From Hodge Index Theorem to the number of points of curves over finite fields

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From Hodge Index Theorem to the number of points of curves over finite fields

Emmanuel Hallouin, Marc Perret

To cite this version:

Emmanuel Hallouin, Marc Perret. From Hodge Index Theorem to the number of points of curves over

finite fields. 2015. �hal-01136433�

From Hodge Index Theorem to the number of points of curves over finite fields

Emmanuel Hallouin & Marc Perret

^∗

July 18, 2014

Abstract

We push further the classical proof of Weil upper bound for the number of rational points of an absolutely irreducible smooth projective curveX over a finite field in term of euclidean relationships between the Neron Severi classes inX×X of the graphs of iterations of the Frobenius morphism. This allows us to recover Ihara’s bound, which can be seen as asecond orderWeil upper bound, to establish a newthird orderWeil upper bound, and usingmagmato produce numerical tables for higher orderWeil upper bounds. We also give some interpretation for the defect of exact recursive towers, and give several new bounds for points of curves in relative situationX →Y.

AMS classification : 11G20, 14G05, 14G15, 14H99.

Keywords : Curves over a finite field, rational point, Weil bound.

Introduction

Let X be an absolutely irreducible smooth projective curve defined over the finite field F

q

with q elements. The classical proof of Weil Theorem for the number of F

q

-rational points ]X( F

q

) rests upon Castelnuovo identity [Wei48], a corollary of Hodge index Theorem for the smooth algebraic surface X × X. The intent of this article is to push further this viewpoint by forgetting Castelnuovo Theorem. We come back to the consequence of Hodge index Theorem that the intersection pairing on the Neron Severi space NS(X × X)

_R

is anti-euclidean on the orthogonal complement E

_X

of the trivial plane generated by the horizontal and vertical classes. Thus, the opposite h·, ·i of the intersection pairing endows E

_X

with a structure of euclidean space. Section 1 is devoted to few useful scalar products computations.

In section 2, we begin by giving a proof of Weil inequality which is, although equiva- lent in principle, different in presentation than the usual one using Castelnuovo Theorem given for instance in [Har77, exercice 1.9, 1.10 p. 368] or in [Sha94, exercises 8, 9, 10 p. 251]). We prove that Weil bound follows from Cauchy-Schwartz inequality for the

∗Institut de Math´ematiques de Toulouse, UMR 5219, hallouin@univ-tlse2.fr, perret@univ-tlse2.fr

orthogonal projections onto E

_X

of the diagonal class Γ

⁰

and the class Γ

¹

of the graph of the Frobenius morphism on X.

The benefit of using Cauchy-Schwartz instead of Castelnuovo is the following. We do not know what can be a Castelnuovo identity for more than two Neron Severi classes, while we do know what is Cauchy-Schwartz for any number of vectors. It is well- known that Cauchy-Schwartz between two vectors is the non-negativity of their 2 × 2 Gram determinant. Hence, we are cheered on investigating the consequences of the non-negativity of larger Gram determinants involving the Neron Severi classes Γ

^k

of the graphs of the k-th iterations of the Frobenius morphism.

For the family Γ

⁰

, Γ

¹

, Γ

²

, we recover the well known Ihara bound [Iha81] which im- proves Weil bound for curves of genus greater than g

₂

=

√q(√ q−1)

2

, a constant appearing very naturally with this viewpoint in section 2.5.2, especialy looking at figure 1 in sec- tion 2.5.2. It follows that the classical Weil bound can be seen as a first order Weil bound, in that it comes from the euclidean constraints between Γ

⁰

and Γ

¹

, while the Ihara bound can be seen as a second order Weil bound, in that it comes from euclidean constraints between Γ

⁰

, Γ

¹

and Γ

²

. Moreover this process can be pushed further: by considering the family Γ

⁰

, Γ

¹

, Γ

²

and Γ

³

, we obtain a new third order Weil bound (The- orem 18), which improves the Ihara bound for curves of genus greater than another constant g

3

=

√q(q−1)

√ 2

.

The more the genus increases, the more the Weil bound should be chosen of high order in order to be optimal. Therefore it is useful to compute higher order Weil bounds.

Unfortunately, establishing them explicitely requires the resolution of high degree one variable polynomial equations over R . For instance, the usual Weil bound requires the resolution of a degree one equation, Ihara and the third order Weil bounds require the resolution of second order equations, while fourth and fifth order Weil bounds require the resolution of third degree equations, and so on. Moreover, a glance at Ihara second order Weil bound and at our explicit third order Weil bound will convince the reader that they become more and more ugly as the order increase.

Hence, we give up the hope to establish explicit formulae for the Weil bounds of order

from 4. We then turn to a more algorithmic point of view. We use an algorithm which,

for a given genus g and a given field size q, returns the best upper order Weil bound

for the number of F

q

-rational points of a genus g curve, together with the corresponding

best order n. The validity of this algorithm requires some results proved in the second

section. In the figure below, we represent the successive Weil bounds (in logarithmic

scales) of order from 1 to 5. Note that taking into account the logarithmic scale for the

y-axis, higher order Weil bounds become significantly better than usual Weil’s one.

Weil bounds of order 1 to 5 for]X(F^q) (here forq= 2). Note that theyaxis is logarithmic.

