aa r X i v : . [ m a t h . N T ] F e b A CHABAUTY-COLEMAN BOUND FOR SURFACES
JERSON CARO AND HECTOR PASTEN
Abstract.
Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit boundfor the number of rational points of a curve C of genus g ≥ F , withJacobian of rank at most g −
1. Namely, in the case F = Q , if p > g is a prime of good reduction,then the number of rational points of C is at most the number of F p -points plus a contributioncoming from the canonical class of C . We prove a result analogous to Coleman’s bound in thecase of a hyperbolic surface X over a number field, embedded in an abelian variety A of rank atmost one, under suitable conditions on the reduction type at the auxiliary prime. This provides thefirst extension of Coleman’s explicit bound beyond the case of curves. The main innovation in ourapproach is a new method to study the intersection of a p -adic analytic subgroup with a subvarietyof A by means of overdetermined systems of differential equations in positive characteristic. Contents
1. Introduction 12. Notation 73. Curves and surfaces 74. ω -integrality 115. Differentials on abelian varieties 166. Zeros of p -adic power series 227. Exponential and logarithm 238. Analytic 1-parameter subgroups 259. The main result 3010. Applications for rational points 3611. Acknowledgments 37References 381. Introduction
The Chabauty-Coleman bound.
Let C be a smooth, geometrically irreducible, projectivecurve of genus g ≥ F , with Jacobian J . In the direction of Mordell’sconjecture, Chabauty [13] proved in 1941 that if rank J ( F ) ≤ g −
1, then the set C ( F ) of F -rationalpoints of C is finite. Let us recall the main ideas for F = Q : After embedding C into J , one has C ( Q ) ⊆ C ( Q p ) ∩ Γ where p is an auxiliary prime and Γ is the p -adic closure of J ( Q ) in J ( Q p ). Thendim Γ ≤ rank J ( Q ) < g = dim J ( Q p ) which leads to the finiteness of C ( Q p ) ∩ Γ because C ( Q p ) isnot contained in a lower dimensional p -adic analytic subgroup of J ( Q p ). Date : February 12, 2021.2010
Mathematics Subject Classification.
Primary 11G35; Secondary 11G25, 14G05, 14J20.
Key words and phrases.
Chabauty, Coleman, surfaces, rational points.J.C. was supported by ANID Doctorado Nacional 21190304. H.P. was supported by FONDECYT Regular grant1190442. oleman [15] proved in 1985 a celebrated explicit version of Chabauty’s theorem, which we recallover Q : With the same conditions, if p > g is a prime of good reduction for C , then(1.1) C ( Q ) ≤ C ′ ( F p ) + 2 g − C ′ is the reduction of C modulo p . While Chabauty’s finiteness result was superseded byFaltings’s proof of Mordell’s conjecture [21], the method of Chabauty and Coleman has led to anumber of striking developments: Explicit determination of the rational points in suitable curves(see [48] for an introduction and references), improvements on (1.1) such as [45, 48, 60, 38], progresstowards the Caporaso-Harris-Mazur conjecture [61, 39], and explicit versions of Kim’s non-abelianapproach [40] such as [2, 3, 4, 5, 6, 20].1.2. Beyond curves.
Despite all these remarkable developments over the last few decades, theproblem of proving a version of (1.1) for the rational points of a higher dimensional variety X contained in an abelian variety A has remained out of reach. Our main results provide such anextension of (1.1) when X is a hyperbolic surface contained in an abelian variety A of dimension n ≥
3, both defined over a number field F , under the assumption rank A ( F ) ≤
1. Although we workover number fields (see Sections 9 and 10), let us keep the discussion over Q in this introductionto simplify the exposition. Our results, when applicable, will imply(1.2) X ( Q ) ≤ X ′ ( F p ) + 4 p · c ( X )where c ( X ) is the first Chern number of the surface X (the self-intersection of a canonical divisor), p is a prime of good reduction satisfying some technical assumptions, and X ′ is the reduction of X modulo p . The coefficient 4 p in (1.2) is in fact a simplification of a slightly more complicatedexpression that gives a better estimate. See Theorem 1.1 and Remark 1.3.We observe that Coleman’s bound (1.1) can be written as C ( Q ) ≤ C ′ ( F p ) + c ( C )where c ( C ) = 2 g − C , i.e. the degree of a canonical divisor. Also,we recall that hyperbolic projective curves are precisely those of genus g ≥
2. We hope that theseremarks clarify the analogy between our results for surfaces and Coleman’s bound for curves.Although the shape of our bound (1.2) is analogous to (1.1), the methods of proof are quitedifferent. Let X be a subvariety of an abelian variety A over Q . Let Γ be the p -adic closure of thegroup A ( Q ) in A ( Q p ). Then X ( Q ) is contained in Γ ∩ X ( Q p ) and one tries to bound the latter.In the case of curves ( X = C embedded in A = J ) Coleman used his theory of p -adic integrationto construct p -adic analytic functions on C ( Q p ) that vanish on C ( Q p ) ∩ Γ and then he boundedthe number of zeros of the relevant one-variable p -adic power series on residue disks.On the other hand, when d = dim X >
1, the analogous approach using p -adic analytic functionson X ( Q p ) quickly encounters difficulties. Such functions are locally given by power series in d variables. To get finiteness of Γ ∩ X ( Q p ) one needs to consider at least d different p -adic analyticfunctions on X ( Q p ) whose zero sets (usually, p -adic analytic sets of dimension d −
1) meet properly,and then there is the problem of giving an upper bound for the number of common zeros. So far,this approach has not succeeded in proving a version of (1.1) other than in the case of curves.We take a different route instead. When rank A ( Q ) = 1 we can give a p -adic analytic parametriza-tion of Γ by power series in one variable. Upon composing with suitably chosen local equations forthe surface X in A and working on residue disks, the problem of bounding ∩ X ( Q p ) is reduced tobounding the number of zeros of a certain power series h ( z ) = P j c j z j ∈ Q p [[ z ]] on a disk. The keydifficulty in doing so (and in the whole approach) it so prove the existence of a small N such that | c N | ≥
1. We achieve this by developing a method based on overdetermined ω -integrality, whichin our case is essentially a study of overdetermined systems of differential equations in positivecharacteristic. See Section 1.4 for a more detailed description. rior to our work, the efforts on extending (1.1) to the higher dimensional setting have focusedon the special case when A = J is the Jacobian of a curve C and X = W d is the image in J of the d -th symmetric power of C via the addition map. Klassen [41] obtained some partial results forthe varieties W d , later improved by Siksek [59]. Although Siksek’s work does not give an explicitbound for W d ( Q ) such as (1.1), it gives a practical method that in many cases computes the setof rational points of W d . Park [55] used tropical geometry to obtain a weak analogue of (1.1) for W d (at least if rank J ( Q ) ≤ W , butthese are also conditional due to the same issue discussed in [31].1.3. Results.
Our main result is Theorem 9.1, which proves an analogue of the Chabauty-Colemanbound (1.1) for surfaces over finite extensions of Q p . For simplicity, let us first state here a versionjust over Q p which follows from the more general Theorem 9.1, see Remark 9.4. As usual, K X denotes a canonical divisor of a smooth projective variety X . We recall that X is said to be ofgeneral type if K X is big. For every field F , we choose an algebraic closure and denote it by F alg . Theorem 1.1 (Main result, case over Q p ) . Let p be a prime. Let X be a smooth, geometricallyirreducible, projective surface contained in an abelian variety A of dimension n ≥ , both definedover Q p and having good reduction. Let X ′ and A ′ be the corresponding reductions modulo p . Let G ≤ A ( Q p ) be a finitely generated group with rank G ≤ and let Γ be its p -adic closure in A ( Q p ) .Suppose that either of the following conditions holds: (i) n = 3 , X is of general type, X ′ contains no elliptic curves over F algp , and p > (128 / c ( X ) . (ii) A ′ is simple over F algp , p > c ( X ) + 2 , and there is an ample divisor H on A such that p > n ! · (3 deg( H .X ) + deg( H.K X )) n n n · deg( H n ) . Then X ( Q p ) ∩ Γ is finite and we have (1.3) X ( Q p ) ∩ Γ ≤ X ′ ( F p ) + p − p − · (cid:16) p + 4 p / + 3 (cid:17) · c ( X ) . Remark . By the Riemann Hypothesis for surfaces over finite fields [17] we have the estimate | X ′ ( F p ) − ( p +1) | ≤ b p / + b p + b p / where b j is the j -th Betti number of X ( C ). In particular,one can use X ′ ( F p ) ≤ p + b p / + b p + b p / + 1 in (1.3) to get a more uniform bound. Remark . When Theorem 1.1 applies, p ≥ X ( Q p ) ∩ Γ < X ′ ( F p )+4 p · c ( X ).Also, Remark 1.2 shows that X ′ ( F p ) is roughly of size p , say, for large p and fixed Betti numbersof X . In this way, X ′ ( F p ) can be seen as the main term in the upper bound (1.3). Remark . As it can be relevant in practical applications, we mention that Proposition 9.14 givesa precise upper bound on the number of points of Γ ∩ X ( Q p ) in a given residue disk, as well as acriterion for this number to be at most 1.The following two results on rational points of surfaces are deduced from Theorem 1.1 by basechange to Q p and choosing G as the group of Q -rational points of the corresponding abelian variety.We also obtain similar results over any number field, not just Q ; see Section 10.1. Theorem 1.5.
Let X be a smooth, geometrically irreducible, projective surface of general typecontained in an abelian threefold A , both defined over Q . Let p > (128 / · c ( X ) be a prime ofgood reduction for X and A , and let X ′ be the reduction of X modulo p . If rank A ( Q ) ≤ and X ′ contains no elliptic curves over F algp , then X ( Q ) is finite and X ( Q ) ≤ X ′ ( F p ) + p − p − · (cid:16) p + 4 p / + 3 (cid:17) · c ( X ) . emark . A compact complex manifold M is hyperbolic if every holomorphic map f : C → M is constant. If a complex projective surface is hyperbolic, then it is of general type (see Lemma3.13). So, in Theorem 1.5 we may require that X ( C ) be hyperbolic instead of requiring general type—depending on the application, this might be easier to check. In fact, there is no loss of generalityin doing so: Under the assumptions of Theorem 1.5, the surface X contains no elliptic curves over Q alg , hence, over C (by specialization). Furthermore, X is not an abelian surface because it is ofgeneral type. Since X ⊆ A , a result of Green [29] implies that X ( C ) is hyperbolic. Theorem 1.7.
Let X be a smooth, geometrically irreducible, projective surface contained in anabelian variety A of dimension n ≥ , both defined over Q . Let H be an ample divisor on A andlet p be a prime of good reduction for X and A satisfying p > max (cid:26) c ( X ) + 2 , n ! · (3 deg( H .X ) + deg( H.K X )) n n n · deg( H n ) (cid:27) . Let X ′ and A ′ be the corresponding reductions modulo p of X and A . If rank A ( Q ) ≤ and A ′ issimple over F algp , then X ( Q ) is finite and X ( Q ) ≤ X ′ ( F p ) + p − p − · (cid:16) p + 4 p / + 3 (cid:17) · c ( X ) . Remark . Since A ′ is geometrically simple, so is A . Thus, X ( C ) is hyperbolic by [29] as it wasin Theorem 1.5 (see Remark 1.6). Deep conjectures by Bombieri and Lang predict that if V is asmooth projective variety over Q such that V ( C ) is hyperbolic, then V ( Q ) is finite. For curves thisis Faltings’s theorem since hyperbolic projective curves are precisely those of genus g ≥
2. When V is contained in an abelian variety and V ( C ) is hyperbolic, finiteness of V ( Q ) was proved by Faltings[22] extending methods of Vojta [63]. Hence, hyperbolicity of X ( C ) is natural in our context. Remark . It is expected that the rank of abelian varieties over Q of a fixed positive dimensionis 0 or 1 a positive proportion of the time each —ordering the abelian varieties, for instance, by(Faltings or Theta) height— and this is proved for elliptic curves [9, 10]. Thus, one can expect thatthe rank assumption in Theorems 1.5 and 1.7 is often satisfied in examples. Remark . For a variety X contained in an abelian variety A over Q , heuristically, one sees thatthe limit of applicability of an analogue of Chabauty’s classical approach should be dim X +dim Γ ≤ dim A where Γ is the p -adic closure of A ( Q ) in A ( Q p ). In our results in the case dim A = 3, thislimit rank condition is in fact reached.A simple case where our results are applicable is given by the following. Corollary 1.11.
Let A be an abelian threefold over Q with rank A ( Q ) ≤ and End( A C ) = Z .There is a set of primes P of density in the primes such that the following holds:Let X be a smooth, geometrically irreducible, projective surface defined over Q and contained in A , and let p ∈ P be a prime of good reduction for X with p > (128 / · c ( X ) . Let X ′ be thereduction of X modulo p . Then X ( Q ) ≤ X ′ ( F p ) + p − p − · (cid:16) p + 4 p / + 3 (cid:17) · c ( X ) . This is deduced in Section 10.2 from Theorem 1.5 and results of Chavdarov [14] on absolutelysimple reduction of abelian varieties. Chavdarov’s results together with Theorem 1.7 imply analo-gous corollaries when dim A ≥ Remark . Given n ≥
1, a general abelian variety B over C of dimension n satisfies End( B ) ≃ Z .In view of Remark 1.9, abelian threefolds satisfying the conditions in Corollary 1.11 should berather common. And in fact, they are easy to find; the Jacobian of the genus 3 hyperelliptic curve y = x − x − emark . For any abelian threefold as in Corollary 1.11, our bounds for the number of rationalpoints apply to any smooth surface contained in A , e.g. by embedding A in a projective space andintersecting with a general hyperplane (by Bertini’s theorem). This gives plenty of examples.For a curve C over a field, we let C ( n ) be its n -th symmetric power. If C is defined over Q , the Q -rational points of C (2) are in bijection with Galois orbits of quadratic points and unordered pairsof Q -rational points. If C is a hyperelliptic curve over Q , then we certainly have that C (2) ( Q ) isinfinite. As a direct application of our results, we can bound C (2) ( Q ) for non-hyperelliptic curveswhose Jacobian has rank 0 or 1, under some conditions on the reduction type at p . Corollary 1.14.
Let C be a smooth, geometrically irreducible, projective curve over Q of genus g ≥ which is not hyperelliptic over Q alg and such that its Jacobian J has rank J ( Q ) ≤ . Let p > (8 g − g be a prime of good reduction for C . Let C ′ and J ′ denote the reduction of C and J modulo p respectively. Suppose that C ′ is not hyperelliptic over F algp and that J ′ is geometricallysimple. Then C (2) ( Q ) is finite and C (2) ( Q ) ≤ C ′ ) (2) ( F p ) + p − p − · (cid:16) p + 4 p / + 3 (cid:17) · (4 g − g − . This is directly obtained from Theorem 1.7 in Section 10.3 after some computations in intersectiontheory. In a similar fashion, we will prove the following strengthening of Corollary 1.14 for curvesof genus 3, by applying Theorem 1.5 instead.
Corollary 1.15.
