FAMILIES OF CURVES OVER ANY FINITE FIELD ATTAINING THE ...iml.univ-mrs.fr/editions/preprint2010/files/ballet_rolland_gdvb15.pdf · Résumé.— Nous prouvons l’existencede familles

FAMILIES OF CURVES OVER ANY FINITEFIELD ATTAINING THE GENERALIZED

DRINFELD-VLADUT BOUND.

by

Stéphane Ballet & Robert Rolland

Abstract. — We prove the existence of asymptotically exact sequencesof algebraic function fields defined over any finite field Fq having anasymptotically maximum number of places of a degree r where r is an in-teger ≥ 1. It turns out that these particular families have an asymptoticclass number widely greater than all the Lachaud - Martin-Deschampsbounds when r > 1. We emphasize that we exhibit explicit asymptoti-cally exact sequences of algebraic function fields over any finite field Fq,in particular when q is not a square, with r = 2. We explicitly constructan example for q = 2 and r = 4. We deduce of this study that for anyfinite field Fq there exists a sequence of algebraic function fields definedover any finite field Fq reaching the generalized Drinfeld - Vladut bound.

Résumé.— Nous prouvons l’existence de familles asymptotiquementexactes de corps de fonctions algébriques définis sur un corps fini Fq

qui ont un nombre maximum de places de degré r où r est un entier≥ 1. Il s’avère que pour ces familles particulières le nombre de classesest asymptotiquement beaucoup plus grand que la borne générale deLachaud - Martin-Deschamps quand r > 1. Pour r = 2 nous construisonsexplicitement des suites de corps de fonctions algébriques sur tout corpsfini Fq qui sont asymptotiquement exactes et ceci même lorsque q n’estpas un carré. Nous construisons un exemple pour r = 4 dans le cas oùq = 2. Nous déduisons de cette étude que pour tout corps fini Fq il existeune suite de corps de fonctions algébriques sur Fq qui atteint la bornede Drinfeld - Vladut généralisée.

Key words and phrases. — finite field, function field, asymptotically exactsequence, class number, tower of function fields.

2 STÉPHANE BALLET & ROBERT ROLLAND

1. Introduction

1.1. General context and main result. — When, for a given finiteground field, the sequence of the genus of a sequence of algebraic func-tion fields tends to infinity, there exist asymptotic formulae for differentnumerical invariants. In [10], Tsfasman generalizes some results on thenumber of rational points on the curves (due to Drinfeld-Vladut [13],Ihara [5], and Serre [8]) and on its Jacobian (due to Vladut [12], Rosem-bloom and Tsfasman [7]). He gives a formula for the asymptotic numberof divisors, and some estimates for the number of points in the Poincaréfiltration.

For this purpose, he introduced the notion of asymptotically exactfamily of curves defined over a finite field. It turns out that this notionis very fruitful. Indeed, for such sequences of curves, we can evaluateenough precisely the asymptotic behavior of certain numerical invari-ants, in particular the asymptotic number of effective divisors and theasymptotic class number. Moreover, when these sequences are maximal(cf. the basic inequality (1)) they have an asymptotically large classnumber. In particular, it is the case when they have a maximal numberof places of a given degree. Unfortunately the existence of such sequencesis not known for any finite field Fq, in particular when q is not a square(cf. Remark 5.2 in [11]). In this paper, we mainly explicitly constructexamples of asymptotically exact sequences of algebraic function fieldsfor any finite field, so proving the existence of maximal asymptoticallyexact sequences of curves defined over any finite field Fq where q is not asquare. So, we answer in Corollary 3.4 the question asked by Tsfasmanand Vladut in [11, Remark 5.2] .

1.2. Notation and detailed questions. — Let us recall the notionof asymptotically exact family of curves defined over a finite field in thelanguage of algebraic function fields.

Definition 1.1 (Asymptotically Exact Sequence)Let F/Fq = (Fk/Fq)k≥1 be a sequence of algebraic function fields Fk/Fq

defined over a finite field Fq of genus gk = g(Fk/Fq). We suppose thatthe sequence of the genus gk is an increasing sequence growing to infinity.The sequence F/Fq is said to be asymptotically exact if for all m ≥ 1the following limit exists:

βm(F/Fq) = limk→+∞

Bm(Fk/Fq)gk

FAMILIES OF CURVES OVER FINITE FIELDS 3

where Bm(Fk/Fq) is the number of places of degree m on Fk/Fq.The sequence β = (β1, β2, ...., βm, ...) is called the type of the asymp-

totically exact sequence F/Fq.

