1 Holomorphic mappings into compact Riemann

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1 The Equi-dimension Theory In this notes, we ll prove the Second Main Theorme for holomorphic mappings f : C n M where n = dim M. We call it the Equi-dimension Theory. The theory is developed by Griffiths and his school in early 8s. Holomorphic mappings into compact Riemann Surfaces We first consider the case dim M =, i.e. M is a compact Riemann surfaces. In this case, we show that the curvature plays an important roles. We first recall the following Theorem.(Green-Jensen s Formula) Let g be a function of class C 2 on D(r) or a sub-harmonic (resp. super-harmonic) function on D(r). Then r dd c [g] = 2π g(re iθ ) dθ t ζ <t 2 2π g(), where the notation dd c [ ] indicates that the differentiation is taken in the sense of distribution. We also need the so-called calculus lemma. Lemma. (Calculus Lemma) Let T be a strictly nondecreasing function of class C defined on (, ). Let γ > be a number such that T (γ) e. Let φ be a strictly positive nondecreasing function such that e tφ(t) = c (φ) <. Then the inequality T (r) T (r)φ(t (r)) holds for all r γ outside a set of Lebesgue measure c (φ). Proof Let A [γ, ) be the set of r such that T (r) T (r)φ(t (r)). Then T (r) meas(a) = dr A γ T (r)φ(t (r)) dr = e tφ(t) = c (φ), which proves the lemma.

2 Lemma.2 Let T be a function of class C 2 defined on (, ). Assume that (i) there exists γ such that T (r) e for all r γ and (ii) that both T (r) and T (r) are strictly nondecreasing functions of r. Let b be a number such that brt (r) e for all r (such number clearly exists). Then r d dr dt (r ) T (r)φ(t (r))φ[brt (r)φ(t (r))] dr for all r γ outside a set of measure 2c (φ). Proof The assumptions guarantee that we may apply the Calculus lemma twice, first to the function brt (r) and then to the function T (r). The typical use of the calculus lemma ia as follows: Let Γ be a nonnegative function on C, define r T Γ (r) = Γ dz d z. t z <t 2π Then we have, for every ɛ >, π 2 Γ(re iθ ) dθ 2π exc (T Γ (r)) +ɛ (brt Γ (r)tγ(r)) ɛ ɛ, where exc means that the inequality holds for all r outside a set of finite measure. So π log Γ(re iθ ) dθ 2π exc O(log T Γ (r)) + log r). To see how to get the above conclusion, we take, in the calculaus lemma, φ(t) = t ɛ, and notice that, using polcar coordinate, dθ dz d z = 2rdr 2π 2π, Hence r dt π ( r Γ dr = ( d r dt ) Γ = 2 r dr dr 2 ) dθ Γ(te iθ )t 2π, π Γ(re iθ ) dθ 2π.

3 The calculaus lemma,thus gives the above conclusion. To get the basic idea of the proof of the SMT, we consider the simple case: the genus of M is 2, in the case, we want to prove that f : C M must be constant. Since the genus of M is 2, there is a a positive (,) form ω on M such that Ricω ω (.i.e the Gauss curvature is ), where while we write ω = h dz d z, then Ricω = 2π ddc log h. To prove that f is constant, we estimate T f,ω (r) := To do so, we use Ricω ω, hence T f,ω (r) By Green-Jensen s formula, r t Since ζ t[dd c log h f] = 2 r 2π t r t ζ t ζ t f ω. f (dd c log h). log h(f)(re iθ ) dθ 2π 2π 2 log h(f)(re iθ ) dθ 2π. [dd c log h f] = dd c log h f + [Ram f ], and applying the calculaus lemma (see the above argument), we get which implies that f is constant. T f,ω (r) exc O(log T f,ω (r) + log r) We now prove, using the idea presented above, the SMT as follows Theorem (SMT for Compact Riemann surfaces) Let M be a compact Riemann surface. Let ω = h dz d z be a positive (,) form on M. Let 2π f : C M be a non-constant holomorphic map. Let a,..., a q be distinct points on M. Then, for every ɛ >, m f (r, a j ) + T (r) + N f,ric(ω) f,ram(r) ɛt f,ω (r) + O(log r) 3