Let C be a smooth, geometrically irreducible, projective curve over Q of genus which is not hyperelliptic over Q alg and such that its Jacobian J has rank J ( Q ) ≤ . Let p ≥ be a prime of good reduction for C and denote by C ′ the reduction of C modulo p . Suppose that C ′ is not hyperelliptic over F algp and that ( C ′ ) (2) does not contain elliptic curves over F algp . Then C (2) ( Q ) is finite and C (2) ( Q ) ≤ C ′ ) (2) ( F p ) + 6 · p − p − · (cid:16) p + 4 p / + 3 (cid:17) < C ′ ) (2) ( F p ) + 7 . · p. Finally, we mention that the finiteness aspect of Theorem 1.1 (and more generally, Theorem9.1) does not follow from Faltings’s theorem for subvarieties of abelian varieties [22], as Γ is nota finite rank group when rank ( G ) = 1. Regarding bounds for the number of rational points, theDiophantine approximation method of Vojta [63] and Faltings [22, 23] led to explicit bounds suchas [57, 58] for subvarieties of abelian varieties over number fields, outside the special set. However,as it is the case for the classical Chabauty-Coleman method on curves compared to Diophantineapproximation bounds, our p -adic approach for surfaces leads to sharper estimates when it applies.1.4. Sketch of the method: overdetermined ω -integrality. We conclude this introductionby presenting an outline of the proof of Theorem 1.1 (and more generally, Theorem 9.1). Tosimplify the notation, let us focus in the case dim A = 3 since the key features already appear here.Furthermore, enlarging G we may assume that rank G = 1.First we note that, at least heuristically, the Chabauty-Coleman p -adic approach has a chanceto succeed since dim X + dim Γ = 2 + 1 = dim A .Consider the reduction map red : A ( Q p ) → A ′ ( F p ) and for each x ∈ A ′ ( F p ) let U x = red − ( x ) ⊆ A ( Q p ) be the corresponding residue disk. We want to bound ∩ X ( Q p ) ∩ U x and then addthese upper bounds as x varies in X ′ ( F p ). Let us parametrize Γ ∩ U x by a p -adic analytic map γ : p Z p → U x ⊆ A ( Q p ). If f is a local equation for X in U x , then ∩ X ( Q p ) ∩ U x is the numberof zeros of the one-variable p -adic analytic function h = f ◦ γ on p Z p . We remark that the idea ofparametrizing Γ can be traced back to work of Flynn [24] and it has been successful in computingthe rational points of curves in examples, although it has not previously led to results such as (1.1). riting h ( z ) = c z + c z + ... ∈ Q p [[ z ]], the number of zeros can be estimated provided that wehave some information on the size of the coefficients c j , see Section 6. Namely, we need:(I) a good upper bound for | c j | for all j , and(II) some small N such that | c N | is not too small, say, | c N | ≥ p -adic analytic map γ very carefully. We developa completely explicit theory of 1-parameter p -adic analytic subgroups in Section 8 and, with somecare in the choice of the local equation f , this allows us to prove the desired upper bound.The key difficulty in the whole argument, however, is (II). We take a somewhat indirect approach.Consider the morphism of p -adic analytic groups Log : A ( Q p ) → Lie( A ( Q p )) ≃ H ( A, Ω A/ Q p ) ∨ constructed by Coleman integration or by classical theory of p -adic Lie groups [11]. As rank G = 1,Log(Γ) is contained in a line of Lie( A ( Q p )) which determines a hyperplane H ⊆ H ( A, Ω A/ Q p ). Wechoose ω , ω ∈ H so that they reduce to independent differentials ω ′ , ω ′ on A ′ which we restrictto differentials u , u on X ′ . For x ∈ X ′ ( F p ) let m ( x ) be the supremum of all integers m such thatthere is a closed immersion φ m : Spec F algp [ z ] / ( z m +1 ) → X ′ supported at x which is ω -integral forboth ω = u , u . Roughly speaking, ω -integrality for a differential ω means that the morphism φ m solves the differential equation determined by ω ; see Section 4 for a detailed discussion.If u and u are independent over the function field of X ′ , then the maps φ m in the definition of m ( x ) are jet solutions to an overdetermined system of differential equations . So, one expects m ( x )to be finite and bounded in terms of u and u . This is indeed correct, but it is far from obvious.Theorem 4.4 gives such a bound in terms of the geometry of the divisor D of the 2-form u ∧ u .Our bound for m ( x ) is somewhat intricate, but Remark 4.7 shows that in some cases it is sharp.Bounding m ( x ) turns out to be crucial, since we will show in Lemma 9.16 that N ≤ m ( x ) + 1with N as in (II). Together with the zero estimates from Section 6 and our upper bounds for | c j | (see (I) above) we will prove the following key estimate in Proposition 9.14(1.4) ∩ X ( Q p ) ∩ U x ≤ m ( x ) · p − p − . Finally, the geometric bound for m ( x ) proved in Theorem 4.4 is applied to (1.4), and then addedover x ∈ X ′ ( F p ). When x is not in the support of D = div( u ∧ u ) we show m ( x ) = 0, thus, ∩ X ( Q p ) ∩ U x ≤
1. The contribution for x in the support of D is more complicated andit corresponds to counting F p -points in the support of D with suitable weights. The divisor D can be non-reduced and highly singular, so this counting problem does not directly follow fromWeil’s estimates for points in curves over finite fields. We address this problem by studying certainmodified Zeta functions in Section 3.5, which allows us to conclude the argument.A technical point that requires attention is that we make heavy use of ω -integrality for schemesover rings or fields of positive characteristic. The classical analytic approach is not useful for usand we need purely algebraic methods. Fortunately such a study was carried out by Garcia-Fritz[26, 27, 28] in the context of her generalization of Vojta’s explicit version of Bogomolov’s approach toquasi-hyperbolicity, see [64, 18]. While Garcia-Fritz’s work is mostly over C , her algebraic methodseasily extend to an arbitrary base. See Section 4.A major technical difficulty is that at various points of the argument we need to restrict differ-ential forms to subvarieties of abelian varieties in a way that preserves non-triviality. For example,while ω ′ and ω ′ are independent on A ′ , is is not clear u ∧ u is not the zero form on X ′ , and this isnecessary even to define the divisor D = div( u ∧ u ). Part of the difficulties are due to the fact that X does not have a particularly convenient presentation that allows for explicit computations withdifferentials, unlike the case of symmetric powers of curves. The required non-vanishing results forrestriction of forms are not difficult to obtain in characteristic zero by analytic means, but we needthem in positive characteristic. This is achieved in Section 5 by means of intersection theory. . Notation
General notation. L is a field and L alg is an algebraic closure of it. The symbol F willdenote a field, a function on a variety, or a power series, and this will be explicitly stated. Thesymbol K will denote a p -adic field or a canonical divisor in a variety, which will be explicitlystated. k will always denote an algebraically closed field and k will always denote a finite field.We write N = Z ≥ . For α = ( α , ..., α n ) ∈ N we define k α k = α + ... + α n . If x = ( x , ..., x n ) isan n -tuple of variables, we write x α = x α · · · x α n n .The exponential function on R is exp and the logarithm on R > is log.2.2. Geometry.
A variety over L is a reduced L -scheme which is separated of finte type over L .We remark that we are not requiring irreducibility as part of the definition. A curve (resp. surface)over L is a variety over L of pure dimension 1 (resp. 2).The normalization of an integral domain A is denoted by e A , and the normalization of a variety Z over k is denoted by e Z .If X is a scheme and x is a (schematic) point in X , the maximal ideal of the local ring O X,x isdenoted by m X,x . The completion of O X,x at m X,x is denoted by b O X,x .If f : X → Y is a morphism of schemes, x is a point of X and y = f ( x ), the induced map onlocal rings is f x : O Y,y → O X,x . If X consists of exactly one point, we just write f instead of f x .For an irreducible projective curve C over the algebraically closed field k , the arithmetic genusof C is denoted by g a ( C ) and the geometric genus is denoted by g g ( C ). Namely, g g ( C ) = g a ( e C ).If L is a perfect field and C is a geometrically irreducible projective curve over L , we define thearithmetic and geometric genus of C by base change to L alg .Given a geometrically irreducible, smooth, projective surface S over L , we have a Z -valuedintersection product on divisors of S , which we write ( D.D ′ ), ( D.D ′ ) S , or D.D ′ for two divisors D and D ′ on S , and we write D = ( D.D ) for a self-intersection. The first Chern number c ( S ) of S is the self-intersection number of a canonical divisor on S .2.3. Local fields.
In the context of local fields, the symbol p denotes a prime number, K is a finiteextension of Q p , R is the ring of integers of K , p is the maximal ideal of R , ̟ ∈ p is a uniformizer, k = R/ p is the residue field of R , q = k , e is the ramification exponent of K/ Q p , and f = [ k : F p ]is the residue degree. We denote by | − | the absolute value on K normalized by | ̟ | = 1 /q , orequivalently, | p | = p − [ K : Q p ] .For n ≥ K n : | v | = max ≤ j ≤ n | v j | , v = ( v , ..., v n ) ∈ K n . For r > B n ( r ) = { v ∈ K n : | v | < r } and B n [ r ] = { v ∈ K n : | v | ≤ r } .3. Curves and surfaces
Curves.
Let C be an irreducible curve over the algebraically closed field k and let x ∈ C ( k ).The order of singularity of C at x is δ ( C, x ) = dim k ( e O C,x / O C,x ), where the quotient e O C,x / O C,x istaken as k -vector spaces. One has the following basic fact: Lemma 3.1. δ ( C, x ) is a non-negative integer. It is equal to if and only if C is regular at x . As a special case of Theorem 2 in [34] we have
Lemma 3.2 (Hironaka) . If C is a projective irreducible curve over k , then g a ( C ) = g g ( C ) + X x ∈ C ( k ) δ ( C, x ) . he multiplicity of C at x ∈ C ( k ) is defined as the multiplicity of the local ring O C,x as defined,for instance, in Ch. 5, Sec. 14 of [46]. We denote it by mult x ( C ). A basic fact: Lemma 3.3. mult x ( C ) is a positive integer. It is equal to if and only if C is regular at x . From Exercise 14.5 in [46] we get a simple expression for mult x ( C ) in a special case. Lemma 3.4.
Let S be a surface over k and let C be a closed irreducible curve in S . Let x ∈ C ( k ) be a point such that S is regular at x . Let F ∈ O S,x be a local equation for C at x . Then mult x ( C ) = max { n ≥ F ∈ m nS,x } . We will also need the following result.
Lemma 3.5 (Hironaka) . Let S be a projective surface and let C be a projective irreducible curvecontained in S . Let x ∈ C ( k ) be a point such that S is regular at x . Then we have δ ( C, x ) ≥ · mult x ( C ) · (mult x ( C ) − . Proof.
By Theorem 1 in [34], after dropping the contribution of higher order neighboring points. (cid:3)
A canonical genus bound.
The following result will be useful to control the genus of thecomponents of certain canonical divisors on surfaces.
Lemma 3.6 (Canonical genus bound) . Let S be a smooth projective irreducible surface over k .Let D be a canonical divisor on S . Assume that D > and that D is numerically effective. Let C , ..., C ℓ be the irreducible components of the support of D and let a , ..., a ℓ be positive integersdefined by D = a C + ... + a ℓ C ℓ . Then we have ℓ X j =1 a j · ( g a ( C j ) − ≤ c ( S ) . Proof.
Let us rearrange the indices so that for certain 0 ≤ r ≤ ℓ we have C j > j ≤ r and C j ≤ j > r . By the adjunction formula we find(3.1) 2 ℓ X j =1 a j · ( g a ( C j ) −
1) = ℓ X j =1 a j · (( D.C j ) + C j ) = D + ℓ X j =1 a j · C j ≤ D + r X j =1 a j · C j ≤ D + r X j =1 a j · C j . For any given j we have ( D − a j C j ) .C j = ( P i = j a i C i ) .C j ≥ C i are different irreduciblecurves. Also, D.C j ≥ D is nef. Therefore D = ℓ X j =1 a j D.C j = r X j =1 a j · C j + r X j =1 a j · ( D − a j C j ) .C j + ℓ X j = r +1 a j D.C j ≥ r X j =1 a j · C j . Together with (3.1), this gives 2 P ℓj =1 a j · ( g a ( C j ) − ≤ · D = 2 c ( X ). (cid:3) Branches.
Let S be a surface over k and let x ∈ S ( k ) be a smooth point of S . Given localparameters s , s ∈ m S,x , the completed local ring at x can be presented as b O S,x ≃ k [[ s , s ]].Let C ⊆ S be a closed irreducible curve passing through x with local equation F ∈ m S,x . Let usfactor F = F · · · F r where each F j ∈ b O S,x is irreducible.
Lemma 3.7.
For each i = j , the power series F i and F j are non-associated. roof. The special case O S,x ≃ k [ s , s ] ( s ,s ) is in [44] Theorem 16.5. As observed in [26] Remark2.58, the general case follows from a result of Chevalley. Indeed, k is perfect since it is algebraicallyclosed. By [66] VIII.13 Theorem 31, the local ring O C,x = O S,x / ( F ) is analytically unramified.Thus, the completion b O C,x = b O S,x / ( F ) does not have nilpotent elements. (cid:3) Hence, the prime ideals q j = ( F j ) ⊆ b O S,x are different. They will be called branches of C at x .Let ν : e C → S be the composition of the normalizaton map e C → C and the inclusion C → S .For each y ∈ ν − ( x ) we have a map induced on local rings ν y : O S,x → O e C,y which extendscontinuously to completed local rings b ν y : b O S,x → b O e C,y . We will need the following:
Lemma 3.8 (Relation between normalization and branches) . With the previous notation, there isa bijection between the set of branches of C at x and the points y ∈ ν − ( x ) . More precisely, theideals ker( b ν y ) ⊆ b O S,x for y ∈ ν − ( x ) are exactly the branches of C at x .Proof. In the special case O S,x ≃ k [ s , s ] ( s ,s ) this is [44] Theorem 16.14; a similar argument worksin general. Alternatively, the general case is [26] Theorem 2.69 assuming that k has characteristic0. This assumption is needed in other parts of [26], but not for this particular result. In fact, theargument in [26] reduces this result to [37] Corollary 5, where characteristic 0 is not required. (cid:3) Lemma 3.9 (Bound for fibres of normalization) . With the previous notation, and assuming that S is a projective surface, we have ν − ( x ) ≤ δ ( C, x ) + 1 .Proof.
From Lemma 3.4, it follows that the number of branches of C at x is at most mult x ( C ). ByLemma 3.8 we get ν − ( x ) ≤ mult x ( C ). The result follows from Lemma 3.5. (cid:3) As the isomorphism b O S,x ≃ k [[ s , s ]] allows us to study branches by means of power seriescomputations, the following result will be useful. It is a direct consequence of [56] Theorem 2.1. Lemma 3.10.
Let f ( x , x ) ∈ k [[ x , x ]] be an irreducible power series. Let φ ( t ) , φ ( t ) ∈ t · k [[ t ]] be such that f ( φ , φ ) = 0 . Then ord t φ ( t ) ≥ ord x f (0 , x ) and ord t φ ( t ) ≥ ord x f ( x , . Surfaces of general type.
Let S be a smooth, irreducible, projective surface over k . Thesurface S is of general type if the canonical sheaf Ω S/k is big, i.e. if dim k H ( S, (Ω S/k ) ⊗ n ) growsquadratically on n . More generally, if X is a smooth, geometrically irreducible, projective surfaceover L , we say that X is of general type if X ⊗ L alg is a surface of general type over L alg .The following result is classical. Here, “minimal” means that S contains no ( − − Lemma 3.11. If S is a minimal smooth projective surface of general type over k , then c ( S ) ≥ and the canonical sheaf Ω S/k is nef.
The following simple condition ensures that the canonical sheaf of a general type surface is ample.
Lemma 3.12.
Let S be a smooth projective surface of general type. If S contains no smooth rationalcurves with self-intersection − or − , then the canonical sheaf of S is ample. In particular, thisis the case when S contains no smooth curve of genus .Proof. This result is well-known. The argument is similar to that in [50]. We present the proof forthe convenience of the reader.Let K be a canonical divisor on S . Since S does not contain ( − K > K is nef. Let C ⊆ X be an irreducible curve. By the Nakai-Moishezon criterion, it suffices to show that K.C = 0.For the sake of contradiction, suppose that K.C = 0. As K >
0, the Hodge Index Theorem givesthat either C < C is numerically trivial. The latter case cannot happen because C.H > ny ample divisor H , so we have C <
0. By the adjunction formula 2 g a ( C ) − C + K · C = C <
0, which gives g a ( C ) = 0. Thus C is smooth and rational by Lemma 3.2. Furthermore, C = C + K · C = 2 g a ( C ) − −
2. Contradiction. So, K is ample. (cid:3) Let us recall that a compact complex manifold M is Brody hyperbolic if every holomorphic map h : C → M is constant. As we are in the compact case, this is equivalent to Kobayashi hyperbolicityby Brody’s theorem [12], and we simply say that M is hyperbolic. It is conjectured that if V is asmooth projective variety over C and V ( C ) is hyperbolic, then V is of general type. This is knownfor surfaces; see [7] Chapter VIII, Theorem 24.1. Lemma 3.13.
Let S be a smooth, irreducible, projective surface defined over C . If S ( C ) is hyper-bolic, then S is of general type. Lemma 3.14.
Let A be an abelian variety over C and let X ⊆ A be a smooth projective surface.If X is not an abelian surface and it does not contain elliptic curves, then X is of general type.Proof. By [65] Theorems 1 and 1’. Alternatively: X is hyperbolic [29], then apply Lemma 3.13. (cid:3) Point counting over finite fields.
In this paragraph we let k be a finite field of characteristic p with q elements. Let D be a projective curve defined over k ; we do not require that D beirreducible, connected, or smooth. Let k ′ be a finite extension of k such that all geometric irreduciblecomponents of D are defined over k ′ , and all singular points of D as well as all points of thenormalization of D ⊗ k ′ mapping to them are k ′ -rational.Let C , ..., C r be the geometric irreducible components of D , which are defined over k ′ . Let ν j : e C j → D ⊗ k ′ be the normalization of C j composed with the inclusion C j → D ⊗ k ′ .Our goal is to show the following estimate Lemma 3.15.
With the previous notation, we have X x ∈ D ( k ) r X j =1 { y ∈ e C j ( k alg ) : ν j ( y ) = x } ≤ ( q + 1) r + 2 q / · r X j =1 g g ( C j ) . More generally, for a finite extension L/ k it will be convenient to define A D ( L ) = X x ∈ D ( L ) r X j =1 { y ∈ e C j ( k alg ) : ν j ( y ) = x } and a D ( L ) = A D ( L ) − D ( L ). Note that a D ( L ) ≥ k ′ we see that thenumber c D := a D ( k ′ ) satisfies a D ( L ) ≤ c D for each finite extension L/ k . Lemma 3.16. If L/ k is a finite extension and k ′ ⊆ L , then a D ( L ) = c D and A D ( L ) = r X j =1 e C j ( L ) . Proof.
The quantity a D ( L ) counts preimages of singularities of D in the normalization, so it stabi-lizes when L contains k ′ . Let C be the disjoint union of the curves C j and let ν : C → D ⊗ k ′ bethe morphism over k ′ determined by all the maps ν j . Then ν is an isomorphism over k ′ away fromthe preimages of singularities of D , all of which are k ′ -rational. Thus, A D ( L ) = C ( L ). (cid:3) Note that D ( k ) ≤ A D ( k ) ≤ D ( k ) + c D . There is related work on the Weil conjectures forsingular curves such as [1] and a natural approach to Lemma 3.15 would be to use an upper boundon D ( k ) together with an upper bound on c D coming from counting singularities. However, wewill consider D ( L ) + c D at once, as this leads to sharper estimates. or an integer n ≥ F a finite field of characteristic p , let F n ⊆ F alg be the only extensionof F of degree n . Given a quasi-projective variety V over F (not necessarily smooth or irreducible)let N V,F ( n ) = V ( F n ) and define the Zeta function on the variable T by Z V,F ( T ) = exp ∞ X n =1 N V,F ( n ) n · T n ! . In this generality, it is a theorem of Dwork [19] that Z V,F ( T ) ∈ Q ( T ).Let N ∗ D, k ( n ) = D ( k n ) + c D and observe that for all n ≥ A D ( k n ) ≤ N ∗ D, k ( n ) . Let us define Z ∗ D, k ( T ) = exp ∞ X n =1 N ∗ D, k ( n ) n · T n ! . Then Z ∗ D, k ( T ) = Z D, k ( T ) / (1 − T ) c D ∈ Q ( T ). Writing d = [ k ′ : k ], we have Lemma 3.17.
