ALGEBRAIC CURVES OVER FUNCTION FIELDS. I

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Ser. Mat. Tom 32 (1968), No. 5 Vol. 2 (1968), No. 5

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I

Α . Ν. PARSIN UDC 513.6

This paper studies the diophantine geometry of curves of genus greater than unity defined

over a one-dimensional function field.

Introduction

Let X be a scheme over Spec Ζ and Κ an extension of finite type of the field Q of rational num-

bers. The fundamental problem in the theory of diophantine equations is the description of the set

X (K) of points of the scheme X whose coordinates belong to the field K. The solution depends both

on X and on K. Thus, for example, if X is an algebraic curve, the answer depends on the value of the

genus g of the curve. The simplest case is g = 0. Either the set X(K) is empty, or the curve is bira-

tionally equivalent to a projective line, which gives a rational parameterization of the set X (K) (for

an arbitrary field K). If g = 1, then, since there are rational points on X, the structure of a commuta-

tive group can be defined on X (K). Mordell proved that the group X (K) has a finite number of genera-

tors if Κ is the field of rational numbers. Finally, when g > 1, numerous examples provide a basis for

Mordell's conjecture that in this case X (Q) is always finite. The one general result in line with this

conjecture is the proof by Siegel that the number of integral points (i.e., points whose affine coordi-

nates belong to the ring Ζ of integers) is finite. These results are also true for arbitrary fields of

finite type over Q. Fundamentally this is because the fields are global, i.e., there is a theory of

divisors with a product formula, which makes it possible to construct a theory of the height of quasi-

projective schemes of finite type over K. Lang's book [ ] contains a description of that theory and

its application to the proof of the Mordell and Siegel theorems. It appears that further progress in

diophantine geometry involves a deeper use of the specific nature of the ground field. This is con-

firmed by Ju. I. Manin's proof of the functional analog of Mordell's conjecture [ ].

Another important problem in diophantine geometry is the classification of algebraic varieties

defined over a given global field Κ (cf. [ ]). We shall consider only the case of algebraic curves.

Every curve X which is nonsingular and geometrically irreducible over Κ has the following invariants:

genus g and type S — the finite set of points of Κ for which X does not have a good reduction (we as-

sume for simplicity that Κ is one-dimensional). We shall say that X has a good reduction at the point

ρ of Κ if there exists a scheme V, smooth and proper over 0p, such that

' ρ — Χ ®κ.Κρ,

1145

1146 Α. Ν. PARSIN

where Up c Kp is the local ring of ρ and Kp its field of quotients. I. R. Safarevic has made the fol-

lowing conjecture [ ].

Conjecture. The set of algebraic curves defined over a global field Κ and having given invariants

g > 1 and S, is finite (in the functional case only non-constant curves are considered).

It can be shown that the problem of classifying algebraic curves is a special case of the problem

formulated above of describing the set of rational points of an algebraic variety. More exactly, let Mg

be the space of moduli of algebraic curves of genus g, itself a scheme over Spec Z. Then the set of

curves in the Safarevic'conjecture is simply the set of S-integral points of the scheme Mg. Hence this

conjecture is most naturally obtained as a corollary of the appropriate generalization of Siegel's theo-

rem to varieties of dimension greater than unity. The only result of this kind to date is the proof of

the finiteness conjecture for hyperelliptic curves [ ]. It uses Siegel's theorem for curves, which in

this case is sufficient because of the simple structure of the space of moduli of hyperelliptic curves.

This paper studies the problems touched on above for curves over function fields. By a function

field we mean the field Κ = k(B) of rational functions on some curve B. Every curve over Κ defines

in a natural way a Α-surface provided with a pencil of curves (cf. Chapter 1). Hence deep results from

the theory of surface [ ' ] can be used in this case. The proofs assume that k = C, but in view of

the Lefschetz principle the basic results hold for an arbitrary algebraically closed field h of charac-

terstic 0.

We pass now to a description of our results. Chapters 1 and 2 contain a proof of the Safarevic

conjecture for a certain class of algebraic curves. As usual in diophantine geometry (cf. [ ]), the

finiteness proof falls into two stages. First it is established that the height of the set of curves in

question is bounded. This is done in Chapter 1 for arbitrary curves (§3, Theorem 2). Sections 1 and

2 of this chapter contain the requisite facts concerning fiberings associated with the curves. To pass

from the boundedness of the height to the finiteness conjecture one needs a theorem on the rigidity of

deformations of fiberings. Such a theorem is obtained for nondegenerate curves in §§1 and 2 of Chap-

ter 2. From this follows the proof of the Safarevic" conjecture for this class of curves (Theorem 3 of

Chapter 2). The problem of nondegenerate curves over fields of genus 0 and 1 is also considered in

Chapter 2 (§3).

In the last chapter a relation is established between Mordell's conjecture and that of Safarevic.

To do this the properties of branched coverings of curves of genus g > 1 are studied in § 1 . From the

results obtained it can be established that the Mordell conjecture is a direct corollary of that of Safarevic j

(see the end of §2). Moreover, it is shown that the boundedness of the height in the Safarevic con-

jecture already implies the boundedness of the height for rational poiuts on curves of genus g > 1.

This leads to a new proof of the functional analog of Mordell's conjecture (Manin's theorem; see

Chapter 3, §2). This proof makes it possible to obtain a bound for the height of rational points on

algebraic curves.

The author wishes to express his deep gratitude to I. R. Safarevic for his constant attention during

the preparation of this paper.

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1147

CHAPTER 1. CURVES OVER FUNCTION FIELDS AND FIBERINGS; GENERAL THEORY

§1. Preliminary facts about fibering

Terminology. In this paper we use essentially the language of schemes due to A. Grothendieck

(cf. [8]). Instead of the term "regular" we use the older term Λ nonsingular." The ground field is de-

noted by k and all subsequent schemes and morphisms are assumed to be of finite type over k. By an

algebraic variety we mean a reduced i-scheme of finite type. A covering is a finite flat epimorphism.

If X is a scheme, X(S) denotes the set of points of X with values in S. If Υ C X is a closed subset,

X — Υ denotes the uniquely defined reduced subscheme whose topological space is the complement

of y in X (cf. [ 8 ], I, 5.2.1). Amplification of all notation used can be found in [8] or [ ].

Definition 1. A fibering f: X -· Β is an object consisting of a nonsingular irreducible projective

surface X, a nonsingular irreducible projective curve B, and a flat projective epimorphism /whose

generic fiber is a nonsingular geometrically irreducible curve.

We shall sometimes denote the fibering by the single letter X if there is no ambiguity. By a

morphism g of the fibering X into the fibering Υ is meant a morphism g : X -> Υ of the β-scheme Χ into

the δ-scheme Y. The fiberings X and Υ are considered identical (or isomorphic) if g is an isomorphism

of the surfaces X and Υ.

By the fibers of the fibering f: X — Β we mean the fibers of the morphism /. All fibers, except

for a finite number, are nonsingular curves of the same genus g (this follows from [ ], IV, 12.2.4 and

the formulas for the arithmetic genus of a curve on a surface). The remaining fibers are called degen-

erate. More precisely, these are those fibers that either are not reduced or have a singular point. The

points on Β corresponding to degenerate fibers are called the points of degeneracy of the fibering and

form a finite set S C B. Thus to every fibering corresponds a set (B, S, g), the type of the fibering.

The types of isomorphic fiberings are clearly identical.

It follows from the smoothness criterion ([ ], IV, 17) that the morphism / is smooth in the open set

.Y — / l(S). Note also that all fibers of a fibering X ate geometrically connected, as follows from the

Main Theorem of Zariski ([ 8 ], III, 4.3, 4.4).

Definition 2. A fibering / : X -» Β is minimal if its fibers contain no exceptional curves of the

first kind.

