Asymptotic methods in number theory and algebraic geometry

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by

PhilippeLebaque &Alexey Zykin

Abstrat. Thepaperisasurveyofreentdevelopmentsintheasymptotitheoryofglobal

eldsandvarietiesoverthem. First,wegiveadetailedmotivatedintrodutiontotheasymptoti

theory ofglobal eldswhihis alreadywellshaped as asubjet. Seond, wetreat ina more

skethyway thehigherdimensional theorywhere muhless isknownandmany newresearh

diretions areavailable.

Résumé. Cetartileestunsurvoldesdéveloppementsréentsdanslathéorieasymptotique

des orps globauxet desvariétés algébriques déniessurles orpsglobaux. Dansun premier

temps,nousdonnonsuneintrodution détailléeetmotivéeàlathéorieasymptotiquedesorps

globaux,théoriedéjàbienétablie.Puisnousaborderonsplusrapidementlathéorieasymptotique

en dimensionsupérieureoùpeudehosessont onnuesetoùbiendes diretionsdereherhe

sontouvertes.

1. Introdution: the origin of the asymptoti theory of global elds

The goal of this artile is to give a survey of asymptoti methods in number theory and

algebrai geometry developed in the last deades. The problems that are treated by the

asymptoti theoryof globalelds (that isnumber eldsor funtion elds) and varieties over

them arequite diverse innature. However, they are onneted by theuse of zeta funtions,

whih playthe keyrole inthe asymptotitheory.

We begin by a very well known problem whih lies at the origin of the asymptoti theory

of global elds. Let

F ^r

^be ^the ^nite ^eld ^with

r

êlements. ^For â ^smooth ^pro^jetive ûrve

C

over

F ^r

^we^let

N r (C)

^be ^the^number^of

F ^r

^-point^on

C.

^W^e ^denote^by

g(C)

^be^the^genus^of

C.

2000 MathematisSubjetClassiation. 11R42,11R29,11G40.

Key wordsand phrases. Towersofglobalelds,

L

^-funtionsⁱⁿ^family,BrauerSiegeltheorem.

TherstauthorwaspartiallysupportedbyEPSRCgrantEP/E049109Twodimensionaladelianalysis". The

seondauthorwaspartiallysupportedbyDmitryZimin'sDynasty"foundation,byAGLaboratorySU-HSE,

RFgovernment grant, ag. 11.G34.31.0023, bythe grantsRFBR 10-01-93110-CNRSa,RFBR11-01-00393-a,

bythegrantoftheMinistry ofEduationandSieneofRussiaNo. 2010-1.3.1-111-017-029. Theindividual

researhprojetN10-01-0054 Asymptotipropertiesof zetafuntions"wasarried outwiththe supportof

theprogramSientifundofHSE".

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Theproblem onsists ofnding the maximum

N _r (g)

^of ^the ^numbers

N _r (C)

^over^all ^smooth

projetive urvesofgenus

g

^over

F r :

N _r (g) = max

g(C)=g N _r (C).

Therst upper bound was disovered by AndréWeil in1940sas adiret onsequene of his

proof of theRiemannhypothesis forurvesovernite elds. He showed that

N _r (C)

^satises

theinequality

N r (C) ≤ r + 1 + 2g √ r.

Weil bound though extremely useful in many appliations is far from being optimal. A

dramatisearhfortheimprovementsofthisboundandfortheexamplesgivinglowerbounds

on

N _r (g)

^has^begun ⁱⁿ^1980s ^with ^the ^disovery ôf ^Goppa ^that ûrvesôver ^nite êlds ^with

many points an be used to onstrut good error-orreting odes. To showhow important

thedevelopments inthis area wereit suesto mention the names ofsome mathematiians

whoturnedtheir attentionto thesequestions: J.-P.Serre,V.Drinfeld, Y.Ihara,H.Stark,R.

Shoof,M. Tsfasman, S.Vl duµ,G. van derGeer, K.Lauter, H.Stihtenoth, A.Garia, et.

As suggested in [Ser85℄ by J.-P. Serre the ases when

g

^is ^small ^and ^that ^when

g

^is ^large

requireompletelydierenttreatment. Thatisthelatterasewhihinterestsusinthisartile.

TherstmajorresultinthisdiretionwasthefollowingtheoremofV.DrinfeldandS.Vl duµ

[DV℄:

Theorem 1.1 (DrinfeldVl duµ). Foranyfamilyofsmoothprojetiveurves

{ C _i }

^over

F ^r

^of ^growing^genus ^we ^have

lim sup

i→∞

N _r (C _i ) g(C _i ) ≤ √

r − 1.

Moreover, in thease, when

r

îs â ^square ^this ^bound ^turns ôut ^to ^be ôptimal. ^The ^families

of urves, attaining this bound are onstruted in many dierent ways: modular urves,

Drinfeldmodularurves,expliititeratedonstrutions, et. Werefer thereader tosetion4

for more details. Thisresult, signiantly improved and then reinterpreted in termsof limit

zetafuntionsbyM. Tsfasmanand S.Vl duµ,liesat theverybaseof theasymptoti theory

of global elds. We will disuss all this in detail in setion 2. It is also possible to extend

theDrinfeldVl duµ inequalitiesto the ase of higherdimensional varieties. Thisservesasa

keystoneintheonstrution of the higherdimensional asymptoti theory(seesetion 5).

Wewillnowturnourattentiontoyetanothersoureofdevelopmentoftheasymptotitheory,

this time in the ase of number elds. Let

K

^be ân âlgebrai ^number êld, ^that îs â ^nite

extension of

Q .

^We ^denote ^by

n _K = [K : Q ]

^its ^degree, ^and ^by

D _K

^its disriminant. An important question (both on its own aount and due to itsappliations in various domains

ofnumbertheory,arithmeti geometry and theoryofsphere pakings)is toknowtherateof

growsofdisriminantsofnumberelds. Therstboundon

D K

^was^obtained^by^H.^Minkowsky

usingthegeometry ofnumbers. Thisboundwasimproved morethan halfa enturylater by

H.Stark,J.-P.SerreandA.Odlyzko([Sta74℄,[Ser75℄,[Odl76℄,[Odl90℄)whousedanalyti

methods involving zetafuntions. Theboundstheyprove areasfollows:

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Theorem 1.2 (Odlyzko). For a familyof number elds

{ K _i }

^we ^have

log | D _K _i | ≥ A · r ₁ (K _i ) + 2B · r ₂ (K _i ) + o(n _K _i ),

where

r ₁ (K _i )

^and

r ₂ (K _i )

^are respetivelythenumber ofrealandomplexplaes of

K _i .

^Unon-

ditionally, we an take

A = log(4π) + γ + 1 ≈ 60.8, B = log(4π) + γ ≈ 22.3

^, ^and, ^assuming

thegeneralized RiemannHypothesis(GRH),onean take,

A = log(8π) + γ + ^π ₂ ≈ 215.3, B = log(8π) + γ ≈ 44.7,

^where

γ = 0.577

îsÊuler's ^gammaônstant.

The fatthat GRHdrastially improvesthe results is omnipresent intheasymptoti theory

ofglobalelds. Fortunately,GRHisknownfor zetafuntionsofurvesoverniteelds(Weil

bounds) and,more generally, of varieties overnite elds (Deligne's theorem), whih allows

to have both strongerresults andsimpler proofsin thease of positive harateristi.

M.TsfasmanandS.Vl duµmanagedto generalizethe above inequalitiestakingintoaount

the ontribution of nite plaes of the elds. In fat, the restrition of the so-alled basi

inequality proven by M. Tsfasmanand S.Vl duµ to innite primes gives us the inequalities

ofOdlyzkoSerre. Ifwe restritthebasi inequalityto nite plaesweobtain ananalogue of

thegeneralized DrinfeldVl duµ inequalityinthease of number elds. The reader will nd

more information onthis inthe next setionof the paper.

