An explicit and uniform Manin-Mumford-type result for function fields
aa r X i v : . [ m a t h . N T ] J a n AN EXPLICIT AND UNIFORM MANIN–MUMFORD-TYPERESULT FOR FUNCTION FIELDS
ROBERT WILMS
Abstract.
We prove that any smooth projective geometrically connectednon-isotrivial curve of genus g ≥ g + 240 g + 380 torsion points for any Abel–Jacobi embeddingof the curve into its Jacobian. The proof basically uses Zhang’s admissiblepairing on curves, the arithmetic Hodge index theorem for function fields andthe metrized graph analogue of Elkie’s lower bound for the Green function. Introduction
Around 1963 Manin and Mumford independently conjectured that for any alge-braically closed field K of characteristic 0, any smooth projective curve X of genus g ≥ K and any divisor D ∈ Div ( X ) of degree 1 on X it holds j D ( X ( K )) ∩ J ( K ) tors < ∞ , where j D : X → J, P P − D denotes the Abel–Jacobi embedding of X into itsJacobian J = Pic ( X ) associated to D . This has been proven by Raynaud [18,Th´eor`eme I.] in 1983. But the following further questions has been left open: Positive characterstic.
One might ask, whether an analogue of this conjecture istrue in positive characteristic. As for K = F p every K -point of a curve over K is atorsion point, the naive analogue is certainly not true. Scanlon [19, Proposition 4.4]and later Pink and Roessler [17, Theorem 3.6] found and proved the right analogueof the Manin–Mumford conjecture in positive characteristic in different ways. Uniformity.
It is reasonable to ask, whether the number of torsion points containedin j D ( X ( K )) is uniformly bounded only in terms of the genus g . This has recentlybeen proven by K¨uhne [11, Theorem 2.] in characteristic 0. Explicitness.
It would be interesting to have an explicit upper bound for the numberof torsion points in j D ( X ( K )). If the curve X is defined over a number field F and J has complex multiplication, Coleman [8, Theorem A.] obtained the explicitbound pg for a prime number p ≥ K/ Q and thereduction of X . Buium [3, Theorem A.] obtained the bound p g g ( p (2 g −
2) + 6 g ) g !without the condition of complex multiplication.In this paper we give a satisfactory answer for all three questions for non-isotrivialcurves over the function field K = k ( B ) of any smooth projective connected curve B over any algebraically closed field k . A curve X over K is called isotrivial ifthere are a finite field extension K ′ of K and a curve C defined over k such that X ⊗ K K ′ ∼ = C ⊗ k K ′ . Date : January 28, 2021.2010
Mathematics Subject Classification. ◦ Theorem 1.1.
Let k be any algebraically closed field, B any smooth projectiveconnected curve over k and K = k ( B ) its function field. For any smooth projectivegeometrically connected non-isotrivial curve X of genus g ≥ defined over K andany Abel–Jacobi embedding j D : X → J, x x − D of X into its Jacobian J = Pic ( X ) , where D ∈ Div ( X ) is any divisor of degree , the number of torsion points in j D ( X ( K )) is uniformly bounded by j D ( X ( K )) ∩ J ( K ) tors ≤ c ( g ) ≤ g + 240 g + 380 , where c (2) = 276 , c (3) = 948 and c ( g ) = j g +16 g − g − g − g − k for g ≥ .Moreover, if X has everywhere potentially good reduction, we may replace c ( g ) by . If only J has everywhere potentially good reduction, we may replace c ( g ) by c tr ( g ) with c tr (2) = 11 and c tr ( g ) = j g − g +2 g +1 g − k ≤ g + 3 for g ≥ . If char k = 0 or X is hyperelliptic, we may replace c ( g ) and c tr ( g ) by j c ( g )+12 k and j c tr ( g )+12 k for g ≥ . The proof makes heavy use of Zhang’s admissible pairing introduced in [23]. If F denotes the divisor of points P , . . . , P s ∈ X ( K ), which are mapped to torsionpoints by j D , then the non-negativity of the N´eron–Tate height of sω − (2 g − F gives a bound of the admissible self-intersection ω a in terms of the Green functionson the metrized reduction graphs in the pairs of the points of F . By an applicationof the arithmetic Hodge index theorem on X we can bound ω a from below byZhang’s invariant ϕ of the metrized reduction graphs. We obtain also an upperbound for the sum of the Green functions in the pairs of points of F in terms of ϕ by the metrized graph analogue of Elkie’s bound for the Green function andan estimate of the supremum of the Green function on metrized graphs. We maysummarize these bounds by max(2 ,g − g +1 ϕ ( X ) ≤ ω a ≤ − g − gs ( s − s X j = k g v ( R v ( P j ) , R v ( P k )) ≤ g − gc ′ ( g ) s − ϕ ( X ) , where ϕ ( X ) = P v ∈| B | ϕ (Γ v ( X )) is the sum of Zhang’s invariant ϕ for all metrizedreduction graphs Γ v ( X ) of X at the closed points v ∈ | B | , g v denotes the Greenfunction on Γ v ( X ) and R v : X ( K ) → Γ v ( X ) is the reduction map. The constant c ′ ( g ) is explicitly given in Lemma 2.2. From this bound it is possible to deduce abound of s in terms of g and independent of K .We obtained the lower bound ω a ≥ g − g +1 ϕ ( X ) of the admissible self-intersectionnumber ω a in the case of number fields already in [21, Theorem 1.2]. It can beproven for function fields by exactly the same arguments. But since the argumentsare widely dispersed over the literature and are often only formulated for numberfields, we decided to include a more self-contained proof here.After completing this paper, the author learned that in September 2020 NicoleLooper announced a similar bound as in Theorem 1.1 under the conditions thatchar k = 0 and X is hyperelliptic in a joint work in progress with Joseph Silvermanat the Arithmetic Dynamics International Online Seminar, see [12] for the notes ofher talk. Their work is also based on Zhang’s admissible pairing. NIFORM MANIN–MUMFORD FOR FUNCTION FIELDS 3
Outline.
