Coverings of singular curves over finite fields

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Yves Aubry, Marc Perret

To cite this version:

Yves Aubry, Marc Perret. Coverings of singular curves over finite fields. manuscripta mathematica, Springer Verlag, 1995, 88 (1), pp.467–478. �10.1007/BF02567835�. �hal-00976489�

manuscripta math. 88, 467 - 478 (1995) m a n u s c r i p t a m a t h e m a t t c a

~) Springer-Verlag 1995

C o v e r i n g s of Singular C u r v e s over F i n i t e Fields

Y v e s A u b r y 1, M a r c P e r r e t 2

1 D 4 p a r t e m e n t d e M a t h 4 m a t i q u e s , U n i v e r s i t 4 de C a e n E s p l a n a d e de la P a i x - 14 032 C a e n C e d e x - France.

2 U n i t 4 de M a t h 4 m a t i q u e s , ]~cole N o r m a l e S u p 4 r i e u r e de L y o n 46, all4e d ' I t a l i e - 69 363 L y o n C e d e x 7 - F r a n c e .

Received April I0, 1995;

in revised form September 14, 1995

We p r o v e t h a t if f : Y ~ X is a finite fiat m o r p h i s m b e t w e e n two r e d u c e d a b s o l u t e l y irreducible algebraic p r o j e c t i v e curves defined over t h e finite field Fq, t h e n

I - 2( y -

w h e r e ~rc is t h e a r i t h m e t i c genus of a curve C. As a p p l i c a t i o n , we give s o m e c h a r a c t e r s u m e s t i m a t i o n o n s i n g u l a r curves.

In this paper, the word curve stands for a reduced abso- lutely irreducible algebraic projective curve defined over the finite field

I~q

with q elements. If X is a smooth curve, it has been shown by Weil in [6] that the number of Fq-rational points of X, denoted by ~X(Fq), is related to the geometric genus g x

by :

[ ~X(Yq) - (q + 1 ) ] < 2 g x x / q (1) (with an improvement of Serre using the integral part, see [5]). In fact, Weil's statement, involving the zeta function of X, is more precise. It implies that if there is a finite morphism f : Y ~ X between two smooth curves X and Y having (geometric) genus g x and gv respectively, then (see for instance [2], proposition 6)

I -

2 ( g . - g x ) v %

(2)

When X is the projective line, this is exactly Weil's bound for Y.

On the other hand, the authors proved in [1] that if X is a singular curve, then Weil's inequality (1) holds if one replaces the geometric genus g x of X by its arithmetic genus ~rx. The aim of this paper is to give a generalization of both (2) and (1) for singular curves. Namely, if f : Y ~ X is a finite flat morphism between two singular curves, then

[ ~ Y ( F q ) - ~X(Fq)1_< 2 ( T r y - 7rx)v~ (3) holds.

We will prove (3) in the third section. The proof goes as follows. Inequality (2) can be applied to the finite morphism f : 1~ ~ )( induced by f on the smooth models of X and Y respectively. Furthermore the number of l~q-rational points of a curve is related to the number of Fq-rational points of its smooth model (first section). Unfortunately, this is not sufficient to prove (3). One has to introduce (in the second section) the auxiliary curve Z = )~ x x Y, birational to Y.

Coverings of singular curves over finite fields 469

Finally, we apply this result in a fifth section to obtain some character sum estimations.

1. A l e m m a

The following lemma relates the number of Fq-rational points of a curve and t h a t of its smooth model. It is given in [1], but in order to be self contained, we give here its short proof. If P is a Fq-rational point of a curve X, we denote by c~p (respectively

ap(OC))

the number of Fq-rational points (respectively of Fq-rational points, where Fq stands for an algebraic closure of Fq) of )(, lying over P in the normalization map Yx : )~ ~ X. Let

Op

be the local ring of X at P, and

Op

its integral closure in the function field Fq(X) of X. The quotient

O p / O p

is a Fq-vector space of finite length : let

5p

be its dimension.

L e m m a 1.

Let X be a reduced absolutely irreducible projec-

tive algebraic curve defined over Fq. Then

I ~:((~q) - ~X(~q) t - < Y~ l a g - X I_<~x - g x 9

PEX(Fq)

Proof.

Let us first prove t h a t if P is a Fq-rational singular point of X, then c~p- 1 <

5p.

