Families of curves over finite fields

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by

Stéphane Ballet& Robert Rolland

Abstrat. Weprovetheexisteneofasymptotially exatsequenesofalgebrai funtion

eldsdenedoveranyniteeld

F ^q

^having^anasymptotiallymaximumnumberofplaesofa degree

r

^where

r

îsânînteger

≥ 1

^. Ît^turnsôut^that^these^partiular^families^haveânâsymptoti

lassnumberwidelygreaterthanalltheLahaud-Martin-Deshampsboundswhen

r > 1

^. ^We

emphasize that we exhibit expliitasymptotially exat sequenesof algebrai funtionelds

overanyniteeld

F q

^,ⁱⁿ^partiular^when

q

^is^not^a^square,^with

r = 2

^. ^We^expliitly^onstrut

anexamplefor

q = 2

^and

r = 4

^. ^We^dedueôf^this^study^that^forâny^niteêld

F ^q

^there^exists

a sequeneof algebrai funtioneldsdenedoverany niteeld

F ^q

^reahing^the generalized Drinfeld-Vladutbound.

Résumé. Nous prouvons l'existene de familles asymptotiquement exates de orps de

fontions algébriques dénis sur un orps ni

F ^q

^qui ^ont ^un ^nombre ^maximum ^de ^plaes ^de

degré

r

^où

r

êst ûn êntier

≥ 1

^. ^Il ^s'avère ^que ^pour ^es ^familles partiulières le nombre de lassesestasymptotiquementbeauoupplusgrandquelabornegénéraledeLahaud-Martin-

Deshamps quand

r > 1

^.^Pour

r = 2

^nousonstruisons expliitement des suites deorps de fontions algébriques surtout orpsni

F ^q

^qui^sont asymptotiquement exates etei même lorsque

q

^n'est ^pas^un ^arré.^Nous onstruisonsun exemplepour

r = 4

^dans^le^as ^où

q = 2

^.

Nousdéduisonsdeetteétudequepourtoutorpsni

F ^q

îlêxisteûne^suite^deôrps^de^fontions

algébriquessur

F ^q

^qui^atteint^la^borne^de^Drinfeld^-^Vladutgénéralisée.

1. Introdution

1.1. General ontext and main result. When, for a given nite ground eld, the

sequene of the genus of a sequene of algebrai funtion elds tends to innity, there exist

asymptoti formulae for dierent numerial invariants. In [10℄, Tsfasman generalizes some

resultsonthenumberofrationalpointsontheurves(duetoDrinfeld-Vladut[13℄,Ihara[5℄,

and Serre [8℄)and on its Jaobian (due to Vladut [12℄, Rosembloom andTsfasman[7℄). He

givesaformulafor theasymptoti numberof divisors,and some estimatesfor thenumberof

pointsinthePoinaré ltration.

Keywordsandphrases. Finiteeld,funtioneld,asymptotiallyexatsequene,lassnumber,tower

offuntionelds.

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For this purpose, he introdued thenotion of asymptotially exat family of urves dened

over a nite eld. It turns out that this notion is very fruitful. Indeed, for suh sequenes

of urves, we an evaluate enough preisely the asymptoti behavior of ertain numerial

invariants, inpartiular theasymptoti numberofeetive divisorsandtheasymptoti lass

number. Moreover,whenthesesequenesaremaximal(f. thebasiinequality(1 ))theyhave

anasymptotiallylarge lassnumber. Inpartiular, itistheasewhenthey haveamaximal

numberofplaesofagivendegree. Unfortunatelytheexisteneofsuhsequenesisnotknown

for any nite eld

F q

^,ⁱⁿ ^partiular ^when

q

îs ^not â ^square ^(f. ^Remark ^5.2 ⁱⁿ^[11℄). În ^this

paper,wemainlyexpliitlyonstrutexamplesofasymptotiallyexatsequenesofalgebrai

funtion elds for any nite eld, so proving the existene of maximal asymptotially exat

sequenesof urvesdened overany niteeld

F q

^where

q

îs ^not â^square. ^So,^we ânswerⁱⁿ

Corollary3.4 thequestionasked byTsfasmanandVladut in[11,Remark5.2℄.

1.2. Notation and detailed questions. Let us reall the notion of asymptotially

exatfamily of urvesdened overa niteeld inthelanguage ofalgebrai funtion elds.

Denition 1.1 (Asymptotially Exat Sequene). Let

F / F q = (F _k / F q ) _k _≥ ₁

^be ^a^se-

queneofalgebraifuntion elds

F _k / F ^q

^denedôverâ^niteêld

F ^q

^of^genus

g _k = g(F _k / F ^q )

^.

We suppose that the sequene of thegenus

g _k

îsân înreasing ^sequene ^growing ^to înnity.

The sequene

F / F ^q

^is ^said ^to ^be asymptotially exat if for all

m ≥ 1

^the ^following ^limit

exists:

β _m ( F / F q ) = lim

k → + ∞

B _m (F _k / F q ) g _k

where

B m (F _k / F ^q )

îs ^the ^numberôf^plaes ôf ^degree

m

^on

F _k / F ^q

^.

Thesequene

β = (β ₁ , β ₂ , ...., β _m , ...)

îsâlled ^the^type ôf ^the asymptotially exat sequene

F / F ^q

^.

Denition 1.2 (Generalized Drinfeld-Vladut Bound). Let

F / F ^q = (F _k / F ^q ) _k _≥ ₁

^be

a asymptotially exat sequene of algebrai funtion elds

F _k / F q

^dened ôver â ^nite êld

F q

^of ^genus

g _k = g(F _k / F q )

^and ^of ^type

β

^. ^The ^sequene

β

(respetively thesequene

F / F q

⁾

is said maximal (of maximal type) when the following basi inequality, alled Generalized

Drinfeld-Vladutbound,

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X ∞ m=1

mβ _m q ^m/2 − 1 ≤ 1

isreahed.

Denition 1.3 (Drinfeld-Vladut Bound of order

r

^). ^Let

B _r (q, g) = max { B _r (F/ F q ) | F/ F q

îsâ ^funtion êldôver

F q

^of ^genus

g } .

Letus set

A _r (q) = lim sup

g → + ∞

B _r (q, g)

g .

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We have theDrinfeld-VladutBoundoforder

r

^:

A _r (q) ≤ 1

r (q ^r ² − 1).

Remark 1.4. Note that iffor a family

F / F ^q

ôf âlgebrai ^funtion êlds,^there êxistsân

integer

r

^suh ^that ^the Drinfeld-Vladut Bound of order

r

îs ^reahed, ^then ^this ^family îs â

maximal asymptotially exat sequene namely attaining the Generalized Drinfeld-Vladut

bound. Moreover,its type is

β = (0, 0, ...., 0, β _r = 1

r (q ^r ² − 1), 0, ...).