In Section 2 we recall the theory of polarized metrized graphs. We givethe definitions of the required invariants and we bound the Green function of apolarized metrized graph in terms of Zhang’s invariant ϕ . We recall the notion ofadelic metrics in Section 3. In Sections 4 and 5 we discuss the admissible adelicmetrics for line bundles on the curve X and on its Jacobian J as well as one oftheir relations. In Section 6 we deduce the lower bound for the admissible self-intersection number ω a from the arithmetic Hodge index theorem on X . Finallyin the last section, we give the proof of Theorem 1.1.2. Polarized metrized graphs
In this section we discuss the notion of polarized metrized graphs as introducedby Zhang [23] and Chinburg–Rumely [5], see also [2]. In particular, we are interestedin the supremum of the canonical Green function of a polarized metrized graph andwe will estimate it in terms of Zhang’s ϕ -invariant.A metrized graph is a compact and connected metric space Γ, which is locallyaround any point p ∈ Γ isometric to the star-shaped set S ( n p , r p ) = { z ∈ C | z = te πik/n p for some 0 ≤ t ≤ r p and k ∈ Z } , for some integer n p ≥
1, the valency of p , and some real number r p >
0, or to apoint, in which case we set n p = 0. A vertex set for Γ is a finite set V ⊆ Γ, suchthat { p ∈ Γ | n p = 2 } ⊆ V and the closure of any connected component of Γ \ V intersects V in exactly two points. We fix a vertex set V and we write e , . . . , e r for the connected components of Γ \ V , which are just open line segments. Let ℓ ( e )be the length of any line segment e and write δ (Γ) = r X i =1 ℓ ( e i )for the total length of Γ.We would like to define the tangent space of Γ at a point p ∈ Γ. By a path in Γwe mean a length-preserving continuous map γ : [0 , l ] → Γ for some l >
0. We saythat two paths γ , γ are equivalent, written by γ ∼ γ , if there exists an ǫ > γ ( t ) = γ ( t ) for all 0 ≤ t < ǫ . The tangent space T p (Γ) of Γ at p is given by T p (Γ) = { γ path in Γ with γ (0) = p } / ∼ . For v ∈ T p (Γ), represented by a path γ , the one-sided directional derivative at p isdefined by d v f ( p ) = lim t → + f ( γ ( t )) − f ( p ) t if the limit exists. For the functionsZh(Γ) = { f : Γ → R | f continuous, piecewise C and d v f ( p ) exists for all p ∈ Γ , v ∈ T p (Γ) } we recall Zhang’s [23, Appendix] definition of the Laplace operator∆( f ) = f ′′ dx + X p ∈ Γ X v ∈ T p (Γ) d v f ( p ) δ p . Here, δ p is the Dirac measure at p and ∆( f ) is considered as a measure on Γ. ROBERT WILMS
For any measure µ of volume 1 on Γ there exists a unique continuous and sym-metric function g µ : Γ × Γ → R , called the Green function associated to µ , suchthat g µ ( x, · ) ∈ Zh(Γ) is characterized by∆ y g µ ( x, y ) = δ y − µ and Z Γ g µ ( x, y ) µ ( y ) = 0for all x ∈ Γ. Further, r ( p, q ) = g δ p ( q, q ) is called the resistance function andmeasures the effective resistance between p and q if we consider Γ as a network,where ℓ ( e ) is the resistance along any edge e .By a divisor on Γ we mean any finite formal sum D = P p ∈ Γ m p p for someintegers m p ∈ Z . A polarization on Γ is an effective divisor K = P p ∈ Γ ( n p + m p − p with m p ≥
0. We call (Γ , K ) a polarized metrized graph of genus g = deg K +1. Inthe following we fix a polarization K of degree deg K = − f : Γ → R and any divisor D = P p ∈ Γ m p p on Γ we write f ( D ) = P p ∈ Γ m p f ( p ).According to [23, Theorem 3.2] there exists a unique measure µ of volume 1 onΓ such that c (Γ , K ) = g µ ( x, K ) + g µ ( x, x )(2.1)is independent of x ∈ Γ. We call g = g µ the canonical Green function associatedto (Γ , K ). Let us recall the following invariants of (Γ , K ): The ǫ -invariant ǫ (Γ , K ) = Z Γ g ( x, x )((2 g − µ + δ K )defined by Zhang in [23, Theorem 4.4] and the ϕ -invariant ϕ (Γ , K ) = − δ (Γ) + Z Γ g ( x, x )((10 g + 2) µ − δ K )defined by Zhang in [26, Theorem 1.3.1]. The following lemma shows, how theconstant c (Γ , K ) can be expressed by these invariants. Lemma 2.1.
Let (Γ , K ) be any polarized metrized graph. The invariant c (Γ , K ) can be expressed by c (Γ , K ) = g (4 ϕ (Γ , K ) + δ (Γ) + ǫ (Γ , K )) . Proof.
If we integrate Equation (2.1) with respect to µ , we obtain by the defintionof the Green function g c (Γ , K ) = Z Γ g ( x, x ) µ. Hence, by the defining equations for the invariants ǫ (Γ , K ) and ϕ (Γ , K ) we get ǫ (Γ , K ) = (2 g − c (Γ , K ) + Z Γ g ( x, x ) δ K , ϕ (Γ , K ) + δ (Γ) = (10 g + 2) c (Γ , K ) − Z Γ g ( x, x ) δ K . Taking the sum of both equations yields12 gc (Γ , K ) = 4 ϕ (Γ , K ) + δ (Γ) + ǫ (Γ , K ) , which proves the lemma after dividing by 12 g on both sides. (cid:3) NIFORM MANIN–MUMFORD FOR FUNCTION FIELDS 5
Next, we recall that we have δ (Γ) ≤ c C ( g ) ϕ (Γ , K )(2.2)with c C (2) = 27, c C (3) = 288 /
17 and c C ( g ) = g (7 g +5)( g − for g ≥ g = 3. If Γ is a tree, we may even replace c C ( g )by c trC ( g ) = g/ (2 g − ϕ -invariant Lemma 2.2.