Let µ d be the set of complex d -th roots of . Then Y ζ ∈ µ d Z ∗ D, k ( ζ · T ) = r Y j =1 Z e C j , k ′ ( T d ) . Proof.
We note that for a given n ≥ S n := X ζ ∈ µ d N ∗ D, k ( n ) n · ζ n T n = ( d ∤ nd · N ∗ D, k ( n ) n · T m if d | n. In the second case we write n = md and observe that S md = m − N ∗ D, k ( md ) T md . Using k md = k ′ m ⊇ k ′ and Lemma 3.16 we get N ∗ D, k ( md ) = D ( k ′ m ) + c D = D ( k ′ m ) + a D ( k ′ m ) = A D ( k ′ m ) = r X j =1 N e C j , k ′ ( m )which proves the result. (cid:3) Proof of Lemma 3.15.
The Weil conjectures for curves (proved by Weil) applied to the curves e C j over k ′ together with Lemma 3.17 show that Z ∗ D, k ( T ) on C has r poles of modulus q , a total of2 · P rj =1 g g ( C j ) zeros of modulus q / , r poles of modulus 1, and no other zeros or poles in C . Astandard computation taking the logarithmic derivative of Z ∗ D, k ( T ), comparing coefficients of T n ,and applying the triangle inequality, shows that for each n ≥ N ∗ D, k ( n ) ≤ r ( q n + 1) +2 q n/ P rj =1 g g ( C j ). We conclude by (3.2) and taking n = 1. (cid:3) ω -integrality ω -integral morphisms. Let T be a scheme and let φ : X → Y be a morphism of T -schemes.Using the canonical morphisms of sheaves Ω Y/T → φ ∗ φ ∗ Ω Y/T and φ ∗ Ω Y/T → Ω X/T , we define φ • : H ( Y, Ω Y/T ) → H ( X, Ω X/T )as the composition H ( Y, Ω Y/T ) → H ( Y, φ ∗ φ ∗ Ω Y/T ) = H ( X, φ ∗ Ω Y/T ) → H ( X, Ω X/T ) . A basic property of this construction is that it respects composition.
Lemma 4.1.
Let φ : X → Y and ψ : Y → Z be morphisms of T -schemes. Then ( ψ ◦ φ ) • = φ • ◦ ψ • . roof. First, we remark that this is not obvious. In particular, we claim that ( ψ ◦ φ ) • and φ • ◦ ψ • areequal, not just equal up to composition with an isomorphism. The result follows from Proposition3.29 in [26]; let us explain the details.We consider the functors φ ∗ , φ ∗ , ψ ∗ , ψ ∗ , ( ψφ ) ∗ , and ( ψφ ) ∗ mapping between the categories of O X -modules, O Y -modules, and O Z -modules. We have ( ψφ ) ∗ = ψ ∗ φ ∗ , while the functors ( ψφ ) ∗ and φ ∗ ψ ∗ are isomorphic.It is a routine exercise to check that for each O Z -module F , the diagram F ψ ∗ ψ ∗ F ( ψφ ) ∗ ( ψφ ) ∗ F ( ψφ ) ∗ φ ∗ ψ ∗ F ψ ∗ φ ∗ φ ∗ ψ ∗ F ≃ = commutes, where the left vertical arrow is the canonical map, and the right vertical arrow isobtained by applying ψ ∗ to the canonical map of O Y -modules ψ ∗ F → φ ∗ φ ∗ ψ ∗ F . In particular,the following diagram commutes(4.1) H ( Z, Ω Z/T ) H ( Z, ψ ∗ ψ ∗ Ω Z/T ) H ( Z, ( ψφ ) ∗ ( ψφ ) ∗ Ω Z/T ) H ( Z, ( ψφ ) ∗ φ ∗ ψ ∗ Ω Z/T ) H ( Z, ψ ∗ φ ∗ φ ∗ ψ ∗ Ω Z/T ) H ( X, ( ψφ ) ∗ Ω Z/T ) H ( X, φ ∗ ψ ∗ Ω Z/T ) H ( Y, φ ∗ φ ∗ ψ ∗ Ω Z/T ) . ≃ = == = ≃ = Applying the functor φ ∗ φ ∗ to the map ψ ∗ Ω Z/T → Ω Y/T we deduce that the diagram(4.2) H ( Z, ψ ∗ ψ ∗ Ω Z/T ) H ( Y, ψ ∗ Ω Z/T ) H ( Y, Ω Y/T ) H ( Z, ψ ∗ φ ∗ φ ∗ ψ ∗ Ω Z/T ) H ( Y, φ ∗ φ ∗ ψ ∗ Ω Z/T ) H ( Y, φ ∗ φ ∗ Ω Y/T ) . == commutes. By Proposition 3.29 in [26], the following diagram commutes(4.3) H ( X, ( ψφ ) ∗ Ω Z/T ) H ( X, φ ∗ ψ ∗ Ω Z/T ) H ( Y, φ ∗ φ ∗ ψ ∗ Ω Z/T ) H ( X, Ω X/T ) H ( X, φ ∗ Ω Y/T ) H ( Y, φ ∗ φ ∗ Ω Z/T ) ≃ == where the middle vertical arrow is obtained by applying φ ∗ to ψ ∗ Ω Z/T → Ω Y/T . The result followsby combining (4.1), (4.2), and (4.3). (cid:3)
Remark . Lemma 4.1 will be used very frequently, so we will not quote it at each application.Given T -schemes X and Y and a differential ω ∈ H ( Y, Ω Y/T ), we say that a morphism of T -schemes φ : X → Y is ω -integral if φ • ( ω ) = 0.Let us consider points x ∈ X , y = φ ( x ) ∈ Y and let t ∈ T be the image of x (and y ) in T .The map φ x : O Y,y → O X,x makes O X,x into an O Y,y -algebra, and both are O T,t -algebras. Thus,we have natural maps Ω O Y,y / O T,t → Ω O Y,y / O T,t ⊗ O Y,y O X,x and Ω O Y,y / O T,t ⊗ O Y,y O X,x → Ω O X,x / O T,t nduced by φ x . Using these morphisms, we define the map φ • x : Ω O Y,y / O T,t → Ω O X,x / O T,t as the composition Ω O Y,y / O T,t → Ω O Y,y / O T,t ⊗ O Y,y O X,x → Ω O X,x / O T,t . The following lemma is a straightforward verification. A more general result (stated over C ) canbe found in [26] Section 3.2. Lemma 4.3.
With the previous notation, the following diagram commutes H ( Y, Ω Y/T ) H ( X, Ω X/T )Ω O Y,y / O T,t Ω O X,x / O T,t φ • φ • x where the vertical arrows are localization maps. Overdetermined ω -integrality. Recall that k is an algebraically closed field. Given asmooth surface S over k and two regular differentials ω , ω on S , in general, one expects thatthere is no curve C over k with a non-constant morphism C → S which is ω i -integral for both i = 1 ,
2. In classical terms, the map C → S would determine a solution to an overdeterminedsystem of differential equations, which is not a likely event. Our goal in this section is to boundthe order of infinitesimal solutions to such an overdetermined system.Let z be a variable. For m ≥ E km = k [ z ] / ( z m +1 ) and V km = Spec ( E km ). The onlypoint of V km is denoted by ζ . Theorem 4.4 (The “overdetermined” bound) . Let S be a smooth irreducible surface over k andlet x ∈ S ( k ) . Let ω , ω ∈ H ( S, Ω S/k ) be such that ω ∧ ω ∈ H ( S, Ω S/k ) is not the zero section.Let D = div S ( ω ∧ ω ) and let us write D = a C + ... + a ℓ C ℓ where a j are positive integers and C j are irreducible curves for each j (possibly, ℓ = 0 if D = 0 ). For each j , let ν j : e C j → S be thenormalization map of C j composed with the inclusion C j → S .Let m ≥ be an integer such that there is a closed immersion φ : V km → S supported at x (i.e.with φ ( ζ ) = x ) which is ω i -integral for both i = 1 and i = 2 . Then for every ω ∈ H ( S, Ω S/k ) ofthe form ω = c ω + c ω with c , c ∈ k , we have (4.4) m ≤ ℓ X j =1 X y ∈ ν − j ( x ) a j · (cid:0) ord y ( ν • j ( ω )) + 1 (cid:1) . Remark . If x is not in the support of D , then the sum in (4.4) is empty and it gives the upperbound m ≤
0, that is, m = 0. This case is simpler and it is proved separately in Corollary 4.11. Remark . If ν • j ( ω ) = 0, then ord y ( ν • j ( ω )) = + ∞ and the upper bound for m is useless. Thus,when we apply Theorem 4.4 it is crucial to make sure that the maps ν j are not ω -integral. Remark . The bound (4.4) is optimal in some non-trivial cases. For instance, assume that k hascharacteristic different from 2 and 3. We take S = A k = Spec k [ s , s ], x = (0 , ω = ds + s ds ,and ω = ds + s ds . Thus, ω ∧ ω = ( s − s )( s + s ) ds ∧ ds . Let C and C be defined by s − s = 0 and s + s = 0 respectively, so that D = C + C . Let m = 2 and let φ : V k → A k bethe map induced by the k -algebra morphism k [ s , s ] → E k = k [ z ] / ( z ), s z mod z , s φ is a closed immersion supported at x which is ω i -integral for i = 1 ,
2, as Ω E k /k =( k [ z ] / ( z, z )) dz = ( k [ z ] / ( z )) dz . For ω = ω , ω , (4.4) becomes m ≤ · (0 + 1) + 1 · (0 + 1) = 2. he rest of this section is devoted to the proof of Theorem 4.4, and we keep the same notationand assumptions as in the statement.4.3. Local expressions.
First we study the map φ ζ : O S,x → E km induced on local rings. Lemma 4.8 (Generators for ker( φ ζ )) . There are local parameters s , s ∈ m S,x at x such that φ ζ ( s ) = z mod z m +1 and φ ζ ( s ) = 0 . Furthermore, ker( φ ζ ) = ( s m +11 , s ) .Proof. As φ is a closed immersion, φ ζ : O S,x → E km is surjective and we can take s ∈ m S,x with φ ζ ( s ) = z mod z m +1 . In fact, s ∈ m S,x r m S,x because φ ζ is a local map.Let s ∈ m S,x with m S,x = ( s , s ), which is possible because S is smooth. One can write φ ζ ( s ) = c z + ... + c m +1 z m +1 mod z m +1 for certain c j ∈ k , and we define s = s − ( c s + ... + c m +1 s m +11 ).Note that m S,x = ( s , s ) = ( s , s ) and φ ζ ( s ) = 0.Clearly ( s m +11 , s ) ⊆ ker( φ ζ ). Equality holds because dim k O S,x / ( s m +1 , s ) = m + 1 anddim k O S,x / ker( φ ζ ) = dim k E km = m + 1, due to smoothness of S . (cid:3) From now on we fix a choice of s , s as in Lema 4.8. With this choice we obtain b O S,x = k [[ s , s ]]. Lemma 4.9 (Local expression for differentials) . Let ω ∈ H ( S, Ω S/k ) be such that the map φ : V km → S is ω -integral. Then there are G , G , G ∈ O S,x such that the germ of ω at x is ω x = ( G s m + G s ) ds + G ds ∈ Ω O S,x /k . Proof.
Since φ • ( ω ) = 0, Lemma 4.3 gives φ • ζ ( ω x ) = 0. Hence, ω x ⊗ E km ∈ ker(Ω O S,x /k ⊗ E km → Ω E km /k ) . Let I = ker( φ ζ ) ⊆ O S,x . As φ is a closed immersion, we have the exact sequence I/I → Ω O S,x /k ⊗ E km → Ω E km /k → ω x ⊗ E km is in the image of the first map. This means that there is f ∈ I suchthat ( df − ω x ) ⊗ E km = 0. Thus, there are g , g ∈ I with df − ω x = g ds + g ds . By Lemma 4.8we have I = ( s m +11 , s ). Since f, g , g ∈ I , the result follows by a direct computation. (cid:3) Let us define F ∈ O S,x by the formula ω ,x ∧ ω ,x = F · ds ∧ ds ∈ Ω O S,x /k . Thus, F is a localequation for D at x . Lemma 4.10 (Local expression for D ) . We have F ∈ ( s m , s ) ⊆ O S,x .Proof.
This follows by applying Lemma 4.9 to ω and ω , and then expanding ω ,x ∧ ω ,x . (cid:3) Corollary 4.11 (The case x / ∈ D ) . If x is not in the support of D , then m = 0 .Proof. In this case F is a unit in O S,x . By Lemma 4.10, this can only happen for m = 0. (cid:3) Bound on a single branch.
For each pair ( j, y ) with y ∈ ν − j ( x ) (if any) let p j,y ⊆ b O S,x = k [[ s , s ]] be the branch of C j at x associated to y by Lemma 3.8. Thus, there is a factorization F = U · ℓ Y j =1 Y y ∈ ν − ( x ) F a j j,y where U ∈ k [[ s , s ]] × and F j,y ∈ k [[ s , s ]] is irreducible with p j,y = ( F j,y ) for each pair ( j, y ) asabove. Here we recall that D = P ℓj =1 a j C j .Let π : k [[ s , s ]] → k [[ s ]] be the quotient by ( s ). emma 4.12 (Individual bound) . Let ω ∈ H ( S, Ω S/k ) be such that φ : V km → S is ω -integral.Take j with ≤ j ≤ ℓ and fix a choice of y ∈ ν − j ( x ) , if any. Then we have ord y ( ν • j ( ω )) ≥ min { m, ord s ( π ( F j,y )) − } . Proof.
Let t ∈ m e C j ,y be a local parameter. Thus, b O e C j ,y = k [[ t ]]. The morphism ν j : e C j → S inducesthe map ν j,y : O S,x → O e C j ,y and the map on completions b ν j,y : k [[ s , s ]] → k [[ t ]].For i = 1 , h i = ν j,y ( s i ) ∈ m e C j ,y ⊆ t · k [[ t ]]. Lemma 4.9 gives ω x = ( G s m + G s ) ds + G ds for some G , G , G ∈ O S,x . By Lemma 4.3 we get( ν • j ( ω )) y = ν • j,y ( ω x ) = (cid:16) ν j,y ( G ) h m + ν j,y ( G ) h (cid:17) h ′ dt + ν j,y ( G ) h ′ dt where h ′ i ∈ O e C j ,y are defined by the relation dh i = h ′ i dt in Ω O e Cj,y /k (as dt generates). It followsthat(4.5) ord y ( ν • ( ω )) = ord t (cid:16) ( ν j,y ( G ) h m + ν j,y ( G ) h ) h ′ + ν j,y ( G ) h ′ (cid:17) . Note that F j,y ( h , h ) = b ν j,y ( F j,y ) = 0 in k [[ t ]], since ( F j,y ) ⊆ k [[ s , s ]] is the branch of C j corresponding to y . As F j,y ∈ k [[ s , s ]] is irreducible, Lemma 3.10 gives(4.6) ord t ( h ) ≥ ord s ( π ( F j,y )) . The relation dh i = h ′ i dt in Ω O e Cj,y /k shows that in b O e C j ,y = k [[ t ]], the power series h ′ i is the formalderivative of h i . Hence, for i = 1 , t ( h ′ i ) ≥ ord t ( h i ) − . Using (4.5), (4.6), and (4.7) we finally deduceord y ( ν • j ( ω )) ≥ min n ord t (cid:16) ν j,y ( G ) h m h ′ (cid:17) , ord t (cid:16) ν j,y ( G ) h h ′ + ν j,y ( G ) h ′ (cid:17)o ≥ min { ( m + 1)ord t ( h ) − , ord t ( h ) − }≥ min { m, ord s ( π ( F j,y )) − } . (cid:3) Proof of the overdetermined bound.
We keep the notation introduced in this section andthe assumptions from the statement of Theorem 4.4.
Proof of Theorem 4.4.
By Corollary 4.11, it is enough to assume that x is in the support of D . Let S = ℓ X j =1 X y ∈ ν − j ( x ) a j · (cid:0) ord y ( ν • j ( ω )) + 1 (cid:1) . Since φ is ω i -integral for i = 1 ,
2, it is also ω -integral. By Lemma 4.12 we have S ≥ ℓ X j =1 X y ∈ ν − j ( x ) a j · min { m + 1 , ord s ( π ( F j,y )) } Since x is in the support of D , the sum is non-empty. If there is at least one ( j, y ) with 1 ≤ j ≤ ℓ , y ∈ ν − j ( x ) and ord s ( π ( F j,y )) ≥ m + 1, then we get S ≥ m + 1 > m . o we can assume that for each j and each y ∈ ν − j ( x ) we have ord s ( π ( F j,y )) ≤ m . In this case S ≥ ℓ X j =1 X y ∈ ν − j ( x ) a j · ord s ( π ( F j,y )) = ord s ( π ( F )) . By Lemma 4.10 we have F ∈ ( s m , s ), which finally gives ord s ( π ( F )) ≥ m . (cid:3) Differentials on abelian varieties
We recall that k is an algebraically closed field. Several results in this section will be stated onlyfor positive characteristic as we just need them in this case and the counterpart for characteristic0 is either easier or well-known.Let A be an abelian variety over k of dimension n ≥
1. The neutral point of A is denoted by e . For a point x ∈ A ( k ), the map of translation by x is denoted by τ x . If Z and Z are algebraiccycles of dimensions r and r , the intersection Z .Z is an algebraic cycle of dimension 2 n − r − r defined up to algebraic equivalence. For 0-cycles we have a Z -valued degree map denoted by deg.If X is a subvariety of A , the smallest translate B of an abelian subvariety of A with X ⊆ B isdenoted by h X i . If e ∈ X then h X i is an abelian subvariety of A . Lemma 5.1.