Thelast,butnotleast,problemthatledtothedevelopmentoftheasymptotitheoryofglobal

eldsandvarietiesoverthemwastheBrauerSiegeltheorem. Let

h K

^denote^the^lass^number

ofanumbereld

K

^and^let

R _K

^be^its^regulator. ^The^lassialBrauerSiegeltheorem, proven by Siegel ([Sie℄) in the ase of quadrati elds and by Brauer ([Bra℄) in general desribes

thebehaviouroftheprodut

h K R K

ⁱⁿ^familiesôf^numberêlds. ^Theînitial^motivation ^forît

wasa onjetureofGauss onimaginary quadratields, however ithasgot many important

appliations elsewhere. The theoreman be stated asfollows:

Theorem 1.3 (BrauerSiegel). Fora familyof number elds

{ K _i }

^we ^have

i lim →∞

log(h _K _i R _K _i ) log p

| D _K _i | = 1

provided the familysatises two onditions:

(i)

lim

i→∞

n _K _i g _K _i = 0;

(ii) either GRH holds,or all the elds

K _i

^are ^normal ^over

Q .

Itispossibletoremovetherstandrelaxtheseondonditionsofthetheorem. Therststep

towards it was made by Y. Ihara in [Iha83℄ who onsidered families of unramied number

elds. A omplete answer(at leastmodulo GRH) was given byM. Tsfasman andS. Vl duµ

in[TV02℄whoshowed howto treatthis probleminthe framework oftheasymptotitheory

of number elds, in partiular using theonept of limit zeta funtions. The orresponding

question for urves overnite eldsis also ofgreat interest sine itdesribes theasymptoti

behaviour of the number of rational points on Jaobians of urvesovernite elds. All this

will be disussed indetailinthesetion3.

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Inourintrodutionwemostlyonsideredtheonedimensionalaseofnumbereldsorfuntion

elds. Here the theory is best developed. However, there is quite a number of results and

onjetures for higher dimensional varieties with partiularly nie arithmetial appliations.

Someoftheresults inthis atively developing areaaredisussed insetion5.

Letus nally saythat, despite of the fat thatthe theory of error orreting odes and the

theory of sphere pakings are just briey mentioned in our introdution their role in the

reation of the asymptoti theory of global elds is fundamental. Indeed many questions

some of whih were mentioned here (maximal number of points on urves, growth of the

disriminants, et.) reeived partiular attention due to their relation to error-orreting

odesor sphere pakings.

2. Basi onepts and results. TsfasmanVl duµ invariants of innite global

elds

Manyauthors onsidered the behaviour of arithmeti data(deomposition of primes, genus,

root disriminant, lass number, regulator et.) in families of global elds. Tsfasman and

Vl duµlaid thefoundationfor theasymptoti theoryofglobal elds inorder not to onsider

elds in a family, but the limit objet (say, a limit zeta-funtion) that would enode the

information onerningthe asymptotisof theinitial arithmetidata.

In this setion we introdue some denitions and give basi properties of families of global

elds.

2.1. TsfasmanVl duµ invariants. Arguments and proofs for the results from this

subsetion an be found in[TV02 ℄. Let us rst dene theobjets we are to work with. Let

r

^be â^powerôfâ ^prime

p,

^and ^let

F r

^denote^the ^algebrai ^losure^of

F r .

Denition 2.1. Afamily ofglobal elds isa sequene

K = { K _n } n ∈N

^suh^that:

1. Either all the

K n

âre ^nite êxtensions ôf

Q

^or ^all ^the

K n

âre ^nite êxtensions ôf

F ^r (t)

with

F r ∩ K _n = F r .

2. if

i 6 = j, K _i

^is^not ^isomorphi ^to

K _j .

A tower of global elds is a family satisfying in addition

K _n ⊂ K _n+1

^for ^every

n ∈ N .

^An

innite global (resp. number, resp. funtion) eld is the limit of a tower of global (resp.

number, resp. funtion) elds,i.e. itis the union

S ∞ n=1

K _n

^.

Denition 2.2. Thegenus

g _K

ôfâ^funtionêldîs^the^genusôf^theorrespondingsmooth projetive urve. We dene the genus ofa numbereld

K

^as

g _K = log p

| D _K | ,

^where

D _K

^is

thedisriminant of

K.

Asthere are(upto an isomorphism)onlynitely manyglobal eldswithgenus smallerthan

axedreal number

g,

^we^have ^the^following proposition.

Proposition 2.3. For any family

{ K _i }

^of ^global ^elds^the ^genus

g _K _i → + ∞ .

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Thus, in the number elds ase, any innite algebrai extension of

Q

îs ân înnite ^number

eld, whereas in the funtion elds ase, we require theinnite algebrai extension of

F r (t)

to ontaina sequeneof funtionelds withgenus going toinnity.

Let us now dene the so-alled TsfasmanVl duµ invariants of a family of global elds.

Throughoutthe paper,weusethe aronymsNFandFFforthenumbereldandthefuntion

eldasesrespetively. Asbefore,theGRHindiation meansthatweassumethegeneralized

Riemann Hypothesis for Dedekindzeta-funtions.

First we introdue some notation to be usedthroughout thepaper:

Q

^the ^eld

Q

^(NF),

F ^r (t)

^(FF);

n _K [K : Q ];

D _K

disriminant of

K

^(NF);

g _K

^the ^genus ^of

K (F F ),

^the^genus ^of

K

^equal ^to

log p

| D _K | (N F )

^;

Pl _f (K)

^the ^set^of^nite ^plaes ^of

K

^;

Np

^the ^norm^of ^a^plae

p ∈ Pl _f (K)

^;

deg p log _r Np (F F )

^;

Φ _q (K)

^the ^number^of^plaes ^of

K

^of ^norm

q

^;

Φ _R (K )

^the ^number^of^real ^plaes ^of

K

^(NF);

Φ _C (K )

^the ^numberôfômplex ^plaes ôf

K

^(NF).

We onsiderthe set ofpossible indiesfor the

Φ _q , A =

( R , C , p ^k | p

^prime,

k ∈ Z >0 (N F )

r ^k | k ∈ Z ^>0 (F F ) ,

and

A _f

^its ^subset ^of ^nite^parameters

p ^k | p

^prime,

k ∈ Z >0 .

Denition 2.4. Wesaythatafamily

K = { K _i }

ôf^globalêldsîsasymptotiallyexat ifthefollowing limitexistsfor any

q ∈ A :

φ _q := lim

i → + ∞

Φ _q (K _i ) g _K _i .

Itissaidtobeasymptotiallygoodifinaddition oneofthe

φ _q

^is^nonzero,^andasymptotially bad otherwise. The numbers

φ _q

^are ^alled^the^TsfasmanVl duµ invariantsof thefamily

K .

This denition has two origins. The rst one is the information theory sine the families

giving good algebrai geometri odes are those for whih

φ _r

êxists ând îs ^big. ^The ^seond

one is more tehnial and an be seen through Weil's expliit formulae. For onveniene we

also put

φ _∞ = lim n _K _i

g _K _i = φ _R + 2φ _C .

Being asymptotiallyexat is not arestritive ondition. To be preise:

Proposition 2.5. 1. Any family of global elds ontains an asymptotially exat sub-

family.

2. Any towerof global eldsis asymptotially exat and the

φ _q

^'s ^depend ^only^on ^the ^limit.

