases, one can prove thatf is rational when Qtdiam(K_v) =1, if one assumes moreover thatf is algebraic.

D. Harbater : Kv =D(0,rv).

3 However : f =^p1−4/x ∈Z[[x]]. Two ramification points, in 0 and 4, one can extendsf to a holomorphic function outside the intervalK = [0,4]. One has tdiam(K) =1 butf is not rational.

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The theorem of BDPB

Proofs

1 Original proof : establish the vanishing of appropriate Hankel/Kronecker determinants.

Using analytical properties off, show that they satisfy Q

v|D|_v <1.

By the product formula,D=0.

2 In his bookG-functions and geometry (1989), Y. André gives a similar criterion for algebraicity/rationality, based on classicaldiophantine approximationtechniques.

3 Small variation : In myBourbaki Seminar (2001) about the work of Chudnovsky, André and Bost, proof of those criteria using the formalism of Arakelov geometry and Bost’sslope inequality.

“Even if all rationality proofs rely on proving the vanishing of the K_N(a)[Kronecker determinants] , the form of the criteria may well disguise that fact.” A. van der Poorten

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Our main theorem

Consider an algebraic curveC over a number fieldF, a point o∈C(F)and a formal functionf ∈O[_C,oat the pointo. Assume that at all placesv ofF,f is induced by an

meromorphic function on av-adic open subsetΩv ofC⊗F_v. Conclusion :if theglobal capacityof the family(Ωv)is at most 1, thenf comes from a meromorphic function onC. The measure of the size of theΩv relies oncapacity theoryfor algebraic curves — generalization of the product of the transfinite diameters.

Proof in two steps :

1 f is algebraic. This part uses diophantine approximation (in the dialect of Arakelov geometry and slopes).

2 f is rational. This part uses the Hodge index theorem in Arakelov geometry.

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The complex case

LetM be a Riemann surface, compact, connected K ⊂M a compact subset,Ω ={K,P∈Ω.

There is a uniquepotentialg_K: Ω→Rsuch that

1 g_K is harmonic onΩ\ {P};

2 for nearly everyx ∈∂K, limz→xg_K(z) =0;

3 ift is a local coordinate aroundP, then g_K =−log(c_K|t−t(P)|) +o(1).

The real numberc_K is thecapacityofK with respect toP; it depends on the choice oft.

On the complex lineT_PM, the hermitian norm defined by

∂

∂t

=c_K

does not depend on any choice :capacitary normonT_PM.

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The p-adic case (after Rumely)

LetM be a projective smooth connected curve over a p-adic fieldF,K ⊂Man affinoid,Ω ={K,P ∈Ω.

There is a fullpotential theoryonM (R. Rumely, A. Thuillier).

Easier definition(inspired by Rumely’s theory and a theorem of Fresnel/Matignon) relying on the following Theorem : Theorem

IfΩis connected, there is a rational function f ∈F(M)regular outside P such that K ={x,|f(x)|61}.

Letd be the order off at the pointP;

theng_K := _d¹max(log|f|,0)is the analogue of thepotential.

Iftis a local coordinate aroundP, define thecapacityc_K such that

g_K =−log(c_K|t−t(P)|) +o(1) and thecapacitary normonT_PMby

∂

∂t

=c_K.

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Global capacity

LetM be an algebraic curve over a number fieldF,P∈M(F). For all placesv choose a compact/affinoid subsetK_v of the analytic curve overFv such thatP6∈Kv.

Assume that there is a rational function onM definingK_v for almost all finite placesv.

Definition

The global capacity is the real number defined by

log cap((K_v);P) = 1 [F :Q]

X

v

log

∂

∂t _v

=−deg[(T_PM,k·k^cap).

(The product formula implies that it does not depend on a chosen local parametert∈ O_M,P onM.)

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Statement of the main rationality theorem

KeepM,P,(K_v)as above.

Theorem

Let f ∈O[_M,P be aformal functionaround P.

Assume

1 “the denominators of the coefficients of the Taylor expansion of f are divisible by only finitely many prime numbers”;

2 for any place v, f defines a v-adic meromorphic function outside Kv;

3 cap((K_v);P)<1.

Then f comes from a rational function on M.

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Examples

The simplest example is that of theprojective line.Our theorem generalizes the classical results of

Borel-Dwork-Pólya-Bertrandias.

Borel-Dwork: F =Q,M =P¹,P=0,t=usual coordinate.

