Analytic and rational sections of relative semi-abelian varieties
aa r X i v : . [ m a t h . C V ] M a y Analytic and rational sections of relativesemi-abelian varieties
P. Corvaja, J. Noguchi ∗ and U. ZannierMay 14, 2020 Abstract
The hyperbolicity statements for subvarieties and complement of hypersurfaces in abelianvarieties admit arithmetic analogues, due to Faltings, Ann. Math. (1991) (and for thesemi-abelian case, Vojta, Invent. Math. (1996); Amer. J. Math. (1999)). In AttiAccad. Naz. Lincei Rend. Lincei Mat. Appl. (2018) by the second author, an analogybetween the analytic and arithmetic theories was shown to hold also at proof level, namelyin a proof of Raynaud’s theorem (Manin-Mumford Conjecture). The first aim of this paperis to extend to the relative setting the above mentioned hyperbolicity results. We shall beconcerned with analytic sections of a relative (semi-)abelian scheme A → B over an affinealgebraic curve B . These sections form a group; while the group of the rational sections(the Mordell-Weil group) has been widely studied, little investigation has been pursued sofar on the group of the analytic sections. We take the opportunity of developing some basicstructure of this apparently new theory, defining a notion of height or order functions for theanalytic sections, by means of Nevanlinna theory. Keywords: Legendre elliptic; Semi-abelian scheme; Diophantine geometry; Nevanlinna theory.AMS Subject Classification: 14K99 ; 14J27; 32H25
Let A be an abelian variety and let f : C → A be an entire curve on it. Then the Zariski-closureof its image is a translate of an abelian subvariety of A (Bloch-Ochiai’s Theorem, c.f., e.g., [17] § D ⊂ A is an ample divisor of an abelian variety, thereis no non-constant entire curve on A \ D . Analogous results hold for semi-abelian varieties (cf.,e.g., [17] Chap. 6). See a relevant Question 7.1 about the topological closure of an entire curve.These hyperbolicity statements for subvarieties and complements of hypersurfaces in abelianvarieties admit arithmetic analogues, due to Faltings [8] (and Vojta [21] for the semi-abeliancase): the rational points on a subvariety of an abelian variety are contained in a finite unionof translates of abelian subvarieties and the integral points on the complement of an ampledivisor in an abelian variety are finite in number. In [16], the second author observed a direct ∗ Supported by Grant-in-Aid for Scientific Research (C) 19K03511. elation between those analytic results and Diophantine properties in a proof of M. Raynaud’sTheorem (Manin-Mumford’s Conjecture), going beyond formal analogies holding at the level ofstatements.The first aim of this paper is to extend to the relative setting the above (nowadays classical)analytic results; in our present situation, the single abelian variety A will be replaced by analgebraic family A → B of abelian varieties over an algebraic base B , and we shall be concernedwith possibly transcendental holomorphic sections B → A . These sections form a group, which isthe complex-analytic analogue of the group of rational sections; while this last group, called theMordell-Weil group of the abelian scheme A → B , has been widely studied, little investigationhas been pursued so far on the group of analytic sections. In this work we take the opportunityof developing some basic structure of this apparently new theory, defining e.g. a notion of heightfor the transcendental sections, by means of Nevanlinna theory.We start by noticing the following: Let π : A → B be a holomorphic family of principallypolarized abelian varieties over a base space B which is algebraic. Then the family π : A → B is algebraic. This is a result by Kobayashi–Ochiai [10], in the spirit of a ‘Big Picard Theorem’.Hence, the initial datum will consist in an algebraic family, but we shall consider (possibly)transcendental sections.More precisely, our main concern will be addressed to the following three problems below,motivated by the known results in the constant case: Let π : A → B be an algebraic familyof semi-abelian varieties. In this paper we always assume the existence of a section B → A ,namely the zero-section. Then: Problem 1.1.
Let D be a relatively big divisor on A over B . Then, every holomorphicsection σ : B → A \ D (omitting D ) is rational. Problem 1.2.
Let σ : B → A be a transcendental holomorphic section, and let X ( σ ) be theZariski-closure of σ ( B ) in A . Then, X ( σ ) contains a translate of a relative non-trivial subgroupof A over B . The conclusion that X ( σ ) is itself a translate of a relative subgroup cannot always hold (see § Problem 1.3.
Let σ : B → A be a strictly transcendental holomorphic section, and let X ( σ ) be the Zariski-closure of σ ( B ) in A . Then, X ( σ ) is a translate of a relative subgroup of A over B . We shall start from the concrete example of the Legendre elliptic scheme: ZY = X ( X − Z )( X − λZ ) ⊂ B × P , where λ varies on the base B = P \ { , , ∞} = C \ { , } . For each λ ∈ B the above curve E λ ,together with its distinguished point (0 : 1 : 0), is an elliptic curve. Removing this point fromeach fiber, so removing a relatively ample divisor, we obtain the affine variety of equation y = x ( x − x − λ ) , B . As an example of a result in the direction of Problem 1.1, weshall prove, by means of Yamanoi’s Second Main Theorem [22], that any holomorphic section B ∋ λ ( x ( λ ) , y ( λ )) of this fibration is reduced to one of the three 2-torsion points with y ( λ ) = 0 (see Theorem 2.2); in the course of the proof we show a rationality criterion ofholomorphic sections for a base-extended family of the Legendre elliptic scheme (see Theorem2.4)We will also see a similar property for hyperelliptic schemes of higher genera by making useof an extension theorem of big Picard type due to Noguchi [14].A general result of that type for sections of a family of semi-abelian varieties will be provedin Theorem 4.22, under the stronger hypothesis that the family admits no bad reduction. Notethat this hypothesis excludes the non-isotrivial elliptic schemes.To prove our main results, we will generalize the Nevanlinna theory of holomorphic curves insemi-abelian varieties ([17]) to a relative setting, and prove a Big Picard Theorem for a localsmooth family; in this context, we have an interesting problem when the family is singular overthe special point. Let ( X : Y : Z ) ∈ P ( C ) denote a homogeneous coordinate system of P ( C ). We consider thesimplest but fundamental Legendre scheme: B = C \ { , } , (2.1) E = { ( λ, ( X : Y : Z )) ∈ B × P ( C ) : Y Z = X ( X − Z )( X − λZ ) } ,π : E ∋ ( λ, ( X : Y : Z )) λ ∈ B, E λ = π − { λ } . We set P = (0 : 0 : 1) , P = (1 : 0 : 1) , Q = (0 : 1 : 0) . We can view E as a hypersurface in B × P ( C ); the natural projection π : E → B together withthe (zero) section B × { Q } gives it the structure of an elliptic scheme over B . We call it theLegendre scheme. It admits three sections of order 2, namely B × { P } , B × { P } , which are abbreviated as P , P , and their sum P := ( P + P ) : B ∋ λ ( λ, ( λ : 0 : 1)) ∈ E \ Q, where Q denotes also the section B × { Q } . Before presenting a theorem in the particular case of the Legendre scheme, we discuss examplesof elliptic schemes and their sections. We keep the notation given above.3 .1.1 Rational sections omitting a relatively ample divisor.