Definition 1.2 (Generalized Drinfeld-Vladut Bound)Let F/Fq = (Fk/Fq)k≥1 be a asymptotically exact sequence of algebraic

function fields Fk/Fq defined over a finite field Fq of genus gk = g(Fk/Fq)and of type β. The sequence β (respectively the sequence F/Fq) is saidmaximal (of maximal type) when the following basic inequality, calledGeneralized Drinfeld-Vladut bound,

(1)∞∑m=1

mβmqm/2 − 1

≤ 1

is reached.

Definition 1.3 (Drinfeld-Vladut Bound of order r)Let

Br(q, g) = max{Br(F/Fq) | F/Fq is a function field over Fq of genus g}.Let us set

Ar(q) = lim supg→+∞

Br(q, g)

g.

We have the Drinfeld-Vladut Bound of order r :

Ar(q) ≤1

r(q

r2 − 1).

Remark 1.4. — Note that if for a family F/Fq of algebraic functionfields, there exists an integer r such that the Drinfeld-Vladut Boundof order r is reached, then this family is a maximal asymptotically ex-act sequence namely attaining the Generalized Drinfeld-Vladut bound.Moreover, its type is

β = (0, 0, ...., 0, βr =1

r(q

r2 − 1), 0, ...).

Tsfasman and Vladut in [11] made use of these notions to obtain newgeneral results on the asymptotic properties of zeta functions of curves.

Note that a simple diagonal argument proves that each sequence ofalgebraic function fields of growing genus, defined over a finite field ad-mits an asymptotically exact subsequence. However until now, we do notknow if there exists an asymptotically exact sequence F/Fq with a maxi-mal β sequence when q is not a square. Moreover, the diagonal extraction


method is not really suitable for the two following reasons. First, in gen-eral we do not obtain by this process an explicit asymptotically exactsequence of algebraic function fields defined over an arbitrary finite field,in particular when q is not a square. Moreover, we have no control onthe growing of the genus in the extracted sequence of algebraic functionfields defined over an arbitrary finite field. More precisely, let us definethe notion of density of a family of algebraic function fields defined overa finite field of growing genus:

Definition 1.5. — Let F/Fq = (Fk/Fq)k≥1 be a sequence of algebraicfunction fields Fk/Fq of genus gk = g(Fk/Fq), defined over Fq. We sup-pose that the sequence of genus gk is an increasing sequence growing toinfinity. Then, the density of the sequence F/Fq is

d(F/Fq) = lim infk→+∞

gkgk+1

.

A high density can be a useful property in some applications of se-quences or towers of function fields. Until now, no explicit examples ofdense asymptotically exact sequences F/Fq have been pointed out unlessfor the case q square and type β = (

√q − 1, 0, · · · ).

1.3. New results. — In this paper, we answer the questions men-tioned above. More precisely, for any prime power q we contruct examplesof asymptotically exact sequences attaining the Drinfeld-Vladut Boundof order 2. We deduce that for any prime power q (in particular when qis not a square), there exists a maximal asymptotically exact sequencesof algebraic functions fields namely attaining the Generalized Drinfeld-Vladut bound. We also construct for q = 2 and r = 4 an exampleof maximal asymptotically exact sequence attaining the Drinfeld-VladutBound of order 4. Then, we show that the asymptotically exact sequencesof considered maximal type have an asymptotically large class numberwith respect to the Lachaud - Martin-Deschamps bounds. The paper isorganized as follows. In section 2, we study the general families of asymp-totically exact sequences of algebraic function fields defined over an arbi-trary finite field Fq of maximal type (0, ..., 1

r(q

r2 − 1), 0, ..., 0, ...) where r

is an integer ≥ 1, under the assumption of their existence. In particular,we study for these general families the behavior of the asymptotic classnumber hk, and we compare our estimation to the general known boundsof Lachaud - Martin-Deschamps (cf. [6]). More precisely, we show that


for such families, if they exist, the asymptotic class number is widelygreater than the general bounds of Lachaud - Martin-Deschamps whenr > 1. Next, in section 3, we constructively prove the existence of asymp-totically exact sequences of algebraic function fields defined over any ar-bitrary finite field Fq of maximal type β = (0, ..., 1

r(q

r2 − 1), 0, ..., 0, ...)

with r = 2. Moreover, we exhibit the example of an asymptotically ex-act sequence of algebraic function fields defined over Fq of maximal typeβ = (0, ..., 1

r(q

r2 − 1), 0, ..., 0, ...) with q = 2 and r = 4. For this pur-

pose, we use towers of algebraic function fields coming from the descentof the densified towers of Garcia-Stichtenoth (cf. [4] and [1]). Note thatall these examples are explicit and consist on very dense asymptoticallyexact towers algebraic function fields (maximally dense tower for q = 2and r = 4).