The importance of this concept is that minimal fiberings are uniquely defined by their generic

fibers if g > 1. More precisely, we have:

Theorem 0. Let X be a nonsingular geometrically irreducible curve of genus g > 1, defined over

the field Κ of rational functions on a nonsingular projective curve B. Then there exists a minimal

fibering f : V -> Β whose generic fiber is isomorphic with X.

If U and V are minimal fiberings, X and Υ their generic fibers, and the genuses of X and Υ are

greater than zero, then the canonical mapping

(U, V) - Η - Isom* (X, Y)

is bijective.

The proof of the first assertion follows from the theorem on resolution of singularities of surfaces

and the Castelnuovo contractibility criterion. The second assertion is proved in [ ] (Chapter VII,

1148 Α. Ν. PAR&N

Theorem 1; it is assumed there that g = I, but the proof goes through without change in the case

g> 1)·

Thus, every curve X defined over the function field h(B), nonsingular and geometrically irredu-

cible (absolutely irreducible in the older terminology), determines a uniquely defined minimal

fibering. We shall call the latter the nonsingular minimal model of the curve X.

Change of base. Let φ: Β' -> Β be a covering of nonsingular curves Β and B', and / : X -> Β a

fibering. Then we have the surface Χ χ g S ' and the canonical morphism f(B'): % x B^' ~* &'• The

minimal fibering associated with the generic fiber of the morphism f(B') will be denoted by / ' : XR>

-> B/ and we shall say that it is obtained by a change of base from the fibering / : X -» B. There is a

natural projection π: Xg> ~* X, which is in general a rational mapping. It is obtained from the morphismpri

X X BB' ——*• X by resolution of the singular points and contraction of the exceptional curves of

first kind.

If Υ and Y' are the generic fibers of the fiberings X and Χβ', then

Y'~Y® K',

where K' = k(B').

Definition 3· The fibering f: X -> Β is called trivial if it is isomorphic with a fibering of the formpr2

C Χ Β *Β, and isotrivial if it becomes trivial after suitable change of base.

Let us describe those curves that correspond to isotrivial fiberings. A curve X defined over the

field Κ = k(B) is called constant if there exist a finite extension K' D Κ and a curve C defined over k,

such that

X® KK' ~C® hK'.

To isotrivial fiberings correspond constant curves. The reverse statement follows from the theorem on

the minimality of trivial fiberings.

Sections and rational points. Let / : V -< Β be a fibering, X a generic fiber, and Κ = k(B). There

is an important relation between the divisors of the curve X which are rational over Κ and the divisors

of the surface V. We consider only the effective divisors. If S is a divisor on V, there corresponds to

it the divisor S.X on the generic fiber. The divisor S.X is rational over K. If Λ is a divisor con-

sisting of components of fibers, then R. X = ($. Given a divisor D on the curve X which is rational

over K, one can single out among the divisors S on V such that S · X = D a uniquely defined divisor

2). The condition defining 3) is the flatness of 3)^over the curve Β ([ ], IV, 2.8.5)- In the present case

the condition is equivalent to the absence from 3) of components contained in the fibers (this follows

from Proposition 6 of Lecture 6 of [ ]). In particular, to every rational point Ρ € Χ (Κ) corresponds

a divisor Ρ which is a section of the fibering V. The curve Ρ is nonsingular and irreducible, and the

morphism / defines an isomorphism

For any closed fiber V ̂ , b € B, the intersection index Fj · Ρ is equal to unity. It follows that Ρ

cannot intersect the singular points of the fibers and consequently the morphism / is smooth in some

neighborhood of the curve Ρ.

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1149

If φ : Β' -> Β is a covering and /: XB'-> δ ' is a change of base for the minimal fibering {: X ->B,

then to every section C of the fibering X corresponds a section C" of the fibering Xg'·

It is not difficult to verify also that the correspondence constructed above commutes with the

operation of changing the base.

§2. The geometry of minimal fiberings

Lemma 1. Let f: X —> Β be a minimal fibering whose generic fiber is a curve of genus g > 1,

while the genus q of the base is also greater than unity. Then:

a) X is a surface of fundamental type;

b) X is a minimal model in the sense of the theory of surfaces.

Proof. By [ ], X is a surface of fundamental type if X has no pencil of rational curves or of el-

liptic curves, the irregularity q (Χ) Φ 0, and X is not a two-dimensional abelian variety. Let us verify

that these conditions hold in the present case. Any curve on X either covers the base, and hence has

genus greater than unity, or is contained in a fiber, and so likewise, with the exception of a finite

number of possibilities (degenerate fiber), has genus greater than unity. The irregularity satisfies

q (X) > q > 2. Finally, if X is an abelian variety, then the canonical class Κχ = 0, and so

(where F is a fiber of the morphism f, g the genus of F), which contradicts the hypothesis. The pre-

ceding implies that exceptional curves of the first kind on X must be contained in fibers, and as-

sertion (b) follows from the minimality of the fibering.

Proposition 1. Let f : X -> Β be a minimal fibering of type (B, S, g), g the genus of the base B,

g > 1, q > 1. Then:

2. K*^3. X(X)4. q(X)

where s = Card(S), pg (X) is the geometric genus, χ the Euler characteristic, Κ the canonical class

and q (X) the irregularity of the surface X.

Proof. Denote by X$ C X the set of points belonging to degenerate fibers, X$ C f"1 (S). We have

the exact sequence of the pair (X,

m (X, x8)—Η* (Χ) Λ m (xs)

where Η is cohomology with complex coefficients. The group H2(Xg) is generated by algebraic cycles

(components of fibers). The group Ω2'0 (X) of double differentials of the first kind is contained in

H2 (X), and since the fibers are one-dimensional it follows that

Hence pg (X) < b2(X, Xs). Using the spectral sequence of the differentiable locally trivial fibering

X \XS - Β \S, w e f ind t h a t χ(Χ, Xs) = χ(Β, S)(2g - 2 ) a n d bl (X, Xs) <2g + 2q + s. F r o m t h i s w e

obtain a bound for b2(X, X$) and so for pg (X). To obtain inequalities 2, 3, and 4, note that K2 > 0

(Lemma 1 and Lemma 5 of §1, Chapter VII, of [ ]) and χ(Χ) > 0 (which is clear from the formula for

1150 Α. Ν. PARSIN

the Euler characteristic and the inequalities g > 1 andq> 1 ([*], Chapter IV, §4, Theorems 6 and 7)).

Hence the required bounds follow from the Noether formula Κ2 + χ = I2pa = 12 (I — q + pg).

Proposition 2. Under the hypothesis of Proposition 1, let n>_6. Then the linear system \nK\ has

no fixed component and defines a regular mapping

φ: Z->P m ,

where:

1. φ is a biregular imbedding on X — f'1(S);

2. nK = φ (L), where L is a hyperplane in P m ;

3. m = l(nK) — 1 ==ξ 50n?(gq + s).

Proof. It follows from Lemma 1 and Theorems 3 and 5 of [ ] that the linear system \nK\ has no

fixed component. Theorem 1 of [10] proves that φ is regular and biregular on the complement of a set

Ε = U C , where the C; are nonsingular rational curves. We assert that C; C f'1 (S). Indeed, if a curve

C C X covers the base B, it is irrational, and the nondegenerate fibers of the morphism /are also ir-

rational (g > 1); hence Ε C f ί (S). Assertion 2 of Proposition 2 follows, as is well known, from the

regularity of φ and the fact that there are no fixed components. Finally, assertion 3 follows from the

Riemann-Roch theorem, Proposition 1 and Theorem 5 of [1"]. This completes the proof.

Remark. For any curve C on the surface X,

K.C S3 0.

Indeed, if C is an irreducible fiber, then K.C > 0; if C is a component of a reducible fiber, then C2

< 0 (I1], Chapter VII, §2) and from the equality K.C + C2 = 2pa(C) - 2 we obtain K.C > - 1 and

K.C = — 1 •<=> pa(C) = 0 , C2 = — 1 . The minimality of X then implies the required result.lNow let

C cover the base and let C2 < 0. Then pa (C) > 1 and

K.C = 2pa (C) - 2 — C1 > 0.