For small genus, red usual first order Weil bound is the best one. Then from genus_√ g2 =

q(√ q−1)

2 , yellow Ihara’s second order Weil bound becomes the best, up to the genus g3 =

√q(q−1)

√

2 where green third order weil bound becomes the best. From some genusg4, light blue fourth order Weil bound is the best up to some genusg5, where dark blue fifth order Weil bound becomes better, and so on. Here, we have chosenq = 2, for which the genera g2, g3, g4, g5 are particularly small, respectively about 0.3,1,2.35 and 4.67. This means for instance that the best bound forq= 2 andg= 3,4 is the fourth order one!

In order to illustrate the efficiency of our algorithm, we display in section 2.6.3 a numer- ical table.

We acknowledge that for any pair (g, q) we have tested, a comparison of our numer- ical results with the table given on the website http://www.manypoints.org/ reveals that we always recover the very same numerical upper bound for N

_q

(g) than those coming from Oesterl´ e bounds! We were not been able to understand this experimental observation.

Nevertheless, we think that the viewpoint introduced in this article is preferable to

Osterl´ e’s one. First, our viewpoint is more conceptual in nature. Any constraints we

use to obtain our bounds come in a quite pleasant way either from algebraic-geometry

or from arithmetic, as explained in the introduction of section 2 in which we outline our

approach. We think moreover that the graph displayed just above is very satisfying.

Second, the viewpoint introduced in this article is perfectly adapted to the study of bounds for the numbers of rational points. As the reader can see, many

¹

known results can be understood with this viewpoint -even asymptotical ones, and new results can be proved. We are pretty sure that we have not extracted all potential outcomes of this viewpoint. Third, new questions can be raised from this viewpoint. We propose a few in Section 4.

To conclude Section 2, we push to the infinite-order Weil bound using the non- negativity of the Gram determinants of any orders, which imply that some symmetric matrix is semi-definite positive. Applied to some very simple vector, this lead us to The- orem 22, a stronger form of [Tsf92] Tsfasman bound in that it gives a new interpretation for the defect of an exact tower as a limit of nice euclidean vectors in (E, h·, ·i).

In Section 3, we study the relative situation. Given a finite morphism f : X → Y between two absolutely irreducible smooth projective curves defined over F

q

, the pull- back functor on divisors from the bottom algebraic surface Y × Y to the top one X × X induces a map from the Neron Severi space NS(Y ×Y )

_R

to NS(X × X)

_R

. Restricting this map to the subspaces F

_X

and F

_Y

generated by the classes of the graphs of iterations of the Frobenius morphisms, and after a suitable normalization (which differs from the normalization chosen in Section 2), we prove that it becomes an isometric embedding. We thus have an orthogonal decomposition F

_X

= F

_Y^∗

⊕ F

_X/Y

, involving a relative subspace F

_X/Y

of F

_X

.

Now, Cauchy-Schwartz inequality applied to the orthogonal projections of Γ

⁰_X

and Γ

¹_X

on the relative space F

_X/Y

is equivalent to the well known relative Weil bound that

|]X ( F

q

) − ]Y ( F

q

)| ≤ 2(g

_X

− g

_Y

) √ q.

With regard to higher orders relative Weil bounds, we encounter a difficulty since the arithmetical constraints ]X(F

q^r

) − ]Y (F

q^r

) ≥ ]X(F

q

) − ]Y (F

q

) for r ≥ 2, similar to that used in Section

²

2 does not hold true! Hence, the only constraints we are able to use are the non-negativity of Gram determinants. This lead to an inequality relating the quantities ]X(F

q^r

) − ]Y (F

q^r

) (corollary 29). We also prove a bound involving four curves in a cartesian diagram under some smoothness assumption (Theorem 33).

As stated above, we end this article by a very short Section 4, in which we raise a few questions.

1 The euclidean space (E , h·, ·i)

Let X be an absolutely irreducible smooth projective curve defined over the finite field F

q

with q elements. We consider the Neron-Severi space NS(X × X)

_R

= NS(X × X) ⊗

_Z

R of the smooth surface X × X. It is well known that the intersection product of two divisors Γ and Γ

⁰

on X × X induces a symmetric bilinear form on NS(X × X)

_R

, whose signature is given by the following Hodge Index Theorem (e.g. [Har77]).

1Other known results can be proved from this viewpoint, for instance the rationality of the Zeta function, using the vanishing of Hankel determinants. In order to save place, we have chosen to skip this.

2See the introduction of Section2

Theorem 1 (Hodge index Theorem). The intersection pairing is non-degenerate on the space NS(X × X)

_R

, and is definite negative on the orthogonal supplement of any ample divisor.

Let H = X × {∗} and V = {∗} × X be the horizontal and vertical classes on X × X. Their intersection products are given by H · H = V · V = 0 and H · V = 1.

The restriction of the intersection product to Vect(H, V ) has thus signature (1, −1).

Moreover Vect(H, V ) ∩ Vect(H, V )

^⊥

= {0}. By non-degeneracy, this gives rise to an orthogonal decomposition NS(X × X)

_R

= Vect(H, V ) ⊕ Vect(H, V )

^⊥

. Let p be the orthogonal projection onto the non-trivial part Vect(H, V )

^⊥

, given by

p : NS(X × X)

_R

−→ Vect(H, V )

^⊥

Γ 7−→ Γ − (Γ · V )H − (Γ · H)V. (1)

Since H + V is ample, the intersection pairing is definite negative on Vect(H, V )

^⊥

by Hodge index Theorem. Hence, Vect(H, V )

^⊥

can be turned into an euclidean space by defining a scalar product as the opposite of the intersection pairing.