2. General results

2.1. Particular families of asymptotically exact sequences. —First, let us recall certain asymptotic results. Let us first give the follow-ing result obtained by Tsfasman in [10]:

Proposition 2.1. — Let F/Fq = (Fk/Fq)k≥1 be a sequence of algebraicfunction fields of increasing genus gk growing to infinity. Let f be afunction from N to N such that f(gk) = o(log(gk)). Then

(2) lim supgk→+∞

1

gk

f(gk)∑m=1

mBm(Fk)

qm/2 − 1≤ 1.

Now, we can easily obtain the following result as immediate conse-quence of Proposition 2.1:

Theorem 2.2. — Let r be an integer ≥ 1 and F/Fq = (Fk/Fq)k≥1 be asequence of algebraic function fields of increasing genus defined over Fqsuch that βr(F/Fq) = 1

r(q

r2 − 1). Then βm(F/Fq) = 0 for any integer

m 6= r. In particular, the sequence F/Fq is asymptotically exact.

Proof. — Let us fix m 6= r and let us prove that βm(F/Fq) = 0. We useProposition 2.1 with the constant function f(g) = s = max(m, r). Thenwe get

lim supk→+∞

1

gk

s∑j=1

jBj(Fk)

qj − 1≤ 1.


But by hypothesis

lim supk→+∞

rBr(Fk)

gk(qr2 − 1)

= limk→+∞

rBr(Fk)

gk(qr2 − 1)

= 1.

Then

lim supgk→+∞

1

gk

∑1≤j≤s;j 6=r

jBj(Fk)

qj − 1= lim

gk→+∞

1

gk

∑1≤j≤s;j 6=r

jBj(Fk)

qj − 1= 0.

ButBm(Fk)

gk≤ (qm − 1)

m

(1

gk

∑1≤j≤s;j 6=r

jBj(Fk)

qj − 1

).

Hencelim

gk→+∞

Bm(Fk)

gk= 0.

Note that for any k the following holds:

B1(Fk/Fqr) =∑i|r

iBi(Fk/Fq).

Then if βr(F/Fq) = 1r(q

r2−1), by Theorem 2.2 we conclude that β1(F/Fqr)

exists and thatβ1(F/Fqr) = (q

r2 − 1).

In particular the sequence F/Fqr reaches the classical Drinfeld-Vladutbound and consequently qr is a square.

If β1(F/Fqr) exists then it does not necessarily imply that βr(F/Fq)exists but only that limk→+∞

Pm|r mBm(Fk/Fq)

gkexists. In fact, this converse

depends on the defining equations of the algebraic function fields Fk/Fq.Now, let us give a simple consequence of Theorem 2.2.

Proposition 2.3. — Let r and i be integers ≥ 1 such that i divides r.Suppose that

F/Fq = (Fk/Fq)k≥1

is an asymptotically exact sequence of algebraic function fields definedover Fq of type (β1 = 0, . . . , βr−1 = 0, βr = 1

r(q

r2−1), βr+1 = 0, . . .). Then

the sequence F/Fqi = (Fk/Fqi)k≥1 of algebraic function field defined overFqi is asymptotically exact of type (β1 = 0, .., β r

i−1 = 0, β r

i= i

r(q

r2 −

1), β ri+1 = 0, . . .).


Proof. — Let us remark that by [9, Lemma V.1.9, p. 163], if P is a placeof degree r′ of F/Fq, there are gcd((r′, i)) places of degree r′

gcd(r′,i)over P

in the extension F/Fqi . As we are interested by the places of degree r/iin F/Fqi , let us introduce the set

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

Then,

Br/i(F/Fqi) =∑r′∈S

ir′

rBr′(F/Fq).

We know that all the βj(F/Fq) = 0 but βr(F/Fq) = 1r(q

r2 − 1). Then

βr/i(F/Fqi) = iβr(F/Fq),

βr/i(F/Fqi) =i

r

(q

r2 − 1

).

2.2. Number of points of the Jacobian. — Now, we are interestedby the Jacobian cardinality of the asymptotically exact sequences F/Fq =(Fk/Fq)k≥1 of type (0, .., 0, 1

r(q

r2 − 1), 0, ..., 0).

Let us denote by hk = hk(Fk/Fq) the class number of the algebraicfunction field Fk/Fq. Let us consider the following quantities introducedby Tsfasman in [10]:

Hinf (F/Fq) = lim infk→+∞

1

gklog hk

Hsup(F/Fq) = lim supk→+∞

1

gklog hk

If they coincides, we just write:

H(F/Fq) = limk→+∞

1

gklog hk = Hinf (F/Fq) = Hsup(F/Fq).

