V. Talamanca* AN INTRODUCTION TO THE THEORY OF HEIGHT ...

Rend. Sem. Mat. Univ. Poi. Torino Voi. 53, 4 (1995) Number Theory

V. Talamanca*

AN INTRODUCTION TO THE THEORY OF HEIGHT FUNCTIONS

Abstract.We give an introduction to the theory of height functions. The following topics are discussed: heights on projective spaces, heights on projective varieties, Néron-Tate heights on abelian varieties, heights on curves.

1. Introduction

Heights have been used as a tool in arithmetic algebraic geometry for quite a long time. Their first appearance dates back to the last third of the nineteenth century in a work by G. Cantor [5] where he defined what today we cali the inhomogeneous height of an algebraic number. It is interesting to note that Cantor used heights to prove that the cardinality of the set of real algebraic numbers is smaller than the cardinality of the real numbers. At the beginning of this century E. Borei in [3] gave the first definition of heights for "système "of rational numbers. However the systematic use of heights in arithmetic algebraic geometry started only with D.G. Northcott ([18] and [19]) and A. Weil [27] in the late forties. Since then Northcott-Weil heights and their generalizations (Arakelov heights and Faltings or modular heights cf.[12]) have proven to be a key tool (especially Northcott's finiteness theorem) for proving major results as the Mordell Conjecture (see [2], [6], [26]). It has to be noticed that heights also play an important role in various conjectures, e.g. Vojta's conjecture [25], the abc conjecture [13] and the Birch and Swinnerton-Dyer conjecture [1].

In this note we give an introduction to the theory of height functions with special regard to its applications in arithmetic algebraic geometry. As Schimdt pointed out in [23], one can define the height associated with any distance function on Mn. On the other hand the heights that are actually used in the literature are those arising from the £P-norms for p = 1, 2 and co. In this note we will use p — oo in order to simplify some of the arguments but we remark that it is possible to treat simultaneously ali the heights arising form the ^-norms cf. [24].

We assume familiarity with algebraic geometry and especially with the theory of

*Partially supported by Fondazione "Francesco Severi".

438 V Talamanca

abelian varieties. We refer the reader to [10], [14], [15], [16].

Conventions and Notations. Wé fìx once and for ali an algebraic closure Q of Q. Ali the algebro-geometrie objects that we will be dealing with (varieties, divisors, morphisms, etc.) are assumed to be defìned over Q.

2. Heights on Projective Spaces

The idea behind the definition of heights is to measure the "arithmetic size" of a point. Let us start with a simple case. Consider P1(Q) as the space of lines in the piane Q2. A good measure of the arithmetic complexity of a straight line V in Q2 is the distance between integrai points, which is nothing else than the distance from the origin of a primitive integrai point P = (ZQ,ZI)£V (primitive means that gcd(z0,zi) = 1)-Suppose that the line V is defìned by the equation ax + by = 0 with a, b positive integers. Then P — ( gcd"^ ^, dfa b) ) is a primitive integrai point of V and so the height of V is

sup{ igcd(a b\i, igodrà MI }» w n e r e I • I denotes the ordinary archimedean absolute value on Q. More generally, given P£lPn(Q) choose integrai coordinates P = [XQ\ ... :xn] and set

H(P)= S U P o < i < " ^ i

gcd(a?o,...a?n)'

Note that H is independent of the particular choice of integrai coordinates for P. How can we generalize this definition to IP" (A), with K a number fìeld? First of ali note that gcd(#o, • • -xn) = Nci;c = #%/ax, where ax denotes the ideal generated by x0,.. .xn. Thus the greatest common divisor ought to be replaced by the absolute norm. Secondly, since there is no way to pick a particular archimedean absolute value on K we have to use ali of them. Precisely let M^ denote the set of equivalence classes of archimedean absolute values on K (for generalities about number fìelds and absolute values see [8]). For any v^M^ we denote by | • \v the representati ve of v normalized by requiring that its restriction to Q coincides with the ordinary archimedean absolute value. Then we defìne the (homogeneous) £°° -height of P = [ XQ '. . . . '.XJI ]eWn(K) tobe

^ ( ^ • = 1 ^ n SUP I^I?V Na veM% -

Ki<n

where a is the fractional ideal generated by XQ, ... ,xn and nv — [Kv : M], Kv being the completion of A' with respect to \-\v. Note that HK is well defìned (i.e. independent of the choice of coordinates for P) because ELeAt^ W"v ~ ^ a A' where d\ is the fractional ideal generated by A. To be able to define a height function on the whole F n ( Q ) , we need to study its behavior under fìeld extension. So let KcL be an extension of number fìelds. Given veMf set M\ - {iveML \ \ • \lu\ , = | • \v}, then Mf = UveM% Ml a n d

An ìntroduction to theory of heiglit functions 439

Y^iueMv n™ ~ \L '• K]nv- Therefore for any veM^- we have:

n / \ n™ / \ y"ì ,-.> , ^ n™ / \[L;K]nv

I I I \ / I l \^W^ML^V) / I I V sup \XÌ\W = sup \XÌ\V = sup \xi\v ioeM (v) Vl<?"<« / \0<ì<n J \0<i<n )

On the other hand for a fractional ideal CICA we have N(ci • OL) — ("Na)^L-K\ so that

HL(x) = / ^ ( z ) ^ ^ . It follows that the function

H : P n ( Q ) —>IK

P i — * ( # t f ( P ) ) [ * : < Q 0 »

where A' is any finite extension of Q such that P£Fn(K), is well defìned, and will be

called the absolute homogeneous £°°-height on P n ( Q ) . The above formulation of the

defìnition of heights is the one used by Northcott. We shall now give Weil's formulation.