Tsfasman andVladut in[11℄ madeuseof these notions to obtain new general results onthe

asymptoti propertiesof zetafuntionsof urves.

Note thata simple diagonal argument proves that eah sequene ofalgebrai funtion elds

of growing genus, dened over a nite eld admits an asymptotially exat subsequene.

However until now, we do not know if there exists an asymptotially exat sequene

F / F ^q

withamaximal

β

^sequene^when

q

îs^notâ^square. ^Moreover,^the^diagonalêxtration^method

isnot really suitablefor the two following reasons. First,ingeneral wedo not obtainbythis

proess an expliitasymptotially exat sequeneof algebrai funtion elds dened over an

arbitrary nite eld, in partiular when

q

îs ^not â ^square. ^Moreover, ^we ^have ^no ôntrol ôn

thegrowing ofthegenus inthe extratedsequeneofalgebraifuntioneldsdened overan

arbitraryniteeld. Morepreisely,letusdenethenotionofdensityofafamilyofalgebrai

funtion eldsdened overa niteeld ofgrowing genus:

Denition 1.5. Let

F / F q = (F _k / F q ) _k≥1

^be â^sequeneôfâlgebrai^funtion êlds

F _k / F q

of genus

g _k = g(F _k / F q )

^, ^dened ^over

F q

^. ^W^e ^suppose ^that ^the ^sequene ^of ^genus

g _k

^is ^an

inreasing sequenegrowing to innity. Then, thedensity ofthesequene

F / F ^q

^is

d( F / F q ) = lim inf

k → + ∞

g _k g _k+1 .

Ahighdensityanbeausefulpropertyinsomeappliationsofsequenesortowersoffuntion

elds. Until now, no expliit examples of dense asymptotially exat sequenes

F / F ^q

^have

beenpointedout unlessfor the ase

q

^square^and ^type

β = ( √ q − 1, 0, · · · )

^.

1.3. New results. In this paper, we answer the questions mentioned above. More

preisely, for any prime power

q

^we ôntrut êxamples ôf asymptotially exat sequenes attaining the Drinfeld-Vladut Boundof order

2

^. ^We ^dedue ^that ^for ^any^prime ^power

q

⁽ⁱⁿ

partiular when

q

îs ^not â ^square), ^there êxists â ^maximal asymptotially exat sequenes of algebrai funtions elds namely attaining the Generalized Drinfeld-Vladut bound. We

also onstrut for

q = 2

^and

r = 4

ân êxample ôf ^maximal asymptotially exat sequene attainingtheDrinfeld-VladutBoundoforder

4

^. ^Then,^we^show^that^theasymptotiallyexat sequenesofonsideredmaximaltypehaveanasymptotiallylargelassnumberwithrespet

to theLahaud -Martin-Deshampsbounds. Thepaperisorganized asfollows. Insetion2,

we study the general families of asymptotially exat sequenes of algebrai funtion elds

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dened over an arbitrary nite eld

F q

^of ^maximal ^type

(0, ..., ¹ _r (q ^r ² − 1), 0, ..., 0, ...)

^where

r

îs ân înteger

≥ 1

^, ûnder ^the âssumption ôf ^their êxistene. În ^partiular, ^we ^study ^for

these general families the behavior of the asymptoti lass number

h _k

^, ând ^we ômpare ôur

estimation to the general known bounds of Lahaud - Martin-Deshamps (f. [6℄). More

preisely,we showthatfor suh families,ifthey exist,the asymptotilass number iswidely

greaterthanthegeneralboundsofLahaud-Martin-Deshampswhen

r > 1

^. ^Next,ⁱⁿ^setion

3,weonstrutivelyprovetheexisteneofasymptotiallyexatsequenesofalgebraifuntion

eldsdenedoveranyarbitraryniteeld

F ^q

^of^maximal^type

β = (0, ..., ¹ _r (q ^r ² − 1), 0, ..., 0, ...)

with

r = 2

^. ^Moreover,^weêxhibit^theêxampleôfânasymptotiallyexatsequeneofalgebrai funtioneldsdenedover

F q

^of^maximal^type

β = (0, ..., ¹ _r (q ^r ² − 1), 0, ..., 0, ...)

^with

q = 2

^and

r = 4

^. ^For ^this ^purpose, ^we ûse^towers ôf âlgebrai ^funtion êlds ôming ^from ^the^desent

of thedensied towers of Garia-Stihtenoth (f. [4℄ and [1℄). Note that all these examples

are expliit and onsist on very dense asymptotially exat towers algebrai funtion elds

(maximallydense towerfor

q = 2

^and

r = 4

^).

2. General results

2.1. Partiular families of asymptotially exat sequenes. First, let us reall

ertainasymptotiresults. Letus rstgivethefollowingresultobtainedbyTsfasmanin[10℄:

Proposition 2.1. Let

F / F q = (F _k / F q ) _k _≥ ₁

^be â ^sequene ôf âlgebrai ^funtion êlds ôf

inreasing genus

g _k

^growing ^to ^innity. ^Let

f

^be ^a ^funtion ^from

N

^to

N

^suh ^that

f (g _k ) = o(log(g _k ))

^. ^Then

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lim sup

g k →+∞

1 g _k

f X (g _k ) m=1

mB _m (F _k ) q ^m/2 − 1 ≤ 1.

Now, we an easily obtainthefollowing resultasimmediate onsequene ofProposition 2.1:

Theorem 2.2. Let

r

^be ^an ^integer

≥ 1

^and

F / F ^q = (F _k / F ^q ) _k _≥ ₁

^beâ ^sequene ôf âlgebrai

funtion elds of inreasing genus dened over

F q

^suh ^that

β _r ( F / F q ) = ¹ _r (q ^r ² − 1)

^. ^Then

β _m ( F / F q ) = 0

^for ^any ^integer

m 6 = r

^. ^In ^partiular, ^the ^sequene

F / F q

^is asymptotially exat.

Proof. Letus x

m 6 = r

^and^let^us^prove ^that

β _m ( F / F ^q ) = 0

^. ^We^useProposition2.1with theonstant funtion

f (g) = s = max(m, r)

^. ^Then^we ^get

lim sup

k → + ∞

1 g _k

X s j=1

jB j (F _k ) q ^j − 1 ≤ 1.

Butbyhypothesis

lim sup

k → + ∞

rB _r (F _k )

g _k (q ^r ² − 1) = lim

k → + ∞

rB _r (F _k )

g _k (q ^r ² − 1) = 1.

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Then

lim sup

g k →+∞

1 g _k

X

1≤j≤s;j6=r

jB _j (F _k )

q ^j − 1 = lim

g k →+∞

1 g _k

X

1≤j≤s;j6=r

jB _j (F _k ) q ^j − 1 = 0.