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WeanthusdenetheTsfasmanVl duµinvariantsofaninniteglobal elds

K

^as^the^invari-

antsofanytowerhavinglimit

K .

^F^rom^nowôn,^weônlyônsiderasymptotiallyexatfamilies, sine theyprovide natural framework for asymptoti onsiderations. One ofthe problemsof

theasymptoti theoryis to understand theset of possible

{ φ _q } .

^In ^the^next propositions we desribesome thegeneral properties ofthe

{ φ _q } .

^Let^us ^start ^with^the^basi inequalities:

Theorem 2.6 (TsfasmanVl duµ). Foranyasymptotiallyexatfamilyofglobal elds,

the followinginequalities hold:

(N F − GRH) X

q

φ _q log q

√ q − 1 + (log √ 8π + π

4 + γ

2 )φ _R + (log 8π + γ)φ _C ≤ 1, (N F ) X

q

φ _q log q

q − 1 + (log 2 √ π + γ

2 )φ _R + (log 2π + γ)φ _C ≤ 1, (F F )

X ∞ m=1

mφ _r ^m r ^m ² − 1 ≤ 1,

where

γ

îs^the Êuler ônstant.

This result is entral in what follows. For instane, it is used to show the onvergene of

thelimit zeta-funtionassoiated tothe family. Itisproven usingtheWeilexpliit formulae,

theeetive Chebotarev density theoremfor number elds and the Riemann hypothesis for

funtionelds.

Intheaseoftowersofnumberelds(andoffuntioneldsifweonsidersuitablequantities),

the degree of the extension gives an upper bound for the number of plaes above a prime

number

p

^:

Proposition 2.7. Foranasymptotiallyexatfamilyofnumbereldsandanyprimenum-

ber

p

^the ^fol^lowing^inequality ^holds:

+ ∞

X

m=1

mφ _p ^m ≤ φ _R + 2φ _C .

Letus nally dene the deieny

δ _K

^of^an asymptotially exat family

K = { K _i }

^of ^global

eldsasthe dierene between thetwo sidesofthebasi inequalitiesunderGRH:

(N F ) δ _K = 1 − X

q

φ _q log q

√ q − 1 − (log √ 8π + π

4 + γ

2 )φ _R − (log 8π + γ )φ _C

and

(F F ) δ _K = 1 − X ∞ m=1

mφ _r ^m r ^m ² − 1 .

A remarkable fat isthat the deieny of innite global elds is inreasing with respet to

theinlusion (see[Leb10℄):

K ⊂ L

^implies

δ _K ≤ δ _L

^. Ône^knows^thatêlds ôf^zero^deieny

existinthe funtion elds ase (.f. setion4). Suh innite globalelds are alledoptimal,

andthey areofpartiular interest for theinformation theory.

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2.2. Ramiation, prime deomposition and invariants. The preise statements

and proofs of the results from this subsetion an be found in [GSR℄ and [Leb10℄. The

TsfasmanVl duµ invariants of innite global elds ontain information on the ramiation

andthedeompositionofplaes intheseelds. Indeed,one sees fromHurwitzgenus formula

thatanynitely ramied and tamely ramiedtower ofnumber eldsis asymptotially good

(beause it has bounded root disriminant). For funtion elds, we have to ask in addition

for theexistene ofa splitplae. Itis not exludedthat thereexistsan asymptotially good

innite global eld with innitely many ramiedplaes and no splitplae, but no examples

havebeen foundsofar. Inthease offuntion elds,A.Gariaand H.Stihtenoth provided

a widely ramied optimal tower and an everywhere ramied tower of funtion elds with

bounded

g/n

^is ^onstruted ⁱⁿ ^{[DPZ ℄.} Unfortunately, we do not know anythingsimilar for numberelds.

In general, we expet asymptotially good towers to have very little ramiation and some

splitplaes. Thenext question,rst raisedbyY.Ihara, ishow manyplaes split ompletely

ina tower

K

ôf ^global êld. Ît ^follows ^from ^the^Chebotarev ^density ^theorem ^that^the ^set ôf

ompletely splitplaes hasingeneral a zeroanalyti density,thatis

s→1 lim ⁺ P

p ∈ D Np ⁻ ^s P

p ∈ Pl f ( Q ) Np ⁻ ^s = 0,

where

D

îs^the^setôf ^plaes ôf

Q

^that^split^ompletelyⁱⁿ

K / Q .

În^theâse ôfasymptotially goodelds,

X

p ∈ D

Np ⁻ ¹

îsêven ^bounded. ^However,ⁱⁿ^theâseôfasymptotiallybad elds,the numeratoranhaveaninnitelimitwhereastheramiationlousisverysmall(butinnite).

We refer thereaderto [Leb10℄for a more detailedtreatment oftheabove questions.

3. Generalized BrauerSiegel theorem and limit zeta-funtions

3.1. Generalizations of the BrauerSiegel theorem. Nowweturnour attention to

theBrauerSiegel theorem. Thein-depth study of mathematial tools involved init leads to

animportant notionof limitzetafuntionswhihplays akeyroleinthestudyof asymptoti

problems.

Whilelooking at the statement oftheBrauerSiegel theorem(theorem 1.3)one immediately

asks a question whether the two onditions present in it are indeed neessary. It is a right

guessthattheseondonditioninvolving normalityistehnial inits nature(thoughgetting

rid of it would be a breakthrough in the analyti number theory sine it is related to the

so-alledSiegelzeroesofzeta-funtionstherealzeroeswhihlieabnormallylose to

s = 1;

of ourse, presumably they do not exist). The seond ondition

n K / log p

| D K | → 0

^looks

muhtrikier. Usingtheinequalities fromproposition2.7it isimmediatethatthis ondition

isequivalent to the fatthatthe family we onsiderisasymptotially bad.

AfundamentaltheoremofM. TsfasmanandS.Vl duµfrom[TV02 ℄allowsbothto treatthe

asymptotially good ase of the BrauerSiegel theorem and to relax the seond ondition.

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We formulate it together with a omplementary result by A. Zykin [Zyk05℄ whih relaxes

the seond ondition in the asymptotially bad ase. Before stating the result we give the

following denition:

Denition 3.1. We say that a number eld

K

îs âlmost ^normal îf ^there êxists â ^tower

K = K _n ⊃ · · · ⊃ K ₁ ⊃ K ₀ = Q ,

^where ^eah^step

K _i /K _i ₋ ₁

^is^normal.

Theorem 3.2 (TsfasmanVl duµZykin). Assume that for an asymptotially exat

familyof number elds

{ K _i }

êither ^GRH ^holdsôr âl^l ^the êlds

K _i

^are ^almost ^normal. ^Then

wehave:

i lim →∞

log(h _K _i R _K _i )

g _K _i = 1 + X

q

φ _q log q

q − 1 − φ _R log 2 − φ _C log 2π,

the sumbeing taken over all prime powers

q

^.

Foranasymptotiallybadfamilyofnumbereldswehave

φ _R = 0

^and

φ _C = 0

^as^well^as

φ _q = 0

forallprimepowers

q,

^so^the ônlusionôf^the^theorem^takes^the^formôf^thatôf^the^lassial

BrauerSiegel theorem. However, there are examples of families of number elds where the

right hand side of the equality in the theorem is either stritly less or stritly greater than

one (see [TV02 ℄). Let us mention one partiularly nie orollaryof thegeneralized Brauer

Siegeltheoremdue to M. Tsfasmanand S.Vl duµ: a boundon theregulators that improves

Zimmert's bound (see [Zim℄, his bound an be written as

lim inf ^log _g ^R ^Ki

Ki ≥ (log 2 + γ)φ _R + 2γφ _C

^).