TakeKv =D(0,Rv), withRv =1 for almost allv. Then c_K_v =1/R_v and cap((K_v);P) = (^QR_v)⁻¹.

Pólya-Bertrandias:M =P¹(say overQ),P =∞, t =1/usual coordinate.

LetK_v be an compact/affinoid subset ofP¹(Cv), thenc_K_v is its transfinite diameter.

The theorem extends readily to more general subsets than affinoid provided that the product of their “outer capacities”

is less than 1.

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Diophantine approximation vs. Slope inequalities

We prove thatf is algebraic usingdiophantine

approximation. In principle, we find a “small” polynomialPin two variables such thatP(x,f(x))vanishes with so high order atPthat it must vanish everywhere.

We use however the method ofslopes inequalitiesin Arakelov geometry due to J.-B. Bost.

The general idea consists in proving thealgebraicity of a suitable formal subscheme:

in (Bost, 2001) : formal leaf of a foliation

here: formal graph off inX =×P¹at the point(P,f(P)).

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Slope inequality

We need toalgebraize the formal graph( ˆV,o)off inX =M×P¹ato= (P,f(P)).

LetLbe an ample line bundle onX with an adelic metric.

Spaces of sectionsE_D = Γ(X,L^⊗D)with L²/sup norms.

Algebraicity ofVˆ means non-injectivity of the restriction map E_D →Γ( ˆV,L^D).Assume injectivity.

The filtration ofΓ( ˆV,L^D)byΓ( ˆV,mⁱ_oL^D)induces a filtration ofE_D by subspacesE_Dⁱ .

Theith jet is a morphism

ϕⁱ_D:E_Dⁱ →Γ(o,mⁱ_oL^D) =Sⁱ(Ω¹_oVˆ)⊗L^⊗D|_o with kernelE_Dⁱ⁺¹.

Slope inequality : deg[E_D6^X

i

rkϕⁱ_Ddeg[(SⁱT_P^∗M) +Ddeg[(L|o) +h(ϕⁱ_D)

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Estimates

Slope inequality deg[E_D6^X

i

rkϕⁱ_Ddeg[(SⁱT_P^∗M) +Ddeg[(L|_o) +h(ϕⁱ_D) The contradiction is obtained by inserting in this inequality generalestimates in Arakelov Geometry :

(rough)arithmetic Hilbert-Samuel formula: deg[E_D >−c₁D³ arithmetic degrees of symmetric powers:

deg[(SⁱT_P^∗M) =ideg[T_P^∗M

height of evaluation morphisms(Schwarz lemma) : h(ϕⁱ_D) +ideg[T_P^∗M 6ilog cap((K_v);P)

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The Schwarz lemma

The height of the evaluation morphisms are estimated using theSchwarz lemma,a central analytic tool in classical diophantine approximation.

Here, it goes as follows :AssumeK_v ={x,|f(x)|_v 61},f having a pole of orderd atP,Ωv ={K_v,ta local parameter atP. Letϕbe holomorphic and bounded onΩ, vanishing at orderi atP; writeϕ=a_itⁱ +. . . aroundP.

The functionθ(x) =f(x)ⁱϕ(x)^d is regular onΩv\ {P}, atP, and bounded bykϕk^d_Ωat the “boundary” ofΩ.

By themaximum principle:

|θ(P)|_v =c_K^−di|a_i|^d 6kϕk^d_Ω, hence |a_i|_v 6cⁱ_Kkϕk_Ω.

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A general algebraization criterion

General result of algebraization of a formal smooth

curve( ˆV,o)in a quasi-projective variety over a number field.

Theorem

IfV is A-analytic andˆ deg[(T_oVˆ,(k·k^can_v ))>0, thenV isˆ algebraic.

Key concepts :

1 canonicalv-adic semi-normsonT_oVˆ defined using the norms of the mapsϕⁱ_D. Namely, foru∈T_oVˆ, one sets

kuk^can_v =lim sup

i D→∞

sup

s∈E_Dⁱ ksk_v61

hϕⁱ_D(s),u^⊗ii

| {z }

∈L^⊗D_o

¹

/i

2 generalized Arakelov degreedeg[(T_oVˆ,(k·k^can_v ))

3 global notion ofA-analyticitybased on thev-adicsizes ofVˆ

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The picture

We know thatf is algebraic.

LetM⁰ be the normalization the Zariski closure of its graph inM×P¹. It is an algebraic curve with a morphismπ:M⁰ →M andsections defined onΩv.