We show in this sub-section that rational sections of abelian schemes can indeed omit relativelyample divisors in a non-trivial way. We first construct a non-isotrivial algebraic family of abelianvarieties over an algebraic variety. Set E λ = E λ \ { Q } = { y = x ( x − x − λ ) } ⊂ C , λ ∈ B, ˜ B = [ λ ∈ B { ( λ, ( x, y )) : ( x, y ) ∈ E λ } ⊂ B × C , A = [ ( λ, ( x,y )) ∈ ˜ B { ( λ, ( x, y ) , ( u, v )) : ( u, v ) ∈ E λ } ⊂ ˜ B × P . Now, let ̟ : A → ˜ B be the natural projection. In this example, there are three sections comingfrom P j , ≤ j ≤
3; i.e., ( λ, ( x, y )) ∈ ˜ B −→ ( λ, ( x, y ) , (0 , ∈ A , ( λ, ( x, y )) ∈ ˜ B −→ ( λ, ( x, y ) , (1 , ∈ A , ( λ, ( x, y )) ∈ ˜ B −→ ( λ, ( x, y ) , ( λ, ∈ A . These omit a relatively ample divisor D defined by D = ˜ B × { Q } ⊂ A . Other than these, we have τ : t = ( λ, ( x, y )) ∈ ˜ B −→ ( λ, ( x, y ) , ( x, y )) ∈ A ,τ ( ˜ B ) ∩ D = ∅ . Note that τ is “non-constant” (the definition is subtle). Note also that by cutting ˜ B we canproduce examples where the base is an affine curve, all its points at infinity are points of badreduction for the elliptic scheme and some rational section omits the divisor at infinity. Theseexamples include the so-called Masser’s sections, e.g. the section λ ( λ ; , , p − λ )) on a(ramified) base change of the Legendre scheme. Given an elliptic curve E λ in the Legendre family, the elliptic exponential is well defined as amap Lie( E λ ) ∼ = C → E λ . As we shall explain in section 3.1, we can identify globally the linebundle of the Lie algebras Lie( E λ ), for λ ∈ ˜ B , with the trivial bundle ˜ B × C . Hence we shallview the exponential map as a map exp λ : C −→ E λ , and set σ : t = ( λ, ( x, y )) ∈ ˜ B −→ ( λ, ( x, y ) , exp λ ( ϕ ( x ))) ∈ A t := π − { t } ⊂ A , where ϕ ( x ) is any non-constant polynomial (or even, entire function) in x ( ∈ C ). Then, σ is a transcendental holomorphic section of ̟ : A → ˜ B . In this example, we have that σ ( ˜ B ) ∩ D = ∅ .4 .1.3 Local sections. One may easily obtain at least locally a holomorphic non-rational section in
E \ Q π −→ B aboutthe boundary points of B , to say, λ = 0. For example, let φ ( λ ) be a holomorphic function in aneighborhood of 0 such that φ (0) = 0. With x = XZ , y = YZ in E \ Q, we then set x ( λ ) = φ ( λ ) λ . We have y ( λ ) = φ ( λ ) λ (cid:18) φ ( λ ) λ − (cid:19) (cid:18) φ ( λ ) λ − λ (cid:19) = φ ( λ )( φ ( λ ) − λ )( φ ( λ ) − λ ) λ . Therefore, taking δ > y ( λ ) = p φ ( λ )( φ ( λ ) − λ )( φ ( λ ) − λ ) λ , | λ | < δ. Even if φ ( λ ) is a polynomial, y ( λ ) is not rational, unless it vanishes identically. Globally, we are going to prove:
Theorem 2.2.
Let E π −→ B be as above in (2.1) . Then there is no holomorphic section B → E \ Q other than P j ( j = 1 , , . We first prove the rationality of the sections, in general even after finite base changes (exten-sions); this is the crucial part of the proof, which makes use of a deep theorem of Yamanoi, andthe result may have an interest of its own.Let φ : ˜ B → B be a finite base change (i.e., a finite proper rational holomorphic map) and let(2.3) ˜ π : ˜ E = ˜ B × B E → ˜ B be the lift of E /B . Then ˜ E / ˜ B carries the natural structure of a group scheme induced from E /B with the zero section ˜ Q induced by Q . In general, ˜ E → ˜ B may carry a non-torsion rationalsection and hence infinitely many rational sections. Theorem 2.4.
Let ˜ π : ˜ E → ˜ B be as above in (2.3) and let γ : ˜ B → ˜ E be a rational sectionof ˜ E → ˜ B . Then, a holomorphic section σ : ˜ B → ˜ E is rational if and only if the intersection σ ( ˜ B ) ∩ γ ( ˜ B ) is finite. roof. It suffices to prove the “if” part. Replacing σ by σ − γ , we may assume that γ = ˜ Q .We set f ( λ ) = λ ( λ −
1) ( λ ∈ B ). By the embedding λ ∈ B → ( λ, /f ( λ )) ∈ { ( λ, µ ) : f ( λ ) µ = 1 } ⊂ C , we identify B with the image, which is a closed affine algebraic curve in C . We consider B asa ramified cover of C via π B : B ∋ λ z = λ + 1 f ( λ ) ∈ C . We set π ˜ B = π B ◦ φ : ˜ B → C . Then, | z | = | π ˜ B ( ζ ) | is an exhaustion function on ˜ B by which wedefine the order function T † ( r ; ⋆ ) of a meromorphic functions on B , and the counting function N k ( r, • ) of a divisor truncated at level k etc. (cf. [11], [17] § B ֒ → ¯˜ B ( ⊂ P N ( C )) with some P N ( C ) and may assume that the polar divisor( φ ) ∞ of φ belongs to the linear system |O ¯˜ B (1) | . We set X = ¯˜ B × P ( C ), which is providedwith the first (resp. second) projection p : X → ¯˜ B (resp. q : X → P ( C )); now every section σ determines a holomorphic map g : ˜ B ∋ ζ ( ζ, x ( ζ )) ∈ X. Using the affine coordinate w of C ⊂ P ( C ), we regard w = x ( ζ ) as a meromorphic functionon ˜ B with poles at those ζ ∈ ˜ B such that σ ( ζ ) = ˜ Q ( ζ ). Then the section σ also provides ameromorphic function y ( ζ ) on ˜ B satisfying(2.5) ( y ( ζ )) = x ( ζ )( x ( ζ ) − x ( ζ ) − φ ( ζ )) . We define the following effective divisor on X : D = { w = 0 } + { w = 1 } + { w = ∞} + { w − φ ( ζ ) = 0 } . Note that π ˜ B ( ζ ) and p ◦ g ( ζ ) = ζ are rational functions on ˜ B . So, the order functions satisfy T g ( r ; L ) = T x ( r ) + O (log r ) , where L = p ∗ O ¯˜ B (1) ⊗ q ∗ O P ( C ) (1) and T x ( r ) = T x ( r ; O P ( C ) (1)). Then we have by Yamanoi[22] Theorem 1.2 with similar notation that T g (cid:16) r ; K X/ ¯˜ B ( D ) (cid:17) ≤ N ( r, { x ( ζ ) = 0 } ) + N ( r, { x ( ζ ) = 1 } ) + N ( r, { x ( ζ ) = ∞} )+ N ( r, { x ( ζ ) − φ ( ζ ) = 0 } ) + ǫT g ( r ; L ) + O (log r ) || , where ǫ is an arbitrary small positive number and the symbol || is used for the standard sensein Nevanlinna theory, while the implicitly mentioned exceptional set depends on ǫ .By the First Main Theorem N ( r, { x ( ζ ) = 0 } ) ≤ T x ( r ) + O (1) ,N ( r, { x ( ζ ) = 1 } ) ≤ T x ( r ) + O (1) . { w − φ ( ζ ) = 0 } is an element of | L | , N ( r, { x − λ = 0 } ) ≤ T g ( r ; L ) + O (1) = T x ( r ) + O (log r ) , and since K X/ ¯˜ B = q ∗ O P ( C ) ( − T g ( r ; K X/ ¯˜ B ( D ))) = 2 T x ( r ) + O (log r ) . By the assumption, { ζ ∈ ˜ B : σ ( ζ ) ∈ ˜ Q } = { ζ ∈ ˜ B : g ( ζ ) = ∞} is a finite set, so that N ( r, { x ( ζ ) = ∞} ) = O (log r ) . Since all zeros of x ( ζ ) , x ( ζ ) − , x ( ζ ) − φ ( ζ ) have order ≥
2, we have(2 − ǫ ) T x ( r ) ≤ (cid:0) N ( r, { x ( ζ ) = 0 } ) + N ( r, { x ( ζ ) = 1 } ) + N ( r, { x ( ζ ) − φ ( ζ ) = 0) (cid:1) + O (log r ) ||≤ T x ( r ) + O (log r ) || . Therefore, by taking ǫ < / T x ( r ) = O (log r ) || ;this implies the rationality of x ( ζ ) and hence that of y ( ζ ).The next lemma finishes the proof of Theorem 2.2. Lemma 2.7.