Theorem 3.3 (TsfasmanVl duµ). For a family of almost normal number elds

{ K _i }

(orany number elds under the assumptionof GRH) we have

lim inf log R _K _i

g _K _i ≥ (log √

πe + γ/2)φ _R + (log 2 + γ)φ _C .

Theproofof this bound isfar frombeingtrivial, it anbe foundin[TV02℄.

The funtion eld versionof the BrauerSiegel theorem is both easier to prove and requires

no supplementary onditions (like normality or GRH). In fat, it was obtained before the

orresponding theorem for number elds and allowed to guess what the result for number

eldsshould be (fora proofsee[Tsf92℄ or [TV97 ℄).

Theorem 3.4 (TsfasmanVl duµ). For an asymptotially exat family of smooth pro-

jetive urves

{ X _i }

ôver â ^nite êld

F r

^we ^have:

i lim →∞

log h _i

g _i = log r + X ∞ f =1

φ _r f log r ^f r ^f − 1 ,

where

h _i = h(X _i ) = | (JacX _i )( F r ) |

îs^the ârdinality ôf ^the ^Jaobian ôf

X _i

^over

F r .

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Let

κ K = Res

s=1 ζ _K (s)

^be ^the^residue^of^the^Dedekind ^zeta^funtion

ζ _K (s) = Q

q

(1 − q ⁻ ^s ) ⁻ ^Φ ^q ^(K)

of the eld

K

^at

s = 1.

^Using ^the ^residue ^formula ^(see ^[Lan94, ^Chapter ^VIII℄ ^and ^{[TVN ,}

ChapterIII℄)

κ K = 2 ^Φ ^R ^(K) (2π) ^Φ ^C ^(K) h _K R _K w _K p

| D _K |

(NF ase);

κ K = h _K r ^g

(r − 1) log r

^(FF^ase)

(here

w _K

îs ^the ^number ôf ^roots ôf ûnity ⁱⁿ

K

⁾ ône ân ^see ^that ^the ^question âbout ^the

behaviouroftheratiofromtheBrauerSiegeltheoremisreduedtotheorrespondingquestion

for

κ K .

^To ^put ît înto â ^more ^general ^framework, ^we ^rst ^seek ân interpretation of the arithmetiquantitieswewouldliketostudyintermsofspeialvaluesofertainzetafuntions,

then we study the behaviour of these speial values in families using analyti methods. We

willseeinsetion5anotherappliationsofthispriniple. Onealsonotiesthatthisredution

stepexplains the appearane ofthe GRH inthe statement of theBrauerSiegel theorem.

Let us formulate yet another version of the generalized BrauerSiegel theorem proven by

Lebaque in [Leb07 ,Theorem 7℄. Ithas theadvantage of beingexpliit withrespetto the

error terms, thus giving information about theBrauerSiegel ratio on thenite level.

Theorem 3.5 (Lebaque). Let

K

^be ^a ^global ^eld. ^Then

(i) in the funtion eld ase

log( κ K log r) = X N f =1

Φ _r f log r ^f

r ^f − 1 − log N − γ + O g _K N r ^N/2

+ O

1 N

;

(ii) in the number eld ase assumingGRH

log κ ^K = X

q ≤ x

Φ q log q

q − 1 − log log x − γ + O

n _K log x

√ x

+ O g _K

√ x

,

where

γ = 0.577 . . .

îs ^the Êuler ônstant. ^The ônstants ⁱⁿ

O

âre âbsolute ând êetively

omputable (and, in fat, notvery big).

Thistheoremanalso beregarded asageneralizationof theMertens theorem(see[Leb07 ℄).

Aslight improvement oftheerrorterm(asbefore,assumingGRH) wasobtainedin[LZ ℄. An

unonditional number eldversionof this resultis also available but isa little more diult

to state ([Leb07 , Theorem 6℄). We should also note that Lebaque's approah leads to a

uniedproofofthe asymptotiallybadandasymptotiallygoodasesoftheorem3.2withor

withoutthe assumption ofGRH.

3.2. Limit zeta-funtions. Forthemomenttheasymptotitheoryofglobaleldslooks

likeaolletionofsimilarbutnotdiretlyrelatedresults. Thesituationislariedimmensely

by meansof the introdution of limitzeta funtions.

(10)

Denition 3.6. Thelimit zetafuntion ofan asymptotiallyexatfamilyofglobalelds

K = { K _i }

^is^dened ^as

ζ _K (s) = Y

q

(1 − q ^−s ) ^−φ ^q ^(K) ,

theprodutbeingtakenover allprimepowersinthenumbereldaseandoverprimepowers

oftheform

q = r ^f

ⁱⁿ^the âseôfûrvesôver

F r .

The basi inequalities from theorem 2.6 give the onvergene of the above innite produt

for

Re s ≥ ¹ ₂

^with^the âssumption ôf ^GRH ând ^for

Re s ≥ 1

^without ^it ⁽ⁱⁿ^partiular, ⁱⁿ^the

funtioneldase theinnite produtonverges for

Re s ≥ ¹ ₂

^). ^In^fat, ^the^basiinequalities themselvesan berestated interms ofthevalues of limit zetafuntions. To formulate them

weintrodue theompletedlimit zeta funtion:

ζ ˜ _K (s) = e ^s 2 ⁻ ^φ ^R π ⁻ ^sφ ^R ^/2 (2π) ⁻ ^sφ ^C Γ s 2

φ _R

Γ(s) ^φ ^C ζ _K (s)

^(NF ^ase);

ζ ˜ _K (s) = r ^s ζ _K (s)

^(FF^ase).

Let

ξ ˜ _K (s) = ˜ ζ ^′ _K (s)/ ζ ˜ _K (s)

^be^the ^logarithmi ^derivative ^of ^the^ompleted ^limit ^zeta ^funtion.

Thenthe basi inequalitiesfrom setion2 take thefollowing form:

Theorem 3.7 (Basi inequalities). For an asymptotially exat family of global elds

K = { K _i }

^we ^have

ξ ˜ _K ( ¹ ₂ ) ≥ 0

ⁱⁿ ^the ^funtion êld âse ând âssuming ^GRH ⁱⁿ ^the ^number

eldase and

ξ ˜ _K (1) ≥ 0

^without ^the ^assumption^of ^GRH.

Letusgiveaninterestinginterpretationofthedeienyintermsofthedistributionofzeroes

ofzetafuntions onthe ritialline. In fat,theresults we aregoing to state areinteresting

ontheir own. Toaglobaleld

K

^we^assoiate^the^ounting^measure

∆ _K = _g ¹

K

P

ρ

δ _t(ρ) ,

^where

t(ρ) = Im ρ

ⁱⁿ ^the ^number êld âse ând

t(ρ) = _log ¹ _r Im ρ

ⁱⁿ ^the ^funtion ^ase; ^the ^sum ^is

taken overall zeroes

ρ

^of

ζ _K (s)

ⁱⁿ ^the ^number êld âse ând ôver âll ^zeroes

ρ

^of

ζ _K (s)

^with

t(ρ) ∈ ( − π, π]

ⁱⁿ^the ^funtion êldâse ⁽ⁱⁿ ^the âseôf ^funtion êlds

ζ _K (s)

^is ^periodi ^with

theperiodequalto

2π/ log r

^),

δ t

^is^the^Dira^(atomi)^measure^at

t.

^Thus^we^get^a^measure^on

R

ⁱⁿ^the^numberêld âseând ôn

R / Z

ⁱⁿ^the^funtion êldâse. ^The âsymptoti^behaviour

of

∆ _K

^was ^rst ^onsidered ^by ^Lang ^[Lan71℄ ⁱⁿ ^the asymptotially bad ase. The following resultisproven in[TV02,Theorem 5.2℄ and[TV97, Theorem2.1℄.