Under the assumption that the global

capacity cap((K_v);P)<1, we need to prove thatπ is an isomorphism.

The pointPcorresponds to a line bundleO(P), which one endows with the metrics defined by the potentialsg_K_v —

“v-adic Green functions”.

Its first arithmetic Chern classc^c₁(O(P),k·k^cap)can be defined as an object ofCH^d¹(M), Bost’s Arakelov-Chow group with L²

1-regularity on an adequate modelMofM.

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Positivity of an Arakelov divisor

Lemma

The Arakelov divisor classcc₁(O(P),k·k^cap)is nef and big.

By the very construction of potential Green functions, this class intersects nonnegatively :

vertical divisors inM(at finite or infinite places);

horizontal effective divisors inMwhose trace onMis not equal toP.

Consequently,this Arakelov divisor class is nef if and only if its self intersection is nonnegative.

In fact, under the assumptions of the theorem, one has the equality

cc₁(O(P),k·k^cap)²=[deg(T_PM,k·k^cap) =−log cap((Kv);P)>0, so thatthis class is big.

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Construction of positive classes on M

⁰

LetP^b =cc₁(O(P),k·k^cap).

We decomposeπ^∗O(P)asO(E₁)⊗ O(E₂),

whereE₁=f(P)andE₂is an effective divisor onM⁰disjoint fromE₂.

Similarly, we decomposeπ⁻¹(Ωv) = Ω₁_,vtΩ₂_,v where Ω_1,v =fv(Ωv).

This induces a decomposition

π^∗P^b =^cE₁+^cE₂,

in the Arakelov-Chow groupCH^d₁(M⁰)of an adequate model ofM⁰.

This decomposition is orthogonal reflecting the fact that the potential Green functions ofE₁andE₂have disjoint supports.

Moreover,π^∗P^b is big and nef.

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Application of the Hodge index theorem

Recall :

π^∗P^b =E^c₁+E^c₂is big and nef cb₁(^cE₁)cb₁(E^c₂) =0.

SinceP^b is nef andE^c₁is effective,

06c^b₁(P)^b ·π∗cb₁(^cE₁) =π^∗cb₁(P)^b ·cb₁(^cE₁) =cb₁(E^c₁)². Similarlyc^b₁(^cE₂)²>0.

Consequently, the quadratic form

q(x,y) = (xcb₁(^cE₁) +ycb₁(E^c₂))²=x²cb₁(E^c₁)²+y²cb₁(^cE₂)²

is nonnegative and theHodge index theoremin Arakelov geometry implies thatc^b₁(E^c₁)andc^b₁(E^c₂)are proportional.

Since they are orthogonal and their sum is big, one of them is zero, hence^cE₂=0.

This impliesπ^∗P^b =E^c₁, hence deg(π) =1.

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Modular forms and overcovergent modular forms

Amodular formof levelN, weightk is a rule (E/S, α_N),→f(E/S, αN)∈ω_E^⊗k_/S

whereS is a scheme,E/Sis an elliptic curve,αN a levelN structure, which commutes with base change.

Anoverconvergent modular formof growth conditionris a similar rule which is only defined on thep-adic subset of the modular curve obtained by removing supersingular disks of radius|r|around singular moduli.

Using the fact that the Eisenstein seriesE_p−₁lifts the Hasse invariant ifp>3, N. Katz defines them inAnvers IIIas a rule

(E/S, α_N,Y),→f(E/S, αN,Y)∈ω_E^⊗k_/S

whereE/Sis an elliptic curve,αN a levelN structure, and Y ∈ω¹_E/S^−psatisfiesYE_p−₁(E/S) =r, which commutes with base change.

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Application to modular forms

Theorem

Let F be a number field,o_F its ring of integers.

Let f ∈o_F[[q]]and assume :

for any embeddingσ:F ,→C, the radius of convergence of f^σ ∈C[[q]]is at least1;

at some finite place v, f extends to anoverconvergent modular form of level N, weight k.

Then f is algebraic.

If, moreover, the growth condition r of f satisfies

|r|_p <e⁻²^π[F:Q]/N, then f is a true modular form.

The proof is an application of our general criteria, together with two computations :

injectivity radius of the uniformization map (mod.

parabolic elements) aroung the cusp at infinity;

capacity of the supersingular disks.

Formal functions on arithmetic surfaces A generalization of the criterion of Borel-Dwork-Pólya-Bertrandias