Definition 2. Let E = Vect(H, V )

^⊥

. We define on E a scalar product, denoted by h·, ·i, as

γ, γ

⁰

= −γ · γ

⁰

, ∀γ, γ

⁰

∈ E . The associated norm on E is denoted by k · k.

In the sequel of this article, all the computations will take place in the euclidean space (E, h·, ·i). To begin with, let us compute this pairing between the iterates of the Frobenius morphism.

Lemma 3. Let F : X → X denotes the q-Frobenius morphism on X, and F

^k

denotes the k-th iterate of F for any k ≥ 0, with the usual convention that F

⁰

= Id

_X

. We denote by Γ

^k

the Neron Severi class of the graph of F

^k

. Then

( p(Γ

^k

), p(Γ

^k

)

= 2gq

^k

∀k ≥ 0

p(Γ

^k

), p(Γ

^k+i

)

= q

^k

(q

ⁱ

+ 1) − ]X( F

ⁱq

)

∀k ≥ 0, ∀i ≥ 1.

Proof — Since the morphism F

^k

is a regular map of degree q

^k

, one has Γ

^k

· H = q

^k

and Γ

^k

·V = 1. Now, let k ∈ N and i ∈ N

^∗

. We consider the map π = F

^k

×Id : X ×X → X × X, which sends (P, Q) to (F

^k

(P ), Q). We have π

^∗

(∆) = Γ

^k

and π

∗

Γ

^k+i

= q

^k

Γ

ⁱ

, so that by projection formula for the proper morphism π:

Γ

^k

· Γ

^k+i

= π

^∗

(∆) · Γ

^k+i

= ∆ · π

∗

Γ

^k+i

= q

^k

∆ · Γ

ⁱ

.

If i ≥ 1, then ∆ · Γ

ⁱ

= ]X( F

qⁱ

). If i = 0, then Γ

⁰

= ∆ and ∆ · Γ

⁰

= ∆

²

= −2g. The

results follow since p(Γ

^k

) = Γ

^k

− H − q

^k

V by (1).

Remark – Of course, dimE ≤ dimN S(X×X)_R −2 ≤ 4g². But as the reader can check, we are working along this article only on the subspace F of E generated by the familyp(Γ^k), k ≥ 0. By ([Zar95] chapter VII, appendix of Mumford), any trivial linear combination of this family is equivalent to the triviality of the same linear combination of the family ϕ^k<sub>`</sub>, k ≥0, of the iteration of the Frobenius endomorphismϕ` on the Tate moduleV`(JacX) for any prime`∧q= 1. We deduce that for a given curve X over F^q of genusg, we actually are working in an euclidean space FX of dimension equal to the degree of the minimal polynomial ofϕ`onV`(JacX).

2 Absolute bounds

We observe, as a consequence of Lemma 3, that the non-negativity 0 ≤ Gram(p(Γ

⁰

), p(Γ

¹

)) =

2g q + 1 − ]X( F

q

) q + 1 − ]X( F

q

) 2gq

= (2g √

q)

²

− (q + 1 − ]X ( F

q

))

²

is nothing else than Weil inequality. In this Section, we give other bounds using larger Gram determinants. For this purpose, we need preliminary notations, normalizations and results.

Normalizations in Section 2.1 play two parts. They ease many formulae and calcu- lations. They also make obvious that several features of the problem, such as the Gram matrix Gram(p(Γ

ⁱ

); 0 ≤ i ≤ n), or the forthcoming n-th Weil domains W

_n

, are essentially independent of q. The authors believe that these features deserve to be emphasized.

With regard to results, let n ∈ N

^∗

. To obtain the Weil bound of order n, let X be an algebraic curve of genus g defined over F

q

. We use both geometric and arithmetic facets of X as follows. We define from X in (4) a point P

n

(X) ∈ R

ⁿ

, whose abscissa x

1

given in (3) is essentially the opposite of ]X ( F

q

), hence have to be lower bounded. The algebraic-geometric facet of X implies, thanks to Hodge index Theorem, that some Gram determinant involving the coordinates of P

n

(X) is non-negative. This is traduced on the point P

n

(X) in Lemma 12 in that it do lies inside some convex n-th Weil domain W

_n

, studied in Subsection 2.3. The arithmetic facet of X is used via Ihara constraints, that for any r ≥ 2 we have ]X (F

q^r

) ≥ ]X(F

q

). We then define in Subsection 2.4 the convex n-th Ihara domain H

^g_n

in genus g. We state in Proposition 16 that for any curve X of genus g defined over F

q

, we have P

_n

(X) ∈ W

_n

∩ H

^g_n

. We are thus reduced to minimize the convex function x

1

on the convex domain W

_n

∩ H

^g_n

. To this end, we establish in Subsection 2.4 the optimization criteria 17, decisive for the derivations of higher order Weil bounds.

This criteria is used to prove new bounds in Subsection 2.6. Finally, we obtain in Subsection 2.7 a refined form of Tsfasman upper bound for asymptoticaly exact towers.

2.1 The Gram matrix of the normalized classes of iterated Frobenius

In this Subsection, we choose to normalize the Neron-Severi classes of the iterated Frobe-

nius morphisms as follows. For any n ≥ 1, and any curve X defined over F

q

of genus

g 6= 0, we define a point P

n

(X) ∈ R

ⁿ

whose x

i

-coordinates is very closely related to ]X( F

qⁱ

).