For any polarized metrized graph (Γ , K ) it holds sup x,y ∈ Γ g ( x, y ) ≤ (12 g − δ (Γ) + 4 ϕ (Γ , K ) − ǫ (Γ , K )12 g ≤ c ′ ( g ) ϕ (Γ , K ) , where c ′ (2) = 55 / , c ′ (3) = 4028 / and c ′ ( g ) = g +8 g − g − g − g (2 g +1)( g − for g ≥ . If Γ is a tree, we may replace c ′ ( g ) by c ′ tr ( g ) = g − g +22 g (2 g − .Proof. By construction we have sup y ∈ Γ g ( x, y ) = g ( x, x ). Hence, it is enough tobound sup x ∈ Γ g ( x, x ). Due to Moriwaki [15, Lemma 4.1] we have g ( x, x ) = c (Γ , K ) − g ( x, K ) = c (Γ , K ) + r ( x, K ) − ǫ (Γ , K )2 g . As r ( x, y ) can be seen as the effective resistance between x and y in the networkΓ, it is bounded from above by the total length δ (Γ) of Γ, see [1, Exercise 12] formore details on this argument. As K is effective of degree 2 g −
2, we concludesup x,y ∈ Γ g ( x, y ) ≤ c (Γ , K ) + (2 g − δ (Γ) − ǫ (Γ , K )2 g . Now the first inequality in the lemma follows by an application of Lemma 2.1.To bound this expression in terms of ϕ (Γ , K ) the direct way would be to useCinkir’s inequality (2.2) and ǫ (Γ , K ) ≥
0. To obtain a slightly better bound, wealso recall from Cinkir’s work [6, Theorem 2.13] that the invariant λ (Γ , K ) = g − g +1) ϕ (Γ , K ) + ( ǫ (Γ , K ) + δ (Γ))satisfies (8 g + 4) λ (Γ , K ) ≥ gδ (Γ). This can be rewritten as − (2 g + 1) ǫ (Γ , K ) ≤ g − ϕ (Γ , K ) − ( g − δ (Γ) . Finally, we may estimate(12 g − δ (Γ) + 4 ϕ (Γ , K ) − ǫ (Γ , K )12 g ≤ (8 g − g − δ (Γ) + 2(3 g − ϕ (Γ , K )4 g (2 g + 1) ≤ (8 g − g − c C ( g ) + 2(3 g − g (2 g + 1) ϕ (Γ , K ) . If we denote the coefficient in the last expression by c ′ ( g ), we can explicitly writefor it c ′ (2) = 55 / c ′ (3) = 4028 /
357 and c ′ ( g ) = g +8 g − g − g − g (2 g +1)( g − for g ≥
4. IfΓ is a tree, we can compute(12 g − δ (Γ) + 4 ϕ (Γ , K ) − ǫ (Γ , K )12 g = (2 g − δ (Γ) − ϕ (Γ , K )2 g ≤ g − g + 22 g (2 g − ϕ (Γ , K ) , ROBERT WILMS as 2 ϕ (Γ , K ) = δ (Γ) + ǫ (Γ , K ) for any tree Γ. This follows for example from [9,Equation (1.4)] as the Jacobian Jac(Γ) of a tree Γ is trivial. (cid:3) We would like to apply the lemma to obtain an explicit lower bound for the sumof the Green functions in pairs of different points in a given set. Let x , . . . , x s ∈ Γbe any points on the metrized graph Γ. Baker and Rumely have proven the followinganalogue of Elkies’ lower bound for the Green function in [2, Proposition 13.7] s X j = k g ( x j , x k ) ≥ − s · sup x ∈ Γ g ( x, x ) . Combining this with Lemma 2.2 we obtain s X j = k g ( x j , x k ) ≥ − sc ′ ( g ) ϕ (Γ , K ) . (2.3) 3. Adelic metrics
We recall the notion of adelic metrics on line bundle in this section. We refer to[24, Section 1] and [4, Section 2] for details. Let K = k ( B ) be the function fieldof a smooth projective connected curve B defined over an algebraically closed field k . Further, let Y be a smooth projective variety over K and L any line bundleon Y . For any closed point v ∈ | B | we denote K v for the algebraic closure of thecompletion of K with respect to v . This is a valuation field and we denote itsvaluation ring by O K v . We write Y v and L v for the pullbacks of Y and L inducedby the embedding K → K v .By a metric k · k on L v we mean a collection of K -norms k · k y on y ∗ L for every y ∈ Y ( K v ). An important example is the model metric k · k f L v associated to anyprojective flat model ( e Y , f L v ) of ( Y, L ⊗ ev ) over O K v for any integer e >
0, which isgiven by k l k f L v ,y = inf a ∈ K v n | a | /e | l ∈ a e y ∗ f L v o for any y ∈ Y ( K v ), where e y ∈ e Y ( O K v ) denotes its unique extension. We call ametric k · k on L v continuous and bounded if there exists a model ( e Y , f L v ), such thatlog k·kk·k g L v is continuous and bounded.We call a collection of metrics k · k = {k · k v | v ∈ | B |} an adelic metric on L if k · k v is a continuous and bounded on L v for all v ∈ | B | and there is an open non-empty subset U ⊆ B and a model ( e Y , e L ) of ( Y, L ) over U such that k · k v = k · k e L v for every v ∈ U , where e L v denotes the pullback of e L along Spec( O K v ) → U . Apair L = ( L , k · k ) of a line bundle L and an adelic metric k · k on L is called an adelic line bundle . An isometry between two adelic line bundles L = ( L , k · k )and L = ( L , k · k ) is an isomorphism of line bundles L ∼ = L , which induces anisometry between the metrized line bundles ( L ,v , k · k ,v ) and ( L v, , c v k · k v, ) on Y v for all v ∈ | B | , where the c v ’s are any constants such that Q v ∈| B | c v = 1. Inparticular, there is up to isometry a unique way to consider a real number r ∈ R asthe trivial bundle O Y equipped with its canonical metric multiplied by a constantsuch that Q v ∈| B | k k v = e − r . NIFORM MANIN–MUMFORD FOR FUNCTION FIELDS 7