Let D be a non-zero effective divisor on A . If supp( D ) contains no translate ofpositive dimensional abelian subvarieties of A , then D is ample.Proof. Define H ( D ) = { x ∈ A ( k ) : τ ∗ x D = D } where τ ∗ x D = D is an equality of divisors, not justup to equivalence. Then H ( D ) is the set of k -points of a Zariski-closed subgroup of A which wealso denote by H ( D ). Taking any x ∈ supp( D ) we get H ( D ) + x ⊆ supp( D ). Our assumptionon supp( D ) implies that H ( D ) is zero dimensional, hence, finite. By Application 1 in p.60 of [51]we get that D is ample. (cid:3) Degrees.
Let D be a divisor on A . It induces a morphism φ D : A → A ∨ by the rule φ D ( x ) = [ τ x ( D ) − D ]. We have deg( φ D ) = (cid:18) deg( D n ) n ! (cid:19) . See [51] p.150. If D is ample, then φ D is an isogeny.Given ψ ∈ End( A ) and a prime ℓ different from the characteristic of k , we consider the polynomial P ψ ( t ) = det( t · Id A − ψ | T ℓ ( A )) where T ℓ ( A ) is the ℓ -adic Tate module of A . The polynomial P ψ ( t ) isin Z [ t ], it is monic, independent of ℓ , and it has degree 2 n . For r ∈ Z one has P ψ ( r ) = deg([ r ] A − ψ )where [ r ] A ∈ End( A ) is the map of multiplication by r . Thus, P ψ (0) = deg( ψ ). See [51] p.180.We define tr( ψ ) = tr( ψ | T ℓ A ), which is the negative of the coefficient of t n − in P ψ ( t ).Let S be the addition map from the group of 0-cycles on A to A ( k ). For a 1-cycle Z and a divisor D we define α A ( Z, D ) ∈ End( A ) by α A ( Z, D )( x ) = S ( Z. ( τ x ( D ) − D )), see [49, 47]. By [47] p.8,(5.1) tr( α A ( Z, D )) = 2 · deg( Z.D ) . For a variety X and an irreducible curve C ⊆ X we let ν C/X : e C → X be the composition of thenormalization map e C → C and the inclusion C → X .Let C ⊆ A an irreducible curve. Fix a point x ∈ e C and consider the associated jacobianembedding j x : e C → J e C . By functoriality, we get maps ν C/A, ∗ : J e C → A and ν ∗ C/A : A ∨ → J e C where ν C/A = ν C/A, ∗ ◦ j x . Given a divisor D on A , from [47] Lemma 3 we get(5.2) α A ( C, D ) = ν C/A, ∗ ◦ ν ∗ C/A ◦ φ D . The following result follows from Lemma 3.6 in [16]. emma 5.2. Let C be an irreducible curve with h C i = A and let D be an ample divisor on A .There is a monic polynomial Q C,D ( t ) ∈ Z [ t ] of degree n such that P α A ( C,D ) ( t ) = Q C,D ( t ) . Theroots of Q C,D ( t ) are all real and positive. If Z is an r -cycle on A , the degree of Z relative to a divisor H is the intersection numberdeg H ( Z ) = deg( H r .Z ). If X ⊆ A is a smooth subvariety of dimension d ≥
1, the canonical degree of X relative to H is cdeg H ( X ) = deg( H d − .K X ), where K X is a canonical divisor on X consideredas an algebraic cycle in A . For instance, if B is an abelian subvariety of A , then cdeg H ( B ) = 0. Lemma 5.3 (Degrees for surfaces in abelian 3-folds) . Let X be a smooth projective surface containedin an abelian threefold A . Then X | X defines a canonical divisor on X and deg X ( X ) = cdeg X ( X ) = deg( X ) A = c ( X ) . Proof. As A has trivial canonical sheaf, X | X gives a canonical divisor on X by the adjunctionformula ([32] Proposition II.8.20). Thus, deg( X ) A = deg( X.K X ) A = deg( K X .K X ) X = c ( X ). (cid:3) Restriction of -forms. We assume that k has positive characteristic p . Lemma 5.4 (Injectivity for curves) . Let H be an ample divisor on A . Let C be an irreduciblecurve in A with h C i = A . Suppose that p > n ! n n · deg H ( C ) n deg( H n ) . Then the map ν • C/A : H ( A, Ω A/k ) → H ( e C, Ω e C/k ) is injective.Proof. Fix a point x ∈ e C and consider the associated embedding j : e C → J = J e C . We havean isomorphism j • : H ( J, Ω J/k ) → H ( e C, Ω e C/k ) which is canonical in the sense that it does notdepend on the choice of x (because the regular differentials on J are translation invariant).Let α = α A ( C, H ) and β = ν C/A, ∗ ◦ ν ∗ C/A : A ∨ → A . Thus, α = β ◦ φ H by (5.2).Since H is ample, Lemma 5.2 gives P α ( t ) = Q ( t ) for a monic polynomial Q ( t ) ∈ Z [ t ] of degree n , whose roots γ , ..., γ n (counting multiplicity) are real and positive. Hence deg( α ) = P α (0) = Q (0) = ( γ · · · γ n ) >
0. It follows that α : A → A is an isogeny, and so is β : A ∨ → A .Furthermore, deg( φ H ) = deg( H n ) /n ! so we getdeg( β ) = deg( α )deg( φ H ) = (cid:18) n ! Q (0)deg( H n ) (cid:19) . This integer is the square of a rational number, so M = n ! | Q (0) | / deg( H n ) is a positive integer. Bythe arithmetic-geometric mean inequality we have M = n !deg( H n ) · γ · · · γ n ≤ n !deg( H n ) (cid:18) γ + ... + γ n n (cid:19) n = n !deg( H n ) (cid:18) deg( C.H ) n (cid:19) n where the last equality uses tr( α ) = − γ + ... + γ n ) and (5.1). Therefore, p > M .Since M is a positive integer and deg( β ) = M , we conclude that p does not divide deg( β ).So, β is a separable isogeny, giving that β • : H ( A, Ω A/k ) → H ( A ∨ , Ω A ∨ /k ) is an isomorphsm.As β = ν C/A, ∗ ◦ ν ∗ C/A , we get that ( ν C/A, ∗ ) • : H ( A, Ω A/k ) → H ( J, Ω J/k ) is injective. Since ν C/A = ν C/A, ∗ ◦ j and j • is an isomorphism, we conclude that ν • C/A is injective. (cid:3)
We remark that if n = 3 and C is smooth, then the condition on p can be dropped by a resultof Nakai [53]. However, this is not enough for us because we will consider possibly singular curves. emma 5.5 (Semi-injectivity on canonical divisors) . Let X be a smooth surface contained in A .Let D be an effective canonical divisor of X and let C ⊆ X be an irreducible curve contained inthe support of D . Consider the map ν • C/A : H ( A, Ω A/k ) → H ( e C, Ω e C/k ) . We have the following: (i) If n = 3 , X is an ample divisor on A containing no elliptic curves, and (5.3) p > c ( X ) , then dim ker( ν • C/A ) ≤ . (ii) If A is simple and there is an ample divisor H on A satisfying (5.4) p > n ! n n · cdeg H ( X ) n deg( H n ) , then ν • C/A is injective.Proof.
If (i) holds we consider the ample divisor H = X and note that Lemma 5.3 givesdeg H ( X ) = deg( X ) A = cdeg( X ) = c ( X ) . Thus, condition (5.4) in case (i) simplifies to p > c ( X ) /
3, which is implied by (5.3).Let B = h C i . If (ii) holds then B = A , while if (i) holds then B = A or dim B = 2 since C isnot an elliptic curve. Thus, there are two cases to consider:(a) A = B and either (i) or (ii) holds.(b) dim B = 2 and (i) holds.In case (a), (5.3) and (5.4) directly allow us to apply Lemma 5.4 to get injectivity of ν • C/A .In case (b) we note that D is ample since it is linearly equivalent to X | X on X (by Lemma 5.3)and X is ample in A . In particular, D is numerically effective and non-zero, so Lemma 3.6 applies.The curve C is an ample divisor on the abelian surface B because h C i = B . Hence, the adjunctionformula for C in B · deg C ( C ) deg( C.C ) B = deg( C.C ) B g a ( C ) − ≤ c ( X )where the last bound is due to Lemma 3.6. Thus, by (5.3) Lemma 5.4 gives that ν • C/B : H ( B, Ω B/k ) → H ( e C, Ω e C/k ) is injective. Let i B/A : B → A be the inclusion of B in A . Since B is (up to transla-tion) an abelian subvariety of A , the map i • B/A : H ( A, Ω A/k ) → H ( B, Ω B/k ) is surjective, hence,its kernel has dimension 1. This proves dim k ker( ν • C/A ) = 1 in case (b). (cid:3)
We also need an injectivity result for surfaces.
Lemma 5.6 (Injectivity for surfaces) . Let X be a smooth surface contained in A . Let ι : X → A be the inclusion. Assume that either of the following conditions holds: (i) n = 3 and h X i = A . (ii) A is simple and there is an ample divisor H on A such that p > n · n ! n n · deg H ( X ) n deg( H n ) . Then the map ι • : H ( A, Ω A/k ) → H ( X, Ω X/k ) is injective.Proof. If (i) holds, the desired injectivity follows from [54] Theorem 5(I), since the latter resultcovers the case of generating hypersurfaces in abelian varieties.Let us assume (ii). By [51] p.163, 3 H is a very ample divisor on A . By Bertini’s theorem ([32]II.8.18 and III.7.9.1) we can choose a divisor D ∼ H on A such that C = D.X is a smooth rreducible curve. Let ν : C → A be the inclusion; this is the same as ν C/A because C = e C . Since ν factors through ι : X → A , it suffices to show that ν • : H ( A, Ω A/k ) → H ( C, Ω C/k ) is injective.As A is simple, h C i = A . We note that deg H ( C ) = deg( H.C ) = deg(3
H.X.H ) = 3 deg H ( X ), sothe condition on p given by (ii) allows us to conclude by Lemma 5.4. (cid:3) Restriction of -forms. We keep the assumption that k has positive characteristic p . Lemma 5.7 (Non-vanishing of 2-forms on surfaces) . Assume n ≥ . Let X be a smooth surface in A , let ι : X → A be the inclusion, and let ω , ω ∈ H ( A, Ω A/k ) be differentials satisfying ω ∧ ω = 0 in H ( A, Ω A/k ) . Assume that either of the following conditions holds: (i) n = 3 , X is an ample divisor on A containing no elliptic curves, and (5.5) p > (128 / · c ( X ) . (ii) A is simple and there is an ample divisor H on A satisfying (5.6) p > n ! n n · (3 deg H ( X ) + cdeg H ( X )) n deg( H n ) . Then ι • ( ω ) ∧ ι • ( ω ) ∈ H ( X, Ω X/k ) is not the zero section.Proof. Let us write u = ι • ( ω ) and u = ι • ( ω ), which are elements of H ( X, Ω X/k ). We note thatif (i) holds, then h X i = A since X is ample. Thus, in cases (i) and (ii) we see that u and u arelinearly independent over k , by Lemma 5.6.For the sake of contradiction, suppose that(5.7) u ∧ u = 0 in H ( X, Ω X/k ) . Then, there is a rational function f ∈ k ( X ) such that u = f · u and we have that f is non-constantbecause u , u ∈ H ( X, Ω X/k ) are linearly independent over k .Let C be an irreducible curve in the support of the divisor of zeros of f , which is non-trivialbecause f is non-constant. As C is in the zero locus of f and u = f · u , we get that C is containedin the zero locus of u , and in particular(5.8) ν • C/A ( ω ) = ν • C/X ( u ) = 0 . If (i) holds, we make the notation more uniform by choosing H = X on A . Lemma 5.3 yields(5.9) deg H ( X ) = deg( X ) A = cdeg H ( X ) = c ( X )and since n = 3 when (i) holds, we see that in this case the conditions (5.5) and (5.6) are the same.Let us construct an auxiliary curve D . Since H is ample, 3 H is very ample ([51] p.163) andby Bertini ([32] Theorem II.8.18) there is a smooth irreducible curve D ⊆ X linearly equivalentto 3 H | X as divisors in X . Given a non-zero reduced effective divisor Z in X which contains C (possibly Z = C ) we can require that D meets Z transversely at smooth points of Z and that D does not pass through points in the intersection of C with the divisor of poles of f . We remarkthat the intersection D.C is non-empty because H is ample. A precise choice of Z will be madelater, but this free parameter will not affect the computations below.By the adjunction formula we have(5.10) 2 g g ( D ) − D + K X ) .D ) X = deg(3 H. ( D + K X )) A = 9 deg H ( X ) + 3 cdeg H ( X ) . We point out that if (i) holds, then (5.9) shows that this expression simplifies to(5.11) 2 g g ( D ) − c ( X ) . We claim that h D i = A . This holds in case (ii) because A is simple. In case (i) note that h D i isnot an elliptic curve since D ⊆ X , so, if h D i 6 = A , then h D i is an abelian surface. The latter would mply that h D i .D = 0 in A since we can translate the abelian surface h D i so that it does not meet D ⊆ h D i . But it is not possible that h D i .D = 0 because in case (i) we have D = 3 H.X = 3
X.X where X is ample. Thus, h D i = A in case (i) and (ii).We note that by (5.5) if (i) holds, and by (5.6) if (ii) holds, we have n ! n n · deg H ( D ) n deg( H n ) = n ! n n · deg( H n − . H.X ) n deg( H n ) = n ! n n · n deg H ( X ) n deg( H n ) < p. In view of this bound and the fact that h D i = A , Lemma 5.4 gives injectivity of the map(5.12) ν • D/A : H ( A, Ω A/k ) → H ( D, Ω D/k ) . Since the map ν • D/A from (5.12) is injective, we see that 0 = ν • D/A ( ω ) = ν • D/X ( u ). Since C iscontained in the zero locus of u , we see that ν • D/X ( u ) vanishes at each point of D ∩ C ⊆ D . Asthis intersection is transverse and ν • D/X ( u ) = 0, from (5.10) we deduce(5.13) deg(( D.C ) X ) = D ∩ C ≤ deg D ( ν • D/X ( u )) = 2 g g ( D ) − H ( X ) + 3 cdeg H ( X ) . Note that if (i) holds, then (5.11) gives(5.14) deg((
D.C ) X ) ≤ c ( X ) . Let B = h C i . As in the proof of Lemma 5.5, we note that if (ii) holds then B = A , while if (i)holds then B = A or dim B = 2 since C is not an elliptic curve. This leaves two cases:(a) A = B and either (i) or (ii) holds.(b) dim B = 2 and (i) holds.First we consider case (a), and let us recall that if (i) holds then we are choosing H = X . Since( D.C ) X = (3 H.C ) A , the bound (5.13) impliesdeg H ( C ) = ( H.C ) A ≤ H ( X ) + cdeg H ( X ) . By (5.5) and (5.6), it follows that n ! n n · deg H ( C ) n deg( H n ) ≤ n ! n n · (3 deg H ( X ) + cdeg H ( X )) n deg( H n ) < p. Since A = B (we are in case (a)), we can apply Lemma 5.4 to conclude that ν • C/A : H ( A, Ω A/k ) → H ( e C, Ω e C/k )is injective. This is a contradiction by (5.8). Therefore we cannot have (5.7) in case (a).Let us now consider case (b). We begin by applying Lemma 5.4 to C and the abelian surface B = h C i , choosing the ample divisor in B given by the 1-cycle X ′ = ( X.B ) A . Using (5.14) we finddeg(( X ′ .C ) B ) = deg(( X.C ) A ) = deg(( K X .C ) X ) = 13 deg(( D.C ) X ) ≤ c ( X ) . On the other hand, since C is a component of the effective 1-cycle X ′ = ( X.B ) A we getdeg(( X ′ .X ′ ) B ) ≥ deg(( X ′ .C ) B ) . From (5.5) we deduce2!2 · deg X ′ ( C ) deg(( X ′ .X ′ ) B ) = 12 · deg(( X ′ .C ) B ) deg(( X ′ .X ′ ) B ) ≤
12 deg(( X ′ .C ) B ) ≤ c ( X ) < p. Thus, Lemma 5.4 implies the injectivity of the map(5.15) ν • C/B : H ( B, Ω B/k ) → H ( e C, Ω e C/k ) . et i B/A : B → A be the inclusion. This is a closed immersion, hence it induces a surjection oncotangent spaces at each point. By translation with the group structure of A we see that the map i • B/A : H ( A, Ω A/k ) → H ( B, Ω B/k )is surjective. Hence, dim k ker( i • B/A ) = 1. Moreover, ω ∈ ker( ν • C/A ) by (5.8). This, together withthe injectivity of (5.15) implies(5.16) ker (cid:16) i • B/A : H ( A, Ω A/k ) → H ( B, Ω B/k ) (cid:17) = h ω i . This, together with the injectivity of (5.15), gives(5.17) ker (cid:16) ν • C/A : H ( A, Ω A/k ) → H ( e C, Ω e C/k ) (cid:17) = h ω i . Let B be a translate of B such that 0 A ∈ B . Then B is an abelian subvariety of A . Definethe elliptic curve E = A/B and the quotient map π : A → E . Since ker( π ) = B is smooth over k , the quotient map π is smooth and we deduce that(5.18) π • : H ( E, Ω E/k ) → H ( A, Ω A/k )is injective. Let ω E ∈ H ( E, Ω E/k ) be a generator. Since π | B : B → E is constant, we have i • B/A ( π • ( ω E )) = ( π | B ) • ( ω E ) = 0 . From (5.16) we deduce π • ( ω E ) ∈ ker( i • B/A ) = h ω i and by injectivity of π • (from (5.18)) we getim (cid:16) π • : H ( E, Ω E/k ) → H ( A, Ω A/k ) (cid:17) = h ω i ⊆ H ( A, Ω A/k ) . Thus, replacing ω E by a suitable non-zero multiple we may assume π • ( ω E ) = ω and in particular(5.19) ( π | X ) • ( ω E ) = ι • ( π • ( ω E )) = ι • ( ω ) = u . We claim that ν C/X : e C → X is u -integral.Let us choose Z as the (reduced) support of B ∩ X . Note that C ⊆ Z and that Z has dimension1 since X is not contained in B ( X is ample in A ). As explained in the construction of D , theintersection C ∩ D is non-empty. Let x ∈ C ∩ D and recall from the construction of D that x is asmooth point of Z (in particular, of C ) and it is not contained in the pole divisor of f in X .Let s , s ∈ O X,x be a system of local parameters for X at x such that s is a local equation for C at x . Then f = s a · θ for some a ≥ θ ∈ O × X,x and in particular, u = s a · θ · u in Ω X/k,x .Let y = π ( x ) ∈ E and let t ∈ O E,y be a local parameter for E at y . Then ω E = σdt for certain σ ∈ O × E,y because the invariant differential ω E has no zeros. Let ϕ = ( π | X ) x : O E,y → O X,x . By(5.19) we have u = ϕ ( σ ) dϕ ( t ) in Ω X/k,x .Note that π ( B ) = { y } , so, Z is the support of B ∩ X = ( π | X ) − ( y ). As x is a smooth point of Z and C is a component of Z , we get ϕ ( t ) = s b · γ for certain b ≥ γ ∈ O × X,x . Thus, s a · θ · u = u = ϕ ( σ ) dϕ ( t ) = ϕ ( σ ) (cid:16) bs b − γds + s b dγ (cid:17) where we recall that a, b ≥
1. Since θ and σ are units in their respective local rings, we have that τ = θ · ϕ ( σ − ) ∈ O × X,x . With this notation,(5.20) s a · τ · u = s b − · ( bγds + s dγ ) . Let us consider two cases depending on the values of a and b : a < b . In this case s b − − a is regular at x and we get(5.21) u = τ − · s b − − a · ( bγds + s dγ ) . Note that C is smooth at x and O C,x ≃ O X,x / ( s ). Let x ′ ∈ e C be the only preimage of x under ν C/X . Then Ω e C/k,x ′ ≃ Ω C/k,x ≃ Ω X/k,x / ( s , ds ) and under this isomorphism thelocal map ν • C/X,x ′ : Ω X/k,x → Ω e C/k,x ′ becomes just the quotient Ω X/k,x → Ω X/k,x / ( s , ds ).The image of bγds + s dγ under this quotient is 0, so ν • C/X,x ′ ( u ) = 0 by (5.21). By Lemma4.3 and injectivity of H ( e C, Ω e C/k ) → Ω e C/k,x ′ we deduce ν • C/X ( u ) = 0. • a ≥ b . In this case (5.20) shows that s divides bγds in Ω X/k,x . As s is part of systemof local parameters at x and γ is invertible in O X,x , this means that b ≡ p . Inparticular, a ≥ b ≥ p because a, b ≥
1. Recall that D is smooth, C is smooth at x , and D, C meet transversely at x . Let x ∈ D be the only preimage of x under ν D/X . By (5.11)and recalling that u = s a · θ · u we find a = ord x ( ν ∗ D/X ( s a )) ≤ ord x ( ν • D/X ( u )) ≤ deg D ( ν • D/X ( u )) = 12 c ( X ) < p by (5.5). This is not possible, so the case a ≥ b cannot occur.This proves that a < b and ν • C/X ( u ) = 0. Thus, ν C/X is u -integral as claimed.Finally, we have u ∈ ker( ν • C/X ) which implies ω ∈ ker( ν • C/A ). By (5.17), this implies ω ∈ h ω i in H ( A, Ω A/k ). This is the desired contradiction in case (b). Therefore, (5.7) cannot hold. (cid:3) Zeros of p -adic power series We keep the notation from Section 2.3.6.1.
