Arithmetic, geometry, cryptography and coding theory

On the fourth moment of theta functions at their central point On the construction of Galois towers Codes defined by forms of degree 2 on quadric varieties in P4Fq Curves of genus 2 with

Gilles Lachaud Christophe Ritzenthaler Michael A Tsfasman

Editors

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Arithmetic, Geometry, Cryptography and Coding

Theory

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American Mathematical Society

Providence, Rhode Island

487

Arithmetic, Geometry, Cryptography and Coding

Theory

International Conference November 5–9, 2007 CIRM, Marseilles, France

Gilles Lachaud Christophe Ritzenthaler Michael A Tsfasman

Editors

Library of Congress Cataloging-in-Publication Data

Arithmetic, geometry, cryptography and coding theory / Gilles Lachaud, Christophe Ritzenthaler, Michael Tsfasman, editors.

p cm — (Contemporary mathematics ; v 487)

Includes bibliographical references.

ISBN 978-0-8218-4716-9 (alk paper)

1 Arithmetical algebraic geometry—Congresses 2 Coding theory—Congresses 3 tography—Congresses I Lachaud, Gilles II Ritzenthaler, Christophe, 1976– III Tsfasman, M.A (Michael A.), 1954–

Cryp-QA242.5.A755 2009

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Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09

On the fourth moment of theta functions at their central point

On the construction of Galois towers

Codes defined by forms of degree 2 on quadric varieties in P4(Fq)

Curves of genus 2 with elliptic differentials and associated Hurwitz spaces

A note on the Giulietti-Korchmaros maximal curve

Subclose families, threshold graphs, and the weight hierarchy of Grassmannand Schubert codes

Characteristic polynomials of automorphisms of hyperelliptic curves

Breaking the Akiyama-Goto cryptosystem

Hyperelliptic curves, L-polynomials, and random matrices

On special finite fields

Borne sur le degr´e des polynˆomes presque parfaitement non-lin´eaires

How to use finite fields for problems concerning infinite fields

On the generalizations of the Brauer-Siegel theorem

v

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The 11th conference on Arithmetic, Geometry, Cryptography and Coding ory (AGC2T 11) was held in Marseilles at the “Centre International de RencontresMath´ematiques” (CIRM), during November 5-9, 2007 This international confer-ence has been a major event in the area of applied arithmetic geometry for morethan 20 years and included distinguished guests J.-P Serre (Fields medal, Abelprize winner), G Frey, H Stichtenoth and other leading researchers in the fieldamong its 77 participants

The-The meeting was organized by the team “Arithm´etique et Th´eorie de l’Information”(ATI) from the “Institut de Math´ematiques de Luminy” (IML) The program con-sisted of 15 invited talks and 18 communications Among them, thirteen wereselected to form the present proceedings Twelve are original research articles cov-ering asymptotic properties of global fields, arithmetic properties of curves andhigher dimensional varieties, and applications to codes and cryptography The fi-nal article is a special lecture of J.-P Serre entitled “How to use finite fields forproblems concerning infinite fields”

The conference fulfilled its role of bringing together young researchers and cialists During the conference, we were also happy to celebrate the retirement ofour colleague Robert Rolland with a special day of talks

spe-Finally, we thank the organization commitee of CIRM for their help during theconference and the Calanques for its inspiring atmosphere

vii

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On the fourth moment of theta functions at their central

point

Amadou Diogo BARRY and St´ ephane R LOUBOUTIN

Abstract. Let χ be a Dirichlet character of prime conductor p ≥ 3 Set A =

(χ(1) −χ(−1))/2 ∈ {0, 1} and let θ(x, χ) =Pn≥1 n A χ(n) exp( −πn2x/p) (x >

0) be its associated theta series whose functional equation is used to obtain

the analytic continuation and functional equation of the L-series L(s, χ) =

P

n≥1 χ(n)n −s These functional equations depend on some root numbers

W (χ), complex numbers of absolute values equal to one. In particular, if

θ(1, χ) = 0, then numerical approximations to W (χ) = θ(1, χ)/θ(1, χ) can

be efficiently computed, which leads to a fast algorithm for computing class numbers and relative class numbers of real or imaginary abelian number fields (see the bibliography) According to numerical computations, it is reasonable

to conjecture that θ(1, χ) = 0 for any primitive Dirichlet character χ One way

to prove that this conjecture at least holds true for infinitely many primitive characters is to study the moments P