The Legendre scheme admits no rational sections other than P j ( j = 1 , , and Q ; i.e. the Mordell-Weil group of E → B consists of the -torsion group.Proof. The family of cubic curves E λ = π − ( λ ) , λ ∈ B ⊂ P ( C ), forms a pencil having as basepoints the three points P , P , Q ∈ P ( C ). Let σ : λ ∈ B → ( λ, ( X ( λ ) : Y ( λ : Z ( λ )) ∈ E λ ⊂ E be a rational section. Let C ⊂ P ( C ) denote the closure of the projection of its image σ ( B ) in P ( C ). Then, C intersects each curve E λ in three fixed points P , P , Q and possibly a fourthmoving point σ ( λ ) = ( X ( λ ) : Y ( λ ) : Z ( λ )). We shall prove that this point is of 2-torsion.Suppose that C does not reduce to a point (i.e. σ ( λ ) is not identically equal to Q , nor P nor P ) and let F ( X, Y, Z ) = 0 be an equation for C , where F is a homogeneous form of degree d >
0. Set f := FZ d ∈ C ( P ( C )) . Consider any value of λ ∈ B and view f as a rational function on E λ . The support of the divisor( f ) of f is then contained in { Q, P , P , σ ( λ ) } . Identifying E λ with its Jacobian Pic ( E λ ), weobtain that the sum of the elements in ( f ), in the sense of the group law on E λ , must vanish. Itfollows that σ ( λ ) belongs to the group generated by P , P , i.e. to the 2-torsion group. Remark S obtained by blowing-up P ( C ) over the base locus of thepencil of cubics E λ , λ ∈ B , i.e. over P , P , Q in the above notation. The surface S isendowed with a natural projection ̟ : S → P ( C ), which is a well-defined morphism,7hose fibers are the curves E λ for λ ∈ P ( C ). By taking for the zero-section the naturalmap inverting the ̟ on the exceptional divisor over Q one obtains a structure of ellipticsurface on S . Recall the Shioda-Tate formula (cf. Shioda [19]) for the rank r of theMordell-Weil group: r = ρ − − X λ ∈ P ( C ) ( n λ − , where ρ is the Picard number of S and for each point λ ∈ P ( C ), n λ is the number ofcomponents of the fiber ̟ − λ . In our case ρ = 4 and the only reducible fiber is the fiberof λ = ∞ , which has three components. It follows that r = 0, i.e. the Mordell-Weil groupis torsion.(ii) In the quoted paper, Shioda proves more: the Mordell-Weil group of an elliptic schemeobtained from the Legendre scheme by any unramified base change is torsion. The in-terested reader is addressed to [5] for a history, motivations and generalizations of thisresult, as well as different approaches to its proof.Related to the above problem, N. Katz asked in a conversation with U. Zannier for the caseof the hyperelliptic scheme of genus g > P B defined by(2.9) y = h ( x )( x − λ )in terms of an affine coordinate system ( x, y ) ∈ C ⊂ P ( C ), where h is a given polynomialwith complex coefficients (i.e., independent of λ ) of degree 2 g > B = C \ { h = 0 } . Note that for each λ ∈ B the equation defines a smooth affine curve with asingle point at infinity. This family is relevant to Katz’ work on monodromy.In this context we can prove: Theorem 2.10.
Let X → B be the hyperelliptic scheme defined by completing the curves ofequation (2.9) above. Let σ : λ ∈ B → ( λ, ( x ( λ ) , y ( λ ))) ∈ X be an arbitrary holomorphic section. Then, σ is a rational section such that either y ( λ ) ≡ or y ( λ ) ≡ ∞ .Proof. This is a case to which a big Picard theorem (a holomorphic extension theorem)obtained by Noguchi [14] is applicable, since X is a family of compact curves of genus ≥ x ( λ ) and y ( λ ) are rational functions.To conclude the proof, we need to prove an analogue of Lemma 2.7, namely that: Claim. The only rational points of X over C ( λ ) are those with y = 0 . The argument below is a variation on the elementary proof of “ abc ” over function fields. Wemay suppose that h is monic and let ξ , . . . , ξ d ∈ C be its distinct roots.Let ( u ( λ ) , v ( λ )) be a solution of (2.9) in rational functions, where v = 0. This last conditionimplies that u, v are in fact both non-constant.8ny pole on C of u or v must appear with even order 2 e in u and order ( d + 1) e in v for someinteger e , and hence we may write u ( λ ) = a ( λ ) q ( λ ) , v ( λ ) = b ( λ ) q d +1 ( λ )for complex polynomials a, b and q = 0, which are pairwise coprime. This yields the equation(2.11) b ( λ ) = ( a ( λ ) − λq ( λ )) d Y i =1 ( a ( λ ) − ξ i q ( λ )) . Since a, q are coprime, the d factors in the product on the right are non-zero and pairwisecoprime, whereas the gcd(( a ( λ ) − λq ( λ )) , a ( λ ) − ξ i q ( λ )) divides λ − ξ i . Hence, since the wholeproduct is a (non-zero) square, we must have(2.12) a ( λ ) − ξ i q ( λ ) = ( λ − ξ i ) µ i c i ( λ ) , i = 1 , . . . , d, for µ i ∈ { , } and suitable non-zero polynomials c i ( λ ). Note that the c i ( λ ) are pairwise coprime.Let m = deg u = max(deg a, q ) >
0. Note that each factor in the product on the rightof (2.11) (namely, each side of equation (2.12)) has degree ≤ m and at most one factor can havedegree < m .Suppose now that m is even. Then, since as remarked above all factors but at most one havedegree m , we should have µ i = 0 for at least three of the factors, corresponding say to i = α, β, γ .But this would give a rational parameterization of the elliptic curve y = ( x − ξ α )( x − ξ β )( x − ξ γ ),a contradiction.Therefore m is odd, which forces m = deg a > q , and in particular all of the said factorshave the same degree m , so µ i = 1 for all i in (2.12); note that this implies in particular that λ − ξ i divides a ( λ ) − λq ( λ ) for each i .Now, since the degree of the whole product is 2 deg b , we must have deg( a ( λ ) − tq ( λ )) even,which implies deg a = 1 + 2 deg q = 1 + 2 h , say. It also follows that deg c i = h .Let now s i := ( λ − ξ i ) (cid:16) c i ( λ ) q ( λ ) (cid:17) . We have that s i − s = ξ − ξ i is constant; hence,(2.13) s ′ i = s ′ , i = 1 , . . . , d. We compute s ′ i = c i q (( c i + 2( λ − ξ i ) c ′ i ) q − λ − ξ i ) c i q ′ ) = c i q φ i , say, where φ i are non-zero polynomials of degree ≤ h (in fact = 2 h , as may be checked). From(2.13) we find c i φ i = c φ . But the c i are pairwise coprime; hence Q di =2 c i divides φ . But φ is not zero, since s is not constant, whence ( d − h ≤ deg φ ≤ h , which implies h = 0. Butthen q is constant and deg a = 1, giving a contradiction with the fact that all λ − ξ i divide a ( λ ) − tq ( λ ). This concludes the proof. 9 Transcendence of sections and the logarithms