If C2 > 0, then K.C > 0 since I (K) = pg > 0 (Theorem 5, Chapter IV of t 1 ] and Lemma 1). It can be

shown, using Hodge's index theorem, that K.C = 0-$=$~pa (C) = 0 and C2 = — 2.

Proposition 3. Let f: X -> Β be a minimal fibering of type (B, S, g), q the genus of the base B, g

> 1, q > 1, Ν > 0 an integer. Then there exists a covering l:B' -> Β for which:

1. The fibering Χβ has at least Ν different sections C,, . . . , C/y.

2. The fibering Xg is of type (B'', S', g), where S' C t'1 (S), and

q' < 100 NHgq + s),

deg t ==S 200 Nz(gq + s).

3. f.(C().^.Y^100^(irg + *),

where f is the natural projection of Χ β onto X, /*(C£) the proper image of the section Cj, and q' the

genus of B''.

Proof. By Proposition 2, there exists in the linear system \GNK\ an irreducible nonsingular curve

B" not contained in the fibers. Let L = k(B): L" = k(B"), L' / L" be the least normal extension of

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1151

L containing L", B' the normalization of Β in L', and t: Β' -> Β the natural covering. Denote by X'

= Xg, the nonsingular minimal model of the fibering Χ χ gB''. Then

where the C ; are sections, # is an effective divisor consisting of fiber components, and /* (G;) = S".

The number of sections is equal to the degree of B" over B, and since

[A(B") :k(B)]= B".F = 6NK.F = QN(2g - 2) > N,

we obtain assertion 1. Since extensions conjugate to L" are branched wherever L" is, and deg ί

^Γ [Ζ/' : Ζ]^" :Ι^!, to obtain bounds for q' and deg ί it is sufficient to know the degree \L" : L] and

the number of branch points of the curve B" over B. Using the Hurwitz formula, Proposition 1, and the

bound

2pa{B") — 2 = {Β")2 + Β"Κ < 2MN2K2,

we obtain the required result. In a similar manner, assertion 3 follows from the equalities

and Proposition 1.

§3. The main theorem

In this section we prove

Theorem 1. Suppose given a nonsingular projective curve Β of genus q > 2, a finite set S of

closed points of B, and an integer g >_ 2. Then there exist nonsingular algebraic varieties V and W

and a smooth proper epimorphism Θ : V -* (Β — S) χ W such that:

1. For any closed point w £ W the variety

VW=V x ((B — S)x w)

(B— S)XW

is a nonsingular surface, and the morphism ®\y : Vw — Β — S is smooth.

2. For any minimal fibering f: X -· Β of type (B, S, g) there exists a closed point w 6 W for

which X — f'1(S)c^ Vw over the curve B — S.

To prove Theorem 1 we require a number of lemmas and propositions.

First we require the concept of the height HK (P) of a point Ρ rational over the global field K.

We use the definition of height in Chapter III of Lang's book t 1 ' ] . If /: X -» Β is a fibering, Κ = k(B)

the field of rational functions on B, and Υ a generic fiber — an algebraic curve defined over K, then

there is a one-to-one correspondence between the sections Ρ of the fibering and the points Ρ of Υ

rational over Κ (§1).

Lemma 2. Let f : X -> Β be a fibering with projective base Β, Υ a generic fiber, Κ = k(B), Ρ

€ F (K). If φ: Χ -> P m is a morphism of the surface X into projective space that defines an imbed-

ding

1152 Α . Ν. PARSIN

then

where D - φ (L) and L is a hyperplane in P m .

Proof. The mapping φ κ is constructed as follows. The morphisms φ and /yield the mapping

Φ: X — P m , Φ = fx βφ, and hence a mapping of generic fibers, which is then φχ. Let Q £ P m (K);

then Propositions 5 and 6 of [ ' ] (Chapter III, §3) show that

deg iQ l(LxB)HK(Q) = e

where L is a hyperplane in P m , ig : Β — Ρβ is the section of the direct product corresponding to the

point Q (in [ *] the curve β is the variety W, e = c"1, (̂ = P). Finally we have

H9K(P) = ΗΚ(ΨΚ.(Ρ)) =•

= e(Lx>· .;-££> =

= eO»(LXB) -P _

where

Corollary. In the same situation as in Lemma 2, let Β' -> β be a covering of the curve B, K' D Κ

the corresponding extension of the function field, and Ρ € y(/ i ' ) . Then

where Ρ C XB', I'· XB' -* ^ is the projection, f* the proper image, and Η φ κ Κ is the relative height.

The proof follows from the definition of height, Lemma 2 and standard formulas of intersection

theory.

Lemma 3· Suppose given a projective space P m and an integer d. Then any two irreducible

curves of degree d in P m coincide if they intersect in more than Ν closed points, where Ν = d2.

The proof follows from the inequality deg (λ" Π X') < deg X • deg X', where X and X' are curves

in Pm, proved in Lang's book ([ ], Chapter 3, §3, Lemma 4).

Lemma 4. Let X be a quasi-projective scheme over k with a fixed projective imbedding and sup-

pose given an integer d > 0. Then there exists a quasi-projective scheme Y, called the Hilbert scheme

of one-dimensional cycles of degree d on X, and denoted by Hilb^ (X) with the following properties:

1. There exists a scheme V and a commutative diagram

V -> Hilbd (Χ) χ Χ

Hilbd (X)

where ρ is a flat morphism, q an imbedding, and for any closed point y 6 Hilb^ (X) the fiber Vy is a

one-dimensional cycle of degree d in X with respect to the imbedding pr2

oq.

2. For any commutative diagram

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1153

S

in the category of schemes satisfying the same conditions as diagram 1, there exists a unique mapping

φ: S -> Hilbd(X) such that R en V χ YS, s = q χ YS.

The scheme V is called a universal family of one-dimensional cycles of degree d.

The proof can be found in [ ]. It is clear from the definition that to each closed point of the

Hubert scheme corresponds a one-dimensional cycle in X. Hence if ο € Hilb^ (X), we shall speak of

the cycle (or curve) a. Conversely, if L is a one-dimensional cycle of degree d in X, property 2 ap-

plied to the diagram

Spec k

shows that it corresponds in Hilb^/Y) to a closed point / (viz., that for which Vι ^ L).

Remark. In the scheme Hilb^ (X) there exists an open subscheme Hilb^ (X) defined by the con-

dition that

Ι ΕΞ Hilbd (X) ,-<=>- Vl is a nonsingular geometrically irreducible curve in X.

Lemma 4 holds for Hilb^ (X) if we replace the words "one-dimensional cycle of degree d" by "irreduci-

ble nonsingular curve of degree d" and the scheme V by V = p"1 (Hilb^(/^)).

We prodeed now to the proof of Theorem 1. As we saw in §2, the divisor 6Κχ defines for any

minimal fibering f: X -> Β of type (B, S, g) a morphism of the surface X into projective space Pm,

where m depends only on g, q and s. This mapping, which we denote by φχ, is an imbedding on

X — f'1 (S). For all b € Β the degree of the divisor 6Κχ on the fiber Xb is independent of b. Let it

equal d. By Lemma 4 we have the morphism

The image of a generic point under this morphism defines a point χ (X) in Hilbj(P m ) ® Κ, Κ = k(B).

Since the curve Β is nonsingular and the scheme Hilb^(Pm) projective (see [7]), there exists a unique

morphism

coinciding with \j.:\ on Β — 5. The graph Γ^ of the morphism φχ is a one-dimensional cycle of the

scheme Β χ Τ (here and hereafter Τ = Hilb£;(Pm)), namely, a section of the morphism Β χ Τ -> Β. On

the generic fiber Τ ® Κ there corresponds to it the point x(X). Let us take as fixed a projective im-

bedding of the scheme Τ and with it the height Η κ on Τ ® Κ.