χ |θ(1, χ)|2k for k ∈ Z ≥1 , where χ ranges over all the even or odd primitive Dirichlet characters of conductor p This paper is devoted to proving a lower bound on these moments for 2k = 4.

n ≥1 χ(n)n −s, (s) > 0, stem from the functional equation

satisfied by the associated theta function (see [Dav, Chapter 9]):

x 1/2 θ(1/x, ¯ χ),

where W (χ) = τ (χ)/ √

p (the Artin root number, a complex number of absolute

value equal to 1 ) with τ (χ) =p

k=1 χ(k)e 2πik/p (Gauss sum)

1991 Mathematics Subject Classification 2000 Mathematics Subject Classification Primary

11R42, 11N37 Secondary 11M06.

Key words and phrases Dirichlet characters, Artin root numbers, Theta functions, Divisor

function.

1 1

Define the moments of order 2k:

(p − 3)/2 if b ≡ ±a (mod p) and gcd(a, p) = gcd(b, p) = 1,

−1 if b ≡ ±a (mod p) and gcd(a, p) = gcd(b, p) = 1,

Remark 4 For such a character χ, we have W (χ) = θ(1, χ)/θ(1, χ), by (1)

(hence numerical approximations to W (χ) can be efficiently computed, which leads

to a fast algorithm for computing class numbers and relative class numbers of real

or imaginary abelian number fields (see [Lou98], [Lou02] and [Lou07]).

The aim of this paper is to prove the following new result:

Theorem 5 There exists c > 0 such that S4(p) ≥ cp2log p for p > 3.

According to Proposition 2, Corollary 3, Theorem 5 and extended numerical

computations, we conjecture that θ(1, χ) = 0 for any primitive Dirichlet character

χ = 1, and the more precise behavior (see [Bar]):

Conjecture6 There exists c4> 0 such that S4(p) is asymptotic to c4p2log p

as p → ∞.

As for the second and fourth moments of the values of Dirichlet L-functions

L(s, χ) at their central point s = 1/2, the following asymptotics are known:

(see [HB, Corollary (page 26)]) It is also conjectured that for k ∈ Z ≥1there exists

a positive constant C(k) such that

2 The second moment of the restricted divisor function

Our proof of Theorem 5 is based on a new method: the study of the moment

of order 2 of the restricted divisor function (see Proposition 7 below) It is known

that (see [Ten]):

is asymptotic to π12x log3x as x → ∞ We give an asymptotic for the second

moment of the restricted divisor function:

Proposition7 Fix c > 1 Then,

Remark 8 It holds that Λ(n)

Proof Let us prove the second assertion Fix c > 1 If 1c √

D2<D1 gcd(D1,D2)=1

m=kdD1D2<x d2 D2 i /c2≤m≤c2d2D2 i , i∈{1,2}

D2<D1

m=kdδ2 D1D2<x d2 δ2 D2 i /c2≤m≤c2d2δ2D2 i , i∈{1,2}

δ< c √ x µ(δ)M c (d, δ, x),

where

M c(d, δ, x) =

D1< c

√ x dδ

D2< c √ x dδ

Lemma 9 It holds that M c (d, δ, x) = (2 log2c) x

dδ2D1D2 otherwise

In setting

X =

√ x cdδ ,

we obtain that N c (d, δ, D1, D2, x) is the number of k’s such that

= 2c2d(log2c)X2+ O(dX) + O(X2),

[Bar] A D Barry Moments of theta functions at their central point PhD Thesis, ongoing work.

[Dav] H Davenport Multiplicative Number Theory Springer-Verlag, Grad Texts Math 74, Third

[Lou02] S Louboutin Efficient computation of class numbers of real abelian number fields

Al-gorithmic Number Theory (Sydney, 2002), Lectures Notes in Computer Science 2369 (2002),

134–147.

[Lou07] S Louboutin Efficient computation of root numbers and class numbers of parametrized

families of real abelian number fields Math Comp 76 (2007), 455–473.

[Rama] K Ramachandra Some remarks on a Theorem of Montgomery and Vaughan J Number

Theory 11 (1979), 465–471.

[RS] Z Rudnick and K Soundararajan Lower bounds for moments of L-functions Proc Natl.

Acad Sci USA 102 (2005), 6837–6838.

[Ten] G Tenenbaum Introduction ` a la th´ eorie analytique et probabiliste des nombres Cours

Sp´ ecialis´ es Soci´ et´ e Math´ ematique de France, Paris, 1995.