Then under the assumptions of the previous section, we obtain thefollowing result on the sequence of class numbers of these families ofalgebraic function fields:

Theorem 2.4. — Let F/Fq = (Fk/Fq)k≥1 be a sequence of algebraicfunction fields of increasing genus defined over Fq such that βr(F/Fq) =


1r(q

r2 − 1) where r is an integer. Then, the limit H(F/Fq) exists and we

have:


1

gklog hk = log

qqr2

(qr − 1)1r(q

r2−1)

.

Proof. — By Corollary 1 in [10], we know that for any asymptotically ex-act family of algebraic function fields defined over Fq, the limit H(F/Fq)exists and


1

gklog hk = log q +

∞∑m=1

βm. logqm

qm − 1.

Hence, the result follows from Theorem 2.2.

Corollary 2.5. — Let F/Fq = (Fk/Fq)k≥1 be a sequence of algebraicfunction fields of increasing genus defined over Fq such that βr(F/Fq) =1r(q

r2 −1) where r is an integer. Then there exists an integer k0 such that

for any integer k ≥ k0,hk > qgk .

Proof. — By Theorem 2.4, we have limk→+∞(hk)1

gk = qqr2

(qr−1)1r (q

r2−1)

. But

qqr2

(qr − 1)1r(q

r2−1)

>qq

r2

(qr)1r(q

r2−1)

= q.

Hence, for a sufficiently large k0, we have for k ≥ k0 the following in-equality

(hk)1

gk > q.

Let us compare this estimation of hk to the general lower bounds givenby G. Lachaud and M. Martin-Deschamps in [6].

Theorem 2.6 (Lachaud - Martin-Deschamps bounds)Let X be a projective irreducible and non-singular algebraic curve de-

fined over the finite field Fq of genus g. Let JX be the Jacobian of X andh the class number h = |JX(Fq)|. Then

(1) h ≥ L1 = qg−1 (q−1)2

(q+1)(g+1),

(2) h ≥ L2 =(√

q − 1)2 gg−1−1

g

|X(Fq)|+q−1

q−1,


(3) if g > √q/2 and if B1(X/Fq) ≥ 1, then the following holds:h ≥ L3 = (qg − 1) q−1

q+g+gq.

Then we can prove that for a family of algebraic function fields satisfy-ing the conditions of Corollary 2.5, the class numbers hk greatly exceedsthe bounds Li. More precisely

Proposition 2.7. — Let F/Fq = (Fk/Fq)k≥1 be a sequence of algebraicfunction fields of increasing genus defined over Fq such that βr(F/Fq) =1r(q

r2 − 1) where r is an integer. Then

(1) for i = 1, 3

limk→+∞hkLi

= +∞,

(2) for i = 2 the following holds:(a) if r>1 then

limk→+∞hkL2

= +∞,

(b) if r=1 thenhkL2

≥ 2.5.

Proof. — (1) case i = 1: the following holds

L1 = qgk−1 (q − 1)2

(q + 1)(gk + 1)= qgk

(q − 1)2

q(q + 1)(gk + 1)<

qgk

(gk + 1),

so, using the previous corollary 2.5, we conclude that for k largehkL1

> gk

and consequently

limk→+∞

hkL1

= +∞;

(2) case i = 2:(a) case r = 1: we bound the number of rational points using the

Weil bound. More precisely

L2 = (q + 1− 2√q)

qgk−1 − 1

gk

B1(Fk/Fq) + q − 1

q − 1≤

(q + 1− 2√q)

qgk−1 − 1

gk

2q + 2gk√q

q − 1<


2q + 1− 2

√q

(q − 1)√qqgk

(1 +

√q

gk

);

but for all q ≥ 2

2q + 1− 2

√q

(q − 1)√q

< 0.4,

thenL2 < 0.4

(1 +

√q

gk

)qgk ,

hencehkL2

> 2.5gk

gk +√q

which gives the result;(b) case r > 1: in this case we know that

limk→+∞

B1(Fk/Fq)gk

= β1(F/Fq) = 0.

ButL2 <

q + 1− 2√q

(q − 1)qqgk

B1(Fk/Fq) + q − 1

gk.

ThenhkL2

>(q − 1)q

q + 1− 2√q

gkB1(Fk/Fq) + q − 1

.

We know that

limk→+∞

gkB1(Fk/Fq) + q − 1

= +∞,

thenlim

k→+∞

hkL2

= +∞.

(3) case i = 3:

L3 = (qg − 1)q − 1

q + g + gq<qgk

gk,

then for k largehkL3

> gk

and consequently

limk→+∞

hkL1

= +∞.


As we see, if F/Fq = (Fk/Fq)k≥1 satisfies the assumptions of Theorem2.4, we have hk > qgk > qgk−1 (q−1)2

(q+1)gkfor k ≥ k0 sufficiently large. In fact,

the value k0 depends at least on the values of r and q and we can notknow anything about this value in the general case.