Lemma 5· There exists a constant C, depending only on g, q and s, such that

HK(x(X)) ^C

for any minimal fibering X of type (B, S, g).

Before proving this lemma we shall show how Theorem 1 follows from it. If we choose a projective

imbedding of the scheme Β χ Τ, Lemma 5 indicates that the degrees of the cycles Γ^ are bounded by

1154 Α. Ν. PARSIN

some constant C" when Γ^ runs through the set of fiberings of type (B, S, g) ([ ], Chapter III, §3,

Theorem 2).

Put

Wx

m= Π \H\h,{BxT).

Denote by ρ : Um -> W*m, 8: R -> Τ the universal families over the Hilbert schemes (Lemma 4), by

y : Um -> Β χ Τ χ If},, the canonical imbedding, and put

Vl=UMx (BxRxWi),(BxTxWn)

where the product is taken with respect to the mappings γ and Ι β χ 1^, . Define the mapping

as the composition of the natural morphism of the scheme V^ into Β χ Τ χ Wl

m and the projection onto

the outside factors. Finally, replace Β by Β — S:

The triple (W^, V%

m, @2

m : V2

m-> (B — S) X W2

m) satisfies condition 2 of Theorem 1. To satisfy

condition 1 we must pare down the scheme W2

m. First we replace W2

m by (Ifm)red = ^m> Θ m> ^m a r e

obtained by restriction to the subscheme (β - S) χ (If m) t e d of t n e scheme (B — S) χ W2

m. If W3

m is

reducible and Wl

m = \J (If;3

m)i, we replace the union |J by the direct sum II, and change W3

m, Θ 3

m

i

correspondingly. Thus we obtain a triple (W*m, V4

m, Θ m), in which the scheme W*m is a finite direct

sum of integral schemes. By [ ] (IV, 6.9-1) there exists in each component of V-'*m an open subset over

which the morphism Q"m ° pr2 is flat. Hence ([ ], IV, 6.9-3) we can replace each component of Wm by a

direct sum of subschemes such that in the new triple (Ws

m, V5

m, Θ „) the morphism Qs

m ° pr2 is flat.

Now retain in Ws

m only those components needed to satisfy condition 2 of the theorem. Then by [ ]

(IV, 17.3-7 and 17.5.1) and condition 2 of the theorem, there exists in each of these components an

open subscheme over which the morphism 0 s

m °pr2 is smooth. Since a flat morphism is open [ ], IV,

2.4.6), we can assume that 0m°P r 2 i s a n epimorphism. Finally, by [ ] (IV, 6.12.6) there exists in

each component of W5

m an open subscheme consisting of nonsingular points. As above, we can replace

each component by a direct sum of integral nonsingular subschemes. Thus, we have obtained a

triple (W6

m, V6

m, ©m) where Q6

m °pr2: V6

m -> W6

m is a smooth epimorphism and the connected components

of W6

m ate nonsingular algebraic varieties. Since the set of points of the scheme V6

m at which the

morphism Θ^ ° pr2 is reduced is open ([ ], IV, 12.1.7, vi), it follows from condition 2 of the theorem

and [ ] (IV, 3-3-5, 12.2.1, viii) that we can replace each component of V'6m by an open subscheme such

that after restricting the whole construction we obtain a triple (W7

m, Vm, ®7

m) in which Θ^ ° pr2 is a

smooth reduced morphism, Vm a reduced scheme, and Wm a nonsingular algebraic variety (in general,

not connected). By I ] (IV, 17.11.1 and 17.15-2), V7

m is a nonsingular scheme and hence, by what has

been proved above, integral.

Consider now the diagram

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1155

'~S) x Wn

By [ ] (IV, 1.8.7) and condition 2 of the theorem, there exists in W7m a constructive set of points s

€ W7m such that the mapping (&m)s : (V7

m)s - Β - S is epimorphic. If we replace W7

m by an appropriate

direct sum of subschemes and repeat the argument above, we can ensure that the mapping ®7

m ° pr1 is

an epimorphism. The morphism

eloVTl:Vl^B~S

is flat. This follows from Proposition 6 of [ ] (Lecture 6), the nonsingularity of Β — S, the integrity

of Vm and the fact that &m ° pr^ is an epimorphism. Hence ([ ], IV, 2.1.4, 2.1.7) the morphism Θ ,̂

is a flat epimorphism. It follows from the definition of a smooth morphism ([ ], IV, 17.3-7) and the

fact that &m ° pr2 is open, that there exists in Wm an open subscheme W"m such that when we restrict

Vm and ®7

m to it we obtain a smooth epimorphism Θ^,. Finally, if we put

w=]Jw*mt v = UK, Θ = ΠΘ8™,m m m

we obtain a triple satisfying all the conditions of the theorem. This completes the proof.

Proof of Lemma 5. Let Κ = k(B), let Κ be the algebraic closure of K, and & the set of points of

the form x(X) in Hilbrf(Pm) (K). Choose the integer <V so that Lemma 3 holds with d = 6(2g — 2) and

m = l(6K) - I from Proposition 2 of §2 (re = 6). Let V - Hi lb r f (P m ) be a universal family. Put

U = V χ · • · χ V.

Hilb"(Pm)

We have the canonical imbedding

γ: U-+(Fm)NXHilbd(Pm).

Let Φ = prj ο γ : U—*· (Pm)N. By Lemma 3 we have an imbedding of the sets of X-rational points,

Φ(Ε): U(E)^ (Pm(K))N.

Denote by Ψ the morphism pr2 ° y. In U (K) we shall construct a set G> with bounded height relative

to the imbedding Φ(Κ) and such that Ψ(Χ) (&') 3 te. From this follows the lemma, since the image of

a set of bounded height has bounded height ([ •*], Chapter 4).

If χ €. (b and / : X -> Β is the corresponding fibering, then φχ : X -> P m is a morphism defined by

6Κχ (Proposition 2of §2). By Proposition 3 of §2 there exists a covering Β' -> Β such that the

fibering Xg' has Ν different sections /\, . . . , Ρ /γ and the generic fiber X ® X', K' = k(B'), Ν dif-

ferent K'-points ? ! , . . . , Ρ/γ. Denote by y € U(K) the point (P,, . . . , PN, x{X)). Then W(X)(y)

= x(X) and Φ(Χ)(γ) = ( P 1 ; i . . , PN). We now find a bound for the height of the point Φ(Κ) (γ). Since

the property of a set of X-points of having a bounded height is independent of the choice of the height

we can take as the height the function

ι χ 56 Α. Ν. PARSIN

on (Ρ"1 (Κ))Ν, where h-g is the absolute height [ 1 3 ] . If Ρ 6 P m (L), then

where L is a finite extension of K. Using Proposition 3 of §2 and the corollary to Lemma 2, we ob-

tain a bound for h-g(P j) depending only on g, q and s. This proves the lemma.

Theorem 2. Suppose given a nonsingular projective curve Β of genus q, a finite set S of closed

points of the curve B, and an integer g > 2. Then there exist integers d and m depending only on g, q

and s = card(S), and a set Q> of bounded height,

K = k{B),

such that for any nonsingular geometrically irreducible curve X defined over Κ and of genus g and type

S, there exists a point e € G> such that Ve CH X.

Proof. If q > 1, the theorem is simply a reformulation of Lemma 5, by virtue of Theorem 0 of § 1 .

The general case can be reduced to this one by replacing the ground field by a field of genus q > 1,

since, as is easy to see, the type of the minimal fibering associated with the curve is not "increased"

by such a replacement.