Institut de Math´ ematiques de Luminy, UMR 6206 163, avenue de Luminy Case 907.

13288 Marseille Cedex 9, FRANCE

E-mail address: barry@iml.univ-mrs.fr, loubouti@iml.univ-mrs.fr

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On the Construction of Galois Towers

Alp Bassa and Peter Beelen

Abstract In this paper we study an asymptotically optimal tame tower over

the field with p2 elements introduced by Garcia-Stichtenoth This tower is related with a modular tower, for which explicit equations were given by Elkies.

We use this relation to investigate its Galois closure Along the way, we obtain

information about the structure of the Galois closure of X0(p n ) over X0(p r),

for integers 1 < r < n and prime p and the Galois closure of other modular towers (X0(p n))n.

1 Introduction

Using Goppa’s construction of codes from curves over finite fields, Tsfasman–

Vladut–Zink [13] constructed sequences of codes of increasing length with limit

parameters above the Gilbert–Varshamov bound and hence better than those ofall previously known such sequences Their construction is mainly based on theexistence of curves over a finite field of high genus with many rational points Thisenhanced the interest in towers of curves over finite fields Subsequently, otherapplications of such towers in coding theory and cryptography were discovered, forinstance for the construction of hash functions, low discrepancy sequences etc

A natural idea is to search for such sequences of curves, with some additionalstructure, which would reflect itself in some additional structure of the objects

constructed from them Stichtenoth [12] constructed for example sequences of

self-dual and transitive codes attaining the Tsfasman–Vladut–Zink bound over finitefields with square cardinality This was done by using a tower of function fields

E0⊆ E1⊆ E2⊆ , where all extensions E n /E0 are Galois

Motivated by this, we study Galois closures of the modular towers (X0(p n))n

In particular, we investigate the Galois closure of a towerM over F p2 introduced

by Garcia–Stichtenoth [3], which is recursively defined by

Y2=X

2+ 1

This tower corresponds to the modular tower (X0(2n))n, for which explicit equation

were given by Elkies [2] Using this interpretation of M as a modular tower, we

find the exact degrees of extensions in the Galois closure of it and study the Galois

1991 Mathematics Subject Classification Primary:14H05, Secondary:11R32.

Key words and phrases Function field, modular curve, Galois tower.

groups that appear We show that the function fields of the Galois closure can beobtained as a compositum of three different embeddings of the function fields inthe tower M.

For more definitions and further details about (explicit) towers of algebraic

function fields, we refer to [5].

2 Groups of Galois closure

In this section the field of definition is always assumed to be C, the field of

complex numbers Let p be a prime number and n > 1 an integer The following

group is standard in the theory of modular curves:

Associated to this group is the modular curve X0(p n) which has been studied

extensively in the literature, cf [7, 8].

Let 0 < r < n be integers The Galois closure of X0(p n ) over X0(p r) has Galoisgroup Γ0(p r )/∆r(p n) with

∆r (p n) := 

σ ∈Γ0(p r)

σΓ0(p n )σ −1 .

The group ∆r (p n) is the largest normal subgroup of Γ0(p r) contained in Γ0(p n),

since if H  Γ0(p r ) and H ⊂ Γ0(p n ), then H ⊂σ ∈Γ0(p r)σΓ0(p n )σ −1 = ∆

r (p n).The maximality of ∆r (p n) with respect to the above property will be used later.The goal of this section is to compute the order of the groups Γ0(p r )/∆r(p n)and to obtain information about its group structure We start by describing thegroup ∆r(pn) in more detail

Clearly H ⊂ Γ0(p n), so to prove the proposition it is enough to show that

H  Γ0(p r) and that ∆r (p n) ⊂ H, since then ∆ r (p n) ⊃ H follows from the

We need to check that this an element of H First we show that it is an element of

From property 3) of h we see that this is satisfied if p = 2, while if p = 2, then

2n −r−1 |bp r and 2|α(α − δ), since δ is odd if p = 2.

It remains to check that the third condition is satisfied, but this can easily be

seen to hold as well We conclude that mhm −1 ∈ H and hence that H  Γ0(p r).Now we wish to prove that ∆r(pn)⊂ H In order to do this we introduce the

which implies that if AhA −1 ∈ Γ0(p n ), then p n −r |a−d−bp r Similarly, if A −1 hA ∈

Γ0(p n ), then p n −r |a−d+bp r Therefore, if h ∈ Γ0(p n)∩AΓ0(p n )A −1 ∩A −1Γ

0(p n )A, then p n |c, p n −r |a − d − bp r and p n −r |a − d + bp r, which is equivalent to conditions1),2) and 3) above In other words:

The group ∆r (p n) has some further properties we wish to ascertain For a

group G, we denote by [G, G] its commutator subgroup.