Let M^ denote the set of equivalence classes of non-archimedean absolute values of A'.

Since MQK is in one-to-one correspondence with the set of prime ideals of the ring of

integers of A, for each v£M°K we can choose a representati ve | • \v for v by requiring

that | • \v restricted to Q coincides with the p-adic absolute value, where p is the prime

lying below p, p being the prime ideal corresponding to v. Given v€M°K explicitly, we

set \a\v = (l/p)(° rdp(a)/ep) where ordp is the discrete valuation associated with p and

ep = ordp(p(9#-). With this normalization the absolute norm of the fractional ideal ci

generated by x0 . ..xneK is (Na) = Uv£M°K suPo<*<n Mvnv> w n e r e nv = iKv • QJ -

Therefore the absolute ^°°-height of P — [XQ: ... :xn]£Fn (li) can be expressed as

H(P) = H sup \XÌ\Ì»

where dv — nv/[K : Q] and MK — M°K (jM^. As we shall see, the above formulation

is very convenient when proving inequalities about heights. For example it is now clear

that H(P) > 1 for ali P £ l P n ( Q ) and that H is invariant under the action of Gal( Q / Q ) .

As we said in the Ìntroduction, one of the most important results about heights is

Northcott's finiteness theorem which we shall now prove. Given P — [XQ\ . . . :xn]^¥n ( Q)

we denote by d(P) the degree of the number field generated by the ratios XÌ/XJ (with

Xj ? 0).

THEOREM 2.1 (NORTHCOTT'S FINITENESS THEOREM). The set

N(Fn, C, d) = { P G F ( Q ) | H(P) < C and d(P) < d)

is finite for any choice of the constante C and d.

440 V. Talamanca

Proof. The case d = 1 is immediate, for the points with height less then C are in one-to-one correspondence with primitive points in Z n + 1 with bounded f^-norm. To reduce the general case to the case d = 1 we need the following:

LEMMA 2.2. If F(T)=Td + • • • + ad=(T - 6i) • • -(T- bd)eQ [T], then H([l:a0:...:ad])<2dYl(j:=1H(ll:bj]).

Proof. Let K be a number fìeld containing ali the roots and ali the coefficients of F(T). Denote by Si,... ,Sd the elementary symmetric functions in d variables and by S0

the Constant function with value 1. Then a? = S»(&i,..., bd) for ali i = 1, . . . ,d. Note that standard inequalities of real numbers (cf [9]) yield:

(a) lfveM°K,thensuv0<h<d\Sh(bu..., bd)\dv« = n£=i™P{l. \bi\Ìv}-

(b) If v£M%, then supg=0 \Sh(bu..., bd)\*- < 2d*r[;=1(sup{l, \bi\Ìv})-

For (a) one also needs to use the ultrametric property of | • \v. Then

H([ì:aQ:...:ad])= J ] sup \Si(bu • • •, Mi" v£MK

< n nsup{L h\t-} n 2*,-n«»p{i.iM2-} =2-n^([l:6 i]).

i = i

Now we can readily prove Northcott's Theorem. Suppose PGA/"(lPn, C,d). Choose homogeneous coordinates P — [xo:...:xn] with XJ = 1 for some j . Let Pi(T) = Tdi +a i ) iT d ' - 1 + .. . + ai(d.GQ[T] be the minimal polynomial of x{. Then deg(P,(T)) = [Q(XÌ) : Q] < d. Since H([1:XÌ\) < H(P), the invariance of tf under the action of Gal(Q/Q) combined with thè above lemma yields H([hai:... :ad.]) < 2diCdi. Set Q = \lj/d N(Wl, 2lCl, l ) ; then what the above argument is saying is that with any P£j\f(Wn,C,d) we can associate a point of Qn, the cartesian product of Q with itself n-times, which is a finite set. Since the number of points of PeAf(Fn,C, d) mapping to the same point is at most n\ Y[j/d ' the theorem follows. •

Another important aspect of the theory of heights is the study of the behavior of H under morphisms. Let us start with the Segre

(Tn>m : PU X f • p(n+l) (m+l)- l

([xQ:... :xn], [ve,:... :ym]) *-+ [x0yo : ...XiVj : ... xnym]

An introduction to theory ofheight junctìons 441

and the ór" Veronese embedding

và : Pn —• FN~1

[x0:... :xn] !-»• [FQ(x0i... xn) : . . . : F/v(z0,... a?n)]

where Fo, . . . F/v are ali the monomials of degree d in n variables, so N = (n+d) - 1.

PROPOSITION 2.3.

(a) H(<T(P,Q))=H(P)H(Q).

(b) H(vd(P))=H(P)d.