Let B be a smooth affine algebraic curve over C , and π : E → B be an elliptic scheme. By thiswe mean that each fiber π − { t } = E t with t ∈ B is a smooth elliptic curve; in other words, thebad reduction can arise only at the points at infinity of a completion of B .Every elliptic curve E t , for t ∈ B , has a Lie algebra Lie( E t ), which is a one-dimensionalvector space. The union of these lines constitutes a line bundle Lie( E ) → B over B , which isholomorphically trivial by the Oka-Principle ( B is a one-dimensional Stein manifold; cf., e.g.,[15] Theorem 5.5.3). The exponential map Lie( E t ) → E t has a kernel Λ t , which is a discretegroup of rank two. These groups together define a local system over B , i.e. a sheaf in abeliangroups Λ, which is locally isomorphic to the constant sheaf associated to the group Z .Recalling that E t too has the structure of an abelian group, so that to the family E → B onecan associated the group-sheaf of its holomorphic sections, we have the short exact sequence(3.1) 0 → Λ → Lie( E ) → E → . From another perspective, we can view Λ as a Riemann surface covering B , Lie( E ) as the totalspace of a line-bundle, i.e. an algebraic surface fibered over B , and E as an (open set of an)elliptic surface. Taking the long sequence in cohomology from (3.1), we obtain(3.2) 0 → Γ( B, Λ) → Γ( B, Lie( E )) → Γ( B, E ) → H ( B, Λ) → H ( B, Lie( E )) = 0 . The last zero is due again to the fact that B is a one-dimensional Stein manifold. Now, if theelliptic scheme E → B is not isotrivial, then no non-zero period can be continuously defined in B ; hence the term Γ( B, Λ) also vanishes. We finally get(3.3) Γ( B, E )exp(Γ( B, Lie( E ))) ∼ = H ( B, Λ) . The group Γ( B, E ) consists of the group of holomorphic sections: it properly contains theMordell-Weil group, formed by the rational sections. The latter is a discrete group, since itinjects into the discrete group H ( B, Λ) via the above projection Γ( B, E ) → H ( B, Λ). In otherwords, no non-zero rational section of an elliptic scheme can admit a logarithm , i.e. a lifting toLie( E ).We shall see that a holomorphic section of an abelian scheme A → B in general (and asemi-abelian scheme with an additional condition) admitting a well-defined logarithm is tran-scendental or a constant section in its C ( B ) / C -trace (see Theorems 3.13, 3.15). N.B.
From the above discussion, it follows that the group of holomorphic sections of anelliptic scheme is an extension of a finitely generated group by an infinite dimensional vectorspace. 10 .2 Transcendency of sections
Let A be a semi-abelian variety over C of dimension n ; it is the middle term in the exactsequence:(3.4) 0 → G lm → A → A → , where A is an abelian variety. Let Lie( A ) → A be the Lie algebra of A , endowed with itsexponential map; analytically,(3.5) Lie( A ) ∼ = C n → A = C n / Γfor a discrete subgroup Γ (semi-lattice) of C n .Let B be a smooth affine algebraic curve. We consider the relative setting of (3.4) over B :(3.6) 0 → G lmB → A φ −→ A → ց ↓ π ւ π B Here we assume that π : A → B is smooth without degeneration and also dπ is everywherenon-zero; in this case , we say that π : A → B is smooth . After deleting possibly a finite numberof points of B , we may reduce the initial case to a smooth one.As in (3.5) we have the relative Lie algebra over B and the corresponding semi-abelian expo-nential(3.7) ̟ : Lie( A ) −→ A . For a point t ∈ B we have A t = π − { t } ∼ = C n / Γ t , where Γ t is a semi-lattice. Since, asabove, the vector bundle Lie( A ) → B is analytically trivial by Grauert’s Oka-Principle ( B is aone-dimensional Stein manifold; cf., e.g., [9] Theorem 5.3.1), we can write(3.8) ̟ : Lie( A ) ∼ = B × C n ∋ ( t, x ) [( t, x )] ∈ C n / Γ t = A t ⊂ A . Let σ : B → A be a holomorphic section of π : A → B . If there is a lifting ˜ σ : B → Lie( A )in (3.7) with ̟ ◦ ˜ σ ( t ) = σ ( t ) ( t ∈ B ), we call ˜ σ a logarithm of σ (over B ).As for the case of elliptic schemes, already analyzed, logarithms do not always exist (cf. § t define a local system over B , and the existence of a logarithm for a section σ is obstructed by a cohomology class in the corresponding first cohomology group of this localsystem (cf. (3.3)).With reference to (3.6) we denote by G the C ( B ) / C -trace of A with the quotient morphism q : A → A / G . We have(3.9) 0 → G lmB → G := Ker φ ◦ q → A φ −→ A q −→ A / G → , and hence the exact sequence 0 → G lmB → G → G → . G gives rise to a semi-abelian scheme over B .Note that G is defined over C and G is isomorphic, as a scheme over B , to a product B × G for an abelian variety G ; G → B is the “constant part” of the abelian scheme A → B . Wesay that a holomorphic section σ : B → A is G -valued constant if φ ◦ σ ( B ) ⊂ G ∼ = B × G and φ ◦ σ ( t ) = ( t, x ) with an element x ∈ G .We consider G lmB in (3.6). Since the only complex affine algebraic model of G m is C ∗ , aftera finite base change we have G lmB ∼ = B × ( C ∗ ) l , (3.10) 0 → B × ( C ∗ ) l → G −→ B × G → B ) . We keep this reduction and the notation henceforth.Taking a smooth equivariant toroidal compactification T of ( C ∗ ) l , we have a fiber bundle(3.11) ¯ A −→ T A ( → B ) . We then have the space Ω ( ¯ A , log ∂ A ) of logarithmic 1-forms with ∂ A = ¯ A \ A and T ( ¯ A , log ∂ A )of logarithmic vector fields along the divisor ∂ A .We consider the transcendency problem of a holomorphic section of A → B with a logarithm.If A ∼ = B × A (trivial family), then any constant section of B × A → B is rational and hasa logarithm; this may happen in a subfamily of S → B of A → B , even if A → B is noniso-trivial.It is also to be noticed that a holomorphic section defined in a neighborhood of a point of¯ B \ B may locally have a non-constant logarithm there. But, globally we have: Lemma 3.12.
Let A be an abelian variety with an exponential map, exp : C n → A . Let g : ∆ ∗ = { < | z | < } → A be a holomorphic map with a logarithm f : ∆ ∗ → C n such that g ( z ) = exp f ( z ) . If g ( z ) is holomorphically extendable at as a map into A , then so is f ( z ) asa vector-valued holomorphic function.In particular, if g : B → A is a rational map with a logarithm, then g is constant.Proof. Assume that g : ∆ ∗ → A is holomorphically extendable at 0. Then f : ∆ ∗ → C n is reduced to be bounded in a small punctured neighborhood of 0, and so Riemann’s extensionimplies that f is holomorphically extendable at the puncture 0.Let f be a logarithm of the rational section g : B → A and let ¯ B be a smooth compactificationof B . Since g extends to a holomorphic map ¯ B → A , f extends holomorphically over ¯ B as avector-valued holomorphic function. Hence, f is constant and so is g .In view of Mordell-Weil over function fields (Lang-N´eron) we have Theorem 3.13.