Remark: We can remark that the class number of these families isvery near the Lachaud - Martin-Deschamps bound L2 when r = 1 butis much greater than the Lachaud - Martin-Deschamps bound L2 whenr > 1.

3. Examples of asymptotically exact towers

Let us note Fq2 a finite field with q = pr and r an integer.

3.1. Sequences F/Fq with β2(F/Fq) = 12(q − 1) = A2(q). — We

consider the Garcia-Stichtenoth’s tower T0 over Fq2 constructed in [4].Recall that this tower is defined recursively in the following way. We setF1 = Fq2(x1) the rational function field over Fq2 , and for i ≥ 1 we define

Fi+1 = Fi(zi+1),

where zi+1 satisfies the equation

zqi+1 + zi+1 = xq+1i ,

withxi =

zixi−1

for i ≥ 2.

We consider the completed Garcia-Stichtenoth’s tower T1/Fq2 definedover Fq2 studied in [2] obtained from T0/Fq2 by adjunction of intermediatesteps. Namely we have

T1/Fq2 : F1,0 ⊂ · · · ⊂ Fi,0 ⊂ Fi,1 ⊂ · · · ⊂ Fi,s ⊂ · · · ⊂ Fi+1,0 ⊂ · · ·with s = 0, ..., r. Note that the steps Fi,0/Fq2 = Fi−1,r/Fq2 are the stepsFi/Fq2 of the Garcia-Stichtenoth’s tower T0/Fq2 and Fi,s/Fq2 (1 ≤ s ≤r − 1) are the intermediate steps considered in [2].

Let us denote by gk the genus of Fk/Fq2 in T0/Fq2 , by gk,s the genus ofFk,s in T1/Fq2 and by B1(Fk,s/Fq2) the number of places of degree one ofFk,s/Fq2 in T1/Fq2 .

Recall that each extension Fk,s/Fk is Galois of degree ps with full con-stant field Fq2 . Moreover, we know by [3, Theorem 4.3] that the descent


of the definition field of the tower T1/Fq2 from Fq2 to Fq is possible. Moreprecisely, there exists a tower T2/Fq defined over Fq given by a sequence:

T2/Fq : G1,0 ⊂ · · · ⊂ Gi,0 ⊂ Gi,1 ⊂ · · · ⊂ Gi,r−1 ⊂ Gi+1,0 ⊂ · · ·defined over the constant field Fq and related to the tower T1/Fq2 by

Fk,s = Fq2Gk,s for all k and s,

namely Fk,s/Fq2 is the constant field extension of Gk,s/Fq. First, letus study the asymptotic behavior of degree one places of the functionfields of the tower T2/Fq and more precisely the existence and the valueof β1(T2/Fq). In order to derive a result on the tower T2/Fq we beginby the study of the terms given by the descent of the classical Garcia-Stichtenoth tower T0/Fq2 . Next we will study the intermediate steps.

Lemma 3.1. — Let T0/Fq2 = (Fk/Fq2)k≥1 the Garcia-Stichtenoth towerdefined over Fq2 and T ′0/Fq = (Gk/Fq)k≥1 its descent over the definitionfield Fq i.e such that for any integer k, Fk = Fq2Gk. Then

β1(T′0/Fq) = lim

k→+∞

B1(Gk/Fq)g(Gk/Fq)

= 0.

Proof. — First, note that if the algebraic function field Fk/Fq2 is a con-stant field extension of Gk/Fq, above any place of degree one in Gk/Fqthere exists a unique place of degree one in Fk/Fq2 . Consequently, let ususe the classification given in [4, p. 221] of the places of degree one ofFk/Fq2 . Let us remark that the number of places of degree one which arenot of type (A), is less or equal to 2q2 (see [4, Remark 3.4]). Moreover,the genus gk of the algebraic function fields Gk/Fq and Fk/Fq2 is suchthat gk ≥ qk by the Hurwitz theorem, then we can focus our study onplaces of type (A). The places of type (A) are built recursively in thefollowing way (cf. [4, p. 220 and Proposition 1.1 (iv)]). Let α ∈ Fq2 \{0}and Pα be the place of F1/Fq2 which is the zero of x1 − α. For anyα ∈ Fq2 \{0} the polynomial equation zq2 + z2 = αq+1 has q distinct rootsu1, · · ·uq in Fq2 , and for each ui there is a unique place P(α,i) of F2/Fq2above Pα and this place P(α,i) is a zero of z2 − ui. We iterate now theprocess starting from the places P(α,i) to obtain successively the placesof type (A) of F3/Fq2 , · · · , Fk/Fq2 , · · · ; then, each place P of type (A)of Fk/Fq2 is a zero of zk − u where u is itself a zero of uq + u = γ forsome γ 6= 0 in Fq2 . Let us denote by Pu this place. Now, we want tocount the number of places P ′u of degree one in Gk/Fq, that is to say theonly places which admit a unique place of degree one Pu in Fk/Fq2 lying