Let Q 1 , . . . , Q~p(~) be the Fq-rational points of )~ lying over P (the

ap

first beeing the Fq-rational ones), and r the Fq-linear map

r O p ) ]~qCtp

f , ,

We prove that r is onto : let ( X l , . . . , x ~ p ) E Fq ~v and fi = xi E Fq C Fq(X) if i < a p . For i _> a g + 1, let fi = 0. T h e n by the weak approximation theorem, there exists g C Fq(X) such t h a t

vQ~(g-fi)

>_ 1 for 1 < i <

ag(Oe).

Hence, r : ( X l , . . . ,Xap) and

g E

N

OQ~

: OF.

Since f ( Q 1 ) . . . . = f(Q,~e) for f e Op, it follows that r

is contained in the vector-line L C Fq~e spanned by (1, ..., 1). One obtains a surjective linear map

r

Op/Op

) ~q~P/L.

Taking dimensions, we obtain that a p - 1 <_ 5p.

Now, Lemma 1 follows from the formulas

7rx -- g x = E ~P PeSing X(Fq) and -

x(Fq) =

-

1).

P~X(Fq) [] 2. A n a u x i l i a r y c u r v e

Let X and Y be two curves, and f : Y ) X be a finite flat morphism. In order to give an estimate for the difference between the numbers of Fq-rational points of X and Y, it is convenient to consider the fibre product Z = ) ( x x Y.

L e m m a 2. Z is a reduced absolutely irreducible projective

curve.

Proof. The map f beeing finite, the map Z ) P( is finite and onto, which implies t h a t dim Z = dimP( = 1, and Z is a curve. In order to prove that Z is absolutely integral, one has to prove that given an affine open set SpecA(Xi) of X, the ring A = A(X~) ~A(Xi) A(Y~) is an integral domain, where

stands for the integral closure of a ring A, and A(Yi) is defined by SpecA(Y~) = f - l ( S p e c A ( X i ) ) . Denote by Frac(A) the quotient field of a domain A. By the flatness hypothesis,

C o v e r i n g s of s i n g u l a r c u r v e s over finite fields 471

induce

0 , A , Frac(A(Xi))

|

A(Yi).

Then, t h e injective m a p

Frac(A(X~))

|

A(Yi) ~-*

Frac(A(Yi))

proves t h a t A is an integral domain, as a subring of a field. Finally, Z is projective. Indeed, f is proper since it is finite, a n d Z , ) ( is also proper, so t h a t t h e composite mor- p h i s m Z , ) ( ) Spec]Fq is proper since ) ( is complete. Hence, Z is projective as a complete curve. []

By t h e universal properties of t h e fibre p r o d u c t and of n o r m a l i z a t i o n maps, one can write t h e following diagram, where all triangles and squares are commutative, and all morphisms are finite :

?

/

z , I Y , X

T h e sheaf of O x - m o d u l e s f . OF is coherent because f is finite. It is t h e n locally free since f is flat. After localization, t h e stalks

(f.Oy)p

are

Ox,g-module

of finite type, whose rank d o e s n ' t d e p e n d on P since X is connected. Let

r = dim~q(p)(f.Oy)p |

F q ( P )

= ~ dim~q(p)(Oy, Q | F q ( P ) ) Q e I - I ( P )

be this rank, w h e r e

Fq(P)

denotes t h e residue field of t h e point P . N o t e t h a t if r = 1, t h e n

f - l ( p )

= {Q} contains only one e l e m e n t for any P c X , a n d f induces a m o r p h i s m of local rings

(.gy, Q

~ (9x,p,

so t h a t f is an isomorphism. Hence, one c a n s u p p o s e r _> 2.

P r o p o s i t i o n 3.

(i) Z

is birational

to Y.

g z = g Y .

(ii) The arithmetic genus of Z is given by

~ z = ~ Y - r ( ~ x - g x ) .

In particular,

Proof.

(i) This is trivial since t h e r e are d o m i n a n t m o r p h i s m s I~ , Z a n d Z ~ Y.

(ii) Since b o t h a r i t h m e t i c a n d geometric genus of a curve C a n d of C x~q Fq = C are t h e same, it is sufficient to c o m p u t e 7r g. To simplify n o t a t i o n s , we continue to d e n o t e by X , Y a n d Z t h e curves X , Y a n d Z respectively. T h e exact sequence of O x - m o d u l e sheaves

0 , O x , y x , , O x , ( u x , , O 2 ) / O x , 0

t e n s o r i z e d by t h e fiat O x - m o d u l e

f . O y ,

gives t h e exact se- q u e n c e

0

, f , Oy

~ ( u x , * 0 2 ) |

f , O r

, | y . o y , o

(4)

f r o m w h i c h we d e d u c e a long exact sequence of cohomology. Let us s c r u t i n i z e its different terms.