Definition 4. Let X be an absolutely irreducible smooth projective curve defined over the finite field F

q

. For k ∈ N , we put

γ

^k

= 1

p 2gq

^k

p(Γ

^k

) ∈ E, (2) so that by Lemma 3 we have

γ

^k

, γ

^k

= 1. For i ∈ N

^∗

, we put x

_i^def.

= D

γ

^k

, γ

^k+i

E

_lemma3

= (q

ⁱ

+ 1) − ]X(F

_qⁱ

) 2g √

q

ⁱ

. (3)

For any n ≥ 1, we define the point

P

_n

(X) = (x

₁

, . . . , x

_n

) ∈ R

ⁿ

. (4)

Remark– Note first thatX is Weil-maximal if and only ifx1=−1, and is Weil minimal if and only ifx1= 1. Second, that to giveupperbounds for]X(Fq) amount to givelower bounds forx1.

From this Definition 4 and Lemma 3, the Gram matrix of the family γ

⁰

, . . . , γ

ⁿ

in E is

Gram(γ

⁰

, . . . , γ

ⁿ

) =



















1 x

₁

· · · x

n−1

x

_n

x

1

. .. . .. x

n−1

.. . . .. . .. . .. .. . x

n−1

. .. . .. x

₁

x

n

x

n−1

· · · x

1

1



















(5)

2.2 Some identities involving Toeplitz matrices

The main result of this Subsection is Lemma 5, providing the existence of the factoriza- tion in Definition 6, in which the G

⁻_n

factor plays a fundamental part in the following of the article.

A symmetric Toeplitz matrix is a symmetric matrix whose entries x

_i,j

depend only on |i − j|, that is are constant along the diagonals parallel to the main diagonal [HJ90,

§0.9.7], that is

T

n+1

(x

0

, x

1

, . . . , x

n

) =



















x

₀

x

₁

· · · x

n−1

x

_n

x

₁

. .. . .. x

n−1

.. . . .. . .. . .. .. . x

n−1

. .. . .. x

1

x

n

x

n−1

· · · x

1

x

0



















(6)

The symmetric Toeplitz matrix is said to be normalized if x

0

= 1.

An Hankel matrix is a matrix whose entries x

i,j

depend only on i + j, that is are constant along the anti-diagonals parallel to the main anti-diagonal [HJ90, §0.9.8], that is

H

_n+1

(x

₀

, x

₁

, . . . , x

_2n

) =



















x

0

x

1

x

2

· · · x

n

x

₁

... ... x

_n+1

x

₂

... ... .. .

.. . ... ... x

2n−1

x

_n

x

_n+1

· · · x

2n−1

x

_2n



















(7)

Remark– It should be noticed that in this article, any matrix is indexed by its size, while this doesn’t hold for determinants, for instance see formulae (9) and (10).

Lemma 5. For any n ≥ 1, we abbreviate T

_n

(x

₀

, . . . , x

n−1

) by T

_n

. The determinant of this Toeplitz matrix factorizes as

Det (T

_2n

) = Det (T

_n

+ H

_n

(x

2n−1

, . . . , x

₁

)) × Det (T

_n

− H

_n

(x

2n−1

, . . . , x

₁

)) and

Det (T

_2n+1

) = Det

T

_n

+ H

_n

(x

_2n

, . . . , x

₂

)

^t

X

_n

2X

n

x

0

× Det (T

_n

− H

_n

(x

_2n

, . . . , x

₂

)) , where X

n

= (x

n

, . . . , x

1

).

Proof — Even size. The operations on the columns C

j

← C

j

+ C

2n−(j−1)

for 1 ≤ j ≤ n, followed by the operations on the lines L

n+i

← L

n+i

− L

_n−(i−1)

for 1 ≤ i ≤ n lead to the determinant of a 2 × 2 blocks upper triangular matrix whose value is the expected product.

Odd size. The operations on the columns C

j

← C

j

+ C

_{2n+1−(j−1)}

for 1 ≤ j ≤ n, followed by the operations on the lines L

_n+1+i

← L

_n+1+i

− L

n+1−i

for 1 ≤ i ≤ n lead to

the result in the same manner.

Definition 6. For n = 0, we put G

₀

= G

⁻₀

= G

⁺₀

= 1, and for n ≥ 1, we put

G

n

(x

1

, . . . , x

n

)

^def.

= Det(T

n+1

(1, x

1

, . . . , x

n

))

^Lemma

=

⁵

G

⁻_n

(x

1

, . . . , x

n

) × G

⁺_n

(x

1

, . . . , x

n

), where the last factorization is the one proved in Lemma 5 in the same order.

Remark– The determinantGnis the determinant of a matrix of sizen+ 1. Note that forn = 2m+ 1 odd, both G⁻n and G⁺n are polynomials of degreem+ 1 in x1, . . . , xn, while forn= 2meven,G⁻n has degreem+ 1, whileG⁺n have degreem. We will see later in the article that Weil bound of order n for]X(F^q) depends heavily on the hypersurface {G⁻n = 0}.