Let A → B be an abelian scheme and let G be a C ( B ) / C -trace of A . Let σ : B → A be a holomorphic section with a logarithm. Then σ is either G -valued constant ortranscendental. roof. Let q : A → A ( C ( B )) / G be the quotient map. Then q ◦ σ is a rational section of A / G over B with a logarithm. By Lang-N´eron, A ( C ( B )) / G is finitely generated and hencediscrete; in particular, no non-zero section of A ( C ( B )) / G is infinitely divisible. Now, if arational section ρ : B → A ( C ( B )) / G admits a logarithm, this section is infinitely divisiblein the holomorphic sense, i.e. for every integer n there exists a holomorphic section ρ n with n · ρ n = ρ . This last equation is an algebraic one, so every solution is algebraic; since ρ n iswell-defined on the whole of B , being algebraic it must be rational. It follows that ρ is infinitelydivisible in the Mordell-Weil group, and hence it is the 0-section.Applying this fact to ρ = q ◦ σ we obtain that q ◦ σ = 0 so σ : B → G ∼ = B × G . We write σ ( t ) = ( t, exp f ( t )) , where exp : Lie( G ) ∼ = C n → G is an exponential map and f : B → C n is a vector-valuedholomorphic function. By Lemma 3.12, f ( t ) ≡ a ∈ C n and σ ( t ) = ( t, x ) with x = exp a .To generalize the above results to semi-abelian varieties we need: Lemma 3.14.
Let g ( z ) be a holomorphic function on a punctured disk ∆ ∗ = { z ∈ C : 0 < | z | < } . If g ( z ) is not extendable at z = 0 as a holomorphic function, then e g ( z ) has an essential(isolated) singularity at .Remark. In function theory, “ e transcendental = algebraic” does not happen, while in numbers, e πi = − Proof.
We distinguish two cases, according to the type of singularity of g at 0:(i) g has a pole at 0. Then in every punctured neighborhood of 0, the real part ℜ g ( z ) of g ( z )takes arbitrarily large positive numbers and arbitrarily small negative numbers, so the function e g ( z ) tends to infinity on a sequence converging to 0 and it also tends to 0 on another suchsequence. This can happen only if e g ( z ) has an essential singularity at 0.(ii) g has an essential singularity at 0. Then the image by g of any punctured neighborhood of0 is dense, so again g tends to two different values on sequences converging to 0 (say it tends to0 and to 1) so e g ( z ) has two limits on different sequences. Thus, e g ( z ) has an essential (isolated)singularity at 0. Theorem 3.15.
Let π : A → B be a smooth semi-abelian scheme and let G be as in (3.9) .Assume that G ∼ = B × G with a semi-abelian variety G over C . If a holomorphic section σ : B → A has a logarithm,then σ is either transcendental or G -valued constant, i.e., σ ( t ) = ( t, x ) with an element x ∈ G through G ∼ = B × G .Proof. By Theorem 3.13 and Lemmata 3.12, 3.14.
We would like to transpose the Nevanlinna theory for holomorphic curves into semi-abelianvarieties (cf. [13], [17] Chap. 6) to a relative setting.13 .1 Jet space of holomorphic local sections
Let ∆ be the unit disk of the complex plane C with center 0 ∈ C . Let t ∈ ∆ be the naturalcomplex coordinate. We consider a smooth family π : A → ∆ of semi-abelian varieties ofdimension n with its zero section: ∆ ∋ t t ∈ A t = π − { t } , t ∈ ∆.Let ¯ A be a relative toroidal compactification of A (cf. (3.11)). Let J k ( ¯ A , log ∂ A ) denotethe k th logarithmic jet space over ¯ A along ∂ A , and let π k : J k ( ¯ A , log ∂ A ) → ¯ A be the natural projection.Let J k ( A / ∆)( ⊂ J k ( A )) denote the space of k -th jets of holomorphic local sections f of π : A → ∆ such that π ◦ f ( t ) = t .In (3.8) we write x = ( x , . . . , x n ) with the natural complex coordinates. Then, η j := dx j (1 ≤ j ≤ n ) give rise to elements of the space Ω ( ¯ A , log ∂ A ) of logarithmic 1-forms and(4.1) { dt, η , . . . , η n } forms the frame over ¯ A .For a jet element j k ( f ) ∈ J k ( A / ∆) f ( t ) ( t ∈ ∆) we set(4.2) f ∗ η j = f ′ j dt, ≤ j ≤ n. Then we have j ( f )( t ) = ( f ( t ); 1 , f ′ ( t ) , . . . , f ′ n ( t )) , (4.3) j ( f )( t ) = ( j ( f )( t ); 0 , f ′′ ( t ) , . . . , f ′′ n ( t )) , ... j k ( f )( t ) = ( j k − ( f )( t ); 0 , f ( k )1 ( t ) , . . . , f ( k ) n ( t )) . In this way we have the trivializations(4.4) J k ( A / ∆) ∼ = A × { (1 , , . . . , } × C nk ∼ = A × C nk . Let I k : J k ( A / ∆) → C nk be the jet projection, which extends holomorphically to the relative logarithmic jet space J k ( ¯ A / ∆ , log ∂ A ). We set the relative jet projection with respect to the frame (4.1)(4.5) ˜ I k = ( π k , I k ) : J k ( ¯ A / ∆ , log ∂ A ) → ∆ × C nk . Note that ˜ I k is proper. 14 .2 A relative exponential map We keep the notation above. We consider the abelian integration(4.6) x t ∈ A t → (cid:18)Z x t t η , . . . , Z x t t η n (cid:19) ∈ C n . We denote by Γ t the semi-lattice generated by the periods of (4.6). We then have a relativeexponential mapexp ∆ : Lie( A ) ∼ = ∆ × C n ∋ ( t, x ) ( t, [ x ]) ∈ { t } × C n / Γ t = A t ⊂ A ,π ◦ exp ∆ ( t, x ) = t. For an element w ∈ C n we have an action “ w · ” associated with (4.1) by(4.7) w · : ( t, [ x ]) ∈ { t } × C n / Γ t = A t → ( t, [ x + w ]) ∈ { t } × C n / Γ t = A t . The notation is kept. We follow [13].Let ∆ ∗ = ∆ \ { } be the punctured disk. We consider a holomorphic section f : ∆ ∗ −→ A , π ◦ f ( t ) = t, t ∈ ∆ ∗ . Let now ω be a real (1 , A and let r > r > r wedefine the order function of f with respect to ω by(4.8) T f ( r ; ω ) = Z rr dss Z { /s< | t | < /r } f ∗ ω, r > r . Let ω be a hermitian metric form on ¯ A . Then there is a constant C > C − T f ( r ; ω ) ≤ T f ( r ; ω ) ≤ CT f ( r ; ω ) , r > r . The above C depends on the choice of r > Proposition 4.10.