But

B m (F _k )

g _k ≤ (q ^m − 1) m



 1 g _k

X

1 ≤ j ≤ s;j 6 =r

jB j (F _k ) q ^j − 1



 .

Hene

g _k lim → + ∞

B _m (F _k ) g _k = 0.

Notethat forany

k

^the ^following ^holds:

B ₁ (F _k / F q ^r ) = X

i | r

iB _i (F _k / F q ).

Thenif

β _r ( F / F ^q ) = ¹ _r (q ^r ² − 1)

^,^by^Theorem ^2.2 ^we^onlude ^that

β ₁ ( F / F ^q ^r )

^exists^and^that

β ₁ ( F / F ^q ^r ) = (q ^r ² − 1).

In partiular the sequene

F / F q ^r

^reahes ^the ^lassial Drinfeld-Vladut bound and onsequently

q ^r

^is ^a^square.

If

β ₁ ( F / F q ^r )

êxists ^then ît ^does ^not ^neessarily împly ^that

β _r ( F / F q )

^exists ^but ^only ^that

lim _k _→ ₊ _∞

P

m | r mB m (F k / F ^q )

g _k

êxists. În ^fat, ^this ônverse ^depends ôn^the ^dening êquations ôf

thealgebrai funtionelds

F _k / F q

^.

Now, letus givea simple onsequene ofTheorem 2.2 .

r

^and

i

^be ^integers

≥ 1

^suh^that

i

^divides

r

^. ^Suppose ^that

F / F q = (F _k / F q ) _k≥1

is an asymptotially exat sequene of algebrai funtion elds dened over

F ^q

^of ^type

(β 1 = 0, . . . , β _r−1 = 0, β _r = ¹ _r (q ^r ² − 1), β _r+1 = 0, . . .)

^. ^Then ^the ^sequene

F / F q ⁱ = (F _k / F q ⁱ ) _k _≥ ₁

^of

algebrai funtion eld dened over

F q ⁱ

^is asymptotially exat of type

(β ₁ = 0, .., β ^r

i −1 = 0, β ^r

i = _r ⁱ (q ^r ² − 1), β ^r

i +1 = 0, . . .)

^.

Proof. Let us remark that by [9, Lemma V.1.9, p. 163℄, if

P

îs â ^plae ôf ^degree

r ^′

^of

F/ F ^q

^,^there^are

gcd((r ^′ , i))

^plaes^of^degree

_gcd(r ^r ^′ _′ _,i)

^over

P

ⁱⁿ^the^extension

F/ F q ⁱ

^. ^As^we^are

interested bythe plaes of degree

r/i

ⁱⁿ

F/ F q ⁱ

^,^let ^us^introdue ^the^set

S = { r ^′ ; r gcd(r ^′ , i) = i r ^′ } = { r ^′ ; lcm (r ^′ , i) = r } .

Then,

B _r/i (F/ F q ⁱ ) = X

r ^′ ∈ S

ir ^′

r B _r ′ (F/ F ^q ).

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Weknowthatall the

β _j (F/ F q ) = 0

^but

β _r (F/ F q ) = ¹ _r (q ^r ² − 1)

^. ^Then

β _r/i (F/ F q ⁱ ) = iβ r (F/ F ^q ),

β _r/i (F/ F q ⁱ ) = i r

q ^r ² − 1 .

2.2. Number of points of the Jaobian. Now, we are interested by the Jaobian

ardinality oftheasymptotially exat sequenes

F / F q = (F _k / F q ) _k≥1

^of ^type

(0, .., 0, ¹ _r (q ^r ² − 1), 0, ..., 0)

^.

Letus denote by

h _k = h _k (F _k / F ^q )

^the^lass^numberôf ^the âlgebrai^funtion êld

F _k / F ^q

^. ^Let

usonsider thefollowingquantities introdued by Tsfasmanin[10℄:

H _inf ( F / F ^q ) = lim inf

k→+∞

1 g _k log h _k H _sup ( F / F q ) = lim sup

k → + ∞

1 g _k log h _k .

Iftheyoinides, we justwrite:

H( F / F ^q ) = lim

k → + ∞

1 g _k log h _k = H _inf ( F / F ^q ) = H _sup ( F / F ^q ).

Then under the assumptions of the previous setion, we obtain the following result on the

sequeneoflass numbers of thesefamilies ofalgebrai funtion elds:

Theorem 2.4. Let

F / F q = (F _k / F q ) _k≥1

^be â ^sequene ôf âlgebrai ^funtion êlds ôf în-

reasing genus dened over

F ^q

^suh ^that

β _r ( F / F ^q ) = ¹ _r (q ^r ² − 1)

^where

r

îs ân înteger. ^Then,

the limit

H( F / F ^q )

^exists^and ^we ^have:

H( F / F q ) = lim

k → + ∞

1 g _k log h _k = log q ^q ^r ² (q ^r − 1) ¹ ^r ^(q

r 2 − 1) .

Proof. ByCorollary

1

ⁱⁿ^[10℄,^we^know^that^for^anyasymptotiallyexatfamilyofalgebrai funtionelds dened over

F q

^,^the^limit

H( F / F q )

^exists^and

H( F / F q ) = lim

k → + ∞

1 g _k log h _k = log q + X ∞ m=1

β _m . log q ^m q ^m − 1 .

Hene,theresult follows fromTheorem 2.2.

Corollary 2.5. Let

F / F q = (F _k / F q ) _k _≥ ₁

^be â ^sequene ôf âlgebrai ^funtion êlds ôf în-

reasing genus dened over

F q

^suh ^that

β _r ( F / F q ) = ¹ _r (q ^r ² − 1)

^where

r

îs ân înteger. ^Then

there existsan integer

k ₀

^suh ^that ^for ^any ^integer

k ≥ k ₀

^,

h _k > q ^g ^k .

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Proof. By Theorem2.4, we have

lim _k→+∞ (h _k ) ^gk ¹ = ^q ^q

r 2

(q ^r − 1) ¹ ^r ^(q

r 2 −1)

. But

q ^q ^r ² (q ^r − 1) ¹ ^r ^(q

r 2 −1) > q ^q ^r ² (q ^r ) ¹ ^r ^(q

r 2 −1) = q.

Hene, for asuiently large

k ₀

^,^we ^have ^for

k ≥ k ₀

^the^following ^inequality

(h _k ) ^gk ¹ > q.