Proof. Let A' be a field of defìnition for P = [xo:... :a?n] and Q — [yo:... :ym]-Then for every v£MK we have sup0<?;j.<n \xiyj\v = sup0<,-<n \XÌ\V sup0<j-<„ \yj\v, from which (a) follows. To prove (b) it suffices to note that for v€MK, we have sup0<i<N \Fi(xo,... a?n)|t, = I \d — ( • | v̂ *̂ \x3v\v — \suPo<i<n\xi\v) • •

Now we pass to the study of the behavior of H under arbitrary morphisms between projective spaces. Any non Constant morphism Pm and an automorphism of P m . The integer d will be called the degree of (p. We have already seen how H behaves under the Veronese embedding so we now have to study its behavior under linear projections and automorphisms.

LEMMA 2.4. Let <p : P"—*Pm be a morphism defined by linear polynomials. Then

H{9(P))<CH(P)

for some positive Constant C. In particular if a : Pn-+Pm is an automorphism then C'H(P) < H((p(P)) < CH{P) for two constants C,C > 0.

Proof. Let /o, •. • fm be the linear polynomials defining ip. Given P — [xo:... :xn] in P n ( Q ) , choose a finite extension K of Q so that PeFn(l<) and K contains ali the coefficients of the /; 's. Denote by T/ C K the set of the coefficients of the /^'s. For v€MK set cv(f) = sup a € T / \a\v. If v£M°K then

\fi(x0,...xn)\v < cv(f) sup \XÌ\V, 0<i<n

while if veM]?, then \fj(x0ì...xn)\v < (n + l)cw(/)sup0<^n |^ | v . Therefore

H(f(P)) < (n + ì)c(f)NH(P)d where c(/) = (l\vzMK cv(fTv) ^ ™ independent

of the field K. •

442 V. Talamanca

LEMMA 2.5. Let A C Pn be a linear subvariety of dimension d. Let us denote by 7r : Pn — A—>H?n~d~1 the linear projection defined by A, let X C Pn be a closed projective subvariety not intersecting A. Then the re exist two positive constants 'Ci, C2 such that

C\H(P) < H(TT(P)) < C2H(P)

forallPeX(Q).

Proof Note that it suffìces to prove the case d — 0. For the image of a closed subvariety not meeting the subspace from which we are projecting is again closed. Suppose first that d — 0, hence A is a point. By Lemma 2.4 we can assume that the point is En — [0 : . . . : 0 : 1]. Moreover we can replace X by a hypersurface that contains it and which does not contain EN. So we assume that X is the zero locus of a homogeneous polynomial F(TQ, . . . , Tn) of degree m not vanishing at En: Then the coeffìcient of T™ is not zero and so T^n is a linear combination of the other monomials. In other words if we let fi : Pn—s-F^-1 be the map defined by ali the monomials of degree m with exception of T7"\ there exists a morphism j> : p ^ - 1 - ^ ^ ^ , defined by linear polynomials, such that vm(P) = <f)(p(P)). By Proposition 2.4 there exists C > 0 such that

H(vm(P)) < C'H(p(P)) for ali PeX(®).

On the other hand the coordinates of g(P) = crrn_i(7/m_i(P), 7r(P)) are exactly, up to repetitions, the coordinates of p(P). Then by Proposition 2.3

H(p(P)) < CH(vm-i(P))H(*(PJ).

Combining these inequalities with the transformation property of H with respect to the Veronese embeddings yields the lower bound for H (ir(P)). The upper bound being clear, the proof is completed. •

As a consequence of the above results we have

PROPOSITION 2.6. Let f : Fn —• Fm be a morphism of degree ci defined over Q. Then there are two constants Ci, Cf2GR+ such that

dH{P)d < H(f(P)) < C2H(P)d.

3. Heights on Projective Varieties

Let X be a smooth projective variety and C an invertible sheaf on X which is generated by global sections. Recali that the morphisms n(l) ^ C correspond to sets of global sections which generate C. Moreover any two morphisms (p : X-*W>n, ip : X-^¥n (m > 11) with the above property differ by a

An introduction to theory of height functions 443

suitable linear projection P m — A—>]Pn and an automorphism of'IP", where A has dimension

m — n — 1. In particular the quotient (Ho<p)/(Hoil>) is a positive bounded function on

X(Q) by Lemmata 2.4 and 2.5.

The above argument can be rephrased by saying that we can assign to any

invertible sheaf £ an element S)cen(X(Q))=f(X(Q),R+)/BT(X(Q), R+), where J7(X(Q)) R + ) is the multiplicative group of functions on X(Q) with values

in the positive real numbers and #.F ( Ar ( Q ), R+) C ^ ( A : ( Q ) , R + ) is the subgroup of

bounded functions. Note also that Sic depends only on the isomorphism class of C.

LEMMA 3.1. Suppose that C and M are two invertible sheaves generated by

global sections. Then S^c®M = S)C¥)M •

Proof. Choose bases for T(X, C) , T(X, M), and let <p and ip be the associated

morphisms to the projective space. Then the morphism

X ^ t P n x JPm - ^ p(w+i)(m+i)-1

corresponds to the basis for T(X, C<S)M) given by the tensor product of the bases chosen

for T(X, C) and T(X, M). The result then follows from the fact that height behaves

multiplicatively with respect to the Segre product (cf. Proposition 2.3). •

The above lemma enables us to defìne S)c even for invertible sheaves that are not

generated by global sections. In fact given any £ there exist A^o and Mi generated by

global sections, such that C = MQ <S> M^1. It is then naturai to set Sic = SÌM0/$ÌMI-

Before stating the next proposition, which is essentially due to Weil (cf. [27]) and

summarizes the properties of this construction, we need three more pieces of notation. Given

Fef(X(Q)} R+) we denote by cl(F) its class in n(X(Q)). Given Sie7i(X(Q))

we say that S) satisfies the Northcott fìniteness property if for one (and hence for ali)

representative F^ of S) in ffó^Q), R + ) the following holds: for any Constant C > 0

and any number field K the set Af(X(K),C,H) = {PeX(K) \ H(P) < C} is finite.