over P ′u. First, note that it is possible only if u is a solution in Fq ofthe equation uq + u = γ where γ is in Fq \ {0}. Indeed, if u is not inFq, there exists an automorphism σ in the Galois group Gal(Fk/Gk) ofthe degree two Galois extension Fk/Fq2 of Gk/Fq such that σ(Pu) 6= Pu.Hence, the unique place of Gk/Fq lying under Pu is a place of degree 2.But uq + u = γ has one solution in Fq if p 6= 2 and no solution in Fqif p = 2. Hence the number of places of degree one of Gk/Fq which arelying under a place of type (A) of Fk/Fq2 is equal to zero if p = 2 andequal to q − 1 if p 6= 2. We conclude that

limk→+∞

B1(Gk/Fq)g(Gk/Fq)

= 0.

Let us remark that in any case, the number of places of degree one ofGk/Fq is less or equal to 2q2.

Now we can get a similar result for the descent T2/Fq of the densifiedtower T1/Fq2 of T0/Fq2 .

Lemma 3.2. — The tower T2/Fq is such that:

β1(T2/Fq) = limg(Gk,s/Fq)→+∞

B1(Gk,s/Fq)g(Gk,s/Fq)

= 0.

Proof. — As Gk+1 = Gk+1,0 is an extension of Gk,s we get B1(Gk,s/Fq) ≤B1(Gk+1/Fq). Using the computation done in the proof of Lemma 3.1 andRemark 3.4 in [4] we have B1(Gk+1/Fq) ≤ 2q2, then we get B1(Gk,s/Fq) ≤2q2. By [2, Corollary 2.2] we know that

liml→+∞

g(Gl+1/Fq)g(Gl/Fq)

= p

where g(Gl/Fq) and g(Gl+1/Fq) denote the genus of two consecutive al-gebraic function fields in T2/Fq. Then for k sufficiently large we get

gk,s ≥ gk,0 = gk.

We conclude that β1(T2/Fq) = 0.

Let us prove a proposition establishing that the tower T2/Fq is asymp-totically exact with good density.

Proposition 3.3. — Let q = pr. For any integer k ≥ 1, for any integers such that s = 0, 1, ..., r, the algebraic function field Gk,s/Fq in the towerT2 has a genus g(Gk,s/Fq) = gk,s with B1(Gk,s/Fq) places of degree one,B2(Gk,s/Fq) places of degree two such that:


(1) g(Gk,s/Fq) ≤ g(Gk+1/Fq)

pr−s + 1 with g(Gk+1/Fq) = gk+1 ≤ qk+1 + qk.

(2) B1(Gk,s/Fq) + 2B2(Gk,s/Fq) ≥ (q2 − 1)qk−1ps.

(3) β2(T2/Fq) = limgk,s→+∞B2(Gk,s/Fq)

gk,s= 1

2(q − 1) = A2(q).

(4) d(T2/Fq) = liml→+∞g(Gl/Fq)

g(Gl+1/Fq)= 1

pwhere g(Gl/Fq) and g(Gl+1/Fq)

denote the genus of two consecutive algebraic function fields in T2/Fq.

Proof. — By Theorem 2.2 in [2], we have g(Fk,s/Fq2) ≤g(Fk+1/Fq2 )

pr−s + 1

with g(Fk+1/Fq2) = gk+1 ≤ qk+1 + qk . Then, as the algebraic functionfield Fk,s/Fq2 is a constant field extension of Gk,s/Fq, for any integer kand s = 0, 1 or 2, the algebraic function fields Fk,s/Fq2 and Gk,s/Fq havethe same genus. So, the inequality satisfied by the genus g(Fk,s/Fq2) isalso true for the genus g(Gk,s/Fq). Moreover, the number of places ofdegree one B1(Fk,s/Fq2) of Fk,s/Fq2 is such that B1(Fk,s/Fq2) ≥ (q2 −1)qk−1ps. Then, as the algebraic function field Fk,s/Fq2 is a constant fieldextension of Gk,s/Fq of degree 2, it is clear that for any integer k and s, wehave B1(Gk,s/Fq) + 2B2(Gk,s/Fq) ≥ (q2 − 1)qk−1ps. Moreover, we knowthat β1(T2/Fq) = 0 by Lemma 3.2. But B1(Gk,s/Fq) + 2B2(Gk,s/Fq) =B1(Fk,s/Fq2) and as by [4], β1(T1/Fq2) = A(q2), we have β2(T2/Fq) =12(q − 1).