4 holds for all r outside a set E (, + ) with finite Lebesgue measure. By the uniformization theorem, M is either biholomorphic to the Riemann sphere P, the torus or the surface of genus 2. So, before we give the proof, we discuss the consequences of the Theorem for each case. When M = P, the Fubini-Study form ω on P is given in terms of an affine coordinate w by ω = ( + w 2 ) 2 2π dw d w = ddc log( + w 2 ). Thus Ric(ω) = 2ω. So, for any meromorphic function f on C (also being regarded as a holomorphic map f : C P ), where T f,ω (r) = r T f,ric(ω) (r) = T f, 2ω(r) = 2T f,ω (r), t ζ t f ω = r t ζ t f 2 dζ d ζ. ( + f 2 ) 2 2π The characteristic function T f,ω (r) above is called the Ahlfors-Shimizu characteristic function. T f,ω (r) differs from the Nevanlinna s characteristic function defined earlier only by a constant. For the torus (elliptic) case, the canonical metric is a flat metric, i.e. there exists a positive (,) form ω such that Ric(ω) =. So in this case, Theorem A2.3.2 implies that m f (r, a j ) + N f,ram(r) ɛt f,ω (r) + O(log r) holds for all r outside a set E (, + ) with finite Lebesgue measure. Finally, for the surface of genus 2, there exists a positive (,) form ω such that Ric(ω) is also a positive (,) form, so that T f,ric(ω) (r). Thus we have T (r) ɛt f,ric(ω) f,ω(r) + O(log r) = ɛ T f,ric(ω) (r) + O(log r) 4

5 holds for all r outside a set E (, + ) with finite Lebesgue measure. This implies that T f,ric(ω) (r) is bounded, hence f is constant. So there is no non-constant holomorphic map from C into M if its genus 2. Proof of the SMT: Note that we don t have Ricω ω, however, motivated the Poincare metric on the punctured disc, we consider Ψ = ω q ( s j 2 (log s j 2 ) 2 ) where ω is a positive (,) form on M, and D j = a j, j q, are the divisors corresponding to the points a j M, s j are the canonical section associated with D j. Write f Ψ = Γ dζ d ζ. 2π Thne, by the Poincare-Lelong formula, dd c [log Γ] = dd c [log f s j 2 ] + f Ric(ω) + D f,ram dd c [log(log f s j 2 ) 2 ]. Applying the integral operator r t ζ t to the identity above and applying the Green-Jensen s formula), we get (log Γ)dθ = 2 ζ =r m f (r, a j ) + T f,ricω (r) + +N f,ram(r) r dd c [log(log f s j 2 ) 2 ]. t ζ t Note that r ( ) 2 dd c [log(log f s j 2 ) 2 ] = log C t ζ t ζ =r log f s j 2 5

6 if we assume that s j δ (after rescalling). Using the Calculus lemma argument, we have So our goal is to estimate (log Γ)dθ exc O(log(T Γ (r) + O(log r). 2 ζ =r T Γ (r) = r t ζ t r Γ 2π dζ d ζ = f Ψ. t ζ t Similar the csae when Ricω ω, we can prove that RicΨ Ψ. However, we prove use and prove the following intermediate result: Claim: ( ) 2 dd c cω log 2{ log s j 2 s j 2 (log s j 2 ) ɛω} 2 for some positive constant c. Assuming the proof is true (will be given later), then T Γ (r) ɛt f,ω (r) + C However, r ɛt f,ω (r) + C t 2 log S r log r t ζ =r ζ t ( log dd c log[ log f s j 2 ]2 log f s j 2 log log S r f s j σ = 2 2 log m f(r, D j ) 2 log T f,ω(r). B t (dd c (log(log f s j 2 ) 2 ) ω n = 2 ) 2 ( ) 2 ɛt f,ω(r) + C log. ζ =r log f s j 2 ( ) σ f s j 2 Thus, Hence the SMT is proved. T Γ (r) 2ɛT f,ω (r). It remains to prove the claim: 6

7 Lemma A Let τ be a non-negative function of class C 2 when τ >. Then dd c log ( ) 2 = 2{ log τ τ(log τ) 2 ddc τ + log τ (log τ) 2 ddc log τ}. It can be proved by the direct computation. Second, we use the connection. We notice that for a section {s α }, since s α = g αβ s β, ds α = g αβ ds β +dg αβ s β. So, usually the differentiation of a section does not yield a section because of the term dg αβ s β. To overcome this difficulty, one introduces a correction term and defines covariant differentiation Ds = {Ds α } where Ds α = ds α + (log h α )s α = s α + (log h α )s α. Then Ds obeys the transition law that Ds α = g αβ Ds β. For a (, ) form A (resp. (,) form), we write, A 2 = 2π A Ā. So A 2 becomes a real (.) form. We write Ds 2 = Ds α 2 h α. Lemma B Let L be a metrized line bundle and s be a holomorphic section of L. Then dd c s 2 = Ds 2 s 2 c (L). Proof Write L = {U α, g αβ } and the metric h = {h α }. definition, s 2 = s α 2 h α = s α s α h α. So Then, by the s 2 = s α sα h α + s α s α hα = s α ( s α + s α log hα )h α = s α D s α h α =< s, Ds >, using the property that s α = since s α is holomorphic. Thus, dd c s 2 = 2π s 2 = 2π (s αd s α h α ) = 2π s α D s α h α + 2π s α (D s α )h α + 2π s α h α D s α = 2π s α D s α h α + 2π s α (D s α )h α + 2π s αh α (log h α ) D s α = 2π s α D s α h α + 2π s αh α (log h α ) D s α + 2π s α (D s α )h α 7