We call an adelic line bundle L nef if there is a sequence of models ( e Y n , e L n )with e L n nef on e Y n , such that log k·k e L n,v k·k v converges to 0 uniformly in Y ( K v ) forall v ∈ | B | . Further, we call an adelic line bundle L integrable if there exist nefadelic line bundles L and L and an isometry L ∼ = L ⊗ L ⊗− . We will use in thesubsequent sections the additive notations L + L = L ⊗ L and n L = L ⊗ n for n ∈ Z .Next we define intersection numbers of integrable line bundles. Let Z ⊆ Y bea subvariety of dimension d = dim Z and L , . . . , L d integrable line bundles on Y . We choose local sections l , . . . , l d of L , . . . , L d such that their common zerolocus does not intersect Z . Fix a v ∈ | B | . If the metric k · k v on L i is induced by amodel ( e Y v , e L i,v ) of ( Y, L ⊗ ei ) for all i and some e >
0, we define the local intersectionnumber at v by the usual intersection number on e Y v (cid:16) c div( l ) · · · c div( l d ) · [ Z ] (cid:17) v = div f Y v ( l | Y v ) · . . . · div e Y v ( l d | Y v ) · [ e Z v ] /e d +1 , where e Z v denotes the Zariski closure of Z v in e Y v . In general, we assume the local in-tersection number (cid:16) c div( l ) · · · c div( l d ) · [ Z ] (cid:17) v at v to vary continuously with respectto the metric on the line bundles. As the metric on L i at v is the limit of metricsinduced by models for all v ∈ | B | and all i , we can define the global intersectionnumber by L · · · L d · Z = X v ∈| B | (cid:16) c div( l ) · · · c div( l d ) · [ Z ] (cid:17) v . If dim Z = 0 we also write d deg( L | Z ) = L · Z and if Z = Y we shortly write L · · · L d = L · · · L d · Z .If moreover div( l d ) is a prime divisor on Y such that l i | div( l d ) = 0 for all i < d ,the intersection number can be computed recursively by the formula L · · · L d = L | div( l d ) · · · L d − | div( l d ) − X v ∈| B | Z Y ( K v ) log k l d k v c ( L ) · · · c ( L d − ) , (3.1)where the integral is defined as follows: First, let each adelic line bundle L i beinduced by a model ( e Y , e L i ) of ( Y, L ⊗ ei ). We write e l d for the section of e L d extending l ⊗ ed . Then V = div( e l d ) − e · div( l d ) is a Weil divisor V = P v ∈| B | V v supported inthe closed fibers of e Y and we set Z Y ( K v ) log k l d k v c ( L ) · · · c ( L d − ) = c ( e L ) · · · c ( e L d − )[ V v ] /e d +1 . In general the integral is defined by continuity and taking limits.Let f : Y → S be any proper smooth morphism of smooth projective varietiesover K and L , . . . , L d integrable line bundles on S . Then we have a projectionformula L · · · L d · f ∗ ( Z ) = f ∗ L · · · f ∗ L d · Z. (3.2)For model metrics this formula follows from the projection formula in classicalintersection theory, see for example [10, Example 2.4.3]. In general it follows by ROBERT WILMS taking limits. If s = dim S is the dimension of S and M , . . . , M s are integrableline bundles on S , we obtain another projection formula f ∗ M · · · f ∗ M s · L · · · L d − s = M · · · M s · ( c ( L ) . . . c ( L d − s )[ f ]) , (3.3)which one also proves first by the classical projection formula for model metricsand by taking limits in general. Here, c ( L ) . . . c ( L n )[ f ] denotes the multidegreeof the generic fiber of f with respect to the line bundles L , . . . , L n .4. Admissible metrics on the curve
In this section we recall Zhang’s admissible pairing on curves introduced in [23].We use the notation as in Theorem 1.1, where we additionally assume, that X hassemistable reduction over B and that X ( K ) is non-empty.First, let us recall the notion of the metrized reduction graph. Let π : X → B bethe minimal regular model of X over B . The polarized metrized reduction graphΓ v ( X ) of X at v ∈ | B | is defined as follows: The underlying graph of Γ v ( X ) isthe dual graph of the special fiber X v = π − ( v ), which vertex set V consists of theirreducible components of X v and which edge set consists of the singular points in X v , such that any edge e connects the two vertices corresponding to the irreduciblecomponents meeting in the singular point corresponding to e . We assign to everyedge e the length ℓ ( e ) = 1 obtaining a metrized graph Γ v ( X ). Finally, we equipΓ v ( X ) with the polarization K v = X p ∈ V ( n p + g p − p, where n p denotes the valency of the vertex p and g p denotes the genus of the normal-ization of the irreducible component corresponding to the vertex p . The polarizedmetrized graph Γ v ( X ) = (Γ v ( X ) , K v ) has genus g . We write g v for the canonicalGreen function associated to Γ v ( X ). There is a canonical map R v : X ( K ) → Γ v ( X )given as follows: We can uniquely extend any point P ∈ X ( K ) to a section e P : B → X of π and we set R v ( P ) to be the vertex of Γ v ( X ) corresponding tothe irreducible component of