In order to raise any ambiguity and for later purpose, let us write down these deter- minants for n = 1, 2 and 3:

G

₁

(x

₁

) =

1 x

₁

x

1

1

= (1 + x

₁

)

| {z }

G⁻₁(x1)

× (1 − x

₁

)

| {z }

G⁺₁(x1)

(8)

G

₂

(x

₁

, x

₂

) =

1 x

1

x

2

x

₁

1 x

₁

x

2

x

1

1

=

1 + x

₂

x

₁

x

₁

+ x

₁

1

| {z }

G⁻₂(x1,x2)

× (1 − x

₂

)

| {z }

G⁺₂(x1,x2)

(9)

G

₃

(x

₁

, x

₂

, x

₃

) =

1 x

1

x

2

x

3

x

₁

1 x

₁

x

₂

x

2

x

1

1 x

1

x

3

x

2

x

1

1

=

1 + x

₃

x

₁

+ x

₂

x

1

+ x

2

1 + x

1

| {z }

G⁻₃(x1,x2,x3)

×

1 − x

₃

x

₁

− x

₂

x

1

− x

2

1 − x

1

| {z }

G⁺₃(x1,x2,x3)

(10)

Lemma 7. For any n ≥ 1 and = ± or nothing, we abbreviate G

_n

(x

₁

, . . . , x

_n

) by G

_n

. Then one has

2G

n

= G

⁺_n−1

× G

⁻_n+1

+ G

⁻_n−1

× G

⁺_n+1

.

Proof — Let S

_n+1⁻

be the matrix obtained from T

n

by adding

^t

(x

n

, . . . , x

1

) to the first column, and S

_n+1⁺

be the matrix obtained from T

n

by removing

^t

(x

n

, . . . , x

1

) to the first column. Then by multilinearity, there exists a polynomial R

_n

such that:

Det S

_n+1⁻

= G

n

+ R

n

and Det S

_n+1⁺

= G

n

− R

n

.

On the other hand, using the same transformations as in the proof of Lemma 5, one get Det(S

_n+1⁻

) = G

⁺_n−1

G

⁻_n+1

and Det(S

_n+1⁺

) = G

⁻_n−1

G

⁺_n+1

Adding Det(S

_n+1⁻

) and Det(S

_n+1⁺

) allows us to conclude.

2.3 The domain of positive definite normalized Toeplitz matrices In this Subsection, we study for any n ≥ 1 some domain W

_n

⊂ R

ⁿ

, whose closure W

_n

also plays a fundamental part in this article.

Definition 8. Let n ∈ N

^∗

. We denote by W

_n

the set of (x

1

, . . . , x

n

) ∈ R

ⁿ

, such that the symmetric normalized Toeplitz matrix T

_n+1

(1, x

₁

, . . . , x

_n

) is positive definite.

The domain W

_n

can be characterized in several useful ways.

Proposition 9. For n ≥ 1, the domain W

_n

is a convex subset of ]−1, 1[

ⁿ

, which can be

written as follows.

1. First, one has:

W

_n

= {(x

₁

, . . . , x

_n

) ∈ R

ⁿ

| G

_i

(x

₁

, . . . , x

_i

) > 0, ∀ 1 ≤ i ≤ n} . 2. Recursively, W

₁

= ]−1, 1[ and for any n ≥ 2

W

_n

= {(x

₁

, . . . , x

n−1

, x

n

) ∈ W

n−1

× R | G

n

(x

1

, . . . , x

n

) > 0}

=

(x

₁

, . . . , x

n−1

, x

_n

) ∈ W

n−1

× R | G

⁻_n

(x

₁

, . . . , x

_n

) > 0 and G

⁺_n

(x

₁

, . . . , x

_n

) > 0 . 3. W

_n

is also the set of points between the graphs of two functions from W

_n−1

to R :

for ε = ±, there exists a polynomial G e

^ε_n

∈ Q [x

₁

, . . . , x

n−1

], such that G

_n

=

−G

_n−2

x

n

+ G e

n

and W

_n

=

(

(x

1

, . . . , x

n−1

, x

n

) ∈ W

_n−1

× R | G e

⁻_n

(x

1

, . . . , x

n−1

)

G

⁻_n−2

(x

₁

, . . . , x

n−2

) < x

n

< G e

⁺_n

(x

1

, . . . , x

n−1

) G

⁺_n−2

(x

₁

, . . . , x

n−2

)

) . Proof — To prove the convexity, we remark that both sets of normalized symmetric

Toeplitz matrices and of symmetric positive definite matrices are convex, so that W

_n

is convex. Moreover, W

_n

⊂ ]−1, 1[

ⁿ

because if T

n+1

(1, x

1

, . . . , x

n

) is positive definite, then all its principal minors are positive. In particular, for any 1 ≤ i ≤ n, the 2 × 2 minors

_x^{1 xi}

i 1

is positive.

Item (1) follows from the well known fact that an n × n symmetric matrix is definite positive if and only if all its n leading principal minors (obtained by deleting the i last rows and columns for 0 ≤ i ≤ n − 1) are positive [HJ90, Theorem 7.2.5]. Hence, the first characterization of item (2) follows from item 1 . To prove the second characterization of item (2), we use Lemma 7 stating that 2G

n−1

= G

⁻_n−2

G

⁺_n

+ G

⁺_n−2

G

⁻_n

. By induction, this allow us to prove that both factors are positive if, and only if, the product

G

_n

(x

₁

, . . . , x

_n

) = G

⁻_n

(x

₁

, . . . , x

_n

)G

⁺_n

(x

₁

, . . . , x

_n

) is positive for any (x

₁

, . . . , x

n−1

) ∈ W

_n−1

.