8 = 2π ( s α + s α log h α ) D s α h α + 2π s α (D s α )h α = Ds α 2 h α + 2π s α (D s α )h α = Ds 2 + 2π s α (D s α )h α. () However, 2π (D s α) = 2π ( s α + s α (loghα )) = s α 2π (log h α ) = s α c (L). So 2π s α ( D s α )h α = s α s α c (L)h α = s 2 c (L). (2) Combining (2.4) and (2.5) proves the lemma. We are now ready to prove the claim. By Lemma A, ( ) 2 dd c log = 2{ log s j 2 s j 2 (log s j 2 ) 2 ddc s j 2 + log s j 2 (log s j 2 ) 2 ddc log s j 2 }. By definition, dd c log s j 2 = c (L j ), and by Lemma B, dd c s j 2 = Ds j 2 s j 2 c (L j ). So ) 2 ( dd c log log s j 2 { Dsj 2 s j 2 c (L j ) = log s j 2 } s j 2 (log s j 2 ) 2 (log s j 2 ) c (L 2 j ) { Ds j 2 = 2 s j 2 (log s j 2 ) + c } (L j ) 2 log s j 2 Recall on the covering {U α } of M, the covariant differential operator is locally defined by Ds = s α + s α (log h α ), for a non-zero holomorphic section s = {s α }. Hence Ds 2 = s α + s α (log h α ) 2 h α ( 2 s α 2 s α (log h α ) 2 )h α = 2 s 2 s 2 2π A Ā = 2 s 2 s 2 A 2 (4) (3) 8

9 where A is the differential form A = (log h α ). Assume that the zero set of s is without multiplicity, then s 2 + s 2 c (L) is a positive (,) form on M, and because M is compact, 2 s 2 + s 2 c (L) > cc (L) (5) where c is a positive constant. Also, since L is positive and M is compact, we have c c (L) A 2 c c (L) (6) for some positive constant c and c. Combining (2.), (2.2) and (2.3), we have Ds 2 + s 2 c (L) cc (L) c s 2 c (L), or this simply means, Ds 2 cc (L) c s 2 c (L), where c = +c >. The above inequality applies for holomorphic sections s j. That is Ds j 2 cc (L j ) c s j 2 c (L j ). Combining (2.) and (2.4) gives ( ) 2 dd c log log s j 2 { cc (L j ) 2 s j 2 (log s j 2 ) c c (L j ) 2 (log s j 2 ) + c } (L j ). (7) 2 log s j 2 When the metric of L j is rescaled by a constant, so that s becomes λ s, the covariant differentiation is not affected by the rescaling and c (L) is also unchanged. Choose a rescaling such that c (log s j 2 ) 2 log s j ɛ. 2 Then ( ) 2 dd c cc (L) log 2{ log s j 2 s j 2 (log s j 2 ) ɛc (L 2 j )}. (8) 9

10 Since ω, c (L) are both positive and M is compact, there are positive constants b, b 2 such that b ω < c (L j ) < b 2 ω, so (2.6) can be written as This proves the claim. ( ) 2 dd c cω log 2{ ɛω}. (9) log s j 2 s j 2 (log s j 2 ) 2 2 Diophantine approximation In this section, we ll descripe the corresponding results of the previous section in Diophantine approximation. Let k be a number field. As we described earlier, a number field k has a canonical set of places, denoted by M k. The set of Archimedean places of k is denoted by Mk, and the non-archimedean places is denoted by Mk. Let X be an algebraic curve defined over k. Denote by X(k) the set of k-rational points on X. We define the height for points on X. We first consider heights for points on P n. Let x P n (k), x = [x :... : x n ], with x i k not all zero. We put and If L/k is a finite extension, H k (x) = υ M k max i n x i υ, h(x) = [k : Q] log H k(x). () H L (x) = H k (x) [L:k], so h is independent of the ground field k, and thus extends to k, the algebraic closure of k. We have thus defined a logarithmic height h : P n ( k) R with values. The following theorem is the work of D.G. Northcott [Nor]. Theorem B2..2 (Northcott) Let n, d, N be integers. There are only finitely many points of P n (Q) of height N and of degree d. Here, for

11 x = (x,..., x n ) P n (Q), the degree of x is the degree of the field generated by the x i /x j, i, j n, x j. Corollary B2..3 Given a number field k, there are only finitely many points of P n (k) of height N. A morphism of degree d between projective spaces is a map F : P N P M, F (P ) = [f (P ) :... : f M (p)], where f,..., f M Q[X,..., X N ] are homogeneous polynomials of degree d without common zeros in Q other than X = = X N =. If f i, i M, have coefficients in k, then F is said to be defined over k. We have the following property of the height regarding the morphism. Theorem B2..4 Let F : P N P M be a morphism of degree d. Then there are constants C and C 2, depending on F, so that for all points P P N (Q), dh(p ) + C h(f (P )) dh(p ) + C 2. Proof This can be directly verified by the definition (see [Lang], Chapter 4 Theorem.8 or [Sil2], Chapter VIII Theorem 5.6 for details). For a given algebraic curve X defined over k, let φ : X P n be an embedding. For x k, we put H φ (x) = H k (φ(x)), and h φ (x) = h(φ(x)). If ψ : X P n is another embedding, then we have c h φ +c 2 h φ (x) c 3 h φ +c 4 on X( k) for some positive constants c,..., c 4. Hence h φ and h ψ have roughly the same size. So we often just denote the height by h. Corollary B2..2 tells us that the height h(x) measures the growth of x, as T f (r) does in Nevanlinna theory. Let D be a divisor on X. D is called very ample if there is an embedding φ : X P n with D = φ H, where H is a hyperplane on P n. Note that the embedding φ can be obtained as follows: Let f,..., f N be a basis of space