One variable.
Let z be a variable. Given a power series h ( z ) = P ∞ n =0 a j z j ∈ K [[ z ]] theradius of convergence of h is defined by ρ h = sup { r : lim j | a j | r j = 0 } .If ρ h > h ( z ) converges on B ( ρ h ), where it defines an analytic function. In particular,the vanishing order ord z ( h ) is defined at each z ∈ B ( ρ h ). For each 0 < r < ρ h we define | h | r = max j | a j | r j , ν ( h, r ) = max { m : | a m | r m = | h | r } , and n ( h, r ) = X | z |≤ r ord z ( h ) . The following classical bound will suffice for our purposes. See, for instance, Theorem 1.21 in [35].
Lemma 6.1. If h = 0 and ρ h > , then for each < r < ρ h we have n ( h, r ) ≤ ν ( h, r ) . Several variables.
Let n ≥ x = ( x , ..., x n ) be an n -tuple of variables. Given H ( x ) = X α ∈ N n c α x α ∈ K [[ x ]]the radius of convergence ρ H is defined by ρ H = sup { r : lim m max k α k = m | c α | r m = 0 } . We note thatthis definition agrees with the one variable case when n = 1. If ρ H > H converges to ananalytic function on B n ( ρ H ). For each j ≥ P H,j ( x ) = P k α k = j c α x α ; this is the homogeneousdegree j part of H . For u ∈ K n with | u | = 1 we define H u ( z ) = H ( z · u ) = ∞ X j =0 P H,j ( u ) · z j ∈ K [[ z ]] . It follows that ρ H u ≥ ρ H for each u ∈ K n with | u | = 1. emma 6.2 (Multivariable zero estimate) . Let H ( x ) = P α c α x α ∈ K [[ x ]] and let u ∈ K n with | u | = 1 . Suppose that there is an integer N ≥ and a real number M ≥ such that (i) For each α we have | c α | ≤ M k α k− . (ii) max {| P H,j ( u ) | : 0 ≤ j ≤ N } ≥ (in particular, H u = 0 ).Then ρ H ≥ M − and for every < r < M − we have n ( H u , r ) ≤ (cid:18) N − log M log( r − ) (cid:19) (cid:18) − log M log( r − ) (cid:19) − . Proof.
By (i), ρ H ≥ M − . Fix 0 < r < M − , in particular r <
1. From (ii) and r < r N ≤ max ≤ j ≤ N | P H,j ( u ) | r j . Let λ = (log M ) / log( r − ) and note that 0 < λ <
1. For each m > ( N − λ ) / (1 − λ ) we have M m − r m = M − ( M r ) m < M − ( M r ) ( N − λ ) / (1 − λ ) = r N because M r <
1. By (i) and these observations, for each m > ( N − λ ) / (1 − λ ) we get(6.1) | P H,m ( u ) | r m ≤ M m − r m < r N ≤ max ≤ j ≤ N | P H,j ( u ) | r j . Let N ′ = ⌊ ( N − λ ) / (1 − λ ) ⌋ and note that N ′ ≥ N . From (6.1) we deduce | H u | r ≤ max {| P H,j ( u ) | r j : 0 ≤ j ≤ N ′ } and ν ( H u , r ) ≤ N ′ . The result follows from Lemma 6.1. (cid:3) Exponential and logarithm
Let us keep the notation from Section 2.3. Let A be an abelian variety over Spec R of relativedimension n ≥ Local parameters.
Let σ : Spec R → A be a section with image Z = σ (Spec R ). The closedpoint of Z is x = σ ( p ). Let b O A ,x , Z be the completion of O A ,x along Z . That is, if I Z is theideal sheaf of Z , then b O A ,x , Z is the completion of O A ,x with respect to the ideal I Z ,x ⊆ O A ,x .Let y , ..., y n be variables. We say that elements t , ..., t n ∈ m A ,x are R -local parameters along σ (or Z ) if the rule y j t j induces a continuous isomorphism of R -algebras b O A ,x , Z ≃ R [[ y , ..., y n ]]. Lemma 7.1 (Choice of local parameters) . With the previous notation, let X be a regular closedsubscheme of A containing Z which is smooth of relative dimension d ≥ over Spec R . There are t j ∈ m A ,x for ≤ j ≤ n such that t , ..., t n are R -local parameters along Z and t , ..., t n − d are asystem of local equations for X in A along Z . In particular, R -local parameters along Z exist.Proof. By [30] Expos´e II, Th´eor`eme 4.15 with X = A , Y = X , S = Spec R , and x = x we get theexistence of a regular sequence t , ..., t n − d ∈ m A ,x that generates the ideal I X ,x of X in O A ,x .Applying [30] Expos´e II, Corollaire 4.17 with X = A , Y = Spec R , i = σ , and y = p ∈ Spec R weget that b O A ,x , Z is a power series ring over R in n variables. Analyzing the proof, one sees that thevariables of this power series ring can be constructed by extending any regular sequence in I Z ,x ,see [30] Expos´e II, Remarques 4.14. We conclude since I X ,x ⊆ I Z ,x . (cid:3) Let t = ( t , ..., t n ) be a choice of R -local parameters along the identity section e . These R -localparameters determine a formal group F = ( F , ..., F n ) with F j ∈ R [[ X , ..., X n , Y , ..., Y n ]] foreach 1 ≤ j ≤ n , where the X j and Y j are variables, see [33] Lemma C.2.4.Given an integer m , let Ψ [ m ] = (Ψ [ m ]1 , ..., Ψ [ m ] n ) be the power series expansion of the morphism[ m ] : A → A of multiplication by m in terms of t . Then, integrality of the formal group F gives emma 7.2. Ψ [ m ] j ∈ R [[ t ]] for each j = 1 , ..., n . Although the notation does not explicitly indicate so, the coefficients of the power series Ψ [ m ] j depend upon the choice of local parameters t . Nevertheless, since [1] = Id we observe: Lemma 7.3. Ψ [1] j = t j for each j = 1 , ..., n . From [11] III.5.3 Proposition 2(i) we get
Lemma 7.4.
The terms of degree less than m in Ψ [ m ] j vanish for each j = 1 , ..., n . Power series construction.
Let ψ [ m ] j be the homogeneous part of degree m in Ψ [ m ] j . Thus, ψ [1] j = t j by Lemma 7.3. Let x , ..., x n and y , ..., y n be variables and for each j = 1 , ..., n we defineLog t j ( y ) = ∞ X m =1 ( − m m · Ψ [ m ] j ( y ) ∈ K [[ y ]]Exp t j ( x ) = ∞ X m =1 m ! · ψ [ m ] j ( x ) ∈ K [[ x ]] . These power series depend upon the choice of local parameters t as the notation indicates. Whenthis choice is clear, we will simply writeLog t j ( y ) = X α b j,α · y α and Exp t j ( x ) = X α c j,α · x α . The coefficients b j,α and c j,α also depend on the choice of t , although the notation is not explicit.By [11] III.5.4 Prop. 3, these are the power series expansions of the exponential and logarithmmaps of the p -adic Lie group A ( K ) at the identity point e , with respect to the choice of localparameters t . They converge on some p -adic neighborhood of e , but for our purposes we need amore precise discussion on convergence. Lemma 7.5 (Convergence of logarithm) . For each j = 1 , ..., n , the radius of convergence of Log t j is at least . Furthermore, for each α ∈ N n we have | b j,α | ≤ k α k [ K : Q p ] .Proof. From Lemmas 7.2 and 7.4 we get | b j,α | ≤ max {| m − | : 1 ≤ m ≤ k α k} ≤ k α k [ K : Q p ] . Theclaim on the radius of convergence follows. (cid:3) Lemma 7.6 (Convergence of the exponential) . For each j = 1 , ..., n and each α ∈ N n , we have m ! · c j,α ∈ R where m = k α k . Furthermore, | c j,α | ≤ p [ K : Q p ]( m − / ( p − . In particular, the radius ofconvergence of Exp t j is at least p − [ K : Q p ] / ( p − .Proof. Since c j,α is a coefficient of m ! − ψ [ m ] j ( x ), we get m ! · c j,α ∈ R from Lemma 7.2. Letting s p ( m ) be the sum of the digits of m in base p , we get | c j,α | ≤ | m ! | = p [ K : Q p ]( m − s p ( m )) / ( p − ≤ p [ K : Q p ]( m − / ( p − . The claim on the radius of convergence follows. (cid:3)
Local linearization.
We consider the open set V = p × ... × p ⊆ K n and the additive Liegroup T given by V with the usual addition. The variables x = ( x , ..., x n ) and y = ( y , ..., y n )will be considered as coordinates on T and V respectively.Since A → Spec R is proper, we have the reduction map red : A ( K ) = A ( K ) → A ′ ( k ) where A ′ = A ⊗ k . This is a group morphism. Then U e = ker(red) is a Lie subgroup of A ( K ). he R -local parameters t = ( t , ..., t n ) define functions t j : U e → p which determine a K -analyticlocal chart χ t : U e → V . The map χ t is bijective, although it does not need to be a group morphism. Lemma 7.7.
Suppose that p > max { e + 1 , exp( e / exp(1)) } . Then we have the following: (i) Log t j and Exp t j have radius of convergence larger than /q for each j = 1 , ..., n . (ii) The power series
Log t ( y ) = (Log t j ( x )) nj =1 and Exp t ( x ) = (Exp t j ( y )) nj =1 give analytic maps Log t : V → T and Exp t : T → V which are inverse to each other. (iii) The maps g Log t and g Exp t defined by g Log t = Log t ◦ χ t : U e → T and g Exp t = ( χ t ) − ◦ Exp t : T → U e are isomorphisms of Lie groups, inverse to each other.Proof. (i) Lemma 7.5 implies the result for Log t j . The condition p > e + 1 gives [ K : Q p ] < ( p − f , hence p [ K : Q p ] / ( p − < q . Lemma 7.6 shows that Exp t j has radius of convergence at least p − [ K : Q p ] / ( p − , which is larger than 1 /q .(ii) As open sets of K n , we have V = T = B n [1 /q ]. Thus, (i) gives that Log t and Exp t areanalytic on the domains V and T respectively, as maps to K n .Let us show that Log t maps V to T . Since B n [1 /q ] = B n (1) in K n , it suffices to check that forall v ∈ V and j = 1 , ..., n we have | Log t j ( v ) | <
1. By Lemma 7.5 we have | Log t j ( v ) | ≤ max m ≥ m [ K : Q p ] /q m = max (cid:26) /q, max m ≥ m [ K : Q p ] /q m (cid:27) . By p > exp( e / exp(1)) we have m/ log m > exp(1) > e / log p for all m ≥
2. Thus, [ K : Q p ] log m = ef log m < f m log p = m log q . Hence, m [ K : Q p ] /q m < m ≥
2, proving | Log t ( v ) j | < t maps T to V , it suffices to check that for each v ∈ T and j = 1 , ..., n wehave | Exp t j ( v ) | <
1. Lemma 7.6 gives | Exp t j ( v ) | ≤ sup m ≥ p [ K : Q p ] m − p − /q m = sup m ≥ p [ K : Q p ] m − p − − f m . Since p > e +1, we get ( m − e < ( p − m . This gives [ K : Q p ] m − p − − f m <
0, proving | Exp t j ( v ) | < t and Exp t are inverse to each other is a power series identity, see [11] III 5.4.Finally, (iii) also follows from [11] III 5.4, where a neighborhood of e is identified with an openset of K n by the choice of local parameters made in [11] III 5.3. (cid:3) Analytic -parameter subgroups Using the results on the logarithm and exponential maps from the previous section, in this sectionintroduce analytic 1-parameter subgroups of abelian varieties in the non-archimedian setting andstudy some fundamental properties. The naive idea is that a 1-parameter subgroup of an abelianvariety should be an analytic curve at the identity element which is locally a subgroup. However,it is more convenient for our purposes to take a different approach.Again, we let A be an abelian variety over Spec R of relative dimension n ≥
1. We keep thenotation introduced in Sections 2.3 and 7. We assume p > max { e + 1 , exp( e / exp(1)) } throughout this section, so that Lemma 7.7 applies. .1. Definitions. If G is an abelian K -analytic Lie group, the K -vector space of invariant dif-ferentials will be denoted by Ω( G ). A morphism of abelian Lie groups f : G → G functoriallyinduces a linear map on invariant differentials, which we denote by f • : Ω( G ) → Ω( G ).A 1 -parameter subgroup of A is a pair γ = ( t , u ) where t = ( t , ..., t n ) is a choice of R -localparameters at e ∈ A ( K ) and u ∈ K n satisfies | u | = 1.The space of invariant differentials on the Lie group T is Ω( T ) = L nj =1 K · dx j and the spaceof invariant differentials on A ( K ) is Ω( A ( K )) = H ( A, Ω A/K ).Given t a choice of R -local parameters at e ∈ A ( K ), the isomorphism of Lie groups g Exp t : T → U e (cf. Lemma 7.7) induces an isomorphism of K -vector spaces on invariant differentials(8.1) ( g Exp t ) • : H ( A, Ω A/K ) → Ω( T ) . Given u = ( u , ..., u n ) ∈ K n with | u | = 1, we define the K -linear map ǫ u : Ω( T ) → K by the rule dx j u j . For a 1-parameter subgroup γ = ( t , u ) we get a non-trivial K -linear map ǫ u ◦ ( g Exp t ) • : H ( A, Ω A/K ) → K. Let us define H ( γ ) = ker( ǫ u ◦ ( g Exp t ) • ) and note that it is a hyperplane in H ( A, Ω A/K ).A 1-parameter subgroup γ = ( t , u ) determines a morphism of Lie groups(8.2) ˜ γ : p → U e , ˜ γ ( z ) = g Exp t ( u z, ..., u n z )where z denotes the variable on the additive Lie group p . The space of invariant differentials on p is Ω( p ) = K · dz . One immediately verifies Lemma 8.1.