Finally we remark that any map <j> : Y(Q) —»• X(Q>) gives rise to a homomorphism

7i(X(Q)) ~^'H(Y(Q)) which, by an abuse of notation, will also be denoted by <f).

THEOREM 3.2. The assignment O—>Sic defines a homomorphism

$ : Pic(X) —> 7i(X(Q)) which has the following properties:

(a) If C is generated by global sections and <p : X—rWn is a morphism associated with

some basis of T(X, C), then c l ^ i / ^ ) = Sic, far ali ì<q<oo. Moreover $ is

uniquely characterized by this property.

(b) If ip '• y —* X is a morphism of smooth varieties, and C is an invertible sheaf on

X,.then<t>(S)c) -^)^c-

(e) If C is ampie, then S)c satisfies the Northcott fìniteness property.

444 V Talamanca

(d) If sGT(X, C), there exists H£$Óc such that H > 1 outside the (divisor of) zeroes ofs.

Proof. We only need to prove (e) and (d), since (a) and (b) follow directly from the construction of $. (e) Note that if C is ampie then Cm is very ampie for some m large enough. Thus there exists a basis of T(X, C) such that the associated morphism M~[l. By defìnition S^c — ^)M0I^)MX thus to prove (d) we need to show that we can fìnd representatives F0 and Fi for $5M0 and S)Mi s u ch that Fy < CF0, with C a positive Constant. Let s0,..., sn be a basis for T(X, Mi), <pi the associated morphism to Fn. By construction ss,- is a global section of Mo, so we can fìnd a basis of T(X,Mo) of the form {SSQ, . . . , ssn , r i , . . . rm}. Let <po be the associated morphism to p n + m . Then if we set Fo=Ho<po and Fi=Hoipi the desired inequality is satisfìéd. •

Note that the homomorphism $ extends formally to a homomorphism from Pic(X)M = Pic(X) <8>M to n(X(Q)). One would like to attach to any cGPic(X) a function on X(Q) and not just a class modulo bounded functions. The next lemma, due to Tate (cf. [22, Lemma 3.1] or [4, Theorem 1.1.1] ), gives a sufficient condition for this to happen. In this context it is convenient to use the "0(1)" notation: given two real valued functions / and g on a set S we write / = g + 0(1) if 1/ - g\ is bounded on 5'.

LEMMA 3.3. Let S be a set, ir : S —• S a map and f a real valued function on S such that /o7r = df + 0(1) far some real number d > 1. Then

(a) There exists a unique real valued function f such that

(1) / = / + 0 ( l )

(2) /o7r=.rf/

(3) f(x) = Kmn-¥OQ£.f{irn(x)) \/xeS.

(b) Let <J : S —»• S' be a map and let ir' : S' —>• S' be such that 7r'ocr = (TOTT. Suppose that f is a real valued function on S' such that fon' — df+0(1) and set f = f'o<r. Then foir = df + 0(1) and f — foa.

(e) Let {T,;} be a family of endomorphisms of S commuting with ir, and suppose that foni = d{f + Oi(l) where di£R and Oi(l) means that the bound depends on i. Then /OTT,- = rf,-/.

Proof Let B be the Banach algebra of bounded real valued functions on S equipped with the sup norm. T : B —» B, T(f) - foir is a contraction operator (i.e. ||T|| < 1),

An introduction to theory of height functions 445

therefore 1 - jT is invertible and its inverse is given by (1 - ^T)'1 = Eftio(x^)*• By hypothesis (^)/o7r — / is a bounded function, hence there exists g£B such that

(I-\T)9 = (>-/•

It is immediate to verify that f = f + g has the required properties proving (a), (b) follows from the uniqueness of / , while (e) follows from (b) applied to S = S', <r = 7r? and ir' = ir.

COROLLARY 3.4. Let X be a smooth projective varìety defined over Q and ir : X —» X a morphìsm. Suppose that £ G P Ì C ( X ) ^ is such that ir*C is isomorphic to Cd

for some d > 1. Then there exists a unique function He such that

(a) c\(Hc)=?>c.

(b) HCOTT = Hdc.

He is called the normalized (or canonical) height associated with C, with respect to ir.

Unfortunately there are very few examples of triples (A',£,7r) satisfying the hypothesis of the corollary. As far as we know they are: - (IPn,£,[n]) where 2 and C is any positive invertible sheaf. - (S,C,TT) where 5 c P 2 x F2 is a K3 surface given by the intersection of (2,2)-form and a (l,l)-form, w = (T10CT2 where 0"i,C2 a r e t n e t w o involutions of S induced by the two doublé covers p{S : -+IP2, and finally C = (p*0^(l) + p50p 2(l)) / ?~1 with (3 = 2 + >/3. For more information on this example see [21]. - (A,JC, [n]) where A is an abelian varìety, £ is a symmetric or anti-symmetric invertible sheaf and [n] is the multiplication by n map. In the last example we have a rather large group of commuting endomorphisms and so it is possible to define a normalized height independently of the morphism chosen, see Proposition 3.5 below. There is also another approach to the construction of heights on abelian varieties which is due to Néron and which will be treated in the next section.