Lemma 2.3 Suppose that n > r > 0 We have

Proof A direct calculation shows that [∆r(pn ), ∆r(p n)] ⊂ Γ0(p n+1) Also,since ∆r(pn) Γ0(p r ), we find that for any σ ∈ Γ0(p r) we have

where O(p m ) denotes some number divisible by p m This can be showed directly

using induction on k If k = p, then cp n (a p −d p )/(a −d) ≡ cp n (a −d) p −1 mod p n+1

Since g ∈ ∆ r (p n ) we have that p n −r |a − d − bp r , implying that p |a − d Hence

g p ∈ Γ0(p n+1) By definition of ∆r(pn ), we have that for any σ ∈ Γ0(p r), the

element σ −1 gσ is in Γ0(p n ), implying that (σ −1 gσ) p = σ −1 g p σ ∈ Γ0(p n+1) This

It is the kernel of the reduction modulo p n map: ϕ : SL(2, Z) → SL(2, Z/p nZ) and

one can show that this map is surjective ([9, section 1.6]) Also it is well known [9]

that

Note that by Proposition 2.1 the group Γ(p n) is a (normal) subgroup of ∆r (p n).The goal of this section is to compute the cardinality of the group ∆r (p n )/Γ(p n)

We will start by giving several lemmas

Lemma 3.1 We have that

Lemma 3.2 Let n > r > 0 be integers and suppose that p is an odd prime.

Then we have that

#∆r (p n )/Γ(p n) =



2p r+n if n ≤ 2r,

2p 3r else.

Proof Using Lemma 3.1 and the assumption that p is odd, it is enough to count the number of triples (a, b, d) ∈ (Z/p nZ)3satisfying p n |ad−1, p n −r |a−d and

p n −r |bp r

We claim that the number of (a, d) ∈ (Z/p nZ)2satisfying p n |ad−1 and p n −r |a−

d equals 2p r From the conditions, it is clear that p n −r |a2− 1, which implies

that a ≡ ±1 (mod p n −r ) This leaves exactly 2p r possibilities for a Given any

a satisfying the last congruence, there exists exactly one d ∈ Z/p nZ such that

p n |ad − 1 and by reducing modulo p n −r we see that d ≡ ±1 ≡ a This means that

p n −r |a − d is satisfied for this d as well.

We claim that the number of b ∈ Z/p n Z such that p n −r |bp r is equal to p n if

n ≤ 2r and equal to p 2r if n > 2r Indeed, if n ≤ 2r, the condition p n −r |bp r is

always satisfied, so that all b’s in Z/p n Z are possible If n > 2r, then the condition simplifies to p n −2r |b, meaning that all p 2r multiples of p n −2rinZ/p nZ are solutions

Multiplying the number of possibilities for (a, d) with that for b, the lemma

a2≡ 1 + 2 n −r−1mod 2n −r We now distinguish several cases.

Case 1, n − r = 1 In this case all solutions are characterized by choosing a ∈

Z/2 n Z to be odd, d its multiplicative inverse modulo 2 n and arbitrary b ∈ Z/2 nZ.Thus there are 22n −1= 22r+1 possibilities

Case 2, n −r = 2 We have seen that a2≡ 1 mod 4 or a2≡ 3 mod 4 The latter

is not possible, so we deduce that a2≡ 1 mod 4, which implies that a ≡ d mod 4 and b2 r ≡ 0 mod 4 All in all we get that we can choose a ≡ ±1 mod 4, b2 r ≡ 0 mod 4

and d ≡ a −1mod 2n For r = 1 this gives 16 possibilities for (a, b, d) and for r > 1

exactly 22r+3

Case 3, n − r = 3 First we get that a2 ≡ 1 mod 8 or a2 ≡ 5 mod 8, but

the latter is again not possible, since 8|a2− 1 for any odd number a This means

that b2 r ≡ 0 mod 8 Moreover, the condition that a2 ≡ 1 mod 8 implies that

a ≡ ±1 or ± 3 mod 8 Counting similarly as above, we find that there are 32

possibilities for (a,b,d) if r = 1, 256 if r = 2 and 2 2r+5 if r > 2.