12 of all rational functions over k with (f) D, the map φ : V P N given by φ(x) = [f (x) :... : f N (x)] is a projective embedding. For a very ample divisor, we define h D (x) = h φ (x). Given two divisors D and D 2, we say that D is rationally (linearly) equivalent to D 2 if there is a rational function f on X such that D = D 2 + (f) where (f) is the divisor associated with f, that is (f) = a X ord a (f)a. A basic lemma of algebraic geometry tells us that any divisor D is equivalent to divisors D D 2, where D and D 2 are very ample divisors. We then define h D (x) = h D (x) h D2 (x). h D (x) is then uniquely determined modulo a bounded function on X. Next, we introcuce the concept of Weil-function. There is a unique way (up to a rather involved definition of O()) to assign to each pair (X, D), where X is a complete k-variety and D is a Cartier divisor on X, a Weil function λ D : (X\D)( k υ ) R, υ M k such that (i) normalization: if H P n is a hyperplane at infinity, then λ H,υ (x) = log x υ max i x i υ + O(). (ii) additivity: λ D+D = λ D + λ D + O(), and (iii) functionality: if f : X Y is a morphism and D is a divisor on Y whose support does not contain the image of X, then λ f D(x) = λ D (f(x)) + O(). (iv) continuity: λ D,upsilon is continuous in the υ-topology. Here λ D,υ means the restriction to (X\D)( k υ ), and additivity is only assumed to hold on the intersection of the domains. We give a short recipe for the construction of the Weil functions on projective varieties (for more details, see [Lang], Chapter ). Let X be a non-singular projective variety over k. Let D be a divisor on X. First construct sets of effective divisors X i, (i =,..., n) and Y j, (j =,..., m) such that D + X i is linearly equivalent to Y i 2

13 for every i, j and such that the X i, i n have point in common and the Y j, j m also have no point in common. Let f ij k(x) be such that (f ij ) = Y j X i D for each pair i, j. Extend the valuation to all of k. For each p X( k). The Weil function with respect to the divisor D is then defined by, for each υ M k, λ D,υ (p) = max min log f ij (p) υ. j i Of course, the Weil function defined above depends on the choice of f ij, but they differ only by a bounded function. Example Let D be a hyperplane D = {[x : : x n ] a x + + a n x n = } in P n (k). Take f j (x) = x j a x + + a n x n, j =,..., n. Here, using the notation above, X i is an empty set, Y j = [x j = ]. The pole divisor of f j is precisely D and the zero divisor is [x j = ], so (f j ) = Y j D. For x = [x :... : x n ] P n (k), our Weil function reads λ D,υ (x) = log max j x j υ a x + + a n x n υ. () By the product formula and from (2.2) and (2.22), we have, h k (x) = [k : Q] So, if we define for any divisor D in X, h D (x) = [k : Q] υ M k λ D,υ (x). υ M k λ D,υ (x), (2) then we can prove (see [Lang], chapter ) that this definition agrees with the definition given earlier, up to a bounded term. Remark: In Nevanlinna theory itself, let X be a complete complex variety and let D be a Cartier divisor on X. Choose a hermitian metric on O(D) 3

14 and let s D be the canonical section associated to D, then λ D := 2 log s D 2 gives a Weil function (which is much simpler notion in this context). Given a finite set S M k, we define the proximity function m S (x, D) by, for x X( k), m S (x, D) = λ D,υ (x). (3) [k : Q] υ S The counting function N S (x, D) is defined by N S (x, D) = λ D,υ (x). (4) [k : Q] υ S Note that the sum above is still a finite sum, since the terms all vanish except for finitely many. Integral points: Let k and S as above, let X be a complete variety over k, and let D be an effective divisor on X such that X\SuppD is affine. Naively, a point P X(k) is a (D, S)-integral point on X if thr is a rational map i : X P n for some n, defined and injective on X\SuppD, such that SuppD = i {x = } and i(p ) A n (O k,s. However, this notion is useful only for infinitely many sets (for any finite set you can clear the denominators)(serre). This definition can be expressed by using Weil functions: Definition A subset Σ of X(k) is (D, S)-integral if there is a Weil function λ D for D and constants c υ R for all υ S such that c υ = for almost all c υ = and λ D,υ (P ) c υ for all υ S and P Σ. If D = i ({x = }, then this is easily seen to be equivalent to earlier naive definition if i is actually morphism, by functorially and normalization of Weil functions. Moreover, by additivity of Weil functions this notion depends only on SuppD. Indeed, if D and D are effective divisors with the same support, then D md for some m, and so λ D mλ D + O(). It is also depeonds only on the quasi-projective variety X\SuppD. This definition does not assume that X\D is affine. Also, if D =, then the condition is vacuous and all sets of rational points are integral. 4