CHAPTER 2. SMOOTH FIBERINGS AND NONDEGENERATE CURVES

§ 1 . The isotriviality criterion

A fibering / : V -* Β is called smooth if the mapping / has no degenerate fibers, i.e., if f is a

smooth morphism. If φ: Β' -< Β is an arbitrary covering, B' a nonsingular curve, then the surface

V = V χ BB' is nonsingular and the natural morphism / ' : V -> B' is smooth. This follows from the

general properties of smooth morphisms (see [ ], IV, 17).

Lemma 6. Let f : X — Υ be a smooth proper morphism of connected algebraic k-schemes. Put

for any closed point y € Υ and assume that h{y) = c o n s t . Then for any morphism φ:Υ'^·Υ we have

a smooth morphism f,y,y X' = X x y Y'-> Y', where:

1) The s/ieo// „, (QL.j,,) is locally free.

2· q>7,(Gx/y)^/<y)«(QW')·

3. The mapping f(Y')*(QXx'iY')(g)k(y')->H0(Xy', Ωχ',) is bijective for all γ' Ε Υ'.

Proof. The morphism /( y ) is smooth and the sheaf Ωι

χ,/γ, locally free ([ ], IV, 17.3-3, 17.2.3);

hence it is flat on Y'. Now note that

SW^(«pXrX)^W (1)

([ 8 ], 0 I V , 20.5-5); hence Ω,ί

χ/y\Xy ^Ωι

χ , and Theorem 4 of [6] (§7) asserts that /*(Ω^/ γ) is locally

free and the canonical morphism

is bijective. By [ ] (III, 7.7.10g) the theorem on change of base holds in the present situation ([ ],

III, 7.7.5, II), and assertion (d) of that theorem together with formula (1) imply assertion 2 of the

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1157

lemma; whence also 1 and 3 follow, since the inverse image of a locally free sheaf is locally free.

Lemma 7. Let f : X -> Υ be

1. an abelian Υ -scheme

or

2. a smooth fibering.

Then Lemma 6 holds for the morphism f : X -> Υ.

Proof. In case 1,

, QZJ = d i m Xy = c o n s t

(equidimensionality of a flat morphism ([ ], IV, 6.1.4)). In case 2,

dimft/fo ( χ ν , Q}Xy) = — χ (X v f © X w ) + 1 = const

(invariance of the Euler characteristic of the fibers of a proper flat morphism ([ ], III, 7.9·4).

Proposition 4. Let f : V — Β be a smooth fibering. Then:

1. The sheaf Ω* , „ is locally free.

2. The sheaves Λ'/̂ Ω1^ . „ and Rlf*Qy are locally free on B, i > 0.

3. The sheaves /̂ Ω1^ .„ and Rlf*Qy are dual to each other.

4. f*0v^6B.

Proof. Assertion 1 follows from the definition of a smooth morphism and the nonsingularity of V.

To verify assertion 2 we use Theorem 4 of [ ] (§7), as in the proof of Lemma 6. Note firstly that

/ is flat, and so the sheaves Ω1^ . β and Qy, being locally free, are flat over B. It is easy to see

(formula (1)) that for any point b 6 Β

where Vb is the fiber of /over b. It follows that dim Hl(Vb, Ω1

ι//β|[/

ί)) and dim Hl (V b, Qv\Vb) are in-

dependent of b. The above-mentioned Theorem 4 yields assertion 2. It also follows from that theorem

that the fibers of the fiberings over Β associated with the sheaves /*Ω' . „ and /i'/^Oj/ are isomorphic

respectively with H° (V b, Ω\/b) and Hl (Vb, Qyb). There is a natural pairing

Since the geometric fibers of a fibering are connected, the sheaf Rif^Q,1

v/β has a regular section which

nowhere vanishes ([ ], Chapter III, §11). Hence

Using the duality theorem for the fibers of f, we find that the above pairing is a duality of locally free

sheaves.

Finally, assertion 4 is a general fact relating to fiberings with geometrically connected fibers.

This completes the proof.

1158 Α. Ν. PARKIN

Proposition 5· Let f: V -> Β be a smooth fibering and g the genus of a generic fiber of the map-

ping f, with g > 1. Let

dV,B = -fog W&t.Ovy

Then:

1. dv/B > 0;

2. dy/g = 0 if and only if V is isotrivial.

Proof. The theorem on a flat change of base ([ ], III, 1.4.15) shows that for any covering Β' -> Β

there is a natural isomorphism

where V = V χ gB', f = fx gB', and hence

dv/B' = deg ψ-dv/B-

It is therefore clear that we can restrict ourselves to fiberings with sections, by an appropriate change

of base. Thus, let the fibering / : V -< Β have a section. Then ([ ], Chapter 6, §1) there exists an

abelian scheme p: A -> Β and a δ-morphism i: V -> A with the following properties. For any point

i £ f i the mapping i\y. induces an isomorphism Pic°(F(,) ^^Α^ and also a sheaf isomorphism

This latter mapping is constructed as the composition of the natural mapping

/**'*ΩΑ/Β-*/,»Ων/Β

and the change-of-base mapping

and the fact that it is an isomorphism follows from Lemmas 6 (assertion 3) and 7 and the corresponding

fact for jacobian varieties. Assertion 3 of Proposition 4 shows that

dVIB = deg fx^f&V.B = deg Ais)P*QAjB.

The abelian scheme 4̂ has a canonical polarization ([ ], Chapter 6, §2). Suppose there exists

a covering of the base Β over which /I becomes trivial, as a polarized family. Torelli's theorem shows

that all fibers of the fibering V become isomorphic over some covering of the base. It follows that V

is locally trivial as an analytic fibering, since curves of genus g > 1 have a local space of moduli

[ ]. Since the genus of each fiber is greater than unity, for all b €. Β the group Aut Fj is finite, and

V, being a fibering with finite structural group, is trivial over the appropriate covering of the base.

Thus the isotriviality of the polarized family A is equivalent to that of the fibering V.

Let φ: Β' -> Β be a covering of the base such that the abelian scheme A' = A x gB' has jacobian

rigidity consisting of the choice of basis in the group of sections of given order re > 3·

Applying Lemma 7, we obtain:

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1159

since the sheaf Ω^/β is invariant with respect to change of base (formula (1)). The fundamental

theorem in modulus theory ([ ], Chapter 7, S3) gives a mapping Ψ: Β' -> Μ into the scheme of moduli

of polarized abelian varieties with given rigidity, for which

A' ~ JH χ MB',

where π:·Μ. —> Μ is the canonical family of abelian varieties over M. Again we have

It follows from [5] (e xp. 17) that the sheaf Λ* 8 ' 3 1 · ^J«/M is ample on U* (its sections are modular

forms in the analytic interpretation of the scheme Λί), and we see that always

where equality

deg /\(e)P*&A',B' = 0

is equivalent to the mapping Ψ being a point-mapping. Returning to the sheaf Rlf*Qy, we obtain the

two assertions of the proposition.

§2. The finiteness theorem

Theorem 3· Let Β be a nonsingular proper curve of genus q, q > 2, g > 1. Then up to isomorphism

there exist only a finite number of smooth non-isotrivial fiberings with base Β and generic fiber

genus g.

Proposition 6. Let f: V -> Β be a smooth non-isotrivial fibering, q the genus of B, q > 2. Then

the sheaf Ω1γ . „ is ample on V.

Corollary. Under the hypothesis of the proposition,

The proof follows from Kodaira's theorem on the vanishing of the cohomology of negative fiberings.

Lemma 8. Under the hypothesis of Proposition 6, let Κ be the canonical class of the surface V,

Kg the canonical class of the curve B. Then:

1.

2. Ων/Β' Ω ν; Β ~ i2dvm·

Proof. Assertion I follows from the usual exact s' ^ence

0 - * fQl

B --• Ων -» &VIB -» 0

and assertion 1 of Proposition 4.

We now prove assertion 2. We have:

Ω̂ ,Β-Ων/Β = (K-f{KB)f = IP - 2χ = 12pa{V) - 3χ, (2)

This was established by Grothendieck, as communicated to me by I. R. Safarevic.