In particular, the following result holds:

Corollary 3.4. — For any prime power q, there exists a sequence ofalgebraic function fields defined over the finite field Fq reaching the Gen-eralized Drinfeld-Vladut bound.

3.2. Sequences F/F2 with β4(F/F2) = 34

= A4(2). — We use thenotations of the previous paragraph concerning the towers T1/Fq2 andT2/Fq. Now we suppose that q = p2 and we ask the following question:is the descent of the definition field of the tower T1/Fq2 from Fq2 to Fppossible? The following result gives a positive answer for the case p = 2.

Proposition 3.5. — Let p = 2. If q = p2, the descent of the definitionfield of the tower T1/Fq2 from Fq2 to Fp is possible. More precisely, thereexists a tower T3/Fp defined over Fp given by a sequence:

T3/Fp = H1,0 ⊆ H1,1 ⊆ H2,0 ⊆ H2,1 ⊆ ...


defined over the constant field Fp and related to the towers T1/Fq2 andT2/Fq by

Fk,s = Fq2Hk,s for all k and s = 0, 1 or 2,

Gk,s = FqHk,s for all k and s = 0, 1 or 2,

namely Fk,s/Fq2 is the constant field extension of Gk,s/Fq and Hk,s/Fpand Gk,s/Fq is the constant field extension of Hk,s/Fp.

Proof. — Let x1 be a transcendent element over F2 and let us set

H1 = F2(x1), G1 = F4(x1), F1 = F16(x1).

We define recursively for k ≥ 1

(1) zk+1 such that z4k+1 + zk+1 = x5

k,

(2) tk+1 such that t2k+1 + tk+1 = x5k

(or alternatively tk+1 = zk+1(zk+1 + 1)),(3) xk = zk/xk−1 if k > 1 (x1 is yet defined),(4) Hk,1 = Hk,0(tk+1) = Hk(tk+1), Hk+1,0 = Hk+1 = Hk(zk+1), Gk,1 =

Gk,0(tk+1) = Gk(tk+1), Gk+1,0 = Gk+1 = Gk(zk+1), Fk,1 = Fk,0(tk+1) =Fk(tk+1), Fk+1,0 = Fk+1 = Fk(zk+1).By [3], the tower

T1/Fq2 = (Fk,i/Fq2)k≥1,i=0,1

is the densified Garcia-Stichtenoth’s tower over F16 and the two othertowers T2/Fq and T3/Fp are respectively the descent of T1/Fq2 over F4

and over F2.

Proposition 3.6. — Let q = p2 = 4. For any integer k ≥ 1, for anyinteger s such that s = 0, 1 or 2, the algebraic function field Hk,s/Fp inthe tower T3 has a genus g(Hk,s/Fp) = gk,s with B1(Hk,s/Fp) places ofdegree one, B2(Hk,s/Fp) places of degree two and B4(Hk,s/Fp) places ofdegree 4 such that:

(1) g(Hk,s/Fp) ≤ g(Hk+1/Fp)

pr−s + 1 with g(Hk+1/Fp) = gk+1 ≤ qk+1 + qk.

(2) B1(Hk,s/Fp) + 2B2(Hk,s/Fp) + 4B4(Hk,s/Fp) ≥ (q2 − 1)qk−1ps.

(3) β4(T3/Fp) = limgk,s→+∞B4(Hk,s/Fp)

gk,s= 1

4(p2 − 1) = 3

4= A4(2).

(4) d(T3/Fp) = liml→+∞g(Hl/Fp)

g(Hl+1/Fp)= 1

2where g(Hl/Fp) and g(Hl+1/Fp)

denote the genus of two consecutive algebraic function fields in T3/Fp.


Proof. — By Theorem 2.2 in [2], we have g(Fk,s/Fq2) ≤g(Fk+1/Fq2 )

pr−s + 1

with g(Fk+1/Fq2) = gk+1 ≤ qk+1 + qk . Then, as the algebraic functionfield Fk,s/Fq2 is a constant field extension ofHk,s/Fp, for any integer k ands = 0, 1 or 2, the algebraic function fields Fk,s/Fq2 and Hk,s/Fp have thesame genus. So, the inequality satisfied by the genus g(Fk,s/Fq2) is alsotrue for the genus g(Hk,s/Fp). Moreover, the number of places of degreeone B1(Fk,s/Fq2) of Fk,s/Fq2 is such that B1(Fk,s/Fq2) ≥ (q2 − 1)qk−1ps.Then, as the algebraic function field Fk,s/Fq2 is a constant field extensionof Hk,s/Fp of degree 4, it is clear that for any integer k and s, we haveB1(Hk,s/Fp) + 2B2(Hk,s/Fp) + 4B4(Hk,s/Fp) ≥ (q2− 1)qk−1ps. Moreover,we have shown in the proof of Lemma 3.2 that for any integer k ≥ 1and any 0 ≤ s ≤ 2 the number of places of degree one B1(Gk,s/Fq) ofGk,s/Fq is less or equal to 2q2 and so β1(T2/Fq) = 0. Then, as the al-gebraic function field Gk,s/Fq is a constant field extension of Hk,s/Fp ofdegree 2, it is clear that for any integer k and s, we have B1(Hk,s/Fp) +2B2(Hk,s/Fp) = B1(Gk,s/Fq) and so β1(T3/Fp) = β2(T3/Fp) = 0. More-over, B1(Hk,s/Fp) + 2B2(Hk,s/Fp) + 4B4(Hk,s/Fp) = B1(Fk,s/Fq2) and asby [4], β1(T1/Fq2) = A(q2), we have β4(T3/Fp) = A(p4) = p2 − 1.