15 Also, note that this defintion is well-behaved with respect to morphisms of X\D: given morphism φ : X X and effective divisors D on X and D on X such that φ(x\suppd) X \SuppD, then φ (D ) SuppD, so φ D nd. Therefore, by functorially and linearity of Weil functions, (D, S)-integral points on X are mapped to (D, S)-integral points on X. Of course, (D, S)-integrality implies that N S (D, P ) is bounded, and this corresponds to N f (r, D) being bounded in Nevanlinna thoery. We now prove the First Main Theorem. Combining (2.23), (2.24) and (2.25) we have our First Main Theorem. Theorem B2..5 (First Main Theorem) h D (x) = m S (x, D) + N S (x, D). Theorem B2..6 (Second Main Theorem) Let k be a number field with its set of canonical places M k. Let S M k be a finite set containing all Archimedean places. Let X be a smooth algebraic curve defined over k. Let K be the canonical divisor on X. Then, for any ɛ >, m(x, D) + h K (x) ɛh(x) (5) holds for all x X(k) except for finitely many points. In the following sections, we will discuss Theorem B2..6 according to the genus of X. Curves of Genus Theorem B2.2. Let X be a smooth curve defined over k. If the genus of X is zero, then X is isomorphic to a conic. If it has a k-rational point, then it is isomorphic to P (k), and thus has infinitely number of k-rational points. Proof As the genus of X is zero, the canonical divisor on X has degree 2. Changing the sign, one obtains a divisor of degree 2. This divisor induces an embedding into the projective space, whose image is of degree 2, hence it is a conic. Thus, since by the assumption X(k) is non-empty, X(k) is isomorphic 5

16 to P (k), and the canonical divisor K has degree 2. So h K (x) = 2h(x), thus (2.26) becomes, for D = q a j, m(x, a j ) 2h(x) ɛh(x). Theorem B2..6 in this case is equivalent to Roth s theorem. Theorem B2..6, thus is proved for this case. Rational Points on Curves of Genus, Mordell-Weil Theorem X is called an elliptic curve if X is an algebraic curve of genus. An elliptic curve may have infinitely many rational points. In 9, Poincaré showed that if a rational point on an elliptic curve X is chosen as an origin, the k-rational points of X form a group. In 922, Mordell proved the conjecture of Poincaré that the group of rational points is finitely generated. In his paper, Mordell also conjectures that the set of integral points is finite, and that the set of rational points on a curve of genus 2 is finite. The conjecture about integral points was proved by Siegel and the conjecture about rational points on a curve of genus 2 was recently settled by G. Faltings [Fal]. An elliptic curve may have infinitely many rational points, although the Mordell-Weil theorem at least assures us that the group of rational points is finitely generated. However, Siegel proved that there will only be a finite number of integral points on an affine elliptic curve. Theorem B2.4. (Siegel) Let k be a number field with its set of canonical places M k. Let S M k be a finite set containing all Archimedean places. Let X be a smooth algebraic curve of genus defined over k. Then, for any ɛ >, m(x, D) ɛh(x) (6) holds for all x X(k) except for finitely many points. When X is of genus, its canonical divisor is trivial. So this Theorem is equivalent to Theorem 2..6 in the case where the genus of the curve is equal to. 6

17 To prove Siegel s theorem, we define the υ-distance on X for υ M k. Definition B2.4.2 Let X be a curve defined over k, and P, Q X(k υ ). Let t Q be a locally defined rational function over X with values in k υ such that it vanishes at Q with order e. The υ-adic distance function from P to Q, denoted by d υ (P, Q), is given by d υ (P, Q) = min{ t Q (P ) /e υ, }. The definition above certainly depends on the choice of t Q, so possibly a better notation would be d υ (P, t Q ). However, since we will only use d υ to measure the rate at which two points approach one another, the following result show that all of our theorems make sense. Proposition B2.4.3 Let Q X(k υ ), and let t Q and t Q be functions vanishing at Q. Then we have the notation log d υ (P, t lim Q) P X(k υ),p Q log d υ (P, t Q ) =, here P Q means P X(k υ ) approaches Q in the υ-topology, i.e., d υ (P, t Q ). Proof. Let t Q and t Q have zeros of order e and e respectively at Q. Then the function φ = (t Q) e /(t Q ) e has neither a zero nor a pole at Q. Hence φ(p ) υ is bounded away from and as P Q; so as P Q, log d υ (P, t Q) log d υ (P, t Q ) log φ(p ) /ee υ = + log d υ (P, t Q ). Roth s theorem implies the following theorem. Theorem B2.4.4 Let X be a curve defined over k and Q X( k). Then, for any ɛ >, the inequality d υ (P, Q) 7 H k (P ) 2+ɛ