X v , where e P intersects X v .Zhang defined canonical admissible adelic metrics on the canonical bundle ω on X , on the bundle O X ( D ) associated to any divisor D on X and on the diagonalbundle O X (∆) on X . If 1 ∆ denotes the canonical section on O X (∆), the metricsatisfies log k ∆ ( P, Q ) k v = − i v ( P, Q ) − g v ( R v ( P ) , R v ( Q )) , (4.1)where P, Q ∈ X ( K ) are different K -rational points and i v ( P, Q ) denotes the usualintersection number on X v of the unique sections e P and e Q of π extending P and Q .In a similar way we may obtain log k ∆ ( P, Q ) k v for any different geometric points P, Q ∈ X ( K ) by repeating the above construction for X ⊗ K K ′ instead of X , where K ′ /K denotes a finite field extension such that P, Q ∈ X ( K ′ ) and dividing theresulting number by [ K ′ : K ] afterwards. The canonical admissible metric on ω is chosen such that the canonical isomorphism s ∗ O X (∆) ∼ = ω ⊗− is an isometry,where s : X → X , x ( x, x ) denotes the diagonal embedding. Similarly, theadmissible metric on O X ( P ) for a point P is given such that the canonical isomor-phism s ∗ P O X (∆) ∼ = O X ( P ) is an isometry, where s P : X → X , x ( x, P ) is theembedding associated to P . We denote ω a , D a and ∆ a for the canonical admissible NIFORM MANIN–MUMFORD FOR FUNCTION FIELDS 9 adelic line bundles ω , O X ( D ) and O X (∆). According to [26, Proposition 3.5.1]these are integrable line bundles.In general we call an adelic line bundle L a = L on X admissible if its metricdiffers only by a constant from the metric on D a , where D is a divisor on X satisfying O X ( D ) ∼ = L . The admissible pairing ( L a , M a ) of two admissible linebundles L a , M a on X is defined to be the intersection number L a · M a of theintegrable line bundles. For two different points P, Q ∈ X ( K ) we deduce fromEquation (4.1), that( P a , Q a ) = X v ∈ B ( i v ( P, Q ) + g v ( R v ( P ) , R v ( Q ))) . (4.2)Further, Zhang obtained in [23, Theorem 4.4] for the self-intersection number of ω a ω a = ( ω a , ω a ) = ω X /B − X v ∈ B ǫ (Γ v ( X )) , (4.3)where ω X /B denotes the usual self-intersection number of the relative dualizingsheaf ω X /B . We would also like to recall the adjunction formula [23, Theorem 4.2]( ω a , P a ) = − ( P a , P a )(4.4)for any point P ∈ X ( K ). If L a is an admissible line bundles on X with deg L = 0,then L defines a point in the Jacobian variety Pic ( X ) of X and due to [23, (5.4)]its N´eron–Tate height is equal to h NT ( L ) = − ( L a , L a ) . (4.5) 5. Admissible metrics on the Jacobian variety
Continuing the notation from the last section, we discuss admissible metricson the Jacobian variety J = Pic ( X ) of X . Moreover, we will compare it to theadmissible metrics on the curve. We refer to [24, Section 2] and [22] for more detailsand to [13, Section 1] for the definition of the Jacobian variety.We fix a point P ∈ X ( K ) and we denote Θ for the divisor given by the image ofthe map X g − → J, ( P , . . . , P g − ) [ P + · · · + P g − − ( g − P ] , which is ample by [13, Theorem 6.6]. Hence, we get an ample and symmetricline bundle L = O J (Θ) ⊗ [ − ∗ O J (Θ) on J , where in general [ n ] : J → J denotesthe multiplication-by- n morphism on J for any n ∈ Z . Thus, we can choose anisomorphism φ : L ⊗ ∼ = [2] ∗ L . Let ( e J, e L ) be a model of ( J, L ) over B . As L is ample,we may assume that e L is nef. There exists an open non-empty subset U ⊆ B , suchthat the maps [2] and φ extend to maps [2] : e J U → e J U and φ U : e L ⊗ U → [2] ∗ e L U on the base changes e J U and e L U of e J and e L along the embedding U → B . Weset ( e J , e L ) = ( e J, e L ) and inductively, we construct models ( e J n , e L n ) of ( J, L ⊗ n ),such that [2] : J → J extends to a map f n : e J n +1 → e J n and e L n +1 = f ∗ n e L n . Thenthe sequence of metrics on L associated to the models ( e J n , e L n ) converges to anadelic metric k · k on L , as the restriction of k · k to U is given by the model metricof ( e J U , e L U ) and as Zhang proved in [24, Theorem (2.2)], k · k v is continuous andbounded for all v ∈ | B | . We call ˆ L = ( L , k · k ) an admissible adelic line bundle . Since e L n is nef for all n ≥
0, we obtain that ˆ L is a nef adelic line bundle. Weremark, that we have by construction h NT ( x ) = h ˆ L ( x ) := d deg( ˆ L| x )(5.1)for all x ∈ J ( K ).The following lemma gives a comparison of the admissible metrics obtained onthe curve and on the Jacobian variety. Lemma 5.1.
Consider the map f : X → J, ( P , P ) [( g − P + P ) − ω ] and write p i : X → X for the projection to the i -th factor. There exists an isometry f ∗ ˆ L ∼ = ( g − g + 1)( p ∗ ω a + p ∗ ω a ) − g − ∆ a − ω a . In particular, the adelic metrized line bundle on the right hand side is nef.Proof.