C o v e r i n g s of s i n g u l a r curves over finite fields 473

Since

Hi(X,

f,(gV) ~

Hi(Y, 9

and Y is projective, then we have

d i m ~ q H ~

= 1 and

dim~ H l ( X , f . O y ) = ~rv.

In the same way,

and

dim~ H~

|

f, O r ) = 1

dim~q

H i ( x , (z,x,.02) |

f . O y ) = rcz.

Furthermore, the sheaf

(z,x,.O2)/Ox

is a sum of skyscraper sheaves on the singular points of X. Hence, this is also the case for

((Z,x,.O2)/Ox) |

f . O y ,

and this prove the vanishing of its H 1. Finally, dim~q H ~ X,

( ( Y x , . O 2 ) / O x ) |

f . O y )

= E dim~q

((Ox,e/Ox,e)@Ox.p ( f . O v ) e ) =

PESingX(Fq) E d i m ~

((Ox,v/Ox,p) (g~ (Fq |

( f . O v ) p ) )

PESing X ('Fq)

=

E

dimTq

((Ox,p/Ox,p)N~q (~q)r)

P6SingX(~q)

:

E

dim~ (Ox,p/Ox,p)r

PESingX(~q) = r( rx - g x ) .

The nullity of the alternating sum of the Fq-dimensions of the different terms of the long exact sequence of cohomology given by (4) gives:

1 - 1 + r(rrx - 9x) - try + fez - 0 = O,

3 . T h e m a i n t h e o r e m

T h e o r e m 4. Let X a n d Y be two reduced absolutely irre- ducible projective algebraic curves defined over Fq, with re-

spective arithmetic genus rrx and try, and let f : Y ~ X be

a finite fiat morphism defined over Fq. Then

T h e proof depends on the following lemma.

L e m m a 5 . ] (~Z(Fq) - ~2(Fq)) - (~Y(Fq) - ~X(Fq)) l< (r - 1)(rrx - g x ) (recall that r = d i m F q ( p ) ( f . O y ) p |

~'q(P))"

Pro@ If P C X(Fq), let a n d

~p = ~{P e 2 ( ~ )

I . x ( P ) = P}

/ 3 p = ~ { Q c Y ( F q ) I f ( Q ) = P } . T h e n 1 < Qe/-l(P) Moreover, Z(Fq) = {(/5, Q) e X(Fq) x Y(Fq) hence a n d - E dimt%(p) (Oy, Q | Fq(P)) = r Qef-~(P)

I ~x(P)= f(Q)}

PEX(Fq) PCX(Fq) = E ~ p ( / ~ p -- 1), PEX(Fq) PeX(~) PeX(F~)

= ~

(~-I).

PeX(Fq) O~p

C o v e r i n g s of s i n g u l a r curves over finite fields 475 By l e m m a 1, and since r > 2, we o b t a i n [ (~Z(IFq) -- ~X(]Fq))-(~Y(]Fq) - ~X(]Fq))1~ P6X(Fq) <_ ( r - 1 ) ( T r x - g x ) . []

T h e t h e o r e m follows easily from t h e preceding a n d t h e trian- gular inequality

I ~Y(~q) - ~x(~q) I<l ~ ? ( ~ ) - ~ z ( ~ ) I

+ I ( ~ z ( ~ ) - ~ ( F ~ ) ) - (~Y(Fq) - ~ Z ( ~ ) ) I

+ I ~ X ( ~ ) -

~ ? ( ~ ) I

4. R e m a r k s

4.1. T h e proof of t h e o r e m 4 gives a b e t t e r upper bound, n a m e l y

[ ~Y(Fq)-~X(Fq)I_<

(~y-gy)-(~x-gx)+(gy-gx)[2v~.

See l e m m a 4.1 of [3] for t h e reason of t h e integer part [2v~ ] of

2v~.

4.2. It c o u l d n ' t be e x p e c t e d t h e o r e m 4 to be true for any finite m o r p h i s m f : Y ~ X. For instance, this is false for the n o r m a l i z a t i o n m a p of a singular curve. Here is a n o t h e r e x a m p l e : let X be t h e plane curve Y 2 Z = X 3. This is a singular cubic curve, its arithmetic genus is 7rx = 1, and for any integer n, we have ~X(F2~) = 2 n + 1. Let us consider t h e m o r p h i s m

: ~ 1 ) X

( u : v )

, , ( u 2 v : ~ 3 : ~ 3 )

Suppose t h e o r e m 4 would be true for any finite m o r p h i s m f , a n d let C be any s m o o t h curve of genus g c , defined over F2.