Let us ompare this estimation of

h _k

^to ^the ^general ^lower ^bounds ^given ^by^G. ^Lahaud ^and

M. Martin-Deshamps in[6℄.

Theorem 2.6 (Lahaud - Martin-Deshamps bounds). Let

X

^be ^a ^projetive ^ir-

reduible and non-singularalgebrai urve dened over the nite eld

F ^q

^of ^genus

g

^. ^Let

J _X

be the Jaobian of

X

^and

h

^the ^lass ^numb^er

h = | J _X ( F q ) |

^. ^Then

1.

h ≥ L 1 = q ^g ⁻ ¹ _(q+1)(g+1) ^(q ⁻ ¹⁾ ²

^,

2.

h ≥ L ₂ = √ q − 1 2 g ^g ⁻ ¹ − 1

g | X( F q ) | +q − 1 q − 1 ,

3. if

g > √ q/2

^and^if

B ₁ (X/ F q ) ≥ 1

^, ^then ^the ^following^holds:

h ≥ L ₃ = (q ^g − 1) q − 1 q + g + gq .

Then we an prove that for a family of algebrai funtion elds satisfying theonditions of

Corollary 2.5 , the lassnumbers

h _k

^greatly ^exeeds^the^bounds

L _i

^. ^More^preisely

F / F ^q = (F _k / F ^q ) _k _≥ ₁

^be â ^sequene ôf âlgebrai ^funtion êlds ôf

inreasinggenus dened over

F q

^suh ^that

β _r ( F / F q ) = ¹ _r (q ^r ² − 1)

^where

r

îsân înteger. ^Then

1. for

i = 1, 3

lim _k _→ ₊ _∞ h _k

L _i = + ∞ ,

2. for

i = 2

^the ^fol^lowing^holds:

(a) ifr>1 then

lim _k _→ ₊ _∞ h _k

L ₂ = + ∞ ,

(b) ifr=1 then

h _k L ₂ ≥ 2.5.

Proof. 1. ase

i = 1

^: ^the ^following ^holds

L ₁ = q ^g ^k ⁻¹ (q − 1) ²

(q + 1)(g _k + 1) = q ^g ^k (q − 1) ²

q(q + 1)(g _k + 1) < q ^g ^k (g _k + 1) ,

so, usingthe previous orollary2.5, we onlude thatfor

k

^large

h _k

L 1

> g _k

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and onsequently

k → lim + ∞

h _k

L ₁ = + ∞ ;

2. ase

i = 2

^:

(a) ase

r = 1

^: ^we ^bound ^the^number ^of ^rational^points^using ^the^Weil ^bound. ^More

preisely

L ₂ = (q + 1 − 2 √ q) q ^g ^k ⁻¹ − 1 g _k

B ₁ (F _k / F q ) + q − 1

q − 1 ≤

(q + 1 − 2 √

q) q ^g ^k ⁻ ¹ − 1 g _k

2q + 2g _k √ q q − 1 <

2 q + 1 − 2 √ q (q − 1) √ q q ^g ^k

1 +

√ q g _k

;

but for all

q ≥ 2

2 q + 1 − 2 √ q (q − 1) √ q < 0.4,

then

L ₂ < 0.4

1 +

√ q g _k

q ^g ^k ,

hene

h _k L 2

> 2.5 g _k g _k + √ q

whih gives theresult;

(b) ase

r > 1

^: ⁱⁿ^this ^ase^we^know ^that

k→+∞ lim

B ₁ (F _k / F ^q )

g _k = β 1 ( F / F ^q ) = 0.

But

L ₂ < q + 1 − 2 √ q

(q − 1)q q ^g ^k B ₁ (F _k / F q ) + q − 1

g _k .

Then

h _k

L ₂ > (q − 1)q q + 1 − 2 √ q

g _k

B ₁ (F _k / F q ) + q − 1 .

We know that

k → lim + ∞

g _k

B ₁ (F _k / F q ) + q − 1 = + ∞ ,

then

k→+∞ lim h _k

L ₂ = + ∞ .

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3. ase

i = 3

^:

L ₃ = (q ^g − 1) q − 1

q + g + gq < q ^g ^k g _k ,

thenfor

k

^large

h _k L ₃ > g _k

and onsequently

k → lim + ∞

h _k

L ₁ = + ∞ .

As we see, if

F / F ^q = (F _k / F ^q ) _k _≥ ₁

^satises ^the assumptions of Theorem 2.4 , we have

h _k >

q ^g ^k > q ^g ^k ⁻ ^{1 (} _(q+1)g ^q ⁻ ¹⁾ ²

k

for

k ≥ k ₀

^suiently ^large. ^In^fat, ^the ^v^alue

k ₀

^depends^at ^least ^on^the

valuesof

r

^and

q

ând ^we ân ^not ^knowânythingâbout^this ^valueⁱⁿ^the^general âse.

Remark: We an remark that the lassnumber of these families is very near the Lahaud

- Martin-Deshamps bound

L ₂

^when

r = 1

^but ^is ^muh ^greater ^than ^the ^Lahaud ^- ^Martin-

Deshampsbound

L 2

^when

r > 1

^.

3. Examples of asymptotially exat towers

Letus note

F q ²

^a^nite ^eld^with

q = p ^r

^and

r

^an ^integer.

3.1. Sequenes

F / F q

^with

β ₂ (F/ F q ) = ¹ ₂ (q − 1) = A ₂ (q)

^. ^W^e ^onsider ^the ^Garia-

Stihtenoth'stower

T ₀

^over

F q ²

^onstrutedⁱⁿ^[4℄. ^Reall^that^this^tower^is^dened ^reursively

inthe following way. We set

F 1 = F q ² (x 1 )

^the ^rational ^funtion ^eld ^over

F q ²

^,^and ^for

i ≥ 1

we dene

F _i+1 = F _i (z _i+1 ),

where

z _i+1

^satises^the ^equation

z ^q _i+1 + z _i+1 = x ^q+1 _i ,

with

x _i = z _i

x _i ₋ ₁

^for

i ≥ 2.

We onsiderthe ompletedGaria-Stihtenoth's tower

T ₁ / F q ²

^dened ^over

F q ²

^studied ⁱⁿ^[2℄

obtained from

T ₀ / F q ²

^by^adjuntion ^of intermediatesteps. Namely we have

T ₁ / F q ² : F _1,0 ⊂ · · · ⊂ F _i,0 ⊂ F _i,1 ⊂ · · · ⊂ F _i,s ⊂ · · · ⊂ F _i+1,0 ⊂ · · ·

with

s = 0, ..., r

^. ^Note^that^the^steps

F i,0 / F q ² = F i − 1,r / F q ²

^are^the^steps

F i / F q ²

^of^the^Garia-

Stihtenoth'stower

T ₀ / F q ²

^and

F _i,s / F q ²

⁽

1 ≤ s ≤ r − 1

⁾ ^are^theintermediatesteps onsidered in[2 ℄.

Letusdenoteby

g _k

^the^genus ^of

F _k / F q ²

ⁱⁿ

T ₀ / F q ²

^,^by

g _k,s

^the^genus ^of

F _k,s

ⁱⁿ

T ₁ / F q ²

^and^by

B ₁ (F _k,s / F q ² )

^the ^numberôf^plaes ôf ^degreeône ôf

F _k,s / F q ²

ⁱⁿ

T ₁ / F q ²

^.