18 holds for all P X(k) except for finitely many points. Proof Let t Q be a locally defined rational function over X values in k υ such that it vanishes at Q with order e. Then by definition, we may take d υ (P, Q) = min{ t Q (P ) /e υ, }. Applying Theorem B2..6 (Roth s theorem) to α =, for any ɛ >, min{ t Q (P ) /e υ, } H k (t Q (P )) 2+ɛ holds for all but finitely many P. The theorem is proven. We now prove Theorem B2.4.. Proof Let D = q (Q j ). It is easy to check, by the definition, that m S (x, D) = d υ (x, Q j ). (7) υ S Let m be a positive integer such that (m 2 )/ɛ 3. By the weak Mordell- Weil theorem, X(k)/mX(k) is finite. For any x X(k), there is x, π X(k) such that x = mx + π. Since X(k)/mX(k) is finite, we may assume that π is independent of x. Since the morphism ω mω + π is etale, we have d υ (x, Q j ) d υ (x, Q j) + O() and h(x) = m 2 h(x ) + O(). By Theorem B2.4.4, log d υ (x, Q j) (2 + ɛ)h(x ) holds for all, but finitely many, x X(k). So log d υ (x, Q j ) (2 + ɛ)h(x ) ɛh(x) holds for all, but finitely many, x X(k). By (2.34), this is equivalent to m S (x, D) ɛh(x) 8

19 which holds for all, but finitely many, x X(k). Corollary B2.4.5 Let X/k be an elliptic curve with Weierstrass coordinate functions x and y, let S M k be a finite set of places containing M k. Let O S be the ring of S-integers of k. Then is a finite set. {P X(k) : x(p ) O S } Proof Suppose Corollary B is false. That is there exist distinct points P, P, {P X(k) : x(p ) O S }. Consider the divisor consisting of a single point O. Since x has a pole of order 2 at O, we have, by the definition, Since for υ S we have x(p i ) υ, d υ (P, O) = min{ x(p ) /2 υ, }. d υ (P i, O) =, if υ S. So m S (P i, O) = h(p i ) + O(), which contradicts with (2.33) with D = O. Clearly the above proof can be applied to any rational function f k(x). So we have the following more general corollary. Corollary B2.4.6 Let X/k be an elliptic curve. let S M k be a finite set of places containing M k. Let O S be the ring of S-integers of k. Let f k(x) be a non-constant function. Then is a finite set. {P X(k) : f(p ) O S } Curves of Genus Greater Than or Equal to Two, Theorem of Faltings When the genus of X is greater than or equal to two, Faltings proved the following theorem. Theorem B2.5. (Faltings) Let X be an algebraic curve over Q whose genus of X is greater than or equal to two. Then for any number field k, the set X(k) is always finite. 9

20 This theorem is equivalent to the statement of Theorem B2..6 in the case that the genus of X is greater than or equal to two. To see this, we note that the canonical bundle K in this case is positive. So taking D =, then Theorem B2..6 reads h K (x) ɛh(x). This is equivalent to, by Corollary B2..3, the statement that the set X(k) is finite. 3 The Equi-dimension Theory Now we condier the case when M is a compact complex manifold of dimension n. We consider a non-degenerate holomorphic mapping f : C n M (i.e. the Jacobian J f (z) ). We consider a positive line bundle L on M and q divisors D j of holomorphic secctions s j of the bundle such that (A). D,..., D q are manifolds intersect in general position, (B) qc (L) + c (K M ) > where K M is the canonical bundle. The Second Main Theorem Let M be a compact complex manifold of dimension n, and f : C n M be a non-degenerate holomorphic mapping. Let D = q D j be a divisor on M satisfying (A) and (B). Then m f (r, D) + T f (K M, r) + N(S f, r) exc O(logT f (L, r) + log r). We first recall the defintion of height, proximity and counting functions, similar to above. Recall the normailzied Euclidean form on C n is φ = dd c z 2. Denote by ω = dd c log z 2, and the poincare-form σ = (d c log z 2 ) (dd c log z 2 ) n. Then S r σ =, where S r is the ball of radius r. Define T f (L, r) = r t B t f c (L, h) ω n = 2 r f c t 2n (L, h) φ n, B t