T h e r e is a finite m o r p h i s m ~v ~ : C ) Ii ~1 , hence by composition a finite m o r p h i s m f = ~ o ~r : C ~ X , and t h e o r e m 4 would i m p l y t h a t for q = 2 n,

I c(Fq) - (q + 1) J< 2(gc - 1)v ,

which is false : one can consider for instance the s m o o t h curve C given by X 3 + y 3 ~ Z 3 ___ 0, which has 9 rational points over Fa a n d genus

gc = 1.

4.3. If X is s m o o t h (in particular if X is the projective line), t h e n a n y finite m o r p h i s m Y ~ X is fiat. Indeed, if

f(Q) = P,

t h e n t h e local ring

(gy, Q

is a finitely generated (9x,p-module w i t h o u t torsion element. But

Ox,p

is a principal ideal domain since X is supposed to be smooth, so t h a t

(gy, Q

is a flat

(gx,p-

module. This shows t h a t theorem 4 contains all known results (1) a n d (2) of the introduction.

Note t h a t there are m a n y other situations where Y ) X is fiat. This is the case for instance if X =

Y/G,

where G is a finite group of a u t o m o r p h i s m s of Y (see [4]), or (by stability of flatness by base change) if f is the reduction modulo a prime ideal of a flat m o r p h i s m between two curves defined over a n u m b e r field, or (also by base change, see section 5 below) if f is a K u m m e r , or an Artin-Schreier morphism.

5. A p p l i c a t i o n t o e x p o n e n t i a l s u m s

Let X be a curve over a finite field Fq of odd characteristic and

f E Fq(X)

a function on X . W h e n X is smooth, there is an extensive literature on estimations of the character sums ~ P e X ( F q ) (f--~qP)), where ( q ) i s the Legendre symbol on Fq ( w i t h t h e c o n v e n t i o n t h a t ( f q - ~ ) = 0 i f P i s a z e r o o r a p o l e

of f ) . In fact, this character sum equals to ~Y(Fq) - ~X(Fq), where Y is t h e Zariski closure of the curve {(P, y) E X • p1 I y2 = f ( p ) } in the surface X x p1. Indeed, there are exactly 1 + ( L ~ q P ) points in

Y(Fq)

above a given point

P E X(Fq).

Coverings of singular curves over finite fields 477

The geometric genus g y of Y is given from g x by Hurwitz formula, and its arithmetic genus is given by the following lemma, whose proof is a standard and tedious calculation using Cech cohomology.

L e m m a 6. L e t Y be the Zariski closure of {(P, y) E X • P 1 I y~ = f ( P ) } . T h e n , Try = nzrx + ( n - 1 ) ( d e g ( f ) o - 1), where ( f ) o d e n o t e s t h e divisors o f zeroes o f f .

Since the squaring map P 1 ~ ~1, y, > y2 is fiat, then by

base change Y = X • p1 ~ X is also fiat. Hence, theorem 4 applies, so that the following holds :

T h e o r e m 7.

P e X ( F q )

( f - ~ ) [_< 2(n - 1)(~x + d e g ( f ) o - l)v/-q.

Of course, one can give a better upper bound using remark 4.1., and one can study additive character sums via Artin- Schreier coverings.

R e f e r e n c e s

[1] Aubry Y., Perret M. : A Weil theorem for singular curves, Proceedings of Arithmetic, G e o m e t r y and Coding T h e o r y IV, ed. Pellikaan, Perret, Vl~du~. De Gruyter, (1995)

[2] Lachaud G. : Sommes d'Eisenstein et nombre de points de certaines courbes alg6briques sur les corps finis, C. R. Acad. Sci. Paris 305, 729-732 (1987)

[3] Lachaud G. : Artin-Schreier curves, exponential sums and the Carlitz- Uchiyama bound for geometric codes, J. Number T h e o r y 39, 18-40 (1991) [4] Mumford D. : Abelian varieties, Oxford Univ. Press, Oxford 1970

[5] Serre J.-P. : Sur le n o m b r e de points rationnels d ' u n e c o u r b e alg~brique s u r u n corps fini, C. R. Acad. Sci. Paris 2 9 6 , s~rie I, 397-402 (1983)

[6] Weil A. : Sur les c o u r b e s alg~briques et les vari~t~s qui s'en d6duisent, H e r m a n n , Paris 1948