1160 Α . Ν. PARSIN

where χ = (2g — 2) (2q — 2) is the Euler characteristic of the surface V. From the spectral sequence

of the fibering we obtain:

«'/·<?v).

Since Z?% 0 7 2 i 0 B , R%OV = 0, i > 1, we have

which with (2) yields the desired result.

Remark. The formula pa(V) = dv/B + y/4 occurs in the paper by Tate [ ], from which we have

also borrowed the proof.

Proof of Proposition 6. By virtue of the Moisezon-Nakai criterion, it is sufficient to verify that

(K — fKB)-O0

for any curve C on the surface V. The first inequality follows from Proposition 5 and Lemma 8. Note

now that Κ2 > 2χ > 0. Let F be a fiber of the morphism /. Then if C = F,

(K - fKB) -C = Κ F = 2g - 2 > 0.

If C is mapped by /onto the curve B, consider the determinant

C2 CK CFC-K K2 KFC-F KF F-

That it is nonnegative follows from Hodge's index theorem. Noting that

KF = 2g - 2,

a, a ^ 0,

we obtain

2 (2g -2)n{2p-2- C2) - rPK* - (2g - 2) 2C2 ^ 0,

(2g —2) — 2

Whence

a.

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1161

On the other hand

(K — f*KB)-C = 2p — 2 — C2 — (2g —2)n = a — C2.

Comparing these relations, we obtain the required inequality. (The idea of using Hodge's theorem for

a bound on C2 was prompted by I. R. Safarevic.)

Remark 1. It follows from the proof that on a non-isotrivial fibering any section C has a negative

square.

Remark 2. It would be interesting to determine whether Proposition 6 is valid for any non-isotri-

vial fiberings without multiple components. It can be shown that it holds if the dimension of the trace

of the jacobian variety of the generic fiber is not less than 2.

Proof of Theorem 3. As follows from Theorem 1 of Chapter 1, it is sufficient to prove the fol-

lowing assertion.

Let f: U' -* U χ δ be a smooth proper epimorphism of non-singular connected algebraic varieties

W, U, B, where Β is a proper curve of genus q > 1, and for any point u € U, f\y : Wu -> Β is a smooth

fibering. Then there exist only a finite number of smooth non-isotrivial fiberings of the form Wu.

Denote by

a smooth family of surfaces. Let u0 € U, Wa = p"1 (u0), and /„ = f\w0- Wo -> Β be a non-isotrivial fiber-

ing. The smoothness conditions imply the existence of families of open sets \Wa β\ in W and \Ba\ in

B, and a neighborhood Uo of the point u0, satisfying the following conditions:

1. U ^ α . β Π ^ Ο = ^ ο ,α, β

2. [)Βα = Β;

3. ( ρ * ο /) (Wa, β) = Βα; ρ (Wa, ρ) = Uo.

4. The fibering f\Wa, β: W α, β— ί/0 χ β α is analytically trivial.

Now let t be a tangent vector to the variety U at u0. Then there exist tangent vector fields ξα β

to W defined in Wa β such that

Ρ·1α, β = I-

It follows that the Kodaira cocycle [ ]

lit = \ha, β, ν, δ}, "α, β, V, δ = ζα, β | VV0 — ζν, δ | W,,

which is an element of the group Ηλ(Ψ0, CVo(7V )), belongs to the image of the mapping

where 7V0/B is the relative tangent bundle and 7V0 the tangent bundle. By definition, Οψ (Τψ /g)

- Ωψ / Β and the corollary to Proposition 6 shows that ht = 0 for any vector t. Hence the family

P: W -> U is analytically trivial in the neighborhood of u0 [ ]. Note now that for all u € U the fiber-

ing Wu is non-isotrivial. Indeed, applying Proposition 5 and the formula dy/g = pa(V) — χ/4, we

1162 Α. Ν. PARSIN

see that it is sufficient to show that pa (Wu) and x(Wu) ate independent of the choice of u 6 U. For

the arithmetic genus this is a general property of a smooth (or even flat) proper morphism ([ ], III,

7.9-4). The constancy of the Euler characteristic follows from the equation y = (2g — 2){2q — 2),

valid for any smooth fibering ([ ], Chapter IV, §4).

It now remains to use the connectedness of U and the theorem of de Franchis ([ ], p. 139) to

complete the proof of the theorem.

Remark. As shown by Kodaira, for sufficiently large q there exist smooth non-isotrivial fiber-

ings for which the genus of the base is q (see the Appendix).

§3· Smooth fiberings over algebraic curves of genus 0 or 1

Theorem 4. Let f: V — Β be α smooth fibering and g the genus of its generic fiber, g > 1. If the

genus of the curve Β is 0 or 1, then the fibering V is isotrivial.

Proof. The group Aut Β acts transitively on B. If σ: Β -> Β is an automorphism of the curve,

denote by Vσ the smooth fibering fa: V χ β δ -> B, fa = / χ gB. If b €. B, the fiber of Vσ at the point

b i s Vo-(b)-

Now note that Theorem 3 remains true when the genus of the base is q < 1. Indeed, if the genus

of Β is at most unity, there exists a normal covering Β' -> Β such that the genus of δ ' is greater than

unity. The change-of-base functor gives a mapping of S-fiberings into the set of β'-fiberings, and the

standard constructions of Galois theory in conjuction with the fact that the group AutgF is always

finite (g > 1 and Theorem 0) imply that this mapping is finite. Hence our assertion follows from

Theorem 3 for q > 2.

The set Η = \σ € Aut β, Vσ — V over B\ forms a subgroup having, by what has just been said,

a finite index. Taking into account the structure of the group Aut β, we find that Η is also transitive

on β. Using the relation between the fibers of Vσ and V, we find that

vb ~ vm

for all σ € Η. It follows that all the fibers of V are isomorphic. Hence V is analytically locally triv-

ial [ ]. Since for all b ε Β the group Aut V(, is finite, we find that V is an analytic fiber space with

finite structural group and hence becomes trivial over some (even unbranched) covering of the base.

This proves the theorem.

Remark 1. In this chapter we have used the definition of isotriviality in §1 of Chapter 1. But for

smooth fiberings a more natural definition is Serre's, i.e., the requirement of triviality over some un-

branched covering of the base. It is easy to see that for smooth fiberings the definitions coincide.

Indeed, smoothness and isotriviality imply the isomorphism of all the fibers, and the argument with

which we have just concluded the proof of Theorem 4 shows that a fibering isotrivial in our sense is

isotrivial in Serre's.

Remark 2. For curves β of genus 0 the theorem has been proved by other means by Safarevic [ ].

Remark 3· Arguing as in the proof of Theorem 4, we can obtain from the conjecture mentioned in

the Introduction the result that any fibering over a base of genus 0 has at least three points of degen-

eracy, and over an elliptic base at least two.

We note further that the results obtained in this chapter can be reformulated in the language of

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I U63

curves. Call the curve X nondegenerate if its associated minimal fibering is smooth.

Theorem 3". The set of nondegenerate nonconstant K-curves of given genus g > 1 is finite.

Theorem 4'. Every nondegenerate curve of genus g > 1 defined over a field of genus 0 or 1 is

constant.

CHAPTER 3. RATIONAL POINTS OF CURVES OF GENUS g > 1 DEFINED

OVER A FUNCTION FIELD

§1. Branched coverings of curves of genus g > 1

Let Κ be, as usual, the field of rational functions on a nonsingular projective curve B. Consider

a complete nonsingular curve X of genus g > 1, defined over Κ and geometrically irreducible. Denote

by /: V -> Β the corresponding minimal fibering (§1, Chapter 1).

We shall say that the curve X is nondegenerate at the point b €. Β if the fiber V'j, of the fibering

V is nondegenerate.