21 m f (D, r) = log S r f s σ. Let A C n be an k-dimensioanl analytic set, let n(a, r) = φ k = ω 2k. k A B r A B r With this notion, we can define N f (D, r) is a natural way. Proof of the SMT: The proof is similar to the Riemann surface case. Motivated the Poincare metric on the punctured disc, we consider Ψ = Ω q ( s j 2 (log s j 2 ) 2 ) where Ω is a volume form (a global positive (n,n) form on M). Write f Ψ = ΓΦ ζ, where Φ ζ is the Euclidean volume form in C n. Thne, by the Poincare-Lelong formula, dd c [log Γ] = dd c [log f s j 2 ] + f Ric(Ω) + S f dd c [log(log f s j 2 ) 2 ]. Applying the integral operator r t B t to the identity above, and by Green-Jensen formula (for C n ), we get log Γσ = m f (r, D) + +T f,k (r) + N(S f, r) 2 S r r t (dd c log(log f s j 2 ) 2 ) ω n B t 2

22 By Green-Jensen formula (for C n ) again, we have r (dd c (log(log f s j 2 ) 2 ) ω n = ( ) log log σ t B t 2 S r f s j 2 2 S log log r f s j σ = 2 2 log m f(r, D j ), Hence, m f (r, D) + +T f,k (r) + N(S f, r) 2 Use Calculus lemma argument, we have S r log Γσ + O(log T f (r, L)). log Γσ = n log Γ 2 S r 2 Sr /n σ n 2 log Γ /n σ exc O(log( ˆT (r) + log r), S r where ˆT (r) = r Γ /n φ n t. 2n B t Now the key is to estimate ˆT (r). To do so, we prove the following claim: Claim: (a) RicΨ >, (b) (RicΨ) n > Ψ, (c) M\D (RicΨ)n <. Assuming the cliam holds, we now use the claim to finish the proof. We show that (f RicΨ) φ n cγ /n φ n. In fact, writing Then, by the claim, f RicΨ = 2π n j,k= R j,k dz j d z k ΓΦ z = f Ψ (f RicΨ) n = n!detrφ z, where R = (R jk ). Use det(r) /n n T rr, 22

23 so But Thus ( Γ n! ) /n φ n n n R jj φ n. n f (RicΨ) φ n = n R jj φ n. (f RicΨ) φ n cγ /n φ n. Now, applying the above inequality, we get ˆT (r) ct f,ricψ (r) for some positive constant c, where T f,ricψ (r) = r f (RicΨ) φ n t 2n. B t It remains to estimate T f,ricψ (r). From the defintion, So RicΨ = qc (L) + c(k M ) dd c log(log s j 2 ) 2, T f,ricψ (r) qt f (L, r) + T f (K M, r). This completes the proof of the Seond Main Theorem. Now we prove the claim. Form defintion, on M\D, RicΨ = qc (L) + c(k M ) 2 dd c log(log s j 2 ), By condition (B), qc (L) + c(k M ) >, and by an elementary formula, dd c log(log s j 2 ) = ddc log s j 2 log s j 2 d log s j 2 d c log s j 2 (log s j 2 ) 2. By rescalling the metric, we see the first term can be made to be small enough, so RicΨ c (qc (L) + c(k M )) d log s j 2 d c log s j 2 (log s j 2 ) 2, ( )

24 where c > is a constant. Since dρ d c ρ = (i/2π) ρ ρ for any real function ρ, so (a) is proved. To prove (b), let p M and we choose local coordinates w = (w,..., w n ) such that D j U = {w j = } for j =,... k. In these coordinates, s j 2 = ρ i w j 2, where ρ j is a smooth function. Then d log s j 2 d c log s j 2 = 2π log s j 2 log s j 2 = 2π dw j d w j + Λ j w j 2, where Λ j = w j 2 ( ρj ρ j ρ 2 j + ρ j d w j ρ j w j + w j ρ ) j ρ j w j is a smooth form which is at the point p. Consequently, for j =,..., k, d log s j 2 d c log s j 2 (log s j 2 ) 2 = 2π ρ dw j d w j + Λ j j (log s j 2 ) 2 s j 2 c 2 dw j d w j + Λ j (log s j 2 ) 2 s j 2 for some positive constant c 2. The remianing terms on the right side of (*) makes up a positive (,)-form and thus can be bounded form below by the Euclidean metric form multiplied by a positive constant c 3. Therefore dw j d w j + Λ j n RicΨ > c 2 (log s j 2 ) 2 s j + c 2 3 2π dw ν d w ν, whence ν= ( ) n nν= (RicΨ) n dw ν d w ν ) n + Λ > c 4 ( 2π k log s j 2 ) 2 s j, 2 where Λ is a smooth (n, n)-form which is at the point p, and c 4 > is a constant. By shrinking U and enlarge c 4, we can discard Λ. On the other hand, we have Hence (b) is proved. ( ) n nν= dw ν d w ν ) n Ψ c 5 ( 2π k log s j 2 ) 2 s j. 2 24