The map ˜ γ • : H ( A, Ω A/K ) → Ω( p ) induced by ˜ γ is given by ω ( ǫ u ◦ ( g Exp t ) • )( ω ) · dz .In particular, ker(˜ γ • ) = H ( γ ) . The image of a 1-parameter subgroup γ is defined as im( γ ) = ˜ γ ( p ). We observe that Lemma 8.2. If γ = ( t , u ) is a -parameter subgroup of A , then im( γ ) is a Lie subgroup of U e .The image of im( γ ) under the isomorphism g Log t : U e → T is p · u , which is a closed, saturated R -submodule of T , free of rank over R , and generated by ̟ · u . Equivalence.
Given γ and γ ′ two 1-parameter subgroups of A , we say that they are equivalent if H ( γ ) = H ( γ ′ ). This is denoted by γ ∼ γ ′ . Lemma 8.3.
Let t be a choice of R -local parameters at e and let u , u ′ ∈ K n with | u | = | u ′ | = 1 .Let γ = ( t , u ) and γ ′ = ( t , u ′ ) . We have γ ∼ γ ′ if and only if there is η ∈ R × with u ′ = η · u .Proof. As (8.1) is an isomorphism, we have H ( γ ) = H ( γ ′ ) if and only if ker( ǫ u ) = ker( ǫ u ′ ). Thelatter holds if and only if u ′ = η · u for some η ∈ K × , in which case η ∈ R × as | u | = | u ′ | = 1. (cid:3) Lemma 8.4.
Let γ and γ ′ be -parameter subgroups of A . The following are equivalent: (i) γ ∼ γ ′ . (ii) im( γ ) = im( γ ′ ) . (iii) im( γ ) and im( γ ′ ) have non-trivial intersection, i.e. { e } ( im( γ ) ∩ im( γ ′ ) .Proof. Write γ = ( t , u ) and γ ′ = ( t ′ , u ′ ). Define δ : T → T as δ = g Log t ◦ g Exp t ′ . Then δ is aLie group automorphism of the additive Lie group T . Thus, δ is given by δ ∈ GL n ( R ) acting on T = p × ... × p on the left. fter these remarks, Lemma 8.2 shows that (ii) and (iii) are equivalent, and in fact, both areequivalent to the following condition: (iv) there exists η ∈ R × with δ ( u ′ ) = η · u (note that as anelement of GL n ( R ), δ acts on K n ).Let us show that (iv) is equivalent to (i). Note that the explicit definition of ǫ u and the fact that | u | = | u ′ | = 1 give that (iv) is equivalent to the condition ker( ǫ u ) = ker( ǫ δ ( u ′ ) ). Directly evaluatingon dx i we see that ǫ δ ( u ′ ) = ǫ u ′ ◦ δ • , so, (iv) is equivalent to ker( ǫ u ) = ker( ǫ u ′ ◦ δ • ). Using the factthat δ • ◦ ( g Exp t ) • = ( g Exp t ◦ δ ) • = ( g Exp t ′ ) • and that ( g Exp t ) • is an isomorphism, we finally see that(iv) is equivalent to ker( ǫ u ◦ ( g Exp t ) • ) = ker( ǫ u ′ ◦ ( g Exp t ′ ) • ), which is exactly (i). (cid:3) Lemma 8.5.
The rule γ H ( γ ) defines a bijection between equivalence classes of -parametersubgroups of A and K -linear hyperplanes of H ( A, Ω A/K ) .Proof. By Lemma 8.4, it only remains to show that given any hyperplane H of H ( A, Ω A/K )there is a 1-parameter subgroup γ with H = H ( γ ). Choose R -local parameters t at e . Let H ′ = ( g Exp t ) • ( H ) and note that it is a hyperplane in Ω( T ). We can choose u ∈ K n with | u | = 1such that ker( ǫ u ) = H ′ , hence H = ker( ǫ u ◦ ( g Exp t ) • ) as desired. (cid:3) Lemma 8.6.
Let ξ ∈ U e with ξ = e and let t be a choice of R -local parameters at e . There exists u ∈ K n with | u | = 1 such that ξ ∈ im( γ ) for γ = ( t , u ) . Furthermore, u is unique for this ξ and t ,up to multiplication by elements of R × .Proof. Let v = g Log( ξ ) ∈ T . We have v = 0 because ξ = e and g Log : U e → T is an isomorphismof Lie groups. Take m ∈ Z such that | ̟ m | = | v | . We can choose u = ̟ − m · v .If γ = ( t , u ) and γ ′ = ( t , u ′ ) satisfy that ξ ∈ im( γ ) ∩ im( γ ′ ) then γ ∼ γ by Lemma 8.4, and weconclude by Lemma 8.3. (cid:3) Infinitesimal -parameter subgroups. For an integer m ≥ • E m = R [ z ] / ( z m +1 ), V m = Spec E m , • E m = E m ⊗ R K , V m = Spec E m , • E ′ m = E m ⊗ R k , V ′ m = Spec E ′ m .Let γ = ( t , u ) be a 1-parameter subgroup of A . The power series expansion of the Lie groupmorphism ˜ γ : p → U e (see (8.2)) with respect to z and t induces a morphism of formal schemes b γ : Spf K [[ z ]] → Spf b O A,e determined by the following map on completed local rings:(8.3) b γ : b O A,e → K [[ z ]] , t j Exp t j ( u z, ..., u n z ) . For each m ≥
0, the map b γ induces a map b γ m : V m → Spf b O A,e giving the morphism of K -schemes˜ γ m : V m → A supported at e . Explicitly, b γ m induces(8.4) b γ m : b O A,e → E m , t j Exp t j ( u z, ..., u n z ) mod z m +1 on completed local rings, and the restriction to O A,e is the map ˜ γ m : O A,e → E m induced by ˜ γ m .In summary, the following diagram commutes:(8.5) b O A,e K [[ z ]] O A,e E m . b γ b γ m ˜ γ m he morphism ˜ γ m is the m -th jet of ˜ γ , which might be thought of as an infinitesimal 1-parametersubgroup. We remark that, although ˜ γ is an analytic maps of Lie groups, the maps ˜ γ m : V m → A are scheme morphisms.For each integer h ≥ P t j,h ( x ) ∈ K [ x ] as thehomogeneous part of degree h of Exp t j ( x ), which in the notation of Section 7.2 can be written as P t j,h ( x ) = 1 m ! · ψ [ h ] j ( x ) = X k α k = h c j,α · x α . In particular, P t j, = 0. With this notation, the map b γ from (8.4) can be expressed as(8.6) b γ m ( t j ) = m X h =0 P t j,h ( u ) · z h mod z m +1 . Lemma 8.7 (Integrality and reduction modulo p ) . Let γ = ( t , u ) be a -parameter subgroup of A . If m < p , the morphism ˜ γ m : V m → A extends to an R -morphism V m → A supported alongthe identity section σ e : Spec R → A . Hence, upon base change to k , it determines a k -morphism V ′ m → A ′ supported at e .Proof. Lemma 7.6 shows that | P t j,h ( u ) | ≤ h ≤ m , since m < p . We conclude by applyingthis estimate to (8.6) and the fact that t is a choice of R -local parameters for A along e . (cid:3) Given a 1-parameter subgroup γ and an integer 0 ≤ m < p , the morphisms provided by Lemma8.7 will be denoted by ˜ γ Rm : V m → A and ˜ γ ′ m : V ′ m → A ′ . Lemma 8.8 (Closed immersions) . Let γ = ( t , u ) be a -parameter subgroup of A . For each m ≥ ,the morphism ˜ γ m : V m → A is a closed immersion. If m < p , then the maps ˜ γ Rm : V m → A and ˜ γ ′ m : V ′ m → A ′ are also closed immersions.Proof. For m = 0 we have E = K , E = R , and E ′ = k , so, the result is clear. Assume m ≥ P t j, ( x ) = x j . Thus, P t j, ( u ) = u j and since | u | = 1 we conclude that forsome j = 1 , ..., n we have P t j, ( u ) ∈ R × . By (8.6) and the fact that t j ∈ m A ,e ⊆ O A ,e ⊆ O A,e (cf.Section 7.1), this implies that the image of ˜ γ m contains an element of the form uz + z h mod z m +1 for some u ∈ R × and some h ∈ K [ z ]. As such an element generates E m as a K -algebra, we obtainthat ˜ γ m : O A,e → E m is surjective. Thus, ˜ γ m is a closed immersion.Since u ∈ R × , an analogous argument works for ˜ γ Rm and ˜ γ ′ m if m < p . (cid:3) ω -integrality.Lemma 8.9. We have H ( V m , Ω V m /K ) = ( K [ z ] / ( z m )) dz . Furthermore, if m ≤ p − then H ( V m , Ω V m /R ) = ( R [ z ] / ( z m )) dz and H ( V ′ m , Ω V ′ m / k ) = ( k [ z ] / ( z m )) dz . In particular, for m ≤ p − we have that H ( V m , Ω V m /R ) = Ω E m /R is a free R -module of rank m .Proof. Note that H ( V m , Ω V m /K ) = Ω E m /K = K [ z ] dz/ ( z m +1 , d ( z m +1 )). The first claim followsfrom the fact that d ( z m +1 ) = ( m + 1) z m dz and m + 1 ∈ K × . The rest is proved similarly. (cid:3) Recall that for the additive Lie group p the space of invariant differentials is Ω( p ) = K · dz . Foreach m we define the K -linear map ρ m : Ω( p ) → Ω E m /K = H ( V m , Ω V m /K )by the rule ρ m ( dz ) = dz ∈ Ω E m /K . emma 8.10 (Analytic-algebraic compatibility) . Let γ = ( t , u ) be a -parameter subgroup of A and let m ≥ . The map on invariant differentials ˜ γ • : H ( A, Ω A/K ) → Ω( p ) induced by theLie group morphism ˜ γ and the map ˜ γ • m : H ( A, Ω A/K ) → H ( V m , Ω V m /K ) induced by the schememorphism ˜ γ m : V m → A satisfy ρ m ◦ ˜ γ • = ˜ γ • m .Proof. For a K -algebra B , the module of universally finite differentials over K (cf. [43] Sec. 11)is denoted by e Ω B/K . We recall that in the special case B = K [[ x , ..., x n ]] we have e Ω B/K = L nj =1 B · dx j and the map d : B → e Ω B/K is continuous for the ( x , ..., x n )-topology.Let κ : e Ω b O A,e /K → e Ω K [[ z ]] /K be the map induced by the continuous ring map b γ : b O A,e → K [[ z ]]given by the power series expansion of ˜ γ , cf. (8.3). Since the map ˜ γ • might be computed from thepower series expansion of the Lie group morphism ˜ γ : p → U e by taking differentials term-by-term,we deduce that the following diagram commutes: H ( A, Ω A/K ) Ω( p ) e Ω b O A,e /K e Ω K [[ z ]] /K . ˜ γ • κ The map H ( A, Ω A/K ) → e Ω b O A,e /K factors through Ω O A,e /K . The quotient K [[ z ]] → E m induces amap e Ω K [[ z ]] /K → Ω E m /K and one checks (evaluating on dz ) that this map composed with Ω( p ) → e Ω K [[ z ]] /K is ρ m . From these two observations we obtain the commutative diagram H ( A, Ω A/K ) Ω( p )Ω O A,e /K e Ω b O A,e /K e Ω K [[ z ]] /K Ω E m /K . ˜ γ • ρ m κ The composition Ω O A,e /K → Ω E m /K of the bottom maps is the morphism induced by ˜ γ m , by (8.5).Finally, this map composed with the inclusion H ( A, Ω A/K ) → Ω O A,e /K is ˜ γ • m , by Lemma 4.3. (cid:3) Lemma 8.11.
Let γ = ( t , u ) be a -parameter subgroup of A , let m ≥ , and let ω ∈ H ( γ ) . Then ˜ γ m : V m → A is ω -integral. Furthermore, if m ≤ p − and ω extends to a section e ω ∈ H ( A , Ω A /R ) ,then ˜ γ ′ m : V ′ m → A ′ is ω ′ -integral, where ω ′ is the restriction of e ω to A ′ .Proof. By Lemma 8.1 we have ˜ γ • ( ω ) = 0. Lemma 8.10 gives ˜ γ • m ( ω ) = ρ m (˜ γ • ( ω )) = 0. Therefore,˜ γ m : V m → A is ω -integral.Suppose now that m ≤ p −
2. The morphism ˜ γ Rm : V m → A induces a map on differentials(˜ γ Rm ) • : H ( A , Ω A /R ) → H ( V m , Ω V m /R ). By base change we get a commutative diagram H ( A ′ , Ω A/ k ) H ( A , Ω A /R ) H ( A, Ω A/K ) H ( V ′ m , Ω V ′ m / k ) H ( V m , Ω V m /R ) H ( V m , Ω V m /K ) (˜ γ ′ m ) • (˜ γ Rm ) • ˜ γ • m Since H ( V m , Ω V m /R ) is free as an R -module (cf. Lemma 8.9), the base change morphism H ( V m , Ω V m /R ) → H ( V m , Ω V m /K ) is injective. Hence, (˜ γ Rm ) • ( e ω ) = 0 because ˜ γ • m ( ω ) = 0 andthe right square of the previous diagram commutes. y commutativity of the left square of the previous diagram, we deduce that (˜ γ ′ m ) • ( ω ′ ) = 0.Hence ˜ γ ′ m : V ′ m → A ′ is ω ′ -integral. (cid:3) The main result
Statement and first steps.
We keep the notation from Section 2.3. Let A /R be an abelianvariety of relative dimension n ≥
3. Let A = A K be the generic fibre and A ′ = A k the specialfibre. Let X be an integral subscheme of A which is smooth, proper, of relative dimension 2 over R with geometrically irreducible fibres. Let X = X K and X ′ = X k . Let G ≤ A ( K ) be a finitelygenerated subgroup and let Γ ⊆ A ( K ) be the p -adic closure of G in A ( K ). Our main result is: Theorem 9.1 (Main result) . With the previous notation, suppose that rank ( G ) ≤ , that (9.1) p > max { e + 1 , exp( e / exp(1)) } , and that either of the following conditions holds: (i) n = 3 , X is of general type, X ′ contains no elliptic curves over k alg , and p > (128 / · c ( X ) . (ii) A ′ is simple over k alg and there is an ample divisor H on A such that p > max (cid:26) c ( X ) + 2 , n ! · (3 deg H ( X ) + cdeg H ( X )) n n n · deg( H n ) (cid:27) . Then Γ ∩ X ( K ) is finite and we have (9.2) ∩ X ( K ) ≤ X ′ ( k ) + (cid:18) − e p − (cid:19) − · ( q + 4 q / + 3) · c ( X ) . We keep the notation and assumptions of Theorem 9.1 for the rest of this section. In additionwe let k = k alg . First we observe Lemma 9.2.
It suffices to prove Theorem 9.1 under the assumption rank ( G ) = 1 .Proof. Suppose that rank ( G ) = 0 and let ξ ∈ A ( K ) be a non-torsion point ( A ( K ) is uncountable).We may replace G by the group generated by G and ξ , which has rank 1. (cid:3) Therefore, from now on we assume rank ( G ) = 1 . The hypotheses of Theorem 9.1 imply the following useful facts.
Lemma 9.3.
If either of (i) or (ii) in Theorem 9.1 holds, then X and X ′ are surfaces of generaltype and ≤ c ( X ′ ) = c ( X ) < ( p − / . Furthermore: • If assumption (i) in Theorem 9.1 holds, then X ′ is an ample divisor on A ′ . • If assumption (ii) in Theorem 9.1 holds, then there is an ample divisor H ′ on A ′ such that deg H ′ ( X ′ ) = deg H ( X ) , cdeg H ′ ( X ′ ) = cdeg H ( X ) , and deg(( H ′ ) ) A ′ = deg( H ) A .Proof. If A ′ is simple over k then A is simple over K alg in which case X ⊗ K alg does not containtranslates of positive dimensional abelian subvarieties of A ⊗ K alg , and Lemma 3.14 implies that X is of general type (note that K alg ≃ C as fields). Thus, if either of (i) or (ii) holds, X is ofgeneral type and Theorem 9.1 in [36] implies that X ′ is of general type. Lemma 9.3 in [36] gives c ( X ′ ) = c ( X ). We get c ( X ) ≥ c ( X ) < ( p − / X ′ is not an abelian surface because it is of general type, and since it does notcontain elliptic curves over k , Lemma 5.1 implies that X ′ is an ample divisor on A ′ .On the other hand, suppose that assumption (ii) in Theorem 9.1 holds. The theory developedin the appendix to Expos´e X in [8] (see also Section 20.3 in [25]) gives a specialization map fromthe group of algebraic cycles modulo numerical equivalence of A to that of A ′ which respects the ntersection product. Let H ′ be the specialization of H . Since A ′ is geometrically simple, thedivisor H ′ is ample. Furthermore, since X → Spec R is smooth, the specialization of a canonicaldivisor on X is a canonical divisor on X ′ . The required equalities follow. (cid:3) Remark . Theorem 1.1 follows from Theorem 9.1. Indeed, taking K = Q p in Theorem 9.1,condition (9.1) becomes p ≥
3. Since c ( X ) ≥ p ≥ p ≥ X ′ k . Lemma 9.5.
Let C be an irreducible curve in X ′ k defined over k . Then g g ( C ) ≥ .Proof. Since C ⊆ A ′ k we have g g ( C ) ≥
1. If it is 1, then we get ν C/A ′ k : e C → A ′ k where e C is anelliptic curve. Thus, ν C/A ′ k is, up to translation, a non-constant morphism of abelian varieties, andwe get that its image C is is an elliptic curve. But X ′ k contains no elliptic curves by assumption. (cid:3) Choice of differentials and canonical divisor.
Recall that we are assuming rank ( G ) = 1. Lemma 9.6.