We first show, that both line bundles are isomorphic. This part of the proofis motivated by a similar result in [14, Lemme 4.10.2]. Let U be the universal linebundle on X × J , which is trivial on { P } × J . According to [14, Corollaire 2.5]it holds det R q ∗ U ∼ = O J ( − Θ), where q : X × J → J denotes the projection to thesecond factor. We denote f ′ = [ − ◦ f and we consider the following diagrams X × X × f / / p (cid:15) (cid:15) X × J q (cid:15) (cid:15) X f / / J , X × X × f ′ / / p (cid:15) (cid:15) X × J q (cid:15) (cid:15) X f ′ / / J , where in general p ij : X → X denotes the projection to the i -th and j -th factors.We may compute by the base changes associated to these diagrams f ∗ L ∼ = f ∗ O J ( − Θ) ⊗− ⊗ f ∗ [ − ∗ O J ( − Θ) ⊗− ∼ = f ∗ (det R q ∗ U ) ⊗− ⊗ f ′∗ (det R q ∗ U ) ⊗− ∼ = (det R p ∗ (id × f ) ∗ U ) ⊗− ⊗ (det R p ∗ (id × f ′ ) ∗ U ) ⊗− . Now we consider the line bundle M = p ∗ O X (∆) ⊗ g − ⊗ p ∗ O X (∆) ⊗ g − ⊗ p ∗ ω ⊗− . on X , where p : X → X denotes the projection to the first factor. It has degree0 on every fiber of p and it holds(id × f )( P , P , P ) = (cid:0) P , (cid:2) M| X ×{ P ,P } (cid:3)(cid:1) . As U is the universal bundle which is trivial on { P } × J , we obtain(id × f ) ∗ U ∼ = M ⊗ p ∗ s ∗ P M ⊗− , (id × f ′ ) ∗ U ∼ = M ⊗− ⊗ p ∗ s ∗ P M , where s P : X → X , ( P , P ) ( P, P , P ) is the section of p associated to P .We recall from [14, Section 1] the following two rules for calculating the determinant:First, it holds det R p ∗ ( L ⊗ p ∗ M ) ∼ = det R p ∗ L ⊗ M ⊗ χ p ( L ) , where L is any line bundle on X , M is any line bundle on X and χ p denotesthe relative Euler characteristic of the family p : X → X . Note, that it holds NIFORM MANIN–MUMFORD FOR FUNCTION FIELDS 11 χ p ( M ) = χ p ( M ⊗− ) = 1 − g , as M has degree 0 on every fiber of p . Secondly,for any section s : X → X of p we havedet R p ∗ ( L ⊗ O X ( s )) ∼ = det R p ∗ L ⊗ s ∗ ( L ⊗ O X ( s )) . Let us define the sections s : X → X , ( P , P ) ( P , P , P ) ,s : X → X , ( P , P ) ( P , P , P )of the projection p . Note, that s ∗ O X ( s ) = s ∗ O X ( s ) = O X (∆) and that s ∗ i O X ( s i ) = p ∗ i ω ⊗− as well as p ◦ s i = p i . It followsdet R p ∗ (id × f ) ∗ U∼ = det R p ∗ (cid:0) O X ( s ) ⊗ g − ⊗ O X ( s ) ⊗ g − ⊗ p ∗ ω ⊗− (cid:1) ⊗ s ∗ P M ⊗ g − ∼ = det R p ∗ p ∗ ω ⊗− ⊗ s ∗ (cid:16) O X ( s ) ⊗ g ( g − / ⊗ O X ( s ) ⊗ ( g − ⊗ p ∗ ω ⊗ − g (cid:17) ⊗ s ∗ (cid:16) O X ( s ) ⊗ g ( g − / ⊗ p ∗ ω ⊗ − g (cid:17) ⊗ s ∗ P M ⊗ g − ∼ = ( p ∗ ω ⊗ p ∗ ω ) ⊗− ( g +2)( g − / ⊗ O X (∆) ⊗ ( g − ⊗ s ∗ P M ⊗ g − , where we used for the last line that det R p ∗ p ∗ ω ⊗− ∼ = π ∗ det R π ∗ ω ⊗− ∼ = O X by the base change X p / / p (cid:15) (cid:15) X π (cid:15) (cid:15) X π / / Spec( K ) . In a similar way we getdet R p ∗ (id × f ′ ) ∗ U ∼ = ( p ∗ ω ⊗ p ∗ ω ) ⊗− g ( g − / ⊗ O X (∆) ⊗ ( g − ⊗ s ∗ P M ⊗ − g . Putting everything together, we obtain as desired f ∗ L ∼ = ( p ∗ ω ⊗ p ∗ ω ) ⊗ ( g +1)( g − ⊗ O X (∆) ⊗− g − . Now, we would like to check, that this isomorphism is an isometry up to theconstant ω a . For this purpose, we choose two geometric points P , P ∈ X ( K ). Onthe one hand, we know by Equations (3.2) and (5.1) that d deg (cid:16) f ∗ ˆ L| ( P ,P ) (cid:17) = d deg (cid:16) ˆ L| f ( P ,P ) (cid:17) = h NT (( g − P + P ) − ω ) . On the other hand, we may compute using Equations (4.4) and (4.5) h NT (( g − P + P ) − ω )= − (( g − P ,a + P ,a ) − ω a , ( g − P ,a + P ,a ) − ω a )= ( g − g + 1)( ω a , P ,a + P ,a ) − g − ( P ,a , P ,a ) − ω a = d deg (cid:0) (( g − g + 1)( p ∗ ω a + p ∗ ω a ) − g − ∆ a − ω a ) | ( P ,P ) (cid:1) . Hence, we obtain, that the above isomorphism indeed induces an isometry f ∗ ˆ L ∼ = ( g − g + 1)( p ∗ ω a + p ∗ ω a ) − g − ∆ a − ω a of line bundles on X , as the degree of the adelic line bundles is the same at everygeometric point in X ( K ). (cid:3) An application of the arithmetic Hodge index theorem
The goal of this section is to deduce a lower bound for ω a as an application ofthe arithmetic Hodge index theorem for function fields by Carney [4, Theorem 3.1]for X . This is motivated by the analogue result in the number field case we foundin [21, Theorem 1.2]. We continue the notation from the previous sections. Let usfirst recall the inequality part of the arithmetic Hodge index theorem for functionfields for the variety X . If N and M are integrable line bundles on X such that(i) N is nef,(ii) N is big and(iii) their usual intersection number satisfies M · N = 0,then it holds
M · M · N ≤ Proposition 6.1.