Corollary 3.7. — Let T3/F2 = (Hk,s/F2)k∈N,s=0,1,2 be the tower definedabove. Then the tower T3/F2 is an asymptotically exact sequence of alge-braic function fields defined over F2 with a maximal density (for a tower).

Proof. — It follows from (4) of Proposition 3.3.

4. Open questions

(1) Find asymptotically exact sequences of algebraic function fieldsdefined over any finite field Fq, of type β = (β1, β2, ...., βm, ...) with severalβi > 0, attaining the generalized Drinfeld-Vladut bound (i.e maximal ornot).

(2) Find asymptotically exact sequences of algebraic function fieldsdefined over any finite field Fq, attaining the Drinfeld-Vladut bound oforder r for any integer r > 2 (except case q=2 and r=4 solved in thispaper).

(3) Find explicit asymptotically exact sequences of algebraic functionfields (not Artin-Schreier type) defined over any finite field Fq having thegood preceding properties.


References

[1] Stéphane Ballet. Quasi-optimal algorithms for multiplication in the ex-tensions of F16 of degree 13, 14, and 15. Journal of Pure and AppliedAlgebra, 171:149–164, 2002.

[2] Stéphane Ballet. Low increasing tower of algebraic function fields andbilinear complexity of multiplication in any extension of Fq. Finite Fieldsand Their Applications, 9:472–478, 2003.

[3] Stéphane Ballet, Dominique Le Brigand, and Robert Rolland. On an ap-plication of the definition field descent of a tower of function fields. InProceedings of the Conference Arithmetic, Geometry and Coding The-ory (AGCT 2005), volume 21, pages 187–203. Société Mathématique deFrance, sér. Séminaires et Congrès, 2009.

[4] Arnaldo Garcia and Henning Stichtenoth. A tower of artin-schreier exten-sions of function fields attaining the drinfeld-vladut bound. InventionesMathematicae, 121:211–222, 1995.

[5] Yasutaka Ihara. Some remarks on the number of rational points of al-gebraic curves over finite fields. journal of Fac. Sci. Tokyo, 28, 721-724,1981.

[6] Gilles Lachaud and Mireille Martin-Deschamps. Nombre de points desjacobiennes sur un corps finis. Acta Arithmetica, 56(4):329–340, 1990.

[7] Michael Rosenbloom and Michael Tsfasman. Multiplicative lattices inglobal fields. Inventiones Mathematicae, 17:53–54, 1983.

[8] Jean-Pierre Serre. The number of rationnal points on curves over finitefields, 1983. Notes by E. Bayer, Princeton Lectures.

[9] Henning Stichtenoth. Algebraic Function Fields and Codes. Number 314in Lectures Notes in Mathematics. Springer-Verlag, 1993.

[10] Michael Tsfasman. Some remarks on the asymptotic number of points. InH. Stichtenoth and M.A. Tsfasman, editors, Coding Theory and AlgebraicGeometry, volume 1518 of Lecture Notes in Mathematics, pages 178–192,Berlin, 1992. Springer-Verlag. Proceedings of AGCT-3 conference, June17-21, 1991, Luminy.

[11] Michael Tsfasman and Serguei Vladut. Asymptotic properties of zeta-functions. Journal of Mathematical Sciences, 84(5):1445–1467, 1997.

[12] Serguei Vladut. An exhaustion bound for algebraic-geometric modularcodes. Problems of Information Transmission, 23:22–34, 1987.

[13] Serguei Vladut and Vladimir Drinfeld. Number of points of an algebraiccurve. Funktsional Anal i Prilozhen, 17:53–54, 1983.


October 15, 2010

Stéphane Ballet, Institut de Mathématiques de Luminy, Case 907, 13288 Marseillecedex 9 • E-mail : [email protected]

Robert Rolland, Institut de Mathématiques de Luminy, Case 907, 13288 Marseillecedex 9 et Association ACrypTA • E-mail : [email protected]

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