The set S of points b € Β at which the fibering V is degenerate is finite and is called the type

of the curve X.

Lemma 9· Let φ: B' -> Β be a covering of nonsingular curves, K' D Κ the corresponding extension

of the field of functions. Then for the type S' of the curve X' = X ® K' we have:

The proof follows easily from the definitions.

We proceed now to the fundamental construction of this section. Let Ρ'ι 6 Χ (Κ), i = 1, . . . , η, be

distinct /(-rational points of X. They correspond to sections Pl of the fibering V. Put Ο = Έ,Ρί and con-

sider the set Sp of points of the curve B:

SD= {foe B: Zii}Pt Π Pi Π Vb Φ φ}.

If Ρ € Χ (Κ) and Ρ / Ρi7 i = 1, . . . , η , denote by D f| Ρ the following set of points of B:

Df]P= {bEEB:SiPif)P()Vb^0}.

It is not difficult to see that Sp and D f] Ρ are finite sets of closed points of B.

We now construct a certain covering of X and study its type. To do this we require generalized

jacobian varieties, whose definitions and properties are given in Serre's book [ ]. In the present

situation there is determined the algebraic variety IQ (X). A point Ρ € Χ (Κ) \SuppD determines a

point in lD (X) and turns it into a commutative algebraic group defined over Κ and denoted below by

Ip ρ (Χ). We have the canonical mapping

Ψ: X\SuppD-+lD,P(X).

If m8, m > 1, is the mapping representing multiplication by m,

τηδ: ID,p(X)-+IDiP(X),

its restriction to /Y\SuppD determines a covering

Θ': X'-vX

1164 Α. Ν. PARSIN

defined over Κ. Denoting the normalization of the curve X' by Xr>, ρ ,m, we obtain a geometrically h>

reducible nonsingular complete curve and a covering

h X D,P,m-•χ,

unbranQhed outside SuppD. The covering Θ is normal and its Galois group is (Z/mZ)2 e + ""'. If re > 1,

then Θ is branched in at least one point, since Hl (X, Z) = (Z) 2 g .

Lemma 10. Put

(D[\P).

* s nondegenerate outside R.Then the curve XD,

Proof. We recall firstly some facts about generalized jacobian varieties. Let Υ be a complete non-

singular algebraic curve defined over an algebraically closed field k of characteristic 0, A its jacobian

variety, and Gm the multiplicative group. If Ρl, • • • , Ρη, Ρ are distinct points of Υ, denote by Ip p,

as above, the generalized jacobian variety corresponding to the "modulus" D = ΣΡ ;. Consider also the

group LD

where α is the diagonal injection. Then Ιβ, Ρ is a n extension of Lp by A (the ordinary jacobian

variety) and hence determines a point d in Ext (A, LQ). We have the commutative diagram

• p l c ° y <- oη

Π Pic

where e is the composite of the morphisms (cf. [ ], Chapter VII, no. 16,23)· The set of divisors

η

(Pl-P,. .., Pn - Ρ) determines a point in f j P i c ° Y a n d > a s follows from [ 1 7] (Chapter VII, no. 20),t = t

e(Pl-P,...,Pn-P)=d.

The fiberings over Υ corresponding to the divisors Pi — Ρ have rational sections Sj whose divisors

are respectively equal to Pi — P. The set of these sections defines a rational mapping of Υ into an

ηalgebraic group I'D, which is an extension of JT (Gm)i by A. This extension is determined by the

element d, as is clear from the commutative diagram above. We have an exact sequence

0 , ρ <~ ID' 1

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1165

and the composition a ° ( s , , . . . , s j gives a rational mapping φβ\ Υ -> Ip, ρ. It is regular in Ρ and

is the canonical mapping of the curve Υ into the generalized jacobian variety ([ ], Chapter VII, no. 20).

We turn now to the proof of the lemma. Put

V'=y_f-i(«), B' = B-R, f = f\v,:V'-+B'.

Then we have a smooth fibering / ' : V -> B', a group scheme I over B'', and a morphism φ~ V -> /,

where for any point b € B' the fiber /;, is the jacobian variety of the fiber Fj (Ij, = Pic°F;,), while

(fib = 0|Κί, : f̂c ̂ 'fc i s t n e canonical morphism of the curve into the jacobian ([ ], Chapter 6). The

divisors Di = Ρ ι — Ρ determine sections of the scheme P i c o ( F ' / # ' ) , and hence of Pic° (I/B') (since

φ induces an isomorphism Pic0 F ' / β ' ~ Pic0I/B'). By the generalized Weil-Barsotti formula ([ ],

Theorem 18.1) we obtain a set of points of the group Extg'(/, (G m )a') and, using the exact sequence

ft.

0«-ExtB.(7, (LD)B')«-H Ext f l.(7, GmB-) - ExtB. (7, G m B -)-0 ,

a point of the group Extg'(I, ( ^ Ό ) β ' ) . Thus we have constructed an extension of the group scheme

= Lp χ Β' by /. Denote the group scheme thus obtained by /'. We have the exact sequence

where It is the extension of {Gm)g' corresponding to the divisor Di- Restricting /; to V via the mor-

phism φ, we obtain a one-dimensional fibering over V determined by the divisor £),·. It has a rational

sections siy of which Di is the divisor. The set of sections a'°(sl, . . . , s n) gives a rational mapping

of the surface V into | | 7,·, and the composition a ' ° ( s , , . . . , sn) defines, as above, a morphismi

φΐ)·. F \ S u p p D -> /'. The scheme /' and morphism φρ have the following properties, as follows from

the facts above concerning jacobian varieties:

1. The generic fiber of /' is \ot ρ (Χ).

2. The morphism φυ on the generic fiber of V is Ψ.

3. For all b € B', /j, = / β 4 > ρb (X), where Dj,, P j are the restrictions of D and Ρ to the fiber Fj,.

4. For all b £ B' the morphism 0 β | κ ' : F J -> /j, is the canonical mapping of the curve into theb

generalized jacobian.The restriction of the morphism

τηδ: Γ-*-Γ

defines covering

u: V" -> V — D.

The generic fiber of V" is X', and on the generic fiber the morphism u coincides with Θ'. If δ € Β',

then

Vl= h Xrb(Vb—Db)

1166 Α. Ν. PARSIN

and we see that the genus of the normalization of the curve F£ is independent of b (since it is equal

to 1 + m2S+n-l(g — 1) 4- 1 {m — l)nm2g+r>-2, as follows from the Hurwitz formula).2

Thus, we have constructed a model of the curve Χβ, ρt m, in general incomplete. Its completion

may have singularities. If we resolve these and contract the exceptional curves of the first kind lying

in the fibers, we obtain the nonsingular minimal model W of the curve Xu, Ρ, m· F° r each point bfiB'

the fiber (Tj contains as a component a completion of the curve F£, since the genus of V'{, is greater

than zero. Moreover, the genus of this component is equal to the genus g' of the generic fiber of the

fibering W. We shall show that W is nondegenerate at the point b € B'. Let

where X is birationally equivalent to F£. We have

K- Wb = 2g'-2 = PK-X + K-Y = p(2g' - 2 + δ) - PX2 + K-Y;

here Κ is the canonical class of the surface ff, Κ • Υ > 0 (as we have seen in §2 of Chapter 1), and

δ = 2pa(X) — 2ξ' 25 0, §•' > 1. Hence we find:

pX* > K-Y + pb+ (p- 1) (2g' - 2) 5= 0

and since X2 < 0 always ([ ] , Chapter VII), we have

X2 = 0.

This means, firstly, that ρ = 1, δ = 0 (otherwise X2 > 0); and secondly, that Υ = 0 ([ ] , Chapter VII,

Theorem 2). This proves our assertion, and thereby the lemma.