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Reallthateahextension

F _k,s /F _k

^is^Galois^of^degree

p ^s

^with^full^onstant^eld

F q ²

^. ^Moreover,

weknowby[3,Theorem 4.3℄thatthe desent ofthe denitioneldofthetower

T ₁ / F q ²

^from

F q ²

^to

F ^q

îs ^possible. ^More^preisely, ^there êxists â ^tower

T 2 / F ^q

^dened ^over

F ^q

^given ^by ^a

sequene:

T ₂ / F ^q : G _1,0 ⊂ · · · ⊂ G _i,0 ⊂ G _i,1 ⊂ · · · ⊂ G _i,r ₋ ₁ ⊂ G _i+1,0 ⊂ · · ·

denedovertheonstant eld

F q

^and ^related^to ^the ^tower

T ₁ / F q ²

^by

F _k,s = F q ² G _k,s

^for ^all

k

^and

s,

namely

F _k,s / F q ²

îs^theônstant êldêxtensionôf

G _k,s / F ^q

^. ^First,^let^us ^study^the^asymptoti

behavior of degree one plaes of the funtion elds of the tower

T ₂ / F q

^and ^more ^preisely

theexistene and the value of

β ₁ (T ₂ / F ^q )

^. În ôrder ^to ^derive â ^result ôn ^the ^tower

T ₂ / F ^q

^we

beginbythestudyofthetermsgivenbythedesentofthelassialGaria-Stihtenoth tower

T ₀ / F q ²

^. ^Next ^we ^will ^study^theintermediate steps.

Lemma 3.1. Let

T ₀ / F q ² = (F _k / F q ² ) _k _≥ ₁

^the Garia-Stihtenoth tower dened over

F q ²

and

T ₀ ^′ / F ^q = (G _k / F ^q ) _k _≥ ₁

îts ^desent ôver ^the ^denition êld

F ^q

î.e ^suh ^that ^for âny înteger

k,

F _k = F q ² G _k

^. ^Then

β ₁ (T ₀ ^′ / F ^q ) = lim

k→+∞

B ₁ (G _k / F ^q ) g(G _k / F ^q ) = 0.

Proof. First,note that ifthe algebrai funtion eld

F _k / F q ²

îs â ônstant êld êxtension

of

G _k / F q

^, âbove âny ^plae ôf ^degree ône ⁱⁿ

G _k / F q

^there êxistsâ ûnique ^plae ôf ^degree ône

in

F _k / F q ²

^. Consequently, let us use the lassiation given in [4, p. 221℄ of the plaes of degreeone of

F _k / F q ²

^. ^Let ûs ^remark ^that ^the^number ôf ^plaes ôf ^degree ône ^whih âre^not

of type (A), is less or equal to

2q ²

^(see ^[4, ^Remark ^3.4℄). ^Moreover, ^the ^genus

g _k

^of ^the

algebraifuntionelds

G _k / F ^q

^and

F _k / F q ²

^is^suh^that

g _k ≥ q ^k

^by^the^Hurwitz^theorem,^then

we an fous our study on plaes of type (A). The plaes of type (A) are built reursively

in the following way (f. [4, p. 220 and Proposition 1.1 (iv)℄). Let

α ∈ F q ² \ { 0 }

^and

P _α

be the plae of

F 1 / F q ²

^whih ^is ^the ^zero ^of

x 1 − α

^. ^F^or ^any

α ∈ F q ² \ { 0 }

^the ^polynomial

equation

z ₂ ^q + z ₂ = α ^q+1

^has

q

^distint ^roots

u ₁ , · · · u _q

ⁱⁿ

F q ²

^,^and^for^eah

u _i

^thereîsâûnique

plae

P _(α,i)

^of

F ₂ / F q ²

^above

P _α

^and ^this ^plae

P _(α,i)

îs â ^zero ôf

z ₂ − u _i

^. ^We ^iterate ^now

the proess starting from the plaes

P _(α,i)

^to ôbtain ^suessively ^the ^plaes ôf ^type ^(A) ôf

F ₃ / F q ² , · · · , F _k / F q ² , · · ·

^; ^then, ^eah ^plae

P

^of ^type ^(A) ^of

F _k / F q ²

îs â ^zero ôf

z _k − u

^where

u

îsîtself â ^zero ôf

u ^q + u = γ

^for ^some

γ 6 = 0

ⁱⁿ

F q ²

^. ^Let ^us ^denote ^by

P u

^this ^plae. ^Now,

we want to ount the number of plaes

P _u ^′

^of ^degree ^one ⁱⁿ

G _k / F q

^, ^that ^is ^to ^say ^the ^only

plaeswhih admita unique plaeofdegree one

P _u

ⁱⁿ

F _k / F q ²

^lying^over

P _u ^′

^. ^First,^note^that

itis possible only if

u

^is ^a ^solution ⁱⁿ

F ^q

^of ^the ^equation

u ^q + u = γ

^where

γ

^is ⁱⁿ

F ^q \ { 0 }

^.

Indeed,if

u

^is^not ⁱⁿ

F q

^,^there ^exists^anautomorphism

σ

ⁱⁿ^the ^Galois ^group

Gal(F _k /G _k )

^of

thedegree two Galois extension

F _k / F q ²

^of

G _k / F ^q

^suh ^that

σ(P _u ) 6 = P _u

^. ^Hene, ^the^unique

plaeof

G _k / F q

^lying ^under

P _u

îsâ ^plae ôf^degree

2

^. ^But

u ^q + u = γ

^has ^one ^solutionⁱⁿ

F q

p 6 = 2 F q p = 2 G _k / F q

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whih are lying under a plae of type (A) of

F _k / F q ²

îs êqual ^to ^zero îf

p = 2

^and ^equal ^to

q − 1

^if

p 6 = 2

^. ^We^onlude ^that

k → lim + ∞

B ₁ (G _k / F q ) g(G _k / F q ) = 0.

Letus remark that inany ase, the numberof plaes of degree one of

G _k / F q

îs^less ôr êqual

to

2q ²

^.

Nowweangetasimilarresultforthedesent

T ₂ / F ^q

^of^the^densied^tower

T ₁ / F q ²

^of

T ₀ / F q ²

^.

Lemma 3.2. The tower

T 2 / F ^q

^is ^suh ^that:

β ₁ (T ₂ / F q ) = lim

g(G k,s / F q ) → + ∞

B ₁ (G _k,s / F ^q ) g(G _k,s / F q ) = 0.

Proof. As

G _k+1 = G _k+1,0

îs ân êxtension ôf

G _k,s

^we ^get

B 1 (G _k,s / F ^q ) ≤ B 1 (G _k+1 / F ^q )

^.