Let ω be a hermitian metric form on ¯ A . A holomorphic section f : ∆ ∗ → A is holomorphically extendable at as a map into ¯ A if and only if (4.11) lim r →∞ T f ( r ; ω )log r < ∞ . Let D be a relative effective divisor on A / ∆ which is extendable to a divisor ¯ D on ¯ A / ∆. Wecall such D a relative algebraic divisor on A / ∆. Let L = L ( ¯ D ) denote the line bundle over¯ A / ∆ determined by ¯ D with a section σ such that the divisor ( σ ) defined by σ satisfies ( σ ) = ¯ D .Let k · k be a hermitian metric in L , and let ω L be the Chern curvature form of the hermitian15etric. For a holomorphic section f : ∆ ∗ → A with f (∆ ∗ ) Supp D , we define the countingfunctions of the pull-back divisor f ∗ D by n ( s, f ∗ D ) = X /s< | ζ | < /r deg ζ f ∗ D , s > r ,N ( r, f ∗ D ) = Z rr n ( s, f ∗ D ) s ds, r > r . Replacing deg ζ f ∗ D above by min { deg ζ f ∗ D , k } ( k ∈ N ), we have the corresponding (truncated)counting functions denoted by n k ( s, f ∗ D ) , N k ( r, f ∗ D ) . We set Γ( r ) = { t = 1 / ( re iθ ) : 0 ≤ θ ≤ π } parameterized by θ , and the proximity function m f ( r, D ) = Z Γ( r ) log 1 k σ ◦ f k dθ π . We have:
Theorem 4.12 (First Main Theorem (cf. [13] (1.4))) . Let the notation be as above. Then T f ( r ; ω L ) = N ( r, f ∗ D ) + m f ( r, D ) − m f ( r , D ) + (log r ) Z Γ( r ) d c log k σ ◦ f k (4.13) = N ( r, f ∗ D ) + m f ( r, D ) + O (log r ) , r > r , where d c = ( i/ π )( ¯ ∂ − ∂ ) .Remark f (Γ( r )) ∩ Supp D = ∅ , the last term of (4.13), R Γ( r ) d c log k σ ◦ f k should be taken as a principal-value integration. We may also take r > f (Γ( r )) ∩ Supp D = ∅ . Then the integrand is smooth on Γ( r ).(ii) (Cf. (4.9)) If L = L ( ¯ D ) is relatively big on ¯ A / ∆, then there is a positive constant C such that C − T f ( r ; ω L ) + O (log r ) < T f ( r ; ω ) < CT f ( r ; ω L ) + O (log r ) . Let f : ∆ ∗ → A be a holomorphic section and the frame (4.1) be given. Recall we have definedthe first derivatives f ′ j ( t ) of f by (4.2), and hence the k -th ( k ∈ N , positive integers) derivatives f ( k ) j ( t ), which are holomorphic functions on ∆ ∗ . We then have the “lemma on logarithmicderivatives”: Lemma 4.15 ([12], [13]) . Let the notation be as above. Then we have Z Γ( r ) log + (cid:12)(cid:12)(cid:12) f ( k ) j (cid:12)(cid:12)(cid:12) dθ π = S f ( r ; ω ) , r > r , k ∈ N , where S f ( r ; ω ) = O (log + T f ( r ; ω )) + O (log r ) || , called a small term in Nevanlinna theory. .4 Relative second main theorem Let A → ∆ and ¯ A → ∆ be a smooth family of semi-abelian varieties and its relative toroidalcompactification as above. We consider a relative algebraic reduced divisor D on A / ∆; i.e.,there is a relative reduced divisor ¯ D on ¯ A such that D = ¯ D ∩ A . We identify it with its support.For a given holomorphic section f : ∆ ∗ → A we deal with a problem to obtain a “Second MainTheorem” with respect to D .We refer to the relative Zariski topology on ¯ A in the sense that closed subsets Z ( ⊂ ¯ A ) areanalytic subsets of ¯ A ; hence, the fibers Z t ( t ∈ ∆) are algebraic subsets of ¯ A t . Then, it inducesthe relative Zariski topology on the open subset A of ¯ A . It is noticed that a relative Zariski-closed subset Y of A is not merely an analytic subset of A but the fibers Y t ( t ∈ ∆) are algebraicsubsets of A t . We define similarly the relative Zariski topology on the jet space J k ( ¯ A / ∆ , log ∂ A )and its open subset J k ( A / ∆): Here one notes that the restrictions J k ( ¯ A / ∆ , log ∂ A ) | ¯ A t ( t ∈ ∆)are affine fiber bundles over ¯ A t with some C r as fibers, which is compactified by P r ( C ).Let f : ∆ ∗ → A be a holomorphic section. We denote by X k ( f ) ( ⊂ J k ( ¯ A / ∆ , log ∂ A ) , k ≥
0) the relative Zariski-closure of the image of the k -th jet lift J k ( f ) : ∆ ∗ → J k ( A / ∆) ֒ → J k ( ¯ A / ∆ , log ∂ A ) of f . For k = 0 we set(4.16) X ( f ) = X ( f ) ⊂ ¯ A . Definition f : ∆ ∗ → A is non-degenerate if X ( f ) = ¯ A ; otherwise, f is degenerate .If f : ∆ ∗ → A is non-degenerate, f is not extendable at 0 as a map into ¯ A .To prepare some technical lemmata, we need to fix the frame (4.1). Lemma 4.18.
Assume that f : ∆ ∗ → A is non-degenerate. Then for all sufficiently large k ∈ N ˜ I k ( X k ( f )) ∩ ˜ I k ( D / ∆) = ˜ I k ( X k ( f )) . Proof.
Cf. [13], [17] Lemma 6.3.2.As in the proof of [17] p. 227, we have:
Lemma 4.19.
Assume that f : ∆ ∗ → A is non-degenerate. Then we have m f ( r ; D ) = S f ( r ; ω ) , r > r . Combining this with Theorem 4.12 we obtain
Theorem 4.20 (Second Main Theorem) . Let D be a relatively algebraic big reduced divisor on A / ∆ and let f : ∆ ∗ → A be a non-degenerate holomorphic section.Then there is a relative compactification ¯ A / ∆ of A / ∆ together with L = L ( ¯ D ) , independentof f , and a natural number k ∈ N such that T f ( r ; ω L ) = N k ( r, f ∗ D ) + S f ( r ; ω L ) , r > r . roof. Cf. [17] § Remark T f ( r ; ω L ) depends on the choice of therelative compactification ¯ A / ∆.(ii) The requirement on ¯ A above is such that for a general point t ∈ ∆, A t acts on ¯ A t and ¯ D t contains no fixed point ([17], Corollary 5.6.7); this is an open property in the parameter t . Theorem 4.22 (Big Picard) . Let f : ∆ ∗ → A and D be as in Theorem 4.20. Then, f intersects D infinitely many times in an arbitrarily small (punctured) neighborhood about .Proof. Suppose that f (∆ ∗ ) ∩ D is finite. Since D is a relatively algebraic big reduced divisoron A / ∆, it follows from Theorem 4.20 that T f ( r ; ω ) = S f ( r ; ω ) , r > r . Then, it is immediate that T f ( r, ω ) = O (log r ) || . By Proposition 4.10 we conclude that f isextendable at 0 as a map into ¯ A , and hence f cannot be non-degenerate. Remark D so that the points of the intersections on D is Zariski-dense in D . Together with the smoothness assumption of A → ∆ at 0, it isinteresting to ask: Question 4.24. Is f (∆ ∗ ) ∩ D of Theorem 4.22 relative Zariski-dense in D ? Question 4.25.
Is it possible to allow the above A → ∆ of Theorems 4.20 and 4.22 todegenerate at 0 ∈ ∆? (Cf. Problem 7.2.) Let ¯ B be a smooth projective algebraic curve. Let π : A → ¯ B and π : ¯ A → ¯ B be a smoothfamily of n -dimensional semi-abelian varieties over ¯ B and its relative toroidal compactificationas in § f : B −→ A , where B is an affine open set of ¯ B . Set S = ¯ B \ B . If f is transcendental, then there is a pointof S at which f is not extendable as a map into ¯ A .To deal with the local Nevanlinna theory for such f as above obtained in the previous sub-sections, we introduce a rational function τ on ¯ B such that τ is holomorphic on a Zariski-openneighborhood B ′ of S in ¯ B and has a zero of order 1 at every point of S . Let B ′ ( r ′ ) ( r ′ > { x ∈ B ′ : | τ ( x ) | < /r ′ } containing the points of S . Taking and fixing a large r ′ , we have dτ ( x ) = 0 , x ∈ B ′ ( r ′ ) . B the Zariski-open set B ∩ B ′ , and set B ( r ) = { x ∈ B : 1 /r < | τ ( x ) | < /r } , where r > r ′ is any fixed number.We use τ for the parameter t of (4.1). For η j (1 ≤ j ≤ n ) of (4.1) we define also them byrational differentials on ¯ A with logarithmic poles on ∂ A = ¯ A \ A , so that { dτ, η , . . . , η n } forms a holomorphic frame over ¯ A | B ′ ( r ′ ) , where r ′ is replaced by a larger one if necessary.We then define the order functions, counting functions and proximity functions as in theformer subsections; e.g., the order function T f ( r ; ω ) of f with respect to a hermitian metricform ω on ¯ A is defined by T f ( r ; ω ) = Z rr dss Z B ( s ) f ∗ ω, r > r . Replacing the relative Zariski topology by the (ordinary) Zariski topology, we apply thearguments given in the previous subsections for f : B → A . We then obtain in particular the Second Main Theorem 4.20 for f : B → A . We deal with the semi-abelian case. Let ¯ B be a smooth projective algebraic curve. Here weneed a rather strong assumption such that A → ¯ B is smooth. Theorem 5.1.