Any smooth projective geometrically connected curve X of genus g ≥ defined over K satisfies ω a ≥ max(2 ,g − g +1 X v ∈| B | ϕ (Γ v ( X )) . If char k = 0 or X is hyperelliptic, we may replace max(2 , g − by g − .Proof. If char k = 0, this follows from [26, Section 1.4] and if X is hyperelliptic,we even have an equality due to [26, Corollary 1.3.3]. As all curves of genus g = 2are hyperelliptic, it remains to prove ω a ≥ g − g +1 P v ∈ B ϕ (Γ v ( X )) for g ≥
3. Fora shorter notation, we write ω = p ∗ ω + p ∗ ω , where p i : X → X denotes theprojection to the i -th factor. Further, we set ˆ ω = p ∗ ω a + p ∗ ω a . We would like toapply the arithmetic Hodge index theorem to the integrable line bundles N = ( g + 1)ˆ ω − g − a − ω a g − , M = ( g − a − ˆ ω on X .As ( g − N is nef by Lemma 5.1, N is also nef, such that condition (i) is satisfied.This also implies, that the underlying line bundle N is nef on X , such that wehave vol( N ) = N . We compute the volume explicitlyvol( N ) = (( g + 1)( ω ) − g − O X (∆)) = 2( g + 1) p ∗ ω · p ∗ ω − g − ω · O X (∆) + 4( g − O X (∆) = (cid:0) g + 1) )( g − − g − − g − (cid:1) deg ω = 8 g ( g − > . Hence, also condition (ii) is satisfied. To check condition (iii), we calculate
M · N = (( g + 1)( ω ) − g − O X (∆)) · (( g − O X (∆) − ω ))= − g + 1) p ∗ ω · p ∗ ω + ( g + 3)( g − ω O X (∆) − g − O X (∆) = ( − g + 1)( g −
1) + 2( g + 3)( g −
1) + 2( g − ) deg ω = 0 . Thus, the arithmetic Hodge index theorem implies that
M·M·N ≤
0. We explicitlycompute the left hand side by
M · M · N = − g − ∆ a + ( g + 5)( g − ˆ ω · ∆ a − g + 2)( g − ω · ∆ a (6.1) + ( g + 1)ˆ ω − M · M g − ω a . NIFORM MANIN–MUMFORD FOR FUNCTION FIELDS 13
By a similar computation as above we have
M · M = 2 p ∗ ω · p ∗ ω − g − ω · O X (∆) + ( g − O X (∆) = − g − . By symmetry and by formula (3.3) we obtainˆ ω = 6( p ∗ ω a · p ∗ ω a · p ∗ ω a ) = 6 · ω a · deg ω = 12( g − ω a . In the same way, we obtain p ∗ ω a · p ∗ ω a · ∆ a = ω a . Let s : X → X , x ( x, x )the diagonal embedding. Then we can compute by the recursion formula for theintersection number (3.1) and by the definition of the metric on ∆ a in (4.1) p ∗ ω a · p ∗ ω a · ∆ a = s ∗ p ∗ ω a · s ∗ p ∗ ω a + X v ∈| B | Z X ( K v ) g v c ( p ∗ ω a ) c ( p ∗ ω a ) = ω a , where the integral vanishes by the definition of the Green function g v , see also[26, Section 3.5] where it is also explained in more details why we can replace − log k ∆ k v by g v . Hence, we can conclude using again symmetryˆ ω · ∆ a = 2( p ∗ ω a · p ∗ ω a · ∆ a + p ∗ ω a · p ∗ ω a · ∆ a ) = 4 ω a . In a similar way, we obtain by formulas (3.1) and (4.1) p ∗ ω a · ∆ a · ∆ a = s ∗ p ∗ ω a · s ∗ ∆ a + X v ∈| B | Z X ( K v ) g v c ( p ∗ ω a ) c (∆ a ) = − ω a , where the integral vanishes as in [26, Lemma 3.5.2], and we conclude by symmetryˆ ω · ∆ a = 2( p ∗ ω a · ∆ a · ∆ a ) = − ω a . Finally, the self-intersection number ∆ a can be computed by∆ a = ( s ∗ ∆ a ) + X v ∈| B | Z X ( K v ) g v c (∆ a ) = ω a − X v ∈| B | ϕ (Γ v ( X )) , where the last equality follows from [26, Lemma 3.5.4]. Putting everything intoEquation (6.1) yields0 ≥ M · M · N = 2( g − X v ∈| B | ϕ (Γ v ( X )) − g − (2 g + 1) ω a , which is equivalent to the inequality we wanted. (cid:3) Proof of Theorem 1.1
In this section we give the proof of Theorem 1.1 using the same notation as inthe theorem.By the semistable reduction theorem there exists a finite field extension K ′ of K such that X K ′ = X ⊗ K K ′ has semistable reduction over the normalization B ′ of B in K ′ . As X K ′ ( K ) = X ( K ), we may assume, that X has semistable reductionover B . We write π : X → B for the minimal regular model of X over B .If j D ( X ( K )) ∩ J ( K ) tors ≤
1, we have nothing to prove. Hence, we may assume j D ( X ( K )) ∩ J ( K ) tors ≥ F = { P , . . . , P s } ⊆ X ( K ) is any set of s ≥ j D ( F ) ⊆ J ( K ) tors . Replacing K by a finite fieldextension again, we may assume that F ⊆ X ( K ).For any two points P, Q ∈ F we have P − Q = ( P − D ) − ( Q − D ) = j D ( P ) − j D ( Q ) ∈ J ( K ) tors , and hence by Equation (4.5),0 = h NT ( P − Q ) = − ( P a − Q a , P a − Q a ) = 2( P a , Q a ) − ( P a , P a ) − ( Q a , Q a ) . Applying this to all points in F , we obtain s X j =1 ( P j,a , P j,a ) = 1 s − s X j = k ( P j,a , P k,a ) . (7.1)We also write F for the divisor F = P + · · · + P s . Again by Equation (4.5) it holds0 ≤ h NT ( sω − (2 g − F ) = − ( sω a − (2 g − F a , sω a − (2 g − F a ) . By bilinearity and symmetry of the pairing we may rewrite this inequality as