Theorem 3.8 (TsfasmanVl duµ). For an asymptotially exat family of global elds

K = { K _i } ,

^assuming ^GRH, ^the ^limit

lim

i →∞ ∆ _K _i

êxistsⁱⁿ ân âpprôpriate ^spae ôf ^meâsures ^(to

be preise, in the spae of measures of slow growth on

R

ⁱⁿ ^the ^NF ^ase,and ⁱⁿ ^the ^spae ^of

measures on

R / Z

ⁱⁿ ^the ^FF âse). ^Moreover, ^the ^limit îs â ^measure ^with ôntinuous ^density

M _K (t) = Re ˜ ξ _K ¹ ₂ + it

.

Of ourse, the expression for

M _K (t)

ân ^be ^written êxpliitly ûsing ^the învariants

φ _q

^. ^Let

us note two important orollaries of the theorem. First, we get an interpretation for the

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deieny

δ _K = ˜ ξ _K ¹ ₂

= M _K (0)

âs ^the âsymptoti ^numberôf ^zeroesôf

ζ _K _i (s)

^aumulating

at

s = ¹ ₂ .

^Seond, ^the ^theorem ^shows ^that ^for âny ^family ôf ^number êlds ^zeroes ôf ^their

zeta-funtionsgetarbitrarily loseto

s = ¹ ₂

^(and,ⁱⁿâ ^sense,^we êven^know^the^rateât ^whih

zeroesof

ζ _K _i (s)

^approah ^to ^this ^point).

3.3. Limit zeta-funtions and BrauerSiegel type results. Let us turn our at-

tention to the BrauerSiegel type results. The formulae from theorems 3.2 and 3.4 an be

rewritten as

lim

i →∞

log κ _Ki

g _Ki = log ζ _K (1).

^Furthermore, using the absolute and uniform onvergene of innite produts for zeta funtions for

Re s > 1,

^T^sfasman ^and ^Vl duµ ^prove ⁱⁿ

[TV02 , Proposition 4.2℄ that for

Re s > 1

^the ^equality

lim

i→∞

log ζ _Ki (s)

g _Ki = log ζ _K (s)

^holds. ^In

fat, thisequalityremains valid for

Re s < 1

^(at^leastîf^we âssume^GRH ⁱⁿ^the^numberêld

ase). Theproof ofthe nexttheorem anbefound in[Zy10 ℄ inthenumber eldaseandin

[Zyk11℄inthe funtion eldase (wherethe same problemis treated inabroader ontext).

Theorem 3.9 (Zykin). For an asymptotially exat familyof global elds

K = { K _i }

^for

Re s > ¹ ₂

^we ^have

i lim →∞

log((s − 1)ζ _K _i (s))

g _K _i = log ζ _K (s)

^(NF ^ase ^assuming ^GRH);

i lim →∞

log((r ^s − 1)ζ _K _i (s)) g K i

= log ζ _K (s)

^(FF^ase).

The onvergene is uniform on ompat subsets of the half-plane

{ s | Re s > ¹ ₂ } .

The ase

s = 1

ôf ^theorem^3.9 îs êquivalent ^to ^the BrauerSiegel theorem and urrent teh- niques does not allow to treat it in full generality without the assumption of GRH. Thus

getting unonditional results similar to theorem 3.9 looks inaessible at the moment. The

analogue of the above result for

s = ¹ ₂

^is onsiderably weaker and one has only an upper bound:

Theorem 3.10 (Zykin). Let

ρ _K _i

^be ^the^rst ^non-zero ^oeient ⁱⁿ ^the^Taylor ^series^ex-

pansionof

ζ K i (s)

^at

s = ¹ ₂ ,

^i.^e.

ζ K i (s) = ρ K i s − ¹ ₂ r _Ki

+o s − ¹ ₂ r _Ki

.

^Thenⁱⁿ^the^funtion

eldase orinthe number eldase assumingthat GRHistrue, for anyasymptotiallyexat

familyof global elds

K = { K _i }

^the ^fol^lowing^inequality ^holds:

lim sup

i →∞

log | ρ _K _i |

g _K _i ≤ log ζ _K 1

2 .

The interest in the study of the asymptoti behaviour of zeta funtions at

s = ¹ ₂

^is ^partly

motivated by the orresponding problem for

L

^-funtions ôf êllipti ûrves ôver ^global êlds,

where this value is relatedto deep arithmeti invariants of theellipti urvesvia the Birh

Swinnerton-Dyeronjeture. Werefer thereader to setion 5for more details. The question

whether the equality holds in theorem 3.10 is rather deliate. It is related to the so alled

low-lying zeroes of zeta funtions, that is the zeroes of

ζ _K (s)

^having ^small ^imaginary ^part

(12)

ompared to

g _K .

^It ^might ^well ^happen ^that ^the ^equality

lim

i →∞

log |ρ _Ki |

g _Ki = log ζ _K ( ¹ ₂ )

^does ^not

hold for all asymptotially exat families

K = { K _i }

^sine ^the ^behaviour ^of ^low-lying ^zeroes

is known to be rather random. Nevertheless, it might hold for most families (whatever it

might mean).

To illustrate how hard the problem may be, let us remark that Iwanie and Sarnak studied

a similar question for the entral values of

L

^-funtions ^of ^Dirihlet ^haraters ^[IS99℄ ^and

modular forms [IS00℄. They manage to prove that there exists a positive proportion of

Dirihlet haraters (modular forms) for whih the logarithm of the entral value of the

orresponding

L

^-funtions ^divided ^by^the ^logarithm ôf ^theânalyti ôndutor ^tends ^to ^zero.

Thetehniquesoftheevaluationofmolliedmomentsusedinthesepapersareratherinvolved.

We also note that, to our knowledge, there has been no investigation of low-lying zeroes of

L

^-funtionsôf^growing ^degree. Ît ^seems^that^theânalogous ^problemⁱⁿ^the^funtion êldâse

hasneither been very well studied.

Letusindiatethattheorrespondingquestionforthelogarithmiderivativesofzetafuntions

hasanegative answer. Indeed, the funtional equationimpliesthat

lim

i→∞

ζ _Ki ^′ (1/2)

ζ _Ki (1/2) = 1

^for ^any

family of funtion elds

K i .

^However, ^the ^logarithmi ^derivative ^of ^the ^limit ^zeta ^funtion

ζ _K (s)

^at

s = ¹ ₂

êqualsône ônly ^forasymptotially optimal families(.f. theorem3.7 ).

Asaorollary of theorem3.9one an obtain a resulton the asymptoti behaviour ofEuler

Kroneker onstants.

Denition 3.11. The EulerKroneker onstant of a global eld

K

^is ^dened ^as

γ _K =

c 0 (K)

c ₋ 1 (K) ,

^where

ζ _K (s) = c ₋₁ (K)(s − 1) ⁻ ¹ + c ₀ (K) + O(s − 1).

In[Iha06℄Y.IharamadeanextensivestudyoftheEuler-Kronekeronstantsofglobalelds,

inpartiular,heobtainedanasymptotiformulafortheirbehaviourinfamiliesofurvesover

nite elds. A omplementary result in the number eld setting was obtain in [Zy10℄ asa

orollaryof theorem 3.9. In fatthe theorem3.9 givesthat inasymptotially exat families

theoeientsoftheLaurantseriesat

s = 1

^of^the^logarithmiderivatives

ζ _K ^′ _i (s)/ζ _K _i (s)

^tend

totheorrespondingoeients oftheLaurant seriesexpansionofthelogarithmiderivative

ofthelimit zetafuntion. Forzeroesoeient this beomes:

Corollary 3.12 (IharaZykin). AssumingGRH in the number eld ase andunondi-

tionallyin thefuntion eldase, for anyasymptotiallyexat familyof global elds

{ K _i }

^we

have

i lim →∞

γ _K _i

g _K _i = − X

q

φ _q log q q − 1 .