Let A → ¯ B be a smooth family of semi-abelian varieties over ¯ B , and let D bea relatively big reduced divisor on a relative compactification ¯ A / ¯ B . Let B be any non-emptyZariski-open subset of ¯ B , and let f : B → A be a holomorphic section. If f ( B ′ ) ∩ D is finite,then f is degenerate.Proof. By the Second Main Theorem 4.20 and § Remark A → B may degenerate at a point where f is alreadydefined holomorphically as a map into ¯ A . What is essentially excluded in the theorem is thecase when the family A → B degenerates at a possibly (essentially) singular point of f . (a) Example. We first give the example mentioned after Problem 1.2. Let C be a smoothprojective curve of genus g ≥
2, and let A be its Jacobian variety with an embedding η : C → A .19et exp A : C → A be an exponential map. Take an affine line L ⊂ C such that the imageexp( L ) is Zariski-dense in A . Let C ′ ⊂ C be an affine open subset with a non-constant rationalholomorphic map ϕ : C ′ → C ∼ = L . Let us set A := C ′ × A with the first projection A → C ′ to the base space C ′ . With ψ ( x ) = exp A ( ϕ ( x )) we have a section(6.1) σ : x ∈ C ′ −→ ( x, η ( x ) , ψ ( x )) ∈ A . By definition (cf. [16]), a non-constant holomorphic map f from C ′ into A is strictly transcen-dental if for every abelian subvariety A of A , the composed map q A ◦ f : C ′ → A/A with thequotient map q A : A → A/A is either transcendental or constant .Then, ψ : C ′ → A is strictly transcendental due to [16]. Let G ( η ) ⊂ C × A denote the graphof η and set G ′ ( η ) = G ( η ) ∩ ( C ′ × A ). Then, the Zariski-closure X ( σ ) (cf. (4.16)) of the imageof σ is given by X ( σ ) = G ′ ( η ) × A. Thus, X ( σ ) contains a translate of a subgroup, { w } × A with w ∈ G ′ ( η ), however it is not itselfa translate of a subgroup. (b) We use the notation defined in § π : A → B be an algebraic family of semi-abelianvarieties over an affine curve B and let ¯ A → B be the relative compactification over B . For aZariski-closed subset X ⊂ A we consider the the set-theoretic stabilizer group of X , which is(6.2) [ t ∈ B { a ∈ A t : a + X t = X t } ( X t = X ∩ A t ) . It is a closed subset of A ; each fiber over any point t ∈ B is an algebraic subgroup of the fiber A t ; its dimension is upper semi-continuous, i.e. either constant or admitting jumps at a finiteset of points. Removing these points, we let St ( X ) be the Zariski-closure of the set-theoreticstabilizer outside this exceptional set. It is the total space of a group-scheme over B , so that wehave the quotients,(6.3) X/St ( X ) ⊂ A /St ( X ) . We set the relative dimension of St ( X ) bydim B St ( X ) = dim St ( X ) − dim B = dim St ( X ) − . For a holomorphic section f : B → A we set (cf. (4.16) for X ( f ))(6.4) St ( X ( f )) = St ( X ( f ) ∩ A ) , dim B St ( X ( f )) = dim B St ( X ( f ) ∩ A ) . Theorem 6.5.
Let π : A → ¯ B be a smooth family of semi-abelian varieties over ¯ B . Let f : B → A be a transcendental holomorphic section over an affine open subset B ⊂ ¯ B . Then dim B St ( X ( f )) > .Proof. We take the restriction of the jet projection ˜ I k defined by (4.5) to X k ( f )( ⊂ J k ( ¯ A /B, log ∂ A )) ˜ I k | X k ( f ) : X k ( f ) −→ ∆ × C nk . I k is proper, the image Y k = ˜ I k ( X k ( f )) is an analytic subset of ∆ × C nk which is algebraicin C nk -factor. We fix arbitrarily a reference point t ∈ ∆ ∗ such that j k ( f )( t ) is a non-singularpoint of X k ( f ), and consider the differential between the holomorphic tangent spaces d ˜ I k | X k ( f ) : T ( X k ( f )) j k ( f )( t ) → T (∆ × C nk ) ˜ I k ( j k ( f )( t )) ∼ = C × C nk . By (4.4) we have T ( X k ( f )) j k ( f )( t ) ⊂ T (∆) t × ( T ( A t ) × T ( C nk )) j k ( f )( t ) ∼ = C × T ( A t ) × C nk . With this product structure the kernel Ker d ˜ I k | X k ( f ) satisfies W k := Ker d ˜ I k | X k ( f ) ⊂ T ( A t ) f ( t ) ∼ = C n , where the other tangent components are 0. Since W k ⊃ W k +1 , they stabilize at some W k = W k +1 = · · · . If W k = { } , we then deduce in the same way as in [17] § T f ( r ; ω ) = S f ( r ; ω ) , so that f is extendable over ¯ B as a map into ¯ A ; i.e., f is not transcendental. Hence W k = { } . Since a tangent vector v ∈ W k is tangent to X ( f ) at f ( t ) with infinite order, we seethat v ( ∈ Lie( A /B )) is tangent to X ( f ) ∩ A at all points of X ( f ) ∩ A (cf. [17] § B St ( X ( f )) > Corollary 6.6.
Let f : B → A be as in Theorem 5.1 and let q : A → A /St ( X ( f )) be thequotient map. Then the composite g := q ◦ f : B → A /St ( X ( f )) is rational. Now, let π : A → ¯ B be an abelian scheme defined over a smooth projective algebraic curve¯ B . Let B ⊂ ¯ B be an affine open subset. We consider a transcendental holomorphic section f : B → A . Let X ( f ) be the Zariski closure of f ( B ) in A and let St ( X ( f )) be the stabilizerof X ( f ) defined by (6.4). Let G be an abelian subscheme of A over ¯ B and let q G : A → A / G be the quotient morphism. We consider the induced composite map q G ◦ f : B → A / G . Then,as in Theorem 3.13 it makes sense to say whether q G ◦ f is a C ( B ) / C -trace (of A / G ) valuedconstant section or not. Following to the notion of “strict transcendency” defined in [16], wegive: Definition f : B → A is strictlytranscendental if for every abelian subscheme G of A the induced holomorphic section q G ◦ f : B → A / G is either transcendental or a C ( B ) / C -trace valued constant section. Theorem 6.8.