To prove item (3), for = ±, formulae defining the polynomials G

_n

in Lemma 5 and Definition 6 imply, developing along their first column and taking advantage of the very particular forms (6) and (7), that both polynomials have degree 1 in x

n

, of the form

G

_n

= −G

_n−2

x

n

+ ε G e

_n

(x

1

, . . . , x

n−1

)

for some G e

_n

∈ Q [x

1

, . . . , x

n−1

]. Since by item 2 we have G

^ε_n−2

> 0, the set W

_n

is thus equal to the set of (x

₁

, . . . , x

n−1

, x

_n

) ∈ W

_n−1

× R , such that:

G e

⁻_n

(x

₁

, . . . , x

n−1

)

G

⁻_n−2

(x

1

, . . . , x

n−2

) < x

_n

< G e

⁺_n

(x

₁

, . . . , x

n−1

) G

⁺_n−2

(x

1

, . . . , x

n−2

) ,

and the proof is complete.

The following Proposition is useful for later purpose in Section 2.6, where convexity plays a quite important part.

Proposition 10. The locus {(x

₁

, . . . , x

n

) ∈ W

_n−1

× R | G

⁻_n

(x

1

, . . . , x

n

) > 0} is convex, while the locus {(x

₁

, . . . , x

_n

) ∈ W

_n−1

× R | G

⁺_n

(x

₁

, . . . , x

_n

) > 0} is concave.

Proof — By item 3 of Proposition 9, W

_n

is the set of (x

1

, . . . , x

n−1

, x

n

) ∈ W

_n−1

× R such that x

_n

is between two functions from W

_n−1

to R . By convexity of W

_n

from Proposition 9, the lower function must be concave has a function on W

n−1

while the upper one must be convex. The Proposition follows easily.

The closure of W

_n

is given by the following Proposition.

Proposition 11. The closure W

_n

of W

_n

in R

ⁿ

corresponds to the set of (n + 1)×(n+ 1) normalized symmetric Toeplitz matrices which are positive semi definite. Moreover, we have

W

_n

= {(x

₁

, . . . , x

n

) ∈ R

ⁿ

| G

n,I

(x

1

, . . . , x

i

) ≥ 0, ∀I ⊂ {0, 1, . . . , n}} ,

where, for I ⊂ {0, 1, . . . , n}, we denote by G

n,I

(x

1

, . . . , x

n

) the principal minor of the normalized symmetric Toeplitz matrix T

_n+1

(1, x

₁

, . . . , x

_n

) obtained by deleting the lines and columns whose indices are not in I.

Proof — This is a consequence of the fact that a matrix is positive semi definite if and only if all the principal minors (the ones obtained by deleting the same subset of lines and columns) are non-negative (i.e. ≥ 0) [HJ90, last exercise after Theorem 7.2.5].

2.4 The curves locus

In this Subsection, we introduce Ihara’s constraints and the resulting n-Ihara domain H

^g_n

for genus g. We then gather the key results for the derivation of higher order Weil bounds. The first one is Proposition 16, the second one is criteria 17.

2.4.1 The n-the Weil domain W

_n

Lemma 12. Let X be an absolutely irreducible smooth projective curve defined over F

q

. Then the point P

n

(X) = (x

1

, . . . , x

n

), as defined in (4), belongs to W

_n

.

Proof — By (3), we have x

_i

=

γ

^k

, γ

^k+i

where the γ

^k

are defined in (2), so that it is easily seen that for any I ⊂ {0, 1, . . . , n}, we have

G

_n,I

(x

₁

, . . . , x

_n

) = Gram(γ

ⁱ

, i / ∈ I),

for G

_n,I

s defined in Proposition 11. The non-negativity of G

_n,I

follows as a Gram

determinant in an euclidean space, hence the Lemma by Proposition 11.

Definition 13. W

_n

is called the n-th Weil domain.

As a first illustration of the informations contained in these Weil domains, we prove the following simple result. From (9), the non-negativity of the Gram determinants G

⁻₂

writes x

2

≥ 2x

²₁

− 1. Taking (3) into account, this gives immediately

Proposition 14. For any curve X of genus g 6= 0, we have ]X ( F

q²

) − (q

²

+ 1) ≤ 2gq − 1

g

]X ( F

q

) − (q + 1)

2

.

This Proposition means that, for a given non rational curve X, any lower bound for the deviation of ]X(F

q

) to q + 1 yields to a better upper bound than Weil’s one for ]X( F

q²

). In the same way, for any given order n, the non-negativity of the 3 × 3 determinant Gram(γ

⁰

, γ

¹

, γ

ⁿ

) gives a quite ugly upper bound of similar nature for ]X(F

qⁿ

) in terms of ]X(F

q

) and ]X(F

_qⁿ⁻¹

).

Remark– Note that this Proposition is a refinement of the following well known particular case: if X is either Weil maximal or minimal overFq, then ]X(Fq)−(q+ 1) =±2g√

q, and this Proposition asserts that thenX(Fq²)−(q²+ 1)≤2gq−^4g_g^2q =−2gq, so thatX is Weil minimal overFq². Note also that curves such that this inequality is an equality are those curves such that the corresponding pointP2(X) = (x1, x2)∈ W2 lies on the bottom parabolax2= 2x²₁−1 of the Ihara domain drawn in figure1below. The above particular case is that of curves corresponding to the corner points (−1,1) and (1,1).