Let t be a choice of R -local parameters for A at e . There is a -parameter subgroup γ = ( t , u ) for A such that Γ ∩ U e ⊆ im( γ ) . Moreover, γ is uniquely determined, up to equivalence,by the condition that im( γ ) ∩ Γ strictly contains e .Proof. By Lemma 8.6 and the fact that rank ( G ) = 1, there is a 1-parameter subgroup γ of theform ( t , u ) for the given t , such that im( γ ) ∩ Γ strictly contains { e } . By Lemma 8.2 and the factthat Γ is the p -adic closure of the rank 1 group G , the non-triviality of im( γ ) ∩ Γ is equivalent toΓ ∩ U e ⊆ im( γ ). By Lemma 8.4, the equivalence class of γ is unique for this last property. (cid:3) In view of Lemma 9.6 and Lemma 8.5, the group G uniquely determines a K -linear hyperplane H = H ( γ ) ⊆ H ( A, Ω A/K ). Lemma 9.7.
There are ω , ω ∈ H such that (i) ω , ω are K -linearly independent. (ii) ω , ω ∈ H ( A , Ω A /R ) . (iii) The differentials ω ′ , ω ′ ∈ H ( A ′ , Ω A ′ / k ) obtained by reducing ω , ω modulo p , are k -linearlyindependent.Proof. The ring R is a DVR, hence H ( A , Ω A /R ) ≃ R n . Thus, the problem is reduced to showingthe following: Given n ≥ K -linear map f : K n → K , there are v , v ∈ ker( f ) ∩ R n whichare linearly independent and such that their images in k n are k -linearly independent. For this, wemay assume that f is R -valued on R n , and it follows that rank R (ker( f | R n )) = n −
1. Thus, wecan take v , v ∈ ker( f ) ∩ R n which are K -linearly independent. After scaling, we can assume that v , v ∈ R n are primitive. If there are b , b ∈ R × with b v + b v ∈ p R n ⊆ R n then there are c , c ∈ R such that b v + b v = ̟c v + ̟c v which is not possible since v , v are K -linearlyindependent. Hence the images of v , v in k n are k -linearly independent. (cid:3) From now on we fix a choice of ω , ω as in Lemma 9.7 and let ω ′ , ω ′ ∈ H ( A ′ , Ω A ′ / k ) be theirreductions modulo p . Let ι : X ′ → A ′ be the inclusion map. For j = 1 , u j = ι • ( ω ′ j ) ∈ H ( X ′ , Ω X ′ / k ) be the restriction of ω ′ j to X ′ . Lemma 9.8.
The -form u ∧ u ∈ H ( X ′ , Ω X/ k ) is not the zero section.Proof. Since A ′ is an abelian variety, (iii) in Lemma 9.7 shows that ω ′ ∧ ω ′ ∈ H ( A ′ , Ω A ′ / k ) is notthe zero section. By the assumptions (i) and (ii) in Theorem 9.1 and by Lemma 9.3, we can applyLemma 5.7 to ω ′ , ω ′ on A ′ and the surface X ′ ⊆ A ′ after base change to k . (cid:3) y Lemma 9.8 we can define D = div X ′ ( u ∧ u ) on X ′ . Lemmas 9.3, 9.5, and 3.12 give: Lemma 9.9. D is effective, ample, and it is a canonical divisor of X ′ defined over k . In particular, D is numerically effective and D > . Let Z = supp( D ) and let C , ..., C ℓ be the irreducible components of Z over k . Thus, Z is a(possibly singular and reducible) curve defined over k and we have D = P ℓj =1 a j C j as divisors on X ′ k = X ′ ⊗ k for certain integers a j ≥
1. Let us write ν j = ν C j /X ′ k : e C j → X ′ k for j = 1 , ..., ℓ . Lemma 9.10.
There is b ∈ k such that the differential w = u + bu ∈ H ( X ′ , Ω X ′ / k ) satisfies thatno map ν j : e C j → X ′ k is w -integral.Proof. By Lemmas 9.3 and 9.9 (especially, ampleness of D ) we have ℓ ≤ ℓ X j =1 a j ≤ ℓ X j =1 a j · ( C j .D ) X ′ k = c ( X ′ ) < p Given b ∈ k let w b = u + bu ∈ H ( X ′ , Ω X ′ / k ). If for every b ∈ F p ⊆ k we have that some ν j is w b -integral, the fact that ℓ < p implies that for some j there are b = b ′ in k such that w b , w b ′ ∈ ker( ν • j ).Hence, ω + bω and ω + b ′ ω are in ker( ν • C j /A ′ k ). Since b = b ′ we deduce dim k ker( ν • C j /A ′ k ) ≥
2. Bythe assumptions (i) and (ii) in Theorem 9.1 and Lemma 9.3, this contradicts Lemma 5.5. (cid:3)
From now on, we fix a choice of b ∈ k and the corresponding w = u + bu ∈ H ( X ′ , Ω X ′ / k ) asin Lemma 9.10.9.3. Bounds from overdetermined ω -integrality. Given x ∈ X ′ ( k ) we let m X ′ ,ω ′ ,ω ′ ( x ) be thesupremum (possibly infinite) of all integers m ≥ φ : V km → X ′ k supported at x which is ω -integral for both ω = u , u . Here we recall from Section4.2 that V km = Spec k [ z ] / ( z m +1 ). We observe that, fixing the ambient abelian variety A ′ over k , thequantity m X ′ ,ω ′ ,ω ′ ( x ) only depends on our choices of X ′ , ω ′ , ω ′ , and x . As X ′ is given and ω ′ , ω ′ are fixed, we may simply write m ( x ) = m X ′ ,ω ′ ,ω ′ ( x ) (except in the proof of Lemma 9.15 below,where a reduction argument will require to consider other choices of X ′ ). Lemma 9.11. If x / ∈ Z then m ( x ) = 0 .Proof. By Theorem 4.4. See Remark 4.5 for details in the case x / ∈ Z = supp( D ). (cid:3) Lemma 9.12. If x ∈ Z then m ( x ) ≤ ℓ X j =1 X y ∈ ν − j ( x ) a j · (ord y ( ν • j ( w )) + 1) . Proof.
By Theorem 4.4, after choosing ω = w = u + bu . (cid:3) Lemma 9.13.
For every x ∈ X ′ ( k ) we have m ( x ) ≤ p − . In particular, m ( x ) is finite.Proof. By Lemma 9.11 we may assume x ∈ Z . Lemma 9.12 gives m ( x ) ≤ ℓ X j =1 a j · deg e C j ( ν • j ( w )) + ℓ X j =1 a j · ν − j ( x ) . Since w is chosen as in Lemma 9.10 we get deg e C j ( ν • j ( w )) = 2 g g ( C j ) − ≤ g a ( C j ) −
2. On theother hand, Lemma 9.5 gives g g ( C j ) ≥ j , and in view of Lemmas 3.2 and 3.9 we get ν − j ( x ) ≤ δ ( C j , x ) + 1 ≤ g a ( C j ) − . sing Lemmas 3.6, 9.3, and 9.9 this gives m ( x ) ≤ · P ℓj =1 a j · ( g a ( C j ) − ≤ c ( X ′ ) < p − (cid:3) Bounds on residue disks.
Every ξ ∈ A ( K ) extends to a section σ ξ : Spec R → A because A → Spec R is proper. This determines a reduction map red : A ( K ) → A ′ ( k ) which is a groupmorphism. For x ∈ A ′ ( k ) we let U x = red − ( x ) ⊆ A ( K ) be the corresponding residue disk. Inparticular, U e = ker(red) as in Section 7.3. The goal of this paragraph is to show the followingresult (under the assumptions of Theorem 9.1), which can be of independent interest: Proposition 9.14.
Let ξ ∈ Γ ∩ X ( K ) and x = red( ξ ) ∈ X ′ ( k ) . Then Γ ∩ X ( K ) ∩ U x is finite and ∩ X ( K ) ∩ U x ≤ m ( x ) · (cid:18) − e p − (cid:19) − . In particular, if m ( x ) = 0 , then U x contains at most one point of Γ ∩ X ( K ) . Recall that we are assuming rank ( G ) = 1. In addition, we can make the following reduction inthe proof of Proposition 9.14. Lemma 9.15.
It suffices to prove Proposition 9.14 under the assumption ξ = e .Proof. Let ξ ∈ Γ ∩ X ( K ) and consider the translation Y = X − σ ξ (Spec R ). Note that the genericand special fibres of Y are Y = X − ξ and Y ′ = X ′ − x where x = red( ξ ). The assumptions ofTheorem 9.1 also hold for Y instead of X . Since Γ − ξ = Γ, we find ∩ X ( K ) ∩ U x = ∩ Y ( K ) ∩ U e . On the other hand, as the differentials ω ′ , ω ′ ∈ H ( A ′ , Ω A ′ / k ) are translation invariant, we have m X ′ ,ω ′ ,ω ′ ( x ) = m Y ′ ,τ ∗ ω ′ ,τ ∗ ω ′ ( e ) = m Y ′ ,ω ′ ,ω ′ ( e )where τ : A ′ → A ′ is the translation map P P + x . The differential ω ′ j is determined by ω j ∈ H ( A, Ω A/K ) (for j = 1 , ω ′ , ω ′ only depends on A and G , not on the particular choice of X . Therefore, ifProposition 9.14 holds in the case ξ = e for the given A and G , and for every choice of X as inTheorem 9.1, then we can in particular apply it to Y obtaining ∩ X ( K ) ∩ U x = ∩ Y ( K ) ∩ U e ≤ m Y ′ ,ω ′ ,ω ′ ( e ) = m X ′ ,ω ′ ,ω ′ ( x ) . (cid:3) For the rest of the current Section 9.4, let us assume that e ∈ Γ ∩ X ( K ) in order to showProposition 9.14 for ξ = e . This is enough, by Lemma 9.15.Let t , ..., t n ∈ m A ,e be a system of R -local parameters for A along σ e with the property that t , ..., t n − are a system of local equations for X along σ e . This is possible by Lemma 7.1. Let t ′ j be the restriction of t j to A ′ . Then t ′ , ...t ′ n − is a system of local equations for X ′ ⊆ A ′ at e . Inparticular, t , ..., t n − restricted to A vanish on X ( K ) ∩ U e .Write t = ( t , ..., t n ) and let u ∈ K n with | u | = 1 be such that γ = ( t , u ) be a 1-parametersubgroup of A with Γ ∪ U e ⊆ im( γ ). By Lemma 9.6, the vector u exists, γ is unique up toequivalence, and γ corresponds to the hyperplane H ⊆ H ( A, Ω A/K ) from Section 9.2. We get ∩ X ( K ) ∩ U e ≤ X ( K ) ∩ im( γ ) . By Lemma 7.7 (which is applicable by (9.1)), the definition of the map ˜ γ : p U e (cf. (8.2)),and the fact that t , ..., t n − vanish on X ( K ) ∩ U e , we deduce that for each j = 1 , ..., n − X ( K ) ∩ im( γ ) ≤ { z ∈ p : Exp t j ( z · u ) = 0 } . herefore, for each j = 1 , ..., n − ∩ X ( K ) ∩ U e ≤ n (Exp t j ( z · u ) , /q ) . Let P t j,h ( x ) ∈ K [ x , ..., x n ] be the homogeneous part of degree h in Exp t j ( x ). Note that P t j, ( x ) = 0. Lemma 9.16.
There are positive integers integers j ≤ n − and N ≤ m ( e )+1 with | P t j ,N ( u ) | ≥ .Proof. Let 0 ≤ m ≤ p − | P t j,h ( u ) | < ≤ h ≤ m and for each1 ≤ j ≤ n −
2. We claim that m ≤ m ( e ).Since ω , ω are chosen as in Lemma 9.7, we see that Lemmas 8.8 and 8.11 give a closed immersion˜ γ ′ m : V ′ m → A ′ which is ω ′ i -integral for i = 1 ,
2. Let us prove that ˜ γ ′ m factors through a closedimmersion φ m : V ′ m → X ′ supported at x = e . Indeed, let G j,m ( z ) ∈ K [ z ] be the truncation ofExp t j ( z · u ) up to degree m , that is, G j,m ( z ) = m X h =0 P t j,h ( u ) · z h ∈ K [ z ] . Our assumption | P t j,h ( u ) | < ≤ h ≤ m and 1 ≤ j ≤ n − G j,m ( z ) ∈ R [ z ]for all 1 ≤ j ≤ n −
2, and in fact, in this case G j,m ( z ) ≡ p . In view of Lemma 8.7 whichgives the construction of ˜ γ ′ m , reducing (8.6) modulo p we get that for each 1 ≤ j ≤ n − γ ′ m ) ( t ′ j ) = G j,m ( z ) mod ( z m +1 , ̟ ) = 0 . As t ′ , ..., t ′ n − are local equations for X ′ at e , this shows that ˜ γ ′ m factors through the inclusion ι : X ′ → A ′ . We obtain the claimed closed immersion φ m : V ′ m → X ′ satisfying ι ◦ φ m = ˜ γ ′ m .Since ˜ γ ′ m is ω ′ i -integral for i = 1 ,
2, we obtain that φ m is u i -integral for i = 1 ,
2. The sameholds for the base change of φ m to k , which is a closed immersion V km → X ′ k supported at x = e .Thus, by definition of m X ′ ,ω ′ ,ω ′ ( x ) (cf. Section 9.3) we finally get m ≤ m ( e ), as claimed.Suppose now that | P t j,h ( u ) | < ≤ h ≤ p − ≤ j ≤ n −
2. Taking m = p − p − ≤ m ( e ), which is not possible by Lemma 9.13.Therefore, there is some 0 ≤ h ≤ p − ≤ j ≤ n − | P t j ,h ( u ) | ≥
1. Let N be the least value of such an h and choose any 1 ≤ j ≤ n − | P t j ,N ( u ) | ≥
1. Thus, N exists and N ≤ p −
2. Also, N ≥ P t j, ( u ) = 0 for each j . Taking m = N − ≤ p − | P t j,h ( u ) | < h ≤ m and j ≤ n −
2, and out initial claim implies N − m ≤ m ( e ). (cid:3) Proof of Proposition 9.14.
As proved in Lemma 9.15, we may assume that e ∈ Γ ∩ X ( K ) and that ξ = e . Let j and N be as in Lemma 9.16, let M = p [ K : Q p ] / ( p − , and let r = 1 /q . Let us write λ = e / ( p −
1) and observe that λ <
M >
1, and r < M − because(9.4) log M log r − = log M log q = [ K : Q p ]( p − f = λ < . If c j,α ∈ K is the coefficient of x α in Exp t j ( x ), Lemma 7.6 gives | c j,α | ≤ M k α k− . Hence, we canapply Lemma 6.2 with these choices to the power series Exp t j ( x ) obtaining n (Exp t j ( z · u ) , /q ) ≤ ( N − λ ) (1 − λ ) − where we used (9.4). By (9.3) and Lemma 9.16 we conclude ∩ X ( K ) ∩ U e ≤ ( m ( e ) + 1 − λ ) (1 − λ ) − = 1 + m ( e ) · (1 − λ ) − . (cid:3) .5. Adding over residue disks.
Proof of Theorem 9.1.
By Proposition 9.14 and Lemma 9.11 we obtain(9.5) ∩ X ( K ) = X x ∈ X ′ ( k ) ∩ X ( K ) ∩ U x = X ′ ( k ) + (cid:18) − e p − (cid:19) − X x ∈ X ′ ( k ) m ( x )= X ′ ( k ) + (cid:18) − e p − (cid:19) − X x ∈ Z ( k ) m ( x ) . Recall from Lemma 9.10 that for each j = 1 , ..., ℓ we have that ν • j ( w ) is not the zero differential.By Lemma 9.12 we have have(9.6) X x ∈ Z ( k ) m ( x ) ≤ X x ∈ Z ( k ) ℓ X j =1 X y ∈ ν − j ( x ) a j · (ord y ( ν • j ( w )) + 1) ≤ ℓ X j =1 a j · deg e C j ( ν • j ( w )) + X x ∈ Z ( k ) ℓ X j =1 a j · ν − j ( x ) . Lemma 3.6 (which is applicable by Lemma 9.9) and Lemma 9.3 give(9.7) ℓ X j =1 a j · deg e C j ( ν • j ( w )) = ℓ X j =1 a j · (2 g g ( C j ) − ≤ c ( X ′ ) = 2 c ( X ) . Let D , ..., D s be the irreducible components of Z over k and let J , ..., J s be the partition of { , , ..., ℓ } determined by the condition D i ⊗ k = ∪ j ∈ J i C j . Since each D i is defined over k , thereare integers b i ≥ ≤ i ≤ s such that a j = b i for each j ∈ J i . By Lemma 3.15 we obtain X x ∈ Z ( k ) ℓ X j =1 a j · ν − j ( x ) = s X i =1 b i · X x ∈ D i ( k ) X j ∈ J i ν − j ( x ) ≤ s X i =1 b i · ( q + 1) · J i + 2 q / · X j ∈ J i g g ( C j ) = ( q + 1) ℓ X j =1 a j + 2 q / · ℓ X j =1 a j · g g ( C j )= ( q + 2 q / + 1) ℓ X j =1 a j + 2 q / · ℓ X j =1 a j · ( g g ( C j ) − . From Lemmas 9.5 and 3.6 (applicable by Lemma 9.9) we deduce(9.8) X x ∈ Z ( k ) ℓ X j =1 a j · ν − j ( x ) ≤ ( q + 4 q / + 1) ℓ X j =1 a j · ( g g ( C j ) − ≤ ( q + 4 q / + 1) c ( X ) . Using (9.7) and (9.8) in (9.6) we get X x ∈ Z ( k ) m ( x ) ≤ ( q + 4 q / + 3) c ( X )which together with (9.5) gives the result. (cid:3) Applications for rational points
Rational points on surfaces.
We begin with the following generalization of Theorems 1.5and 1.7 to the case of an arbitrary number field.
Theorem 10.1.