Proposition 7. Let X be a nonsingular complete geometrically irreducible curve of genus g > 1

defined over the function field Κ = k(B), Ρ € X(K) a rational point, and S the type of X. Then there

exist a finite extension K' 3 Κ and a covering φ: X' -> X, defined over K' with the following prop-

erties:

1. K' is unbranched outside S.

2. The degree of the morphism φ and [K' : K] depend only on g.

3. The morphism φ is branched at Ρ and unbranched outside P.

4. φ*(Ρ) =ΣβίΡί, Pie=X'(K').

5. The type of. the curve X' is at most that of the curve X ® K'.

Proof. Let Φ: Υ -> X be the covering induced through multiplication by 2 in the jacobian of the

curve X. The type of Υ is at most that of X. The cycle Φ (Ρ) is rational over K. Let K' 3 Κ be the

extension over which

Clearly, Κ' is unbranched outside S and the degree [K': K] is at most Φ (Ρ)! < 22g\. On the curve

F (g> K' we have the divisor D = 2 ^i a n d t n e P o i n t Ρ £ SuppD. Put

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1167

X'= (Y®K')D,P,2.

Then assertion 5 follows from Lemmas 9 and 10. We have already verified assertions 1 and 4. Finally,

assertions 2 and 3 follow from the general properties of generalized jacobian.

§2. Manin's theorem

Theorem 5 (Manin [ ]). Let X be a complete nons ingular geometrically irreducible nonconstant

curve of genus g > 1 defined over the function field Κ — k(B). Then the set X(K) of K-rational points

is finite.

Proof. Let f:V~>B be the minimal fibering associated with the curve X. We know [ ] that the

finiteness of the set X (K) will follow from the fact that the numbers C .Ky, where C runs through all

sections of the fibering V and Ky is a canonical divisor of V, are bounded above.

Let Ρ € X(K); let Xp denote the covering of X constructed in Proposition 7, and let C be the

section of the fibering V corresponding to P. Let Vρ and V be the nonsingular minimal models of the

curves Xp, X ® K'; and ψ : Vρ -> V the rational mapping equal on the generic fiber to the morphism

φ of Proposition 7. By applying σ-processes to the surface Vρ we can make the mapping ψ regular

([ ], Chapter 1). Let V" be the surface thus obtained, s the number of σ-processes needed. We have

the diagram

V" " V'\ /

vP

where τ is the product of the s σ-processes and u is a morphism. We can assume that the genus of the

base Β is greater than 1. Hence the geometric genus of the surface V is greater than 0 (see Chapter

1, §2), and there exists a regular differential ω ̂ 0 of degree 2 on V. Let us find the canonical di-

visor Ky of the surface V". We have

Jf ill' { ,-k\\

It follows that

Ky,,= U'(KV.) +M,

where Μ is an effective divisor. On the other hand, on the generic fiber Xp of the fibering V" we

have the equations

Κχρ — φ* {Κ χ) 4- Σ(βί — 1)Ρ, = Χ p. u* (Κν·) 4- Σ(^ — 1 ) ^ .

If Ci denotes the section of V" corresponding to the point Ρit we find that the divisor Μ is

where Φ is an effective divisor consisting of fiber components. Finally, note that the exceptional

curves of the first kind occur in Μ with multiplicity 1. Indeed,

Kv..= (τ··ψ*(ω))ν»,

1168 Α. Ν. PARSIN

and we need only observe that ψ* (ω) is a regular differential and apply the formula for the behavior

of a canonical class under a σ-process. Thus we have

AY,, = u*(Kv.) + Σ (e, - l)Ct + θ,

where Κγιι-Ci ^ 0, Κγ»·® ^ —S (as we have seen in §2, Chapter 1, it is always the case that

K. C > — 1 and K. C = - 1 means that C is an exceptional curve of the first kind).

We come now to the fundamental bound:

Kv,.-Kv»^Kr»-u'(Kv) —s^u*(Kv.)-u'(Kv) + Σ(β« - l)Cru*(Kv.) +V-u'(Kv.) -s

^ (deg u)Kv,.Kv. + Σ(e{ - l)u,(C<) -Kv. + ΐι.(θ) ·Kv. - s ̂ Σ (e, - l)«.(Ct) · £ ν . - s..

There exists an i0 for which e;o > I, and we obtain

C'-Kv < Ky-Ky + S < Avp · ̂ Vp,

where C = u* (^i0) i s the section of V obtained from the section C of V.

By conditions 1 and 2 of Proposition 7, there are only finitely many fields K'. Hence if the set

X(K) is infinite, there exist an infinite number of points Ρ € Χ (Κ) such that Kp £*. K'. Each of these

points determines a point P' 6 (X ® K')(K') and a section C of V. But as follows from the above

bound for C · ^i/', there are only finitely many such sections (and points). This contradiction proves

the theorem.

Remark 1. It is easy to show that the proof of Theorem 5 is entirely effective. It leads to the

following bound for the height:

CK^ (qs+ I)exp(exp4g)!

Further considerations can be used to obtain a more exact bound.

Remark 2. We can show that Mordell's conjecture follows from that of Safarevic, formulated in the

Introduction, and Proposition 7. If Ρ € X(K), we have the covering Xp defined over the field Kp

(Proposition 7). It is easy to see that all the curves Xp, as well as X, are nonconstant. Moreover, as

already remarked, there are only a finite number of fields K'. Hence from Proposition 7 (assertions 2

and 5) and the Safarevic* conjecture it follows that the set of curves Xp is finite up to isomorphism.

Since any curve has only a finite number of morphisms onto a curve of genus g > 1, we conclude that

the set of coverings Xp -> X is finite. From this and Proposition 7 (assertion 3) it follows that the

set X(K) is finite.

Appendix

1. Kodaira's examples. The proof in Chapter 2 that the number of smooth non-isotrivial fiberings

is finite left open the question of the existence of such fiberings. The first examples of such fiberings

were constructed by Kodaira (cf. note on Russian p. 174 of [ ]). We now give a construction of these

examples, essentially coinciding with Kodaira's.

Let Υ be an arbitrary nonsingular proper curve defined over k. Assume the genus g(Y) is greater

than 1, and choose a covering φ: Β —> Υ. The graph of the morphism φ defines a section C of the

ALGEBRAIC CURVES OVER FUNCTION FIELDS. I 1169

trivial fibering /: V = Υ χ Β — Β. This section is nonconstant, i.e., it does not have the form Υ χ Β,

γ € Υ. Consider the curve Χ = Υ <S> Κ, Κ = k(B). It is defined over K, and is nonsingular, proper and

geometrically irreducible. The section C determines a rational point on X, which we denote by P.

Now apply Proposition 7. We obtain a covering φ: X' -> X of X, which is branched at P. The minimal

fibering associated with A" is smooth (assertion 5 of Proposition 7). We show that it is not isotrivial.

Denoting it by / ' : V -> B', we find from the proof of Proposition 7 that for each point b ۥ B' the fiber

V'b is a covering of a fiber of V, i.e., of the curve Υ, branched at the point φψ). If the fibering V

were isotrivial, then, as we have seen in §3, Chapter 2, all its fibers would be isomorphic. Our con-

struction would then give a continuous family of subfields of the function field Κ = k(Vl). Since all

these subfields are distinct (this follows from a comparison of the branch points and the choice of the

section C), the contradiction proves the required result.

2. Spaces of moduli. The results obtained in Chapters 1 and 2 give some information on the

spaces of moduli of algebraic curves of given genus g > 1. The definition of the space of moduli,

denoted below by Mg, is given in [ ]. Theorem 2 of Chapter 1 leads to the following fact.

Take a fixed hyperplane section on Mg. Then curves of given genus q contained in Mg have

bounded degree.

If we consider curves on Mg isomorphic with a given curve B, we find from Theorem 3 of Chapter

2 that the set

{X a Me-. X ~ B}

is always finite. Moreover, Mg contains no curves of genus 0 or 1. It goes without saying that the

above is true only for proper curves.

Received 7 MAY 68

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