Using the omputation done in the proof of Lemma 3.1 and Remark 3.4 in [4℄ we have

B ₁ (G _k+1 / F q ) ≤ 2q ²

^,^then^we ^get

B ₁ (G _k,s / F q ) ≤ 2q ²

^. ^By ^[2,^Corollary ^2.2℄ ^we ^know^that

l → lim + ∞

g(G _l+1 / F ^q ) g(G _l / F q ) = p

where

g(G _l / F q )

^and

g(G _l+1 / F q )

^denote^the^genusôf ^twoônseutive âlgebrai^funtion êlds

in

T ₂ / F q

^. ^Then ^for

k

^suiently ^large^we ^get

g _k,s ≥ g _k,0 = g _k .

We onlude that

β ₁ (T ₂ / F q ) = 0

^.

Let us prove a proposition establishing that the tower

T ₂ / F q

^is asymptotially exat with good density.

q = p ^r

^. ^Fôr âny înteger

k ≥ 1

^, ^for ^any ^integer

s

^suh ^that

s = 0, 1, ..., r

^, ^the ^algebrai ^funtion ^eld

G _k,s / F q

ⁱⁿ ^the ^tower

T ₂

^has ^a ^genus

g(G _k,s / F q ) = g _k,s

with

B ₁ (G _k,s / F q )

^plaes ^of ^degree ^one,

B ₂ (G _k,s / F q )

^plaes ^of ^degree ^two ^suh ^that:

1.

g(G _k,s / F q ) ≤ ^g(G _p ^k+1 r−s ^/ ^F ^q ⁾ + 1

^with

g(G _k+1 / F q ) = g _k+1 ≤ q ^k+1 + q ^k .

2.

B 1 (G _k,s / F ^q ) + 2B 2 (G _k,s / F ^q ) ≥ (q ² − 1)q ^k ⁻ ¹ p ^s

^.

3.

β ₂ (T ₂ / F q ) = lim _g _k,s _→ ₊ _∞ ^B ² ^(G _g ^k,s ^/ ^F ^q ⁾

k,s = ¹ ₂ (q − 1) = A ₂ (q)

^.

4.

d(T ₂ / F ^q ) = lim _l _→ ₊ _∞ _g(G ^g(G ^l ^/ ^F ^q ⁾

l+1 / F q ) = ¹ _p

^where

g(G _l / F ^q )

^and

g(G _l+1 / F ^q )

^denote ^the^genus ^of

two onseutive algebrai funtion elds in

T ₂ / F q

^.

Proof. By Theorem 2.2 in[2℄, we have

g(F _k,s / F q ² ) ≤ ^g(F ^k+1 _p ^r − ^/ s ^F ^q ² ⁾ + 1

^with

g(F _k+1 / F q ² ) = g _k+1 ≤ q ^k+1 + q ^k

^. ^Then, âs^theâlgebrai ^funtionêld

F _k,s / F q ²

îsâônstantêld êxtension

of

G _k,s / F q

k

^and

s = 0, 1

^or

2

^, ^the ^algebrai ^funtion ^elds

F _k,s / F q ²

^and

G _k,s / F q

^have ^the ^same ^genus. ^So, ^the ^inequality ^satised ^by ^the ^genus

g(F _k,s / F q ² )

^is ^also

true for the genus

g(G _k,s / F ^q )

^. ^Moreover, ^the ^number ôf ^plaes ôf ^degree ône

B 1 (F _k,s / F q ² )

of

F _k,s / F q ²

^is ^suh ^that

B ₁ (F _k,s / F q ² ) ≥ (q ² − 1)q ^k−1 p ^s

^. ^Then, âs^theâlgebrai ^funtion êld

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F _k,s / F q ²

îs â ônstant êld êxtension ôf

G _k,s / F q

^of ^degree

2

^,ît îs ^lear ^that ^for âny înteger

k

^and

s

^, ^we ^have

B ₁ (G _k,s / F ^q ) + 2B ₂ (G _k,s / F ^q ) ≥ (q ² − 1)q ^k−1 p ^s

^. ^Moreover, ^we ^know ^that

β ₁ (T ₂ / F q ) = 0

^by ^Lemma ^{3.2 .} ^But

B ₁ (G _k,s / F q ) + 2B ₂ (G _k,s / F q ) = B ₁ (F _k,s / F q ² )

^and ^as ^by

[4℄,

β ₁ (T ₁ / F q ² ) = A(q ² )

^,^we ^have

β ₂ (T ₂ / F ^q ) = ¹ ₂ (q − 1)

^.

Inpartiular,thefollowing resultholds:

Corollary 3.4. Foranyprimepower

q

^,^there êxistsâ ^sequene ôfâlgebrai^funtionêlds

dened over the nite eld

F ^q

^reahing ^the Generalized Drinfeld-Vladut bound.

3.2. Sequenes

F / F 2

^with

β ₄ ( F / F 2 ) = ³ ₄ = A ₄ (2)

^. ^Wê ûse^the^notationsôf ^the^previ-

ousparagraph onerningthetowers

T ₁ / F q ²

^and

T ₂ / F q

^. ^Now^we ^suppose^that

q = p ²

^and^we

askthefollowing question: isthe desent of thedenitioneld ofthetower

T ₁ / F q ²

^from

F q ²

to

F p

^possible? ^The^following ^result^givesâ^positive ânswer ^for^theâse

p = 2

^.

p = 2

^. ^If

q = p ²

^, ^the ^desent ôf ^the ^denition êld ôf ^the ^tower

T ₁ / F q ²

^from

F q ²

^to

F p

îs ^possible. ^More ^preisely, ^there êxists â ^tower

T ₃ / F p

^dened ^over

F p

given by a sequene:

T ₃ / F p = H _1,0 ⊆ H _1,1 ⊆ H _2,0 ⊆ H _2,1 ⊆ ...

dened over the onstant eld

F p

^and^related ^to ^the ^towers

T ₁ / F q ²

^and

T ₂ / F q

^by

F _k,s = F q ² H _k,s f or all k and s = 0, 1

^or

2, G _k,s = F ^q H _k,s f or all k and s = 0, 1

^or

2,

namely

F _k,s / F q ²

îs ^the ônstant êld êxtension ôf

G _k,s / F ^q

^and

H _k,s / F ^p

^and

G _k,s / F ^q

^is ^the

onstant eld extension of

H _k,s / F p

^.

Proof. Let

x ₁

^be ^atransendent element over

F 2

^and ^let^us ^set

H ₁ = F 2 (x ₁ ), G ₁ = F 4 (x ₁ ), F ₁ = F 16 (x ₁ ).