For thesake of ompleteness let us mention an expliit analogue of theorem 3.9 obtained in

[LZ ℄:

Theorem 3.13 (LebaqueZykin). Foranyglobal eld

K,

^any^integer

N ≥ 10

^and^any

ǫ = ǫ ₀ + iǫ ₁

^suh ^that

ǫ ₀ = Re ǫ > 0

^we ^have

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(i) in the funtion eld ase:

X N f=1

f Φ _r f

r ⁽ ¹ ² ^+ǫ)f − 1 + 1 log r · Z K

1 2 + ǫ

+ 1

r ⁻ ¹ ² ^+ǫ − 1 = O g _K

r ^ǫ ⁰ ^N

1 + 1 ǫ ₀

+ O

r ^N ²

;

(ii) and inthe number eld ase assuming GRH:

X

q ≤ N

Φ _q log q q ¹ ² ^+ǫ − 1 + Z _K

1 2 + ǫ

+ 1

ǫ − ¹ ₂ =

= O

| ǫ | ⁴ + | ǫ |

ǫ ² ₀ (g _K + n _K log N ) log ² N N ^ǫ ⁰

+ O √ N

.

3.4. Some other topis related to limit zeta-funtions. Let us nally state some

related results on the asymptoti properties of the oeients of zeta funtions. For the

moment they are only available in the funtion eld ase (see [TV97℄). Let

K/ F ^r (t)

^be

a funtion eld and let

ζ _K (s) = P ^∞

m=1

D _m r ^−ms

^be ^the ^Dirihlet ^series ^expansion ^of ^the ^zeta

funtionof

K.

^One^knows^that

D _m

îsêqual^to^the^numberôfêetive ^divisorsôf^degree

m

^on

theorrespondingurve. We have the following resultson theasymptotibehaviourof

D _m :

Theorem 3.14 (TsfasmanVl duµ). For an asymptotially exat family of funtion

elds

K = { K _i }

^and ^any ^real

µ > 0

^we ^have

i lim →∞

log D _[µg] (K _i )

g _K _i = min

s ≥ 1 (µs log q + log ζ _K (s)).

Moreover, the minimum an be evaluated expliitly via

φ _q

^(.f. ^[TV97,Proposition4.1℄).

Theorem 3.15 (TsfasmanVl duµ). For an asymptotially exat family of funtion

elds

K = { K _i } ,

^any

ǫ > 0

^and ^any

m

^suh ^that

^D _g ^m ≥ µ ₁ + ǫ

^we ^have

log D _m (K _i )

h _K _i = q ^m−g+1

q − 1 (1 + o(1))

for

g → ∞ , o(1)

^being ^uniform ⁱⁿ

m.

^Here

µ ₁

îs^the ^largest ôf ^the ^two ^roots ôf ^the êquation

µ

2 + µ log _r µ

2 + (2 − µ) log _r 1 − µ

2 = − 2 log _r ζ _K (1).

We should notethat

o(1)

^from ^theorem^3.15îs âdditive ^whereas ^mostôf ^the^previous ^results

wereestimatesofmultipliativetype(theyontainedlogarithmsofthequantitiesinquestion).

It would be interesting to know whether there exist analogues of the above results in the

numbereldase.

Letusonlude byreferingthereaderto theSetion 6of[TV02℄foralistofopenquestions.

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4. Examples

4.1. Towers of modular urves. Let us begin with the examples of asymptotially

optimal familiesof urves over nite elds oming from towers of modular urves. Therst

onstrutions were arried out by Ihara ([Iha81℄), TsfasmanVl duµZink ([TVZ ℄). The

researhinthisdiretionwasontinuedbyN.Elkiesandmanyothers. Letusdesribeseveral

onstrutions.

4.1.1. Classial modular urves. Letus start withthe onstrution of towers of modular

urveswhihleadstoasymptotially optimalinnitefuntion elds. Forfurtherinformation,

we refer the reader to [TV92 , Chapter 4℄. It is well known that themodular group

Γ(1) = PSL ₂ ( Z )

âts ôn ^the^Poinaré ûpper^half-plane

h

^by

a b c d

· z = az + b

cz + d .

^W^e ^x ^a ^positive

integer

N

ând^we ^dene ^the ^prinipal ôngruene ^subgroup ôf^level

N

^by

Γ(N ) =

γ ∈ Γ(1) | γ ≡ 1 0

0 1

mod N

. Γ(N ) ⊳ Γ(1)

^and

Γ(1)/Γ(N )

^is ^isomorphi^to

P SL ₂ ( Z /N Z ).

^In ^partiular,

[Γ(1) : Γ(N )] =

 



N ³ 2

Q

ℓ | N

1 − ℓ ⁻ ²

si

N ≥ 3

6

^si

N = 2.

We also put

Γ 0 (N ) =

γ ∈ Γ(1) | γ ≡ ∗ ∗

0 ∗

mod N

,

^so ^that

Γ(N ) ⊂ Γ 0 (N ).

^W^e ^have

[Γ(1) : Γ ₀ (N )] = N Y

ℓ | N

1 − ℓ ⁻ ¹ .

Letnow

Γ

^beâôngruene^subgroup,^thatîs,âny^subgroupôf

Γ(1)

^ontaining

Γ(N ).

^The^most

important ase for us is

Γ = Γ(N )

^or

Γ ₀ (N ).

^The ^set

Y _Γ = Γ \ h

îs êquipped ^withân ânalyti

struture, but is not ompat. To ompatify it we add points at innity (named usps):

Γ(1)

^ats ^naturally ^on

P ¹ ( Q )

^and ^we ^put

X _Γ = (Γ \ h) ∪ (Γ \P ¹ ( Q )).

^This ^way ^it ^beomes ^a

onneted Riemann surfae alledmodularurve. We let

X(N ) = X _Γ(N) , X ₀ (N ) = X _Γ ₀ _(N) , Y (N ) = Y _Γ(N ₎

^and

Y ₀ (N) = X _Γ ₀ _(N ₎ .

If

Γ ^′ ⊂ Γ ⊂ Γ(1),

^there ^is^a ^natural ^projetion ^from

X _Γ ′ → X _Γ ,

^whih âllows ûs ^to ômpute

thegenus ofthemodularurve usingthe overing(thefuntion

j

^isⁱⁿ^fat^the

j

^-invariant ^of

theellipti urve

C /( Z + z Z )

^):

X _Γ ^/ ^/ X _Γ(1) ^∼

j

/ / P ¹ ( C )

viatheHurwitz formula. For instane,

g _X _(N) = 1 + (N − 6)[Γ(1) : Γ(N )]

12N .

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It an be shown that

Y (1)

^lassies isomorphism lasses of omplex ellipti urves and that

Y ₀ (N )

^lassies ^pairs

(E, C _N ), E

^being â ômplex êlliptiûrve ând

C _N

^being ^a ^yli ^sub-

groupof

E

^of ^order

N.

Now, to onstrut towers of urves dened over nite elds, we need to take redutions of

our modular urvesmodulo primes. If

S

îs â ^sheme ând

E → S

îsân êlliptiûrve, ^the^set

of setions

E(S)

îs ân âbelian ^group. ^Let

E _N (S)

^denote ^the ^points ^of ^order ^dividing

N

ⁱⁿ

E(S).

^Wê âll â ^level

N

^struture ^an isomorphism

α _N : E _N (S) → ( Z /N Z ) ² .