Case 4, n − r > 3 First we assume that b2 r ≡ 0 mod 2 n −r, which means

that there are 22r possibilities for b if n > 2r and 2 n otherwise Then we found

that a2 ≡ 1 mod 2 n −r , which implies that a ≡ ±1 or ± 1 + 2 n −r−1mod 2n −r,

leaving 4· 2 r possibilities for a Now we can choose d to be the multiplicative inverse of a modulo 2 n and a direct computation shows that a ≡ d mod 2 n −r All

in all we find 2n+r+2 possibilities if n ≤ 2r and 2 3r+2 if n > 2r, still assuming that b2 r ≡ 0 mod 2 n −r Now assume that b2 r ≡ 2 n −r−1mod 2n −r This canonly occur if r ≤ n − r − 1, or equivalently if n > 2r and then the number of

possibilities for b is 2 2r We saw that a2 ≡ 1 + 2 n −r−1mod 2n −r, implying that

a ≡ ±1 + 2 n −r−2or± 1 − 2 n −r−2mod 2n −r As before we choose d to be the inverse of a, but now we find that d ≡ a + 2 n −r−1mod 2n −r, so that condition 2)

is satisfied Condition 3) is satisfied automatically We find 23r+2 possibilities if

n > 2r, but none if n ≤ 2r In total for case 4, we find 2 n+r+2 possibilities for

4 Degrees and structure of Galois closure

Given n > r > 0 and a prime p, we will now determine the degree of the Galois closure of X0(p n ) over X0(p r) We quote the following well-known facts [9, section

1.6]: Let m be a positive integer The degree of the covering X(p m)→ X(1) equals

p 3m −2 (p2− 1)/2, unless p = 2 and m = 1 in which case it equals 6 The degree of

the extension X0(p m)→ X(1) equals (p + 1)p m −1 As a consequence we see thatthe degree of X(p m+1)→ X(p m ) equals p3unless p = 2 and m = 1, in which case it equals 4 Also the degree of X0(p m+1)→ X0(p m ) equals p This together with the

previous results enables us to compute all degrees in the tower obtained by taking

the Galois closure of X0(p n ) over X0(p r ) for running n and fixed r.

Lemma 4.1 Let n > r > 0 be integers, p an odd prime and let ˜ X r

For n > r + 1, the covering ˜ X r (p n)→ ˜ X r (p n −1 ) is elementary abelian.

Proof From Lemma 3.2 we can calculate all degrees of the coverings X(p n)→

˜

X r

0(p n) Indeed, since−I ∈ ∆ r (p n) and−I ∈ Γ(p n), the only thing we need to do

is divide #∆r (p n )/Γ(p n ) by 2 Further, since deg(X0(p r)→ X(1)) = (p + 1)p r −1 and deg(X(p r)→ X(1)) = (p2− 1)p 3r −2 /2, we find that deg(X(p r)→ X0(p r)) =

(p −1)p 2r −1 /2 All in all we now know all degrees of the coverings X(p m)→ ˜ X r (p m)

for m ≥ r Combined with the fact that deg(X(p m+1)→ X(p m )) = p3, the firstpart of the lemma follows The second part follows directly from Lemma 2.3

Lemma4.2 Let n > r > 0 be integers and let ˜ X r(2n ) denote the Galois closure

Lemma4.3 Let n > r > 0 be integers and p a prime The extension X0(p n)→

X0(p r ) is Galois if and only if

(1) p = 2 and n − r = 1,

(2) p = 2, r > 1 and n − r = 2,

(3) p = 2, r > 2 and n − r = 3,

(4) p = 3 and n − r = 1.

In all of these cases the Galois group is cyclic.

Proof Since deg(X0(p n)→ X0(p r )) = p n −r, we can use Lemmas 4.1 and 4.2

to check when this degree is the same as deg( ˜X r (p n) → X0(p r)) Assuming the

covering X0(p n)→ X0(p r ) is Galois of order p n −r, we also see that its Galois group

is Γ0(p r )/∆r(p n ) However, the element A mod ∆r(p n ), with A as in Corollary 2.2,

5 Reduction mod

Let p be a prime Until now, we have assumed that all the modular curves

we considered were defined over the field C However, it is well known that the

curves X0(p n) have a model defined over Q [6] Denote by ζp n a primitive p n-th

root of unity The curve X(p n) has a model defined over Q(ζp n) and the covering

X(p n) → X(1) is still Galois and has the same degree as when working over C.