Wedene reursively for

k ≥ 1

1.

z _k+1

^suh^that

z _k+1 ⁴ + z _k+1 = x ⁵ _k

^,

2.

t _k+1

^suh^that

t ² _k+1 + t _k+1 = x ⁵ _k

(or alternatively

t _k+1 = z _k+1 (z _k+1 + 1)

^),

3.

x _k = z _k /x _k ₋ ₁

^if

k > 1

⁽

x ₁

^is ^yet^dened),

4.

H _k,1 = H _k,0 (t _k+1 ) = H _k (t _k+1 )

^,

H _k+1,0 = H _k+1 = H _k (z _k+1 )

^,

G _k,1 = G _k,0 (t _k+1 ) = G _k (t _k+1 )

^,

G _k+1,0 = G _k+1 = G _k (z _k+1 )

^,

F _k,1 = F _k,0 (t _k+1 ) = F _k (t _k+1 )

^,

F _k+1,0 = F _k+1 = F _k (z _k+1 )

^.

By[3℄,the tower

T ₁ / F q ² = (F _k,i / F q ² ) _k≥1,i=0,1

isthedensiedGaria-Stihtenoth'stowerover

F 16

^and^the^two^other^towers

T ₂ / F q

^and

T ₃ / F p

arerespetively thedesent of

T ₁ / F q ²

^over

F 4

^and ^over

F 2

^.

(13)

q = p ² = 4

^. ^Fôr âny înteger

k ≥ 1

s

^suh ^that

s = 0, 1

^or

2

^,^the^algebrai^funtion^eld

H _k,s / F p

ⁱⁿ^the^tower

T ₃

^has^a^genus

g(H _k,s / F p ) = g _k,s

with

B 1 (H _k,s / F ^p )

^plaes ^of ^degree ^one,

B 2 (H _k,s / F ^p )

^plaes ^of ^de^gree ^two ^and

B 4 (H _k,s / F ^p )

plaes of degree

4

^suh ^that:

1.

g(H _k,s / F p ) ≤ ^g(H _p ^k+1 r − s ^/ ^F ^p ⁾ + 1

^with

g(H _k+1 / F p ) = g _k+1 ≤ q ^k+1 + q ^k .

2.

B ₁ (H _k,s / F p ) + 2B ₂ (H _k,s / F p ) + 4B ₄ (H _k,s / F p ) ≥ (q ² − 1)q ^k ⁻ ¹ p ^s

^.

3.

β ₄ (T ₃ / F p ) = lim _g _k,s _→ ₊ _∞ ^B ⁴ ^(H _g ^k,s ^/ ^F ^p ⁾

k,s = ¹ ₄ (p ² − 1) = ³ ₄ = A ₄ (2)

^.

4.

d(T ₃ / F ^p ) = lim _l _→ ₊ _∞ _g(H ^g(H ^l ^/ ^F ^p ⁾

l+1 /F p ) = ¹ ₂

^where

g(H _l / F ^p )

^and

g(H _l+1 / F ^p )

^denote ^the ^genus

of two onseutive algebrai funtionelds in

T ₃ / F ^p

^.

Proof. By Theorem 2.2 in[2℄, we have

g(F _k,s / F q ² ) ≤ ^g(F ^k+1 _p r − ^/ s ^F ^q ² ⁾ + 1

^with

g(F _k+1 / F q ² ) = g _k+1 ≤ q ^k+1 +q ^k

^. ^Then,âs^theâlgebrai^funtionêld

F _k,s / F q ²

îsâônstantêldêxtensionôf

H _k,s / F ^p

^,^for^any^integer

k

^and

s = 0, 1

^or

2

^,^the^algebrai^funtion^elds

F _k,s / F q ²

^and

H _k,s / F ^p

havethesame genus. So,theinequalitysatisedbythegenus

g(F _k,s / F q ² )

^is^also ^true^for^the

genus

g(H _k,s / F p )

^. ^Moreover, ^the ^number ôf ^plaes ôf ^degree ône

B ₁ (F _k,s / F q ² )

^of

F _k,s / F q ²

^is

suh that

B 1 (F _k,s / F q ² ) ≥ (q ² − 1)q ^k ⁻ ¹ p ^s

^. ^Then, âs ^the âlgebrai ^funtion êld

F _k,s / F q ²

^is ^a

onstanteldextensionof

H _k,s / F p

^of^degree

4

^,îtîs^lear^that^forânyînteger

k

^and

s

^,^we^have

B ₁ (H _k,s / F ^p ) + 2B ₂ (H _k,s / F ^p ) + 4B ₄ (H _k,s / F ^p ) ≥ (q ² − 1)q ^k−1 p ^s

^. ^Moreover, ^we^have ^shownⁱⁿ

theproof of Lemma 3.2 thatfor any integer

k ≥ 1

^and ^any

0 ≤ s ≤ 2

^the ^number^of ^plaes

of degree one

B ₁ (G _k,s / F q )

^of

G _k,s / F q

îs ^lessôr êqual ^to

2q ²

^and ^so

β ₁ (T ₂ / F q ) = 0

^. ^Then, ^as

thealgebrai funtion eld

G _k,s / F ^q

îs â ônstant êld êxtension ôf

H _k,s / F ^p

^of ^degree

2

^,^it ^is

lear that for any integer

k

^and

s

^,^we ^have

B ₁ (H _k,s / F p ) + 2B ₂ (H _k,s / F p ) = B ₁ (G _k,s / F q )

^and

so

β ₁ (T ₃ / F p ) = β ₂ (T ₃ / F p ) = 0

^. ^Moreover,

B ₁ (H _k,s / F p ) + 2B ₂ (H _k,s / F p ) + 4B ₄ (H _k,s / F p ) = B 1 (F _k,s / F q ² )

^and ^as^by^[4℄,

β 1 (T 1 / F q ² ) = A(q ² )

^,^we ^have

β 4 (T 3 / F ^p ) = A(p ⁴ ) = p ² − 1

^.

Corollary 3.7. Let

T ₃ / F ² = (H _k,s / F ² ) _k _∈N _,s=0,1,2

^be ^the ^tower ^dened ^ab^ove. ^Then ^the

tower

T ₃ / F 2

^is ^an asymptotially exat sequene of algebrai funtion elds dened over

F 2

witha maximal density(for a tower).

Proof. It follows from (4 )ofProposition 3.3 .

4. Open questions

1. Find asymptotially exat sequenes of algebrai funtion elds dened over any nite

eld

F ^q

^, ^of ^type

β = (β 1 , β 2 , ...., β m , ...)

^with ^several

β i > 0

^, ^attaining ^the generalized Drinfeld-Vladutbound(i.e maximalor not).

2. Find asymptotially exat sequenes of algebrai funtion elds dened over any nite

eld

F q

^, ^attaining ^the Drinfeld-Vladut bound of order

r

^for ^any ^integer

r > 2

^(exept

ase q=2and r=4 solvedinthis paper).