PROPOSITION 3.5. Let A be an abelian variety. There exists a unique homomorphism e H-» Hcfrom Pie (A) to ^(A^Q),^) such that

(a) cl(Hc) = f)c.

(b) For ali endomorphisms • A is a homomorphism , then H^*c = Hcoip for ali cGPic(A).

Proof. Given cGPic(.4), we have 2c = (e + c~) + (e — c~). Since e + c~ and e — c~ are respectively symmetric and anti-symmetric, it suffices to establish the

446 V. Talamanca

proposition for symmetric and anti-symmetric class. By the theorerh of the cube [n]* e = n2c for e symmetric and [n]*c = ne for e anti-symmetric. Since [2] commutes with ali endomorphisms of A the proposition follows from Lemma 3.4. •

4. Néron-Tate Heights on Abelian Varieties

Let A be an abelian variety. We have seen in the previous section how to construct a homomorphism from F'ic(A) to Jr{A(Q)ì'R

+) using Tate's method to define normalized heights. Now we want to examine Néron construction of normalized heights. Néron approach (cf. [18]) is based on the construction of a symbol (which since then has been called the Néron symbol)

(,)N: Div(A) x Z0(A) -+ M .

where Zo(A) is the group of zero cycles on A and Div(yl) is the group of divisors on A. The Néron symbol is defìned as a sum over ali v£MK (K being a number field where the divisor and zero-cycle, of which we are computing the symbol, are defìned) of locai symbols, which in tiirn are defìned in term of Néron's quasi-functions. We will not dwell here'on the defìnition of the locai symbols, on the contrary we take as a starting point of our discussion the global symbol.

The Néron symbol enjoys the following interesting and useful properties (see [18, Proposition 6]): NSl The symbol (, ) N is bi-additive. NS2 If D is linearly equivalent to zero, then (D, a)N — 0 for ali aEZo(A). NS3 If g : A -* B is a morphism, then (g*D, a)N = (Òfg(a))N.

Following Néron we will construct the canonical height pairing, the height pairing associated with an element of the Néron-Severi group of A and the canonical height associated with a divisor from the Néron symbol. First of ali we need two definitions and a lemma. The kernel of the homomorphism S : 2o(A) —* A(Q), defìned by a = J2mi(ai) HH' S m « a * ' where on the right we are summing on A, is denoted by 2t(A). A divisor D is algebraically equivalent to zero if D -f a is linearly equivalent to D for ali aeA(Q). The set of such divisors is denoted by Diva(A) and the quotient of Diva(A) modulo linear equivalence by Vie0(A).

LÈMMA 4.1. Let DeDiva(A). Then (D, a)N = 0. far ali ae2,(A).

Proof. Suppose first that a = (a + b) - (a) - (b) + (0). Then (D, a)N= (D, (a + b) - (b))N - (D, (a) - (0))N (by bi-additivity)

= (D_6 j (a) - (0))„ - (D, (a) - (0))N (by NS4) = (D-b - D} (a) - {0))N (by bi-additivity) = 0. (byNS2)

An introduction tv theory of height fimctions 447

The generai case follows from the bi-additivity of the Néron symbol. •

Let A be the dual abelian variety of A, which, when convenient, will be identifìed

with Pie0(A). The pairing {, ) : A((Q>) x À(Q) -> M defined by setting

(c,a) = (c,(a)-(0)> N

is well defined (by Lemma 3.1 and NS2) and is called the canonical height pairing on A(Q) X A(Q) . ByNSl and NS3 the canonical height pairing satisfies the following properties: HP1 The pairing (, } is bi-additive. HP2 If / : A -+ B is a homomorphism, then {f{c),a) = (c,/(a)>VaGA(Q) and VCG5(Q").

Next we want to define the height pairing associated with an element f of NS(/1) = Pic(A)/Pic°(.4). Recali that with any £ENS(T4) we can associate a homomorphism <Pt : A -* A, by setting ^ ( a ) = ci (Ai — ^)» where D£Div(A) is any element of f. By the theorem of the square ^ is a homomorphism. The symmetric bi-additive pairing (, >£ : A(Q) x A(Q) —» M, defined by

(a, 6)̂ = fo(a), 6} = - < £ , (6 + a) - (6) - (a) + (0))N>

is called the height pairing associated with £.

Finally the Néron-Tate height associated with a divisor DEDiv(A), is

/ * D : A ( Q ) - + M

a^hD{a) = -{D,{a)-(0))N.

By NS2, /i£> depends only on the linear equivalence class of D and NS3 implies that given a homomorphism f : B -* A, then /*/*(£>) = tao/. Let us check which are the relations among the various height pairings and height functions we just defined.

PROPOSITION 4.2. Let DGDiv(A) and £ be the class of D in NS(-A). Then

1 1 _ hD(a) - - - ( a , a ) ^ + ~(D -D,a).