^One ^an ^prove

thatthere existsa smoothane sheme

Y (N )

^over

Spec Z [1/N ]

^lassifying^theisomorphism lasses ofpairs

(E, α _N )

ônsistingôfânêllipti ûrve

E/Spec Z [1/N ]

^together ^with^a^level

N

struture

α _N

^on

E.

Ôneân ^prove ^that ^this ûrve îsâ ^model ôf

Y (N )

^over

Spec Z [ζ _N , 1/N ],

where

ζ N

^is^a ^primitive

N

^th^-root ^of

1.

^There îsâlso â ^modelôf

Y 0 (N )

^over

Spec Z [1/N ]

^and

thisoarse moduli spaelassiespairsonsistingof anelliptiurve together witha yli

subgroup of order

N.

^Models^for

X(N)

^and

X ₀ (N )

ân âlso ^be ôbtained ⁱⁿ^suh â ^way^that

they beome ompatible withthose for

Y (N )

^and

Y ₀ (N ).

^These^urves^have ^good ^redution

overanyprimeideal not dividing

N.

^Moreover, ^the^urve

X ₀ (N )

^an ^be ^dened ^over

Q

^and

has good redution at any prime number not dividing

N.

^Let

p

^be ^suh ^prime. ^W^e ^denote

by

C _0,N

^the ^urve ^over

F p ²

^obtained ^by ^redution ^of

X ₀ (N )

^mod

p.

^The ^urve

X(N )

^an

be dened over the quadrati subeld of

Q (ζ _N )

ând ^has ^good ^redution ât âll ^the ^primes

not dividing

N.

^Let

C _N

^be ^the ^redution ^of

X(N )

ât â ^prime, î. ê. â ûrve ôver

F p ² .

^One

an see that the genus of

X ₀ (N )

^and ^of

X(N )

^is ^preserved ^under ^redution. ^The ^points

of these urvesorresponding to supersingular ellipti urves are

F p ²

^-rational ^and ^there ^are

[Γ(1) : Γ(N )]

12 (p − 1)

^of^them ^on

C _N .

^This^leads ^to ^the ^following ^theorem:

Theorem 4.1. (Ihara, TsfasmanVl duµZink) Let

ℓ

^be ^a ^prime ^number ^not ^equal ^to

p.

The families

{ C _ℓ ⁿ }

^and

{ C _0,ℓ ⁿ }

^satisfy

φ _p 2 = p − 1

^and^therefore ^are asymptotiallyoptimal.

Notethat theresultfor

C _0,ℓ ⁿ

^an ^be ^deduedimmediately fromthe orrespondingresult for

C _ℓ ⁿ .

4.1.2. Shimura modular urves. Similar results on Shimura urves allow us to onstrut

diretly asymptotially optimal families over

F r

^with

r = q ² = p ^2m , p

^prime. ^To ^do ^so,

following Ihara, we start with a

p

^-adi ^eld

k _p

^with

N (p) = q = p ^m .

^Let

Γ

^be ^a torsion-free disrete subgroupof

G = P SL ₂ ( R ) × P SL ₂ (k _p )

^with^ompat ^quotient ^and ^dense^projetion

to eah of thetwo omponents of

G

^(suh

Γ

^'s^exist). ^Ihara ^proved ^the^following ^results ^that

relate the onstrution of optimal urves to (anabelian) lass eld theory, and therefore are

of greatinterestfor us:

Theorem 4.2. (Ihara [Iha08℄)To any subgroup

Γ

^of

G

^with^theâbove ^propertiesôneân

assoiate a omplete smooth geometrially irreduible urve

X

^over

F ^r

^of ^genus

≥ 2,

^together

witha set

Σ

^onsisting^of

(q − 1)(g − 1) F r

^-rational^points^of

X

^suh ^that ^there îsâânonial

isomorphism (upto onjugay) fromthe pronite ompletion of

Γ

^to

Gal(K ^Σ /K )

^where

K ^Σ

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denotesthe maximalunramiedGaloisextensionof thefuntion eld

K

^of

X

ⁱⁿ ^whih^all^the

plaes orresponding to the pointsof

Σ

^are ^ompletely ^split.

Aneasy omputation leads tothe following result:

Corollary 4.3. For any square prime power

r,

^there îsâ ^tower ôf ûrves ^dened ôver

F r

with

φ _r = √ r − 1.

Infat, the elliptimodular urves

X(N )

^that ^we ^onstruted ⁱⁿ ^the^previous ^setion ^orre-

spondto

Γ = P SL ₂ ( Z [1/p])

ândîts ^prinipal ôngruene ^subgroups ôf^level

N.

4.1.3. Drinfeldmodularurves. TheappliabilityofDrinfeldmodularurvestotheprob-

lemofonstrutionofoptimalurveshasbeenknownsinelate80's. Theresultswearegoing

todisuss next anbefound in[TV92 ℄.

Let

L

^beâ êldôfharateristi

p

^and^let

L { τ }

^denote ^the^ring^of non-ommutative polyno- mialsin

τ,

^onsisting^ofexpressionsoftheform

X n i=0

a _i τ ⁱ , a _i ∈ L,

^withmultipliation satisfying

τ · a = a ^p · τ

^for ^any

a ∈ L.

^Let

A = F r [T ].

ADrinfeld moduleisan

F ^r

-homomorphism

φ : A → L { τ } , a 7→ φ a

^satisfying ^a^few ^tehnial

onditions. Let

γ

^be ^the ^map

γ : A → L

^sending

a ∈ A

^to ^the ^term ^of

φ _a

^of ^degree ^zero.

Notiethat

φ

^is^determined^by

φ _T

^and

γ

^by

γ (T ).

^Wêônsiderônly^Drinfeld^modulesôf^rank

2

^that^is^we^assume^that

φ T

^is^a^polynomialⁱⁿ

τ

^of^degree

2

^and^we^put

φ T = γ(T ) +gτ + ∆τ ² (∆ 6 = 0).

^More^generally^,ône ân ^dene ^Drinfeld^modules ôverâny

A

^-sheme

S.

Just as in the lassial ase, given a proper ideal

I

^of

A

^, ône ân ^dene â ^level

I

^struture

on

φ.

^There îs ân âne ^sheme

M (I )

^of ^nite ^type ^over

A

^that parametrizes pairs

(φ, λ),

where

φ

îs â ^Drinfeld ^module ôver

S

^and

λ

^is ^a ^level

I

^struture. ^The ^sheme

M (I )

^has ^a

anonialompatiation: there existsa unique sheme

M (I)

^ontaining

M (I )

âs ân ôpen

dense subsheme, whose bres over

Spec A[I ⁻ ¹ ]

âre ^smooth ômplete ûrves. ^The ^group

GL ₂ (A/I)

^ats ^naturally ^on

M (I )

^by ôperating ôn ^the ^strutures ôf ^level

I

^and ^this ^ation

extendsto

M(I).

From now on,let

I

^be â ^prime îdeal ^generated ^byâ ^polynomial ôf ^degree

m

^prime^to

q − 1.

Now, onsider the smooth omplete (reduible) urve

X(I) = M (I) ⊗ A F ^q

^over

F ^q .

^Note

thatthe

A

^-algebra ^struture ^on

F q

^is ^obtained ^through ^the^redution

mod T.

^Consider ^the

subgroup

Γ ₀ (I) =

a b c d

∈ GL ₂ (A) | c ∈ I

and let

Γ 0 (I )

^be ^the ^image ^of ^this ^subgroup ⁱⁿ

GL 2 (A/I).

^Finally, ^we ^onsider ^the ^smooth

omplete absolutely irreduible urve

X ₀ (I ) = X(I )/Γ ₀ (I ).