2.4.2 Ihara constraints : the n-th Ihara domain H

^g_n

in genus g

If (x

1

, . . . , x

n

) ∈ R

ⁿ

comes from a curve X over F

q

by formulae (3), then we have seen in Lemma 12 that (x

₁

, . . . , x

_n

) lies on the closure W

_n

of W

_n

. But there are also other constraints, resulting from the arithmetical inequalities ]X(F

_qⁱ

) ≥ ]X(F

q

) for any i ≥ 1.

These inequalities, by (3), write

∀i ≥ 2, x

i

≤ x

₁

q

ⁱ⁻¹²

+ q

ⁱ⁻¹

− 1

2gq

ⁱ⁻²²

. (11)

Let

α = 1

√ q . (12)

For g ≥ 0, and n ≥ 2, i ≥ 2, we define

h

^g_i

(x

1

, x

i

) = x

i

− α

ⁱ⁻¹

x

1

− 1 2gα

1

α

ⁱ⁻¹

− α

ⁱ⁻¹

. (13)

Then it is easily seen that (11) is equivalent to

h

^g_i

(x

1

, x

i

) ≤ 0. (14)

We now define

Definition 15. Let n ≥ 1 and g ≥ 0. We define:

1. The n-th Ihara domain in genus g as

H

^g_n

= {(x

₁

, . . . , x

_n

) ∈ R

ⁿ

| h

^g_i

(x

₁

, x

_i

) ≤ 0, ∀ 2 ≤ i ≤ n} (15) 2. The n-th Ihara line in genus g as

L

^g_n

= {(x

₁

, . . . , x

n

) ∈ R

ⁿ

| h

^g_i

(x

1

, x

i

) = 0, ∀ 2 ≤ i ≤ n} . (16) Each L

^g_n

is a line which is identified with R using the first coordinate x

1

as a param- eter. We denote by

P

_n^g

(x

₁

) =

x

₁

, αx

₁

+ q − 1

2g , α

²

x

₁

+ q

²

− 1 2g √

q , . . . , α

ⁿ⁻¹

x

₁

+ q

ⁿ⁻¹

− 1 2gq

ⁿ⁻²²

(17) the point of L

^g_n

with parameter x

1

.

In the same way, for g = ∞, we define the n-th Ihara infinite line by L

^∞_n

=

(x

1

, . . . , x

n

) ∈ R

ⁿ

| x

i

= α

ⁱ⁻¹

x

1

, 2 ≤ i ≤ n . It follows from this the following Proposition.

Proposition 16. Let X be an absolutely irreducible smooth projective curve defined over the finite field F

q

and n ≥ 2. Then P

n

(X) = (x

1

, . . . , x

n

) ∈ W

_n

∩ H

^g_n

, where the x

i

’s are defined from X by (3).

From this Proposition 16 and the remark below Definition 6, the strategy is the following.

Strategy. — For each n ≥ 1, we minimize the coordinate function x

₁

(P ) for P lying inside the compact convex domain W

_n

∩ H

^g_n

. By Proposition 16, this leads to a lower bound for x

1

(P

n

(X)) for any curve X of genus g over F

q

, hence by (3) to an upper bound for the number ]X( F

q

). It turns that for given q and genus g, this bound is better and better for larger and larger n, up to an optimal one.

We are thus face to an optimization problem. Since the domain W

_n

∩ H

^g_n

on which we have to minimize the convex function x

₁

is also convex by Proposition 9, one has the following necessary and sufficient characterization of the minimum from [HU96, Th´ eor` eme 2.2] where in our case the active constraints are G

⁻_n

= 0 and h

^g_i

= 0 for 2 ≤ i ≤ n, and where h

^g_i

are defined in (13).

Let P

₀

∈ W

_n

∩ L

^g_n

. Suppose that:

• for any I ⊂ {1, . . . , n + 1} and ε = ± such that G

^ε_n,I

(P

₀

) = 0, there exists µ

^ε_I

≥ 0,

• for any 2 ≤ i ≤ n, there exists µ

i

≥ 0, such that

∇x

₁

(P

₀

) − X

I⊂{1,...,n+1}

µ

⁻_I

∇G

⁻_n,I

(P

₀

) − X

I⊂{1,...,n+1}

µ

⁺_I

∇G

⁺_n,I

(P

₀

) +

n

X

i=2

µ

_i

∇h

^g_i

(P

₀

) = 0.

(18) Then, x

1

(P

0

) = min{x

₁

| (x

1

, . . . , x

n

) ∈ W

_n

∩ H

^g_n

}.

We deduce from that the following criteria.

Criteria 17 (for minimizing x

₁

). If P

₀

= (x

₁

, . . . , x

_n

) ∈ W

_n

∩ L

^g_n

satisfies 1. G

⁻_n

(P

₀

) = 0;

2. ∂

i

G

⁻_n

(P

0

) ≥ 0 for all 2 ≤ i ≤ n;

3. P

n

i=1

∂

i

G

⁻_n

(P

0

)α

ⁱ⁻¹

> 0,

then P

₀

minimizes the first coordinate x

₁

on W

_n

∩ H

^g_n

.

Remark– Corollary21bellow states that the first requirement implies the third one.

Proof — Suppose that the assumptions of the criteria hold true. Then the system



 

 



 

 

 P

_n

i=1

∂

i

G

⁻_n

(P

0

)α

ⁱ⁻¹

µ

⁻_∅

= 1 µ

₂

= ∂

₂

G

⁻_n

(P

₀

)µ

⁻_∅

.. .