(14)

3. Find expliit asymptotially exat sequenes of algebrai funtion elds (not Artin-

Shreiertype)dened overanynite eld

F q

^having^the ^good^preeding properties.

Referenes

[1℄ StéphaneBallet.Quasi-optimalalgorithmsformultipliationintheextensionsof

F ¹⁶

^of^degree

13

^,

14

^,^and

15

^.^Journalôf^PureândÂppliedÂlgebra,171:149164,2002.

[2℄ StéphaneBallet.Lowinreasingtowerofalgebraifuntioneldsandbilinearomplexityofmul-

tipliationinanyextensionof

F ^q

^. ^Finite^Fields^and^Their Appliations, 9:472478,2003.

[3℄ StéphaneBallet,DominiqueLeBrigand,andRobertRolland.Onanappliationofthedenition

eld desent of a tower of funtion elds. In Proeedings of the Conferene Arithmeti, Geometry

andCodingTheory(AGCT2005),volume21,pages187203.SoiétéMathématiquedeFrane,sér.

Séminaireset Congrès,2009.

[4℄ Arnaldo Gariaand Henning Stihtenoth. A towerof artin-shreierextensions of funtion elds

attainingthedrinfeld-vladutbound.InventionesMathematiae,121:211222,1995.

[5℄ Yasutaka Ihara. Some remarks on the number of rational points of algebrai urves overnite

elds.journal ofFa.Si. Tokyo,28,721-724,1981.

[6℄ Gilles Lahaud and Mireille Martin-Deshamps.Nombre depointsdes jaobiennessur unorps

nis.AtaArithmetia,56(4):329340,1990.

[7℄ MihaelRosenbloom and Mihael Tsfasman. Multipliative latties in global elds. Inventiones

Mathematiae,17:5354,1983.

[8℄ Jean-PierreSerre.Thenumberof rationnalpoints onurvesovernite elds, 1983. NotesbyE.

Bayer,Prineton Letures.

[9℄ Henning Stihtenoth. Algebrai Funtion Fields and Codes. Number 314 in Letures Notes in

Mathematis.Springer-Verlag,1993.

[10℄ Mihael Tsfasman. Some remarks on the asymptoti number of points. In H. Stihtenoth and

M.A. Tsfasman,editors, Coding Theory andAlgebrai Geometry, volume1518 ofLeture Notes in

Mathematis,pages178192,Berlin,1992.Springer-Verlag.ProeedingsofAGCT-3onferene,June

17-21,1991,Luminy.

[11℄ MihaelTsfasmanandSergueiVladut.Asymptotipropertiesofzeta-funtions.JournalofMath-

ematial Sienes,84(5):14451467,1997.

[12℄ SergueiVladut.Anexhaustionboundforalgebrai-geometrimodularodes.Problemsof Infor-

mationTransmission,23:2234,1987.

[13℄ SergueiVladutandVladimirDrinfeld.Numberofpointsofanalgebraiurve.FunktsionalAnal

iPrilozhen,17:5354,1983.

17mai2010

Stéphane Ballet,Institut deMathématiques de Luminy, Case 907, 13288 Marseille edex 9

E-mail:stephane.balletunivmed.fr

Robert Rolland, InstitutdeMathématiquesdeLuminy,Case907,13288Marseilleedex9etAssoiation

ACrypTA

•

^E-mail^:robert.rollandarypta.fr

Families of curves over finite fields

F q

r

r

≥ 1

r > 1

F q

q

r = 2

q = 2

r = 4

F q

F q

F q

r

r

≥ 1

r > 1

r = 2

F q

q

r = 4

q = 2

F q

F q

F q

q

F q

q

F / F q = (F k / F q ) k ≥ 1

F k / F q

F q

g k = g(F k / F q )

g k

F / F q

m ≥ 1

β m ( F / F q ) = lim

k → + ∞

B m (F k / F q ) g k

B m (F k / F q )

m

F k / F q

β = (β 1 , β 2 , ...., β m , ...)

F / F q

F / F q = (F k / F q ) k ≥ 1

F k / F q

F q

g k = g(F k / F q )

β

β

F / F q

X ∞ m=1

mβ m q m/2 − 1 ≤ 1

r

B r (q, g) = max { B r (F/ F q ) | F/ F q

F q

g } .

A r (q) = lim sup

g → + ∞

B r (q, g)

g .

r

A r (q) ≤ 1

r (q r 2 − 1).

F / F q

r

r

β = (0, 0, ...., 0, β r = 1

r (q r 2 − 1), 0, ...).

F / F q

β

q

q

F / F q = (F k / F q ) k≥1

F k / F q

g k = g(F k / F q )

F q

g k

F / F q

d( F / F q ) = lim inf

F ^q

F ^q

F ^q

F ^q

F ^q

F ^q

F ^q

F / F q = (F _k / F q ) _k _≥ ₁

F _k / F ^q

F ^q

g _k = g(F _k / F ^q )

g _k

F / F ^q

β _m ( F / F q ) = lim

B _m (F _k / F q ) g _k

B m (F _k / F ^q )

F _k / F ^q

β = (β ₁ , β ₂ , ...., β _m , ...)

F / F ^q

F / F ^q = (F _k / F ^q ) _k _≥ ₁

F _k / F q

g _k = g(F _k / F q )

mβ _m q ^m/2 − 1 ≤ 1

B _r (q, g) = max { B _r (F/ F q ) | F/ F q

A _r (q) = lim sup

B _r (q, g)

A _r (q) ≤ 1

r (q ^r ² − 1).

F / F ^q

β = (0, 0, ...., 0, β _r = 1

r (q ^r ² − 1), 0, ...).

F / F ^q

F / F q = (F _k / F q ) _k≥1

F _k / F q

g _k = g(F _k / F q )

g _k

F / F ^q

g _k g _k+1 .

F / F ^q

(0, ..., ¹ _r (q ^r ² − 1), 0, ..., 0, ...)

h _k

F ^q

β = (0, ..., ¹ _r (q ^r ² − 1), 0, ..., 0, ...)

β = (0, ..., ¹ _r (q ^r ² − 1), 0, ..., 0, ...)

F / F q = (F _k / F q ) _k _≥ ₁

g _k

f (g _k ) = o(log(g _k ))

1 g _k

f X (g _k ) m=1

mB _m (F _k ) q ^m/2 − 1 ≤ 1.

F / F ^q = (F _k / F ^q ) _k _≥ ₁

β _r ( F / F q ) = ¹ _r (q ^r ² − 1)

β _m ( F / F q ) = 0

β _m ( F / F ^q ) = 0

1 g _k

jB j (F _k ) q ^j − 1 ≤ 1.

rB _r (F _k )

g _k (q ^r ² − 1) = lim

rB _r (F _k )

g _k (q ^r ² − 1) = 1.

1 g _k

jB _j (F _k )

q ^j − 1 = lim

1 g _k