^The ^image ^of

M (I) − M (I )

ⁱⁿ

X ₀ (I)

^onsists^of^two

F ^q

^-rational ^points. ^Moreover, ^the^following ^result^holds.

Theorem 4.4. The family

{ X ₀ (I ) } ,

^where

I

îs â ^prime îdeâl ôf

A

^generated ^by ^a ^poly-

nomial of degree prime to

q − 1,

^is ^an asymptotially exat family of urves dened over

F q ,

satisfying

φ _q 2 = q − 1

ând ^thus îsôptimal.

(17)

Moreover, N. Elkies proved in [Elk℄ that the family of urves

X ˙ ₀ (T ⁿ )

^whih parametrizes normalized Drinfeld modules

(γ(T ) = 1, ∆ = − 1)

^with^a^level

T ⁿ

^struture ^isasymptotially optimal. He also relatedit to the expliittowers of Garia and Stihtenoth disussed inthe

next subsetion.

4.2. Expliit towers. In the last fteen years, Garia, Stihtenoth and many others

managed to onstrut asymptotially good towers expliitely in a reursive way. Their in-

terest omes from oding theory for suh towers provide asymptotially good odes via the

onstrution of Goppa. Letus give an exampleof suh expliittowers.

Theorem 4.5. (GariaStihtenoth)Let

r = q ²

^be^a^prime^power. ^The^tower

{ F n }

^dened

reursively starting from the rational funtion eld

F ₀ = F r (x ₀ )

^using ^the ^relations

F _n+1 = F _n (x _n+1 ),

^where

x ^q _n+1 + x _n+1 = x ^q n

x ^q n ⁻ ¹ + 1 ,

satises

φ _r = √

r − 1

ând^thus îs ôptimal.

If the ardinalityof the ground eldis not a squareno towers with

φ _r = √

r − 1

^are^known.

However, there existoptimal towers inthesense that theyhave zerodeieny. Suh towers

anbeonstrutedstartingfromanexpliittoweroverabiggereldusingadesentargument

(see BalletRolland [BR℄ forthedetails) or using modular towers.

Letus now saya word about Elkies modularity onjeture. Elkies'workshows that most of

the reursive examples of Garia and Stihtenoth an be obtained by nding equations for

suitable modulartowers. Thismade himformulate the following onjeture:

Conjeture 4.6 (Elkies). Any asymptotially optimal towerismodular.

Finally,letusnotethatthereareotherinteresting onstrutionsleading toexpliitasymptot-

ially goodtowersoffuntion elds. Asan examplewe mentionthepaper [BB℄byP.Beelen

and I. BouwwhouseFuhsian dierential equations to produe optimal towersover

F q ² .

4.3. Classeldtowers. Asitwassaidinsetion

2,

^tamely^ramiedînniteêxtensionsôf

globaleldswithnitelymanyramiedplaesandwithompletelysplitplaesgiveexamples

of asymptotially goodtowers. Given a global eld

K,

îtîs ^natural ^to ônsider ^the^maximal

extension of

K

ûnramied ôutside â ^nite ^set ôf ^plaes

S,

ⁱⁿ ^whih ^plaes ^from ^a ^set

T

^are

ompletelysplit. Buttheseextensionsareveryhardtounderstand. Themaximal

ℓ

-extensions are muh easierto handle. These extensions arethe limits of the

ℓ

^-

S

^-

T

^-lass ^eld ^towers ^of

K.

Foraglobaleld

K

^,^two^sets^of^nite^plaes

S

^and

T (T 6 = ∅ (F F ))

^of

K,

^and^a^prime^number

ℓ

^,^onsider^the^maximal^abelian

ℓ

^-extension

H _S,ℓ ^T (K)

^of

K,

^unramied^outside

S

^andⁱⁿ^whih

theplaesfrom

T

âre^split⁽ⁱⁿ^theâseôf^funtionêlds^theâssumptionôn

T

^to^be^non-empty

is made in order to avoid innite onstant eld extensions). Consider thetower reursively

onstruted as follows:

K ₀ = K, K _i+1 = H _S,ℓ ^T (K _i ).

^All ^the ^extensions

K _i /K

^are ^Galois,

Asymptotic methods in number theory and algebraic geometry

F r

r

C

F r

N r (C)

F r

C.

g(C)

C.

L

N r (g)

N r (C)

g

F r :

N r (g) = max

g(C)=g N r (C).

N r (C)

N r (C) ≤ r + 1 + 2g √ r.

N r (g)

g

g

{ C i }

F r

lim sup

i→∞

N r (C i ) g(C i ) ≤ √

r − 1.

r

K

Q .

n K = [K : Q ]

D K

D K

{ K i }

log | D K i | ≥ A · r 1 (K i ) + 2B · r 2 (K i ) + o(n K i ),

r 1 (K i )

r 2 (K i )

K i .

A = log(4π) + γ + 1 ≈ 60.8, B = log(4π) + γ ≈ 22.3

A = log(8π) + γ + π 2 ≈ 215.3, B = log(8π) + γ ≈ 44.7,

γ = 0.577

h K

K

R K

h K R K

{ K i }

i lim →∞

log(h K i R K i ) log p

| D K i | = 1

lim

i→∞

n K i g K i = 0;

K i

Q .

r

p,

F r

F r .

K = { K n } n ∈N

K n

Q

K n

F r (t)

F r ∩ K n = F r .

i 6 = j, K i

K j .

K n ⊂ K n+1

n ∈ N .

S ∞ n=1

K n

g K

K

g K = log p

| D K | ,

D K

K.

g,

{ K i }

g K i → + ∞ .

F ^r

F ^r

F ^r

N _r (g)

N _r (C)

N _r (g) = max

g(C)=g N _r (C).

N _r (C)

N _r (g)

{ C _i }

F ^r

N _r (C _i ) g(C _i ) ≤ √

n _K = [K : Q ]

D _K

{ K _i }

log | D _K _i | ≥ A · r ₁ (K _i ) + 2B · r ₂ (K _i ) + o(n _K _i ),

r ₁ (K _i )

r ₂ (K _i )

K _i .

A = log(8π) + γ + ^π ₂ ≈ 215.3, B = log(8π) + γ ≈ 44.7,

R _K

{ K _i }

log(h _K _i R _K _i ) log p

| D _K _i | = 1

n _K _i g _K _i = 0;

K _i

K = { K _n } n ∈N

F ^r (t)

F r ∩ K _n = F r .

i 6 = j, K _i

K _j .

K _n ⊂ K _n+1

K _n

g _K

g _K = log p

| D _K | ,

D _K

{ K _i }

g _K _i → + ∞ .

F ^r (t)

n _K [K : Q ];

D _K

g _K

| D _K | (N F )

Pl _f (K)

p ∈ Pl _f (K)

deg p log _r Np (F F )

Φ _q (K)

Φ _R (K )

Φ _C (K )

Φ _q , A =

( R , C , p ^k | p

r ^k | k ∈ Z ^>0 (F F ) ,

A _f

p ^k | p

K = { K _i }

φ _q := lim

Φ _q (K _i ) g _K _i .

φ _q

φ _q

φ _r

φ _∞ = lim n _K _i

g _K _i = φ _R + 2φ _C .

φ _q

{ φ _q } .

{ φ _q } .

φ _q log q

2 )φ _R + (log 8π + γ)φ _C ≤ 1, (N F ) X

φ _q log q

2 )φ _R + (log 2π + γ)φ _C ≤ 1, (F F )

mφ _r ^m r ^m ² − 1 ≤ 1,

mφ _p ^m ≤ φ _R + 2φ _C .

δ _K

K = { K _i }