µ

n

= ∂

n

G

⁻_n

(P

0

)µ

⁻_∅

has a solution µ

⁻_∅

> 0, µ

₂

≥ 0, . . . , µ

_n

≥ 0. As easily seen, this solution also satisfy















 1 0 .. . .. . 0

















− µ

⁻_∅

















∂

1

G

⁻_n

(P

0

)

∂

2

G

⁻_n

(P

0

) .. . .. .

∂

n

G

⁻_n

(P

0

)















 + µ

2















−α 1 0 .. . 0















+ · · · + µ

n















−α

ⁿ⁻¹

0

.. . 0 1















=











 0 .. . .. . 0











 ,

hence (18) holds with the choice µ

_I

= 0 for any I 6= ∅ and µ

⁺_∅

= 0, so that x

₁

(P

₀

) = min{x

₁

| (x

1

, . . . , x

n

) ∈ W

_n

∩ H

^g_n

}, and the criteria is proved.

2.5 Weil bound and Ihara bound as bounds of order 1 and 2

We prove here that our viewpoint enables to deduce Weil bound and Ihara bound. We

think that this Subsection is of interest since it shows on simple cases how this method

works, especialy in the case of Ihara bound thanks to figure 1.

2.5.1 Weil bound as a bound of order 1

With this viewpoint, the usual Weil bound comes from Cauchy-Schwartz inequality applied to γ

⁰

and γ

¹

. Indeed, one have by (8)

Gram(γ

⁰

, γ

¹

) =

1 x

₁

x

₁

1

≥ 0, that is |x

₁

| ≤ 1.

By (3), we have just recovered the Weil bound

|]X( F

q

) − (q + 1)| ≤ 2g √

q (Weil “first order” bound)

Just for fun, we can also easily recover the fact that a curve which is maximal over F

q

must be minimal over the F

q²ⁱ

and maximal of the F

q²ⁱ⁺¹

. Indeed, being maximal over F

q

means by (3) that x

1

= −1, therefore

Gram(γ

⁰

, γ

¹

, γ

²

) =

1 −1 x

₂

−1 1 −1 x

2

−1 1

= −(1 − x

2

)

²

≥ 0, so that x

2

= 1, that is by (3) that X is minimal over F

q²

. Then

Gram(γ

⁰

, γ

²

, γ

³

) =

1 1 x

₃

1 1 −1

x

3

−1 1

= −(1 + x

3

)

²

≥ 0, so that x

3

= −1, that is X is maximal over F

q³

, and so on. . . In the same way, if X is minimal over F

q

, that is if x

1

= 1, then

Gram(γ

⁰

, γ

¹

, γ

²

) =

1 1 x

2

1 1 1

x

2

1 1

= −(x

₂

− 1)

²

≥ 0, that is x

₂

= 1,

and X still minimal over F

q²

. And so on again...

2.5.2 Ihara bound as a bound of order 2

For n = 2, the domain W

₂

recursively corresponds by item 2 of Proposition 9 and (9) to the set of (x

₁

, x

₂

) ∈ R

²

satisfying:



 

 

G

₁

(x

₁

) = 1 − x

²₁

> 0 G

⁺₂

(x

1

, x

2

) = 1 − x

2

> 0 G

⁻₂

(x

₁

, x

₂

) = 1 + x

₂

− 2x

²₁

> 0

, that is

( −1 < x

1

< 1

2x

²₁

− 1 < x

₂

< 1 .

We represent the second Weil domain W

₂

on figure 1.

L

^g₂

, g >

√q(√ q−1) 2

L

^g₂

, g =

√q(√ q−1) 2

L

^g₂

, g <

√q(√ q−1) 2

µ

₂

A

^g₂

x

₁

x

2

−1 1

−1 1

Figure 1: The Weil domain W

₂

. If X is a curve over F

q

of genus g ≥

√q(√ q−1) 2

, then P

2

(X) = (x

1

, x

2

) as defined by (3) lies on the grey domain W

₂

∩ H

₂^g

, so that x

1

is lower bounded by some constant µ

2

= min{x

₁

(P ), P ∈ W

₂

∩ H

^g₂

} > −1, improving Weil bound by Proposition 16.

The second Ihara line L

^g₂

with positive slope α =

^√1_q

meets the domain W

₂

if and only if the genus is greater than the genus g

₂

for which L

^g₂²

contains the point (−1, 1) (see figure 1), that is

1 = α × (−1) + 1 2g

2

α

1 α − α

⇐⇒ g

₂

=

√ q √ q − 1

2 .

For g > g

₂

, the constraint h

^g₂

(x

₁

, x

₂

) ≤ 0 restricts the domain W

₂

to the grey domain W

₂

∩ H

^g_n

, as can be seen on figure 1. The point A

^g₂

such that [A

^g₂

, B

₂^g

] = W

₂

∩ L

^g₂

lies on the curve G

⁻₂

= 0, and minimizes

³

the first coordinate x

1

on W

₂

∩ H

^g₂

.

µ

₂

is thus the solution in [−1, 0] of the quadratic equation G

⁻₂

(x

₁

,

^√x1_q

+

^q−1_2g

) = 0.

Solving it, we find

µ

2

= α

²

−

r α

²

+ 8

q−1 2g

− 1

4 ,

so that using (3), we recover the well known Ihara bound ]X( F

q

) − (q + 1) ≤

p (8q + 1)g

²

+ 4q(q − 1)g − g 2

(Ihara “second order Weil” bound)

3It can also be trivially proved using criteria17since∇G⁻₂ = ^−4x₁¹

andx1(A^g₂)<0.