Assume that A → ¯ B is smooth. Let f : B → A be a strictly transcenden-tal holomorphic section. Then, after a finite base change, X ( f ) is a translate of an abeliansubscheme G of A by a rational section σ : B → X ( f ) such that q G ◦ f : B → A / G is a C ( B ) / C -trace valued constant section. roof. We set X = X ( f ) and G = St ( X ). By the assumption and Corollary 6.6(6.9) q G ◦ f : B −→ A / G is a C ( B ) /B -trace valued constant section. Thus for general t ∈ B except for finitely many, thefibers satisfy X t = f ( t ) + G t . We take a finite base change ˜ B → B so that there is a rational section ˜ σ : ˜ B → ˜ X with the lift˜ X of X . We denote by ˜ A (resp. e G ) the lift of A (resp. G ). Thus, we have˜ X = ˜ σ + e G . It follows that q ˜ G ◦ ˜ σ : ˜ B → ˜ A / e G is the lift of the section (6.9) and hence a C ( ˜ B ) / C -tracevalued constant section.We would like to stress that we are assuming the smoothness of the family π : A → ¯ B over¯ B . It is interesting to ask: Question 6.10. (i) Is it sufficient in Theorem 6.8 to assume the smoothness condition for A only over the affine open B ?(ii) How do we deal with the semi-abelian case? (a) As mentioned in §
1, the Zariski-closure of the image of an entire curve f : C → A in anabelian variety A is a translate of an algebraic subgroup. It is still unknown what should be thetopological closure in the sense of the complex topology, of the image f ( C ). One might ask thefollowing: Question 7.1.
Let f : C → A be an entire curve in a complex abelian variety. Is it true thatthe topological closure of f ( C ) is a translate of a real Lie subgroup of A ( C )? (b) Related to Question 4.25, one may raise:
Problem 7.2.
Extend the results of § π : A → ¯ B degenerates at finitelymany points of ¯ B . Let B be an affine algebraic smooth curve. For the heights of an abelian or elliptic scheme E → B , we have to consider three different settings:(i) The height of a family E → B .(ii) The height of a fiber E b for each b ∈ B .(iii) The height of a section σ : B → E of E → B .22 i) Height of elliptic schemes. Let
E → B be an elliptic scheme with a smooth projectivealgebraic compactification ¯ B of B . The moduli space of elliptic curves is an orbifold complexspace H / Γ(1) with λ : H = { z ∈ C : ℑ z > } → H / Γ(1). We have a holomorphic map φ E : B → H / Γ(1), which has a local lift ˜ φ E U : U → H from a neighborhood U of every point of B . Since the Poincar´e metric ω P (or the Bergman metric in the case of Siegel space) is invariantby Γ(1), we have the pull-back φ ∗E ω P = ˜ φ ∗E U ω P . We define the height of E → B by(8.1) h ( E /B ) = Z B φ ∗E ω P ∈ R ≥ . Note that the integral is finite, because φ ∗E ω P has at most the Poincar´e growth at every pointof ¯ B \ B . Also, we have h ( E /B ) = 0 if and only if the scheme E /B is isotrivial. After thenormalization of the curvature of ω P we have (by Schwarz or by the comparison of Kobayashihyperbolic metrices)(8.2) h ( E /B ) ≤ g ( B ) − B \ B ) = χ ( B ) , where B \ B ) denotes the cardinality, so χ ( B ) is the Euler characteristic of the affine curve B . The same holds for a family of principally polarized abelian varieties, up to replacing theupper half plane by the Siegel space. (ii) Height of E b . Consider a non-isotrivial elliptic scheme
E → B ; identifying the Liealgebra Lie( E ) with B × C , we can identify the period lattice Λ b over each point b with a latticein C , and calculate its volume V ( b ). Then we can define the height of E b to be h ( E b ) = 1 V ( b ) . This function on B depends on the trivialization of the line bundle Lie( E ). A canonical choicecan be done by considering first the Legendre scheme, whose base is B = P \ { , , ∞} .There one disposes of a canonical choice for the fundamental periods ρ , ρ , given by Gauss’hypergeometric series defined in the region of B where | λ | < | − λ | < ρ ( λ ) = π X n ≥ (cid:18) / n (cid:19) λ n , ρ ( λ ) = iπ X n ≥ (cid:18) / n (cid:19) (1 − λ ) n . Then the volume of the lattice equals V ( λ ) = i (cid:18) ρ ρ ρ ρ (cid:19) = i | ρ | τ − τ ) = ℑ ( τ ) · | ρ | . Here τ = τ ( λ ) = ρ /ρ . Recall that the λ function is expressed in terms of τ as λ = 16 q − q + . . . , for q = e πiτ . Now, given any non-isotrivial elliptic scheme, we can suppose, after suitable base extension,that the 2-torsion is rational. Letting B be the new base, the elliptic scheme can be viewed asa pull-back of the Legendre scheme via a finite map B → B . The height of the fiber is thencalculated by looking at the fibers over B . 23 iii) Height of a section σ : B → E . We follow [7], where we gave a closed integral formulafor the height of sections. Given an elliptic scheme π : E → B , and a simply connected open set U ⊂ B , we consider a basis ρ , ρ of the period lattice: we can view them as sections of the linebundle Lie( E ) over U ; each point of E over some point of U admits a logarithm, defined up tointegral multiples of ρ , ρ . This logarithm can be expressed as a linear combination with realcoefficients of ρ , ρ . We then obtain real valued functions β , β on π − ( U ), which are locallywell-defined, and globally defined only up to addition of integer numbers. The differentials dβ , dβ are well defined on the whole of π − ( U ). Finally, the exterior product ω := dβ ∧ dβ is well-defined on the whole of E . Note that its integral on each fiber equals 1. Now, for arational section σ : B → E we define its height to be(8.4) ˆ h ( σ ) = Z B σ ∗ ω. Clearly, it vanishes if and only if a non-trivial linear combination of β , β is constant. By atheorem of Manin (see [6] for a modern presentation and [1] for generalizations), this happensif and only if σ is torsion, or the scheme is isotrivial and the section is constant.In [7], § §
4, the following properties have been established(i) The differential 2-form ω is the only closed 2-form on the total space E such that [2] ∗ ω =4 · ω , where [2] : E → E is the multiplication-by-2 morphism.(ii) It is also the only 2-form on E whose integral on each fiber equals 1, and such that[2] ∗ ω = 4 · ω .(iii) The height ˆ h ( σ ) coincides with the normalized N´eron-tate height as defined e.g. in chap.VI of Silverman’s book [20]: ˆ h ( σ ) = lim n →∞ ( nσ · O ) n , where the numerator denotes the intersection product (on a smooth projective model of E ) between the closure of the image of the section nσ : B → E and the zero section.(iv) The height is also expressible as the limitlim n →∞ ♯ { b ∈ B | nσ ( b ) = 0 } n . (v) In the case of the Legendre scheme, the differential form ω can be expressed as ω = dd c ( ℜ ( zη λ ( z )) , where λ ∈ B is the coordinate on the base, z is the coordinate in the Lie algebra C of E λ , and the function ( λ, z ) η λ ( z ) is defined as follows: (1) for each λ ∈ B , the function z η λ ( z ) is R -linear; (2) for each λ with | λ | < , | λ − | <
1, so that ρ ( λ ) , ρ ( λ ) are welldefined by (8.3), η λ ( ρ ( λ )) and η λ ( ρ ( λ )) are the semi-periods of E λ ; (3) the holomorphicfunctions λ η λ ( ρ i ( λ )) are analytically continued along paths on B (see again chap. VIof [20]). 24henever σ : B → E is a holomorphic section in general, the integral (8.4) does not necessarilyconverge. In [5], § h ( σ ), this time in terms of the‘modular logarithm’, defined therein.In order to extend the notion of normalized (N´eron) height to transcendental sections, wefollow the same pattern as in § ξ on B such that ξ isholomorphic on B and has a pole at every point of ¯ B \ B , where ¯ B is a smooth compactification of B . Then, the modulus | ξ | : B → R is a non-negative exhaustion function such that dd c log | ξ | ≡ { x ∈ B : ξ ( x ) = 0 } . Setting B ( r ) = { x ∈ B : | ξ ( x ) | < r } for r >
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