Since the Galois closure of X0(p n ) over X0(p r ) is contained in X(p n), it also has amodel defined over Q(ζp n) and all degrees computed before are still correct when

working over this field These models have good reduction modulo a prime if

= p The Galois covering X(p n) → X0(p r) is not necessarily Galois after this

reduction, but will be so when we consider it over a field containing a p n-th root ofunity After having done so, the Galois group will be the same as before reducingand in particular its degree is the same All group theoretic arguments used beforeare then still valid for the reductions, as long as the field of definition contains a

p n-th root of unity

The following Lemmas will be useful:

Lemma 5.1 Let F be a function field over a perfect field K and let f (T ) ∈

F [T ] be a separable irreducible polynomial over F Let α ∈ Ω be a root of f(T )

in some fixed algebraically closed field Ω ⊃ F Let K  be a separable algebraic

extension of K Suppose that there exists a place P of F , which splits completely

in the extension F (α)/F Then the polynomial f (T ) is irreducible in F K  [T ] and

G(f, F ) ∼= G(f, F K  ) where G(f, F ) and G(f, F K  ) denote the Galois group of f

over F and F K  , respectively.

Proof Since there exists a place P of F splitting completely in the extension

F (α)/F , the field K is algebraically closed in F (α) So the polynomial f (T ) is

irreducible in F K  [T ] (cf [11, Proposition III.6.6]) Denote by Z (respectively

Z  ) the splitting field of f (T ) over F (respectively F K  ) Let α1, , α n be allconjugates of α over F We have Z = F (α1, , α n) Since f (T ) is irreducible

in F K  [T ], the conjugates of α over F K  are also given by α1, , α n, and hence

Z  = F K  (α1, , α n) = ZK  and therefore

G(f, F K  ) ∼=G(Z  /F K  ) ∼=G(Z/Z ∩ F K  ).

Since the place P of F splits completely in the extension F (α)/F , it will also split

in the Galois closure Z/F So the field K is algebraically closed in Z and hence

E/F respectively EK  /F K  Then

GC(EK  /F K ) =GC(E/F )K ;

i.e., taking the Galois closure of such an extension commutes with extending the field

of constants Moreover the Galois groups of GC(EK  /F K  )/F K  and GC(E/F )/F are isomorphic.

Denote by Fn the function field of the curve X0(p n ) reduced modulo a prime

Note that its constant field isF We would like to use Lemma 5.2 in order to gaininformation onGC(F n /F r) In order to do so, we need that the extension Fn /F r

contains a completely splitting place, but this is not true in general It is well knownhowever, that if we extend the constant field toF2, the tower Fr ⊂ F r+1 ⊂ · · · is

asymptotically optimal So if the constant field is F2, we can expect completelysplitting places The following lemma confirms this for a large class of cases.Lemma 5.3 Suppose that and p are two primes such that ≥ 13 and = p, and let 0 < r < n be two integers The extension F nF2/F rF2 contains a completely splitting place.

Proof Let F −1F2 denote the function field arising by reducing the modular

curve X(1) modulo and then extending the constant field The reason the function fields FnF2 have many rational places is that the supersingular j-invariants in

F −1F2areF2-rational and all places in FnF2 lying above any of these j-invariants

different from 0 and 1728 are F2-rational as well (see Lemma 5.3 in [1]) On the

other hand, it is well known that the only branching in Fn /F −1 occurs at j = 0,

j = 1728 and j = ∞.

In order to prove the lemma it is therefore enough to show that there exists

a supersingular j-invariant different from 0 and 1728 Such a j-invariant always

6 Galois closure of a tame tower

Explicit equations for some of the towers considered above (and also for some

other modular towers) were given by Elkies (see [2]) In particular let p = 2 and

consider the tower

→ → X0(26)→ X0(25)→ X0(24).

Let be a prime such that ... Soc., 2007.

[5] A Garcia and H Stichtenoth (eds.), Topics in geometry, coding theory and cryptography,

Algebr Appl 6, Springer-Verlag, 2007.... Recent trends in coding< /small>

theory and its applications, pp 83–92, AMS/IP Stud Adv Math., 41, Amer Math Soc., 2007.

[5] A Garcia and H Stichtenoth... Therefore, from Lemma 3.2 and the geometricstructure ofX and Q, we get |X ∩ Q| = 4q2+ whenX and Q have the

same vertex Π0 and contain four common