Proof. If aeA(Q), then hD(a) + hD(-a) = -{a,a}^ by the bi-additivity of ( , ) N . On the other hand by NS3 hD(-a) = - ( D , ( - a ) - (0))^ = - ( D " , ( a ) - (0))^ hence

hD(a) - hD(-a) = (D~ - D,(a) - (0))„ - (Z) - D" ,a ) .

Summing the two relations yield the proposition. •

448 V Talamanca

Recali that A x A is equipped with a divisor class V, the Poincaré class, which

is defìned by requiring that V\r •• represents the element of Pic0(A) determined by a \{a}xA \ / J

under the identification of Pic°M.) with A. One also requires that VI - is trivial.

PROPOSITION 4.3. Let V be the Poincaré divisor class on A x A. Then

h-p(c,a) = —(e,a) = /ic(a)

P/*<%>/.* Since V is symmetric, hp(c,a) = — | ( (c , a) , (c ,a) )p . Using the defìning

properties of the Poincaré divisor class one immediately verifìes that ((0,a),(0,a))v —

0 = {(c,0),(cf,0))v. The proposition follows from

(*) ((O.aJ.foO)),, = (e, a) = ((c,0), (0, a))v.

In fact if .(*) holds we have

1 1 hv(c,a) = - - ( (c ,a ) , (c ,a )> 7 ? = - - ( ( ( 0 , a ) , (c ,0)P + ((c,0),(0,a))v) = (c,a).

It remains to prove (*). By the bi-additivity of ( , ) N we have

((e, 0),(0, a))v = -(V,(c, a) -(e, 0))N - (7>,(0, a) + (0, 0 ) ) „ .

Given ceÀ set / c : A -» 1 x A, a H-> (C, a), then /C*("P) = c<EPic°(,4). By NS3

- ), (a) - (0))N = - ( e , (a) - (0))N = (e, a) .

For a£j4 set fa : A —*- A x A, e •-»• (e, a). Since V induces 0 on A x 0 we have

(V, (0, a) + (0, 0))N = (/0*(7>), (a) - (0))jy = <0, W - (0))„ = 0,

completi ng the proof. •

We now come to the relation between Néron-Tate height and normalized height.

THEOREM 4.4. Let cGPic(A). Then exp(/ic) = Hc.

Proof. The map e i-+ exp(/*c) from Pic(.4) to F(A(Q),M+) is a homomorphism

by the bi-additivity of the Ne'ron symbol and by NH 2 it satisfìes property (b) of Proposition

3.5. Therefore to prove the theorem it suffices to show that cl(exp(/ic)) = f j c . Since the

proof of this uses properties of the locai symbols (and we have not even defìned them) it

will not be given here. The interested reader is referred to Ne'ron's article ([18]) for the

proof.

We now derive some consequence of the above theorem.

\

An introductioii to theory of heiglit functions 449

PROPOSITION 4.5.

(a) If D is ampie and symmetric then hjj(a) > 0 with equality if and only if aG^Tors-

(b) If£ is a polarization on A, thenfor every number field K and every Constant C > 0 the set {a£A(K) | -(a,a)^ < C} is finite.

Proof. (a) By the above theorem there exists a Constant C such that hD (a) > C for ali aeA(Q). Since D is symmetric n2hD(a) = hD([n]a) > C. Therefore hi)(a) > C/n2

for ali n, proving hD(a) > 0. If ho{a) = 0, then ho([n]a) = 0 for ali n. Thus, by Northcott's finiteness theorem the set {[n]a,,n£Z} is finite, i.e. a is a torsion point. (b) Let D be an ampie divisor such that [D] = £. Then - ( , L = hD+D- and since D -f D~ is ampie and symmetric (b) follows by combining Theorem 4.4 with Northcott's finiteness theorem. •

Another application of Theorem 4.5 regards the non-degeneracy of the height pairing associated with a polarization. Let V be a vector space over Q and F a real valued symmetric bilinear form on V. Then F is said to be positive definite if for ali finite dimensionai subspaces W of V, F considered as a symmetric bilinear form on W (g> M is positive definite. The following criterion will be used.

LEMMA 4.6. Let V be aQ vector space and F a real valued symmetric bilinear form on V. Iffor anyfinitely generated subgroup F of V and for any CGM there are only finitely many JET such that F(j, j) < C, then F is positive definite.

Proof. Let T be a finitely generated subgroup of V. The assumption implies that F is positive on T and hence on Wr — T <g) M. Suppose that F is not definite. Then F would arise from a form on a proper subspace U of V r̂ by means of some projection p : Wr-+U. But pi is injective by the finiteness hypothesis. It follows that p(T) is not discrete in U and so there is a sequence 7» of distinct elements of T such that p(ji) approaches 0. But then F(ji,ji) < 1 for ali sufficiently large i's, contradicting our hypothesis. •

THEOREM 4.7. Let £ be a polarization on A. Then —(,)c induces a positive definite symmetric bilinear form on V — A(Q) /A(Q) Tors.