25 (c) can be proved by using the fact that dw j d w j 2 (log w j 2 ) 2 w j 2 is finite. Conjectures U j Conjecture (Griffiths) Let f : C M be an algebraically non-degenerate holomorphic curve in a compact projective complex manifold M of dimension n. Let L be a positive line bundle and let D L. Let K M be the canonical line bundle over M. Assume that L + K M is positive. Then the inequality T f,l (r) + T f,c (K M )(r) N (n) f (r, D) + O(log T f,l (r)) holds for all r outside a set E (, + ) with finite Lebesgue measure. We consider the case that M = P n (C). To determine the canonical divisor we consider the differential form Ω = dx dx 2 dx n in the affine coordinates (, x,..., x n ) on U = {x P n x }. There are no zeros or poles on U. But if we rewrite Ω with respect to (x,...,,..., x n ) on U i = {x P n x i } we find Ω = x n+ dx dˆx i dx n. Hence Ω has a pole of order n + along x = and K = (n + )H where H is the hyperplane at infinity. So T f,c (K M )(r) = (n + )T f (r), where T f (r) is the Nevanlinna s characteristic function defined in Chapter 3. Take D = H + + H q, where H,..., H q are hyperplanes in general position, then Conjecture in this case is just Theorem mentioned above. If M is an abelian variety, then K M is trivial. So Griffiths conjecture implies that if the image f omits a divisor on an Abelian variety, then f is degenerate. Furthermore, if D is ample, then f must be constant. This is known as Lang s conjecture. which is solved by Siu-Yueng. Conjecture (Vojta) Let V be a smooth projective variety defined over a number field k and let A be a pseudo ample divisor. Let D be a normal 25

26 crossing divisor defined over a finite extension of k. Let K be a canonical divisor of V. Let S be a finite set of valuation on k and for each υ S let λ D,υ be a Weil function for D. Let ɛ >. Then there exits a Zariski closed variety Z of V such that for all P V (k), P Z we have λ D,υ (P ) + h K (P ) < ɛh A (P ) + O(). υ S As an example consider V = P n. Then K = (n + )H where H is the hyperplane at infinity. So by linearity of heights we find that h K = (n+)h where h is the ordinary projective height. Thus Vojta s conjecture for V = P n reads as follows: for all P V (k), P Z we have λ D,υ (P ) < (n + + ɛ)ɛh(p ) + O(). υ S In the case when D is a union of hyperplanes in general position, this is just Schmi s subspace theorem. We take for D any hypersurface, and for k = Q, the field of rational numbers, and we obtain the following conjecture. Conjecture B5..2 Let D be a hypersurface in P n defined over Q of degree d n+2 with at most normally crossing singularities. Suppose that it is given by the homogeneous equation Q(x,..., x n ) where Q has coefficient in Z. Let S be a finite set of rational primes. Then the set of points (x,..., x n ) Z n+, with gcd(x,..., x n ) = such that Q(x,..., x n ) only contains primes from S, lies in a Zariski closed subset of P n. Proof In fact, we apply Vojta s conjecture with the proximity functions x d i λ D,υ = log max i Q(x,..., x n ) and the set of valuations S { }. Vojta s conjecture implies that for any ɛ > the set of projective n+-tuples (x,..., x n ) Z n+ with gcd(x,..., x n ) = which satisfy υ S { } log max i x d i > (n + + ɛ)h(x Q(x,..., x n ),..., x n ) υ 26 υ

27 lie in a Zariski closed subset of P n. The inequality can be restated as υ S { } log Q(x,..., x n ) υ < (n + + ɛ)h(x,..., x n ) + υ S { } log max x d i i υ The sum on the right is precisely dh(x,..., x n ) since the sum includes the infinite valuation and the max i is for all finite υ, So the set of solutions to υ S { } log Q(x,..., x n ) υ < (d n ɛ)h(x,..., x n ) lies in a Zariski closed subset of P n. Suppose Q(x,..., x n ) is composed of primes only from S. Then log Q(x,..., x n ) υ = for υ S. Thus, by the product formula, υ S { } log Q(x,..., x n ) υ =. So υ S { } log Q(x,..., x n ) υ < (d n ɛ)h(x,..., x n ) holds since d n +. Hence the set of points (x,..., x n ) Z n+, with gcd(x,..., x n ) = such that Q(x,..., x n ) only contains primes from S, lies in a Zariski closed subset of P n. Note that we know that Conjecture holds if Q is decomposable, i.e. Q = L L 2 L d is the product of d linear forms in general position(see Chapter 3, Part B). But it in general is still an open question. Indeed, it does not appear to be known whether the integer solutions to the specific equation are Zariski dense in P 3 (Q). x 5 + y 5 + z 5 + w 5 = Recall that V is said to be of general type if K is pseudo-ample. So taking D =, Vojta s conjecture gives us the following conjecture. Conjecture (Bombieri) Let V be a projective variety over a number field k and suppose that V is of general type. Then V (k) is contained in a Zariski closed subset of V. 27

28 Finally, if V is an abelian variety, then Vojta s conjecture gives the following result (now known as Falting s theorem). Theorem (Faltings) Let A be an Abelian variety over k and E a subvariety, also defined over F. Let h be a height on A and υ a valuation on F. Let ɛ >. Then we have λ E,υ (P ) < ɛh(p ) for almost every point P A(k) E. 28