Let F be a number field, let p be a prime number, let p be a prime of F above p ,let e be the ramification index of p , let k be the residue field of p , and let q = k . We assume that p > max { e + 1 , exp( e / exp(1)) } . Let X be a smooth projective surface contained in an abelian variety A of dimension n ≥ , bothdefined over F and having good reduction at p . Let X ′ and A ′ be the corresponding reductionsmodulo p . Suppose rank A ( F ) ≤ and that either of the following conditions holds: (i) n = 3 , X is of general type, X ′ contains no elliptic curves over k alg , and p > (128 / · c ( X ) . (ii) A ′ is simple over k alg and there is an ample divisor H on A such that p > max (cid:26) c ( X ) + 2 , n ! · (3 deg H ( X ) + cdeg H ( X )) n n n · deg( H n ) (cid:27) . Then X ( F ) is finite and we have X ( F ) ≤ X ′ ( k ) + (cid:18) − e p − (cid:19) − · ( q + 4 q / + 3) · c ( X ) . Proof.
By Theorem 9.1, after base-change to the completion K = F p and taking G = A ( F ). (cid:3) Primes of geometrically simple reduction.
The next result is due to Chavdarov [14].
Lemma 10.2.
Let A be an abelian variety over Q of odd dimension n satisfying End( A C ) = Z .There is a set of primes P of density in the primes such that for every p ∈ P the reduction of A modulo p is geometrically simple. With this at hand, we have:
Proof of Corollary 1.11.
Let P be the set of primes afforded by Lemma 10.2 applied to A (asdim A = 3 is odd), discarding the primes of bad reduction for X . For every p ∈ P we have that X ⊗ F algp does not contain elliptic curves since A ⊗ F algp is simple. We conclude by Theorem 1.5. (cid:3) From the argument it is clear that we can get a similar result over number fields and abelianvarieties of dimension n ≥ A . The general problem regarding abundance of primes of geometricallysimple reduction is a conjecture of Murty and Patankar [52]. See [67] and the references therein forthe formulation of this conjecture and available results.10.3. Quadratic points in curves.
We write ≡ for numerical equivalence.Let C be a smooth, irreducible, projective curve of genus g ≥ J , over analgebraically closed field k of characteristic different from 2. The symmetric square of C , denotedby C (2) , is a smooth, irreducible, projective surface. The quotient map π : C × C → C (2) is finiteof degree 2 ramified along the diagonal ∆ ⊆ C × C . Let D = π (∆) and for ξ ∈ C ( F alg ) we let V ξ = π ( { ξ } × C ). As we vary ξ , all the divisors V ξ are numerically equivalent to each other and thisnumerical equivalence class is denoted by V . Using adjunction for π (see [32] Proposition II.8.20)and intersections on C × C we find Lemma 10.3.
We have K C (2) ≡ (2 g − V − D/ . Moreover, V = 1 , ( D.V ) = 2 , D = 4 − g ,and c ( C (2) ) = (4 g − g − . et d be degree 2 divisor on C . The map C × C → J given by ( x , x ) [ x + x − b ] inducesa map j : C (2) → J birational onto its image (since g ≥ J andrecall that it is ample. We define θ = j ∗ Θ on C (2) . By Lemma 7 in [42] and Lemma 10.3 we have Lemma 10.4. D ≡ g + 1) V − θ ) . Hence, θ ≡ ( g + 1) V − D/ and K C (2) ≡ θ + ( g − V . Corollary 10.5.
For every curve C of genus g ≥ we have that C (2) has ample canonical divisor.In particular, C (2) is a surface of general type.Proof. Since θ is ample and all the V ξ are effective, we get the result from Lemma 10.4. (cid:3) By Lemmas 10.3 and 10.4 (namely, θ ≡ ( g + 1) V − D/
2) we find
Lemma 10.6. ( θ.K C (2) ) = 2 g ( g − . Let W = j ( C (2) ). It is clear that Lemma 10.7. If C is not hyperelliptic, then j is injective, in which case C (2) ≃ W . Now consider a field F of characteristic different from 2, not necessarily algebraically closedand suppose that C is defined over F . If the divisor d is defined over F then so is the map j : C (2) → J and its image W . However, Θ is only defined in some algebraic extension of F ingeneral. Nevertheless, in this setting we have Lemma 10.8.
Assuming the existence of an effective degree divisor d on C defined over F ,there is a divisor H on J defined over F with H ≡ . In particular, H is ample.Proof. Write d = x + x where x , x are either F -rational or of degree 2 over F in which casethey are Galois conjugate. Let Θ i be the ( g − j ( V x i ) ⊆ J with itself, for i = 1 ,
2. Then the divisor H = Θ + Θ has the required properties. (cid:3) Proof of Corollary 1.14.
We keep the previous notation in the case F = Q . We can assume theexistence of the effective degree 2 divisor d on C defined over Q , for otherwise C (2) ( Q ) is empty.Let us apply Theorem 1.7 to A = J , n = g , and X = W ⊆ J , which is a smooth projective surfacesatisfying W ≃ C (2) (Lemma 10.7). Furthermore, the reduction of C (2) modulo p is ( C ′ ) (2) and theisomorphism C (2) ≃ X induced by j reduces to an isomorphism ( C ′ ) (2) ≃ X ′ with X ′ the reductionof X , because C ′ ⊗ F algp is not hyperelliptic. The divisor H is taken as in Lemma 10.8.By the Poincar´e formulas (see for instance [42]) we have deg( H g ) = 2 g deg(Θ g ) = 2 g · g ! anddeg( H .X ) = 4 deg(Θ .W ) = 4 g ( g − H.K X ) = 2 deg(Θ .K W ) =2( θ.K C (2) ) = 4 g ( g − g ! · (3 deg( H .X ) + deg( H.K X )) g g g · deg( H g ) = g ! · (4 g (4 g − g g g · g · g ! = (8 g − g . In view of Lemma 10.3 and the fact that 3 c ( X ) + 2 = 3(4 g − g −
1) + 2 < (8 g − g for g ≥ (cid:3) Proof of Corollary 1.15.
Similar to the previous argument, using Theorem 1.5 instead of Theorem1.7. Here, X = W ≃ C (2) is of general type by Lemma 10.5. Lemma 10.3 and the fact that g = 3give c ( X ) = c ( C (2) ) = 6. Thus, (128 / c ( X ) = 512 and the least prime p >
512 is 521. (cid:3)
Acknowledgments
We would like to thank Natalia Garcia-Fritz for answering numerous questions regarding branchesof curves and ω -integrality. Initially, our results were proved for surfaces contained in abelianthreefolds, and we thank Peter Sarnak for asking a question that led us to address the general case.J.C. was supported by ANID Doctorado Nacional 21190304 and H.P. was supported by FONDE-CYT Regular grant 1190442. eferences [1] Y. Aubry, M. Perret, On the characteristic polynomials of the Frobenius endomorphism for projective curves overfinite fields . Finite Fields and Their Applications, 10(3), (2004) 412-431.[2] J. Balakrishnan, A. Besser, J. M¨uller,
Computing integral points on hyperelliptic curves using quadratic Chabauty .Mathematics of Computation, 86(305), (2017) 1403-1434.[3] J. Balakrishnan, N. Dogra,
Quadratic Chabauty and rational points, I: p -adic heights . Duke Math. J. 167(11),(2018), 1981-2038.[4] J. Balakrishnan, N. Dogra. An effective Chabauty-Kim theorem . Compositio Math. 155 (2019) 1057-1075.[5] J. Balakrishnan, N. Dogra.
Quadratic Chabauty and rational points II: Generalised height functions on Selmervarieties . Int. Math. Res. Not. (2020) doi:10.1093/imrn/rnz362[6] J. Balakrishnan, N. Dogra, J. M¨uller, J. Tuitman, J. Vonk,
Explicit Chabauty–Kim for the split Cartan modularcurve of level 13 . Ann. Math., 189(3), (2019) 885-944.[7] W. Barth, K. Hulek, C. Peters, and A. Van de Ven.
Compact complex surfaces . Ergebnisse der Mathematik undihrer Grenzgebiete, vol. 4, second edition. Springer, 2004.[8] P. Berthelot, A. Grothendieck, L. Illusie, eds.
Theorie des Intersections et Theoreme de Riemann-Roch (SGA6) .1966/67, Lecture Notes in Math. 225, Springer (1971).[9] M. Bhargava, A. Shankar,
Ternary cubic forms having bounded invariants, and the existence of a positive propor-tion of elliptic curves having rank 0 . Ann. Math., 181 (2015), 587-621.[10] M. Bhargava, C. Skinner,
A positive proportion of elliptic curves over Q have rank one . Journal of the RamanujanMathematical Society, 29(2), (2014), 221-242.[11] N. Bourbaki,
Lie groups and Lie algebras . Chapters 1-3, Elements of Mathematics (Berlin), Springer-Verlag,Berlin, 1998.[12] R. Brody,
Compact Manifolds and Hyperbolicity . Trans. Amer. Math. Soc. 235 (1978), 213-219.[13] C. Chabauty,
Sur les points rationnels des courbes alg´ebriques de genre sup´erieur `a l’unit´e . C. R. Acad. Sci. Paris212 (1941), 882-885.[14] N. Chavdarov,
The generic irreducibility of the numerator of the zeta function in a family of curves with largemonodromy . Duke Math. J. 87 (1997), no. 1, 151-180.[15] R. Coleman,
Effective Chabauty . Duke Math. J. 52 (1985), no. 3, 765-770.[16] O. Debarre,
Degrees of curves in abelian varieties . Bulletin de la S. M. F., tome 122, no 3 (1994), p. 343-361[17] P. Deligne,
La conjecture de Weil: I . Publications Math´ematiques de l’IH´ES. 43: 273-307 (1974).[18] M. Deschamps,
Courbes de genre g´eom´etrique born´e sur une surface de type g´en´eral (d’apr`es F. A. Bogomolov) .S´eminaire Bourbaki 30e ann´ee, 1977/78, Lecture Notes in Mathematics 710, Springer, 1978, No. 519.[19] B. Dwork,
On the rationality of the zeta function of an algebraic variety . American Journal of Mathematics,82(3), (1960), 631-648.[20] B. Edixhoven, G. Lido,
Geometric quadratic Chabauty . Preprint, v3 in arXiv: 1910.10752[21] G. Faltings,
Endlichkeitss¨atze f¨ur abelsche Variet¨aten ¨uber Zahlk¨orpern . (German) [Finiteness theorems forabelian varieties over number fields] Invent. Math. 73 (1983), no. 3, 349-366.[22] G. Faltings,
Diophantine approximation on abelian varieties . Ann. Math., 133 (1991), 549-576.[23] G. Faltings,
The general case of S. Lang’s conjecture . Barsotti Symposium in Algebraic Geometry (Abano Terme,1991). Perspect. Math. 15. Academic Press. San Diego. 1994. p. 175-182[24] E. Flynn,
A flexible method for applying Chabauty’s theorem . Compositio Math. 105 (1997), no. 1, 79-94.[25] W. Fulton,
Intersection Theory . Second edition, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete,vol. 2, Springer-Verlag, Berlin, 1998.[26] N. Garcia-Fritz,
Curves of low genus on surfaces and applications to Diophantine problems . PhD Thesis, Queen’sUniversity, 2015.[27] N. Garcia-Fritz,
Sequences of powers with second differences equal to two and hyperbolicity . Trans. Am. Math.Soc. 370(5), 3441-3466 (2018)[28] N. Garcia-Fritz,
Quadratic sequences of powers and Mohanty’s conjecture . International Journal of NumberTheory 14.02 (2018), 479-507.[29] M. Green,
Holomorphic Maps to Complex Tori . American Journal of Mathematics, 100(3), (1978) 615-620.[30] A. Grothendieck, M. Raynaud,
Revˆetement ´etales et groupe fondamental (SGA1) . Lecture Note in Math. 224,Springer (1971).[31] J. Gunther, J. Morrow,
Irrational points on random hyperelliptic curves . Preprint, v3 in arXiv:1709.02041[32] R. Hartshorne.
Algebraic geometry . Graduate texts in Math. vol. 52., Springer Science & Business Media, 2013.[33] M. Hindry, J. Silveman,
Diophantine Geometry, An Introduction . Graduate Texts in Mathematics, 201, Springer-Verlag New York (2000).
34] H. Hironaka, H.
On the arithmetic genera and the effective genera of algebraic curves . Memoirs of the College ofScience, University of Kyoto. Series A: Mathematics, 30(2), (1957) 177-195.[35] P.-C. Hu, C.-C. Yang,
Meromorphic Functions over non-Archimedean Fields . Kluwer Academic Publishers, 2000.[36] T. Katsura, K. Ueno,
On elliptic surfaces in characteristic p . Math. Ann. 272(3), (1985) 291-330.[37] D. Katz, On the number of minimal prime ideals in the completion of a local domain . The Rocky MountainJournal of Mathematics, 16(3), (1986) 575-578.[38] E. Katz, D. Zureick-Brown,
The Chabauty-Coleman bound at a prime of bad reduction and Clifford bounds forgeometric rank functions . Compos. Math. 149 (2013), no. 11, 1818-1838.[39] E. Katz, J. Rabinoff, D. Zureick-Brown,
Uniform bounds for the number of rational points on curves of smallMordell-Weil rank . Duke Math. J., 165(16), (2016), 3189-3240.[40] M. Kim,
The motivic fundamental group of P \ { , , ∞} and the theorem of Siegel . Inventiones Mathematicae,161, (2005) 629-656.[41] M. J. Klassen, Algebraic points of low degree on curves of low rank . Thesis, University of Arizona, 1993.[42] A. Kouvidakis,
Divisors on symmetric products of curves . Transactions of the American Mathematical Society,(1993), 337(1), 117-128.[43] E. Kunz,
K¨ahler Differentials . Advanced lectures in mathematics, Friedr. Vieweg and Sohn, Braunschweig, 1986.[44] E. Kunz,
Introduction to Plane Algebraic Curves . Birkh¨auser Boston, Inc., Boston, MA, 2005.[45] D. Lorenzini, T. Tucker,
Thue equations and the method of Chabauty-Coleman . Invent. Math. 148 (2002), 47-77.[46] H. Matsumura,
Commutative ring theory . Cambridge Studies in Advanced Mathematics (M. Reid, Trans.).Cambridge: Cambridge University Press (1987).[47] T. Matsusaka,
On a characterization of a Jacobian variety . Mem. College Sci. Univ. Kyoto Ser. A Math. 32(1959), no. 1, 1-19.[48] W. McCallum, B. Poonen,
The method of Chabauty and Coleman . Explicit methods in number theory; rationalpoints and diophantine equations, Panoramas et Synth`eses 36, Soci´et´e Math. de France, (2012) 99-117.[49] H. Morikawa,
Cycles and endomorphisms of abelian varieties . Nagoya Math. J. 7 (1954), 95-102.[50] D. Mumford, appendix to “
The Theorem of Riemann-Roch for High Multiples of an Effective Divisor on anAlgebraic Surface ” by O. Zariski. Ann. Math., 76(3), (1962) 560-615[51] D. Mumford,
Abelian Varieties . Tata Inst. Fund. Res. Stud. Math. 5. Tata Institute of Fundamental Research,Bombay; Oxford Univ. Press, London, 1970.[52] V. K. Murty, V. Patankar,
Splitting of abelian varieties . Int. Math. Res. Not., 12 (2008).[53] Y. Nakai,
Notes on invariant differentials on abelian varieties . J. Math. Kyoto Univ. 3 (1963), no. 1, 127-135.[54] Y. Nakai,
On the theory of differentials on algebraic varieties . J. Sci. Hiroshima Univ. Ser. A-I 27 (1963), 7-34.[55] J. Park,
Effective Chabauty for symmetric powers of curves . Preprint, v1 in arXiv:1504.05544[56] A. P loski,
Introduction to the local theory of plane algebraic curves . Krasi´nski T., Spodzieja S. (eds.), Analyticand Algebraic Geometry, L´od´z University Press, L´od´z 2013, 115-134.[57] G. R´emond,
D´ecompte dans une conjecture de Lang.
Inventiones Mathematicae, (2000) 142 (3), 513-545.[58] G. R´emond,
Sur les sous-vari´et´es des tores . Compositio Mathematica 134.3 (2002) 337-366.[59] S. Siksek,
Chabauty for symmetric powers of curves . Algebra & Number Theory, 3(2), (2009), 209-236.[60] M. Stoll,
Independence of rational points on twists of a given curve . Compositio Math. 142 (2006), 1201-1214[61] M. Stoll,
Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank .Journal of the European Mathematical Society, 21(3), (2019) 923-956.[62] S. Vemulapalli, D. Wang,
Uniform bounds for the number of rational points on symmetric squares of curves withlow Mordell-Weil rank . Preprint, v3 in arXiv:1708.07057[63] P. Vojta,
Siegel’s theorem in the compact case . Ann. Math., 1991, p. 509-548.[64] P. Vojta,
Diagonal quadratic forms and Hilbert’s tenth problem . Hilbert’s tenth problem: relations with arithmeticand algebraic geometry (Ghent, 1999), 261-274, Contemp. Math., 270, Amer. Math. Soc., Providence, RI, 2000.[65] S.-T. Yau,
Intrinsic measures of compact complex manifolds . Math. Ann., 212, (1975) 317-329.[66] O. Zariski, P. Samuel,
Commutative Algebra, volume II . Graduate Texts in Mathematics, 29, Springer-VerlagBerlin Heidelberg (1960).[67] D. Zywina,
The splitting of reductions of an abelian variety . Int. Math. Res. Not., (2014) Issue 18, 5042-5083.
Departamento de Matem´aticas, Pontificia Universidad Cat´olica de Chile. Facultad de Matem´aticas,4860 Av. Vicu˜na Mackenna, Macul, RM, Chile
Email address , J. Caro: [email protected]
Departamento de Matem´aticas, Pontificia Universidad Cat´olica de Chile. Facultad de Matem´aticas,4860 Av. Vicu˜na Mackenna, Macul, RM, Chile
Email address , H. Pasten: [email protected]@gmail.com