Proof. Let Te V bea finitely generated subgroup. Let A'be a number field such that A(Q) contains a set of representatives Pu.. .Pn for the generators of T. By Proposition 4.5 there are only finitely many points P in the subgroup generated by Pi,... Pn such that ~(P,P)c < C for any given Constant C. Thus the symmetric bilinear pairing induced by - ( , ) , on K satisfies the hypothesis of Lemma 4.6 and hence is positive definite. •

5. Heights on Curves

In this section C will denote a smooth curve of genus g > 2 defìned over a number

450 V. Talamanca

fìeld A'. We fix a divisor O of degree 1 on C with the property that (2g — 2)0 is a canonical divisor on C. Since abelian varieties are divisible it is possible to fìnd such a divisor at the cost of a finite extension of the base fìeld. Now we embed C into its Jacobian J = J(C) = Pic°(C) by mapping P to cl((P) - 0 ) G P Ì C ° ( C ) . From now on we consider C as a subvariety of J and we denote by i the inclusion. Let Cd be the c/-fold symmetric power of C. The embedding of C into its Jacobian induces birational maps of Cd to «7. In particular the image of Cg-\ is a divisor 0 on J, which can also be described as

0 = {aG7 | a = a\ -f . .a^_i,a,-EC-}.

Note that in general 0~ = 0 + O' where O' = -K + (2g - 2)0 (K. being a canonical divisor on C). The choice we have made for O is justified by observing that with this choice 0~ turns out to be linearly equivalent to 0. If we denote by 0 the divisor class of 0 , we then have 9~ = 9. It can be shown that <p@ : J—+J is an isomorphism and thus J is self-dual. We identify J with J via <̂>© and we want to determine the Poincaré bundle on J x J. Let pi : J x J—>J be the projection onto the i tn factor and s : J x J—»• J the morphism giving the group structure on J. Since

PÌ©|? +2^2©|7 - * * 0 L - 0 + O - r a 0 = ^e(«) I Jxa 1.7 x« U x a

we have V — p\9 -\- p^9 - s*9. As a consequence we get

THEOREM 5.1. The height pairing on a Jacobian variety is positive. Moreover the kernel, on each side, is JTOVS-

Proof. Let 6 : J^J x J denote the diagonal embedding. Then

-d*V = d*(pl9 + pl$ - s*9 = [2]*0 -20 = Z0 + $- - 20 = 20.

By prop. 4.3. (a,b) = —h-p(a,b), therefore

{a,a) - -hv(aìa) - (-hVod)(a) = h-.d*v(a) - lìbici) = 2he{a).

Since 0 is ampie and symmetric, Proposition 4.6 applies, proving the theorem. •

COROLLARY 5.2. Let A be an abelian variety. The left and the right kernels of the height pairing are the torsion subgroups of A and A respectively.

Proof By symmetry it suffìces to prove the corollary for the kernel of one of the two sides only. Let J be a Jacobian variety which admits a surjective homomorphism / to A. Then / : A-+J has finite kernel. Suppose that beA(Q) is such that (6, a) = 0 for ali a<EA(Q). Then 0 = {b, f(c)) = (/(&),e) for ali cGJ(Q). Therefore f(b) belongs to Jtors- But / has finite kernel so ÒG-Ators- •

We will now discuss Mumford's inequality. Let A be the diagonal of C x C. The basic relations among divisors that we will use are (cf. [17]):

An introditction to theory of height functions 451

MI L*e = gO

M2 (t x i)*(V) = A-C xO-OxC.

The translation into a statement about height functions is the following

T H E O R E M 5.3 (MUMFORD'S INEQUALITY). There exists a Constant Mc

depending only on C such that

-(<P)^(Q))e<^(he(i(P)) + he(t(Q)))^Mc)

forallP^QeC(Q). .

Proof. Let / be the real valued function on the Q-valued points of C xC defìned

by

f(P, Q) = exp(-(he (t(P)) + he (i(Q))) + 2(i(P), *«)}©)•

To prove the proposition it then suffices to show that there exists a Constant M > 1 such

that Mf(P, Q) > 1. First of ali note that (MI) yields

On the other hand by Theorem 4.3 h-p(a,b) = — (a ,6) 0 , thus (M2) implies

We know by Proposition 3.2 that there exists a representati ve H for f)A such that H > 1

away from A, proving the theorem. •

To conclude this short introduction to the theory of heights we want to very briefly

examine the role of heights in Bombieri's and Vojta's proofs of Mordell's conjecture.

Vojta's main result [26], from which an explicit bound for the number of K-rational points

is readily obtained, is

T H E O R E M 5.4 (VOJTA'S INEQUALITY). Let C be defìned over a numberfield

K. There exists an effectively determined Constant j(C) > 1 with the following property:

far every pair of points P, QEC(K) satisfying

7(c) < VTh^p), 7(c) VTW) < VlhAQ) we have

(P,Q)o < jVèMJVèMQ).

We like to point out that while Vojta's proof of Theorem 5.4 uses arithmetic

intersection theory as developed by Gillet and Soulé, Bombieri's one [2] uses tools that

were ali available since 1965. We conclude by sketching how one deduces a bound for the

452 V. Talamanca

number of A'-rational points of C from Vojta's inequality, since it is a good example of

the versatility of heights.

T H E O R E M (MORDELL'S C O N J E C T U R E ) . The set C(K) is finite. In particular

we nave

#{P£C(K) | v W ) > 7(C)} < #J(K)tonW(llogy/\og2)

wliere p is the rank of J(K).

Proof. As we have seen in Proposition 4.5 — (, )e gives a structure of euclidean

space to Rp=J(K)/j(K)tors ® H&. It is possible to cover the unit sphere Sp~1 with

less than 10^ spherical caps Bj such that (^i ,^2)5>3/4 for any zi,z2£Bj. Let Tj

be the cone over Bj. Let Po,.. .Pm€C(K) be such that the image of Pi lies in Tj,

\/%he(Pi) > j(C) and y/%he(Pk) > Vè / l«(p0 i f &-> *• BY Mumford's inequality

2y/(Pi-\yPi-i)e < \J±ho(Pi). In particular starting from Pm and iterating we get

2m\/±ho(Po) < \/\he(Pm). On the other hand the images of the P^'s are contained

in Tj and so \/%he(Pi) > ^{Po,Po)e > y(C), so Vojta's inequality implies

But then we must have 2m < j(C), i.e. m < log7(C)/ log2. Thus each cone contains

at most 1 + log7(C)/ log2 points with height larger than j(C). The number of points of

C(K) with height less than j(C) is finite by Propositions 3.4 and 4.4 and so Mordell's

Conjecture follows. •

REFERENCES

[1] BIRCH B., SWINNERTON-DYER H.P.F., Notes on elliptic curves I and II, J. Reine Angew. Math. 212 (1963), 7-25 and 218 (1965), 79-108.

[2] BOMBIERI E., The Morde II. Conjecture Revisited, Ann. Se. Norm. Super. Pisa CI. Sci., (4) 17, 4 (1993), 615-640.

[3] BOREL E., Contribution à l'analyse arìthmétique du contimi, J. Math. Pures Appi. 9 (1903), 329-356, reprinted in (Euvres de Émile Borei, tome III, Editions du Centre National de la Recherche Scientifique, 1439-1485.

[4] CALL G., SILVERMAN J. H., Canonical heights on varieties, Compos. Math. 89 (1993), 163-206.

[5] CANTOR G., Uber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen, J. Reine Angew. Math. 77 (1874), 258-262".

[6] FALTINGS G., Endlichkeitssàtze fur abelsche Varietàten iiber Zahlkorpen, Invent. Math. 73 (1983), 366-224, reprinted in Arithmetic Geometry (Eds. G. Cornell and J.H. Silverman), Springer-Verlag, 1986.

[7] FALTINGS G., Diophantine approximation an abelian varieties, Annals of. Math. 133 (1991), 549-576.

[81 FRÒHLICH A., TAYLOR M.J., Algebraic Number Theory, Cambridge University Press, Cambridge, 1991.

An introduction to theory of height fitnctions 453

[9] HARDY G.H., LITTLEWOOD J .E., PO'LYA G., Inequalitìes, Cambridge University Press, Cambridge, 1952.

[10] HARTSHORNE R., Algebraic Geometry, Springer-Verlag, Berlin, 1977. [11] LANG S., Lesfonnes bilinéaires de Néron et Tate, Se'minaire Bourbaki Exp. No. 274, Secretariat

Mathématique, Paris, 1963/64. [12] LANG S., Fundamentals of Diophantine Geometry, Springer-Verlag, Berlin, 1983. [13] LANG S., Number Theory III, Encyclopaedia of Mathematical Sciences 60, Springer-Verlag,

Berlin, 1991. [14] MILNE J. S.,Abelian Varieties, in: Arithmetic Geometry (Eds. G. Cornell and J. H. Silverman),

Springer-Verlag, Berlin, 1986. [15] MILNE J. S.,Jacobian Varieties, in: Arithmetic Geometry (Eds. G. Cornell and J. H. Silverman),

Springer-Verlag, Berlin, 1986. [16] MUMFORD D., Abelian Varieties, Oxford University Press, Oxford, 1970. [17] MUMFORD D., A remark on Mordell Conjecture, Amen J. Math. 87 (1965), 1007-1016. [18] NE'RON A., Quasi-fonctions et Hauteurs sur les variétés abéliennes, Ann. of Math. 82 (1965),

249-331. [19] NORTHCOTT D. G., An inequality in the theory of arithmetic on algebraic varieties, Proc.

Cambridge Philos. Soc. 45 (1949), 502-509. [20] NORTHCOTT D. G., Afurther inequality in the theory of arithmetic on algebraic varieties, Proc.

Cambridge Philos. Soc. 45 (1949), 510-518. [21] SILVERMAN J. H., Rational points on K3 surfaces: a new canonical height, Invent. Math. 105

(1991), 347-373. [22] SERRE J. P. , Lectures on the Mordell-Weil Theorem, Vieweg, Braunschweig, 1989. [23] SCHMIDT W. M., On heights of subspaces and diophantine approximations, Ann. of Math. 85

(1991), 430-472. [24] TALAMANCA V., Height preserving transformations on linear spaces, Ph.D thesis, Brandeis

University, Waltham, 1995. [25] VOJTA P., A Higher Dimensionai Mordell Conjecture, in: Arithmetic Geometry (Eds. G. Cornell

and J. H. Silverman), Springer-Verlag, Berlin, 1986. [26] VOJTA P., Siegel's Theorem in the compact case, Ann. of Math. 133 (1991), 509-548. [27] W E I L A., Arithmetic on algebraic varieties, Ann. of Math. 53 (1951), 412-444.

Valerio TALAMANCA Dipartimento di Matematica Università degli studi di Roma Tre Via C. Segre, 2 00146 Rome, Italy

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