ω a ≤ g − s s X j =1 ( ω a , P j,a ) − g − s s X j =1 ( P j,a , P j,a ) + s X j = k ( P j,a , P k,a ) . As ( ω a , P j,a ) = − ( P j,a , P j,a ) by the adjunction formula (4.4), we obtain by anapplication of Equation (7.1) ω a ≤ (cid:16) − g − s ( s − − g − s − g − s ( s − (cid:17) s X j = k ( P j,a , P k,a ) = − g − gs ( s − s X j = k ( P j,a , P k,a ) . By Equation (4.2) we have( P j,a , P k,a ) = X v ∈| B | ( i v ( P j , P k ) + g v ( R v ( P j ) , R v ( P k ))) ≥ X v ∈| B | g v ( R v ( P j ) , R v ( P k )) . Thus, we conclude using Proposition 6.1 and Equation (2.3) max(2 ,g − g +1 X v ∈| B | ϕ (Γ v ( X )) ≤ ω a ≤ g − gc ′ ( g ) s − X v ∈| B | ϕ (Γ v ( X ))with c ′ ( g ) as in Lemma 2.2.If X has everywhere good reduction, we have ω a = ω X /B by Equation (4.3).The Noether formula states ω X /B = 12 deg π ∗ ω X /B in this case and it has beenshown by Parˇsin [16, Proposition 5] for char k = 0 and by Szpiro [20, Theorem 1]for char k > π ∗ ω X /B > X is non-isotrivial. On the other hand wehave P v ∈| B | ϕ (Γ v ( X )) = 0 in this case, contradicting the above inequality. Hence, j D ( X ( K )) ∩ J ( K ) tors ≤ X has everywhere good reduction.If X has not everywhere good reduction, we have P v ∈| B | ϕ (Γ v ( X )) > s ≤ (cid:22) g − g (2 g + 1) c ′ ( g )max(2 , g − (cid:23) + 1 = c ( g ) . (7.2)Let us compute this number c ( g ) explicitly. If g = 2, we obtain c (2) = 276. For g = 3, we have c (3) = 948. In general, it holds c ( g ) = j g +16 g − g − g − g − k for g ≥ X and J have semistable reduction, they have good reduction at a place ifthey have potentially good reduction at this place. As already mentioned above,we have j D ( X ( K )) ∩ J ( K ) tors ≤ X has everywhere good reduction. If J has everywhere good reduction, every reduction graph Γ v ( X ) is a tree. Hence byLemma 2.2, we may replace c ′ ( g ) in (7.2) by c ′ tr ( g ), such that we obtain s ≤ c tr ( g ) NIFORM MANIN–MUMFORD FOR FUNCTION FIELDS 15 with c tr (2) = 11 and c tr ( g ) = j g − g +2 g +1 g − k for g ≥
3. If char( K ) = 0 or X ishyperelliptic, we may replace max(2 , g −
1) in (7.2) by 2 g − g ≥ s ≤ j c ( g )+12 k , respectively s ≤ j c tr ( g )+12 k if J has everywheregood reduction. References [1] Baker, M. and Faber, X.:
Metrized graphs, Laplacian operators, and electrical networks . In:Quantum graphs and their applications, Contemp. Math. (2006), 15–33.[2] Baker, M. and Rumely, R.:
Harmonic analysis on metrized graphs . Canad. J. Math. (2007), no. 2, 225–275.[3] Buium, A.: Geometry of p-jets . Duke Math. J. (1996), 349–367.[4] Carney, A.: The arithmetic Hodge index theorem and rigidity of dynamical systems overfunction fields . Pacific J. Math. (2020), 71–102.[5] Chinburg, T. and Rumely, R.:
The capacity pairing . J. Reine Angew. Math. (1993),1–44.[6] Cinkir, Z.:
Zhang’s conjecture and the effective Bogomolov conjecture over function fields .Invent. Math. (2011), no. 3, 517–562.[7] Cinkir, Z.:
Admissible invariants of genus 3 curves . Manuscripta Math. (2015), no. 3-4,317–339.[8] Coleman, R.:
Torsion points on curves and p-adic abelian integrals . Ann. of Math. (1985), 111–168.[9] de Jong, R. and Shokrieh, F.:
Tropical moments of tropical Jacobians . Preprint,arXiv:1810.02639 (2018).[10] Fulton, W.:
Intersection theory . Second edition. Ergeb. Math. Grenzgeb. , Springer-Verlag,Berlin, 1998.[11] K¨uhne, L.: Equidistribution in families of abelian varieties and uniformity . Preprint,arXiv:2101.10272 (2021).[12] Looper, N.:
The admissible pairing and quantitative Bogomolov for curves . Notes from atalk at the Arithmetic Dynamics International Online Seminar on Sept. 2, 2020, available at https://drive.google.com/file/d/1d2_GC4d7-emUvEjEy8jLYFbppmeNjip9/view .[13] Milne, J. S.:
Jacobian varieties . In: Arithmetic geometry (Storrs, Conn., 1984), Springer,New York, 1986, 167–212.[14] Moret-Bailly, L.:
M´etriques permises . In: S´eminaire sur les Pinceaux Arithm´etiques: LaConjecture de Mordell, (Szpiro ed.), Ast´erisque No. (1985), 29–87.[15] Moriwaki, A.:
A sharp slope inequality for general stable fibrations of curves . J. Reine Angew.Math. (1996), 177–195.[16] Parˇsin, A. N.:
Algebraic curves over function fields. I . Math. USSR-Izvestiya (1968), no.5, 1145–1170.[17] Pink, R. and Roessler, D.: On ψ -invariant subvarieties of semiabelian varieties and theManin–Mumford conjecture . J. Algebraic Geom. (2004), 771–798.[18] Raynaud, M.: Courbes sur une vari´et´e ab´elienne et points de torsion . Invent. math. (1983), 207–233.[19] Scanlon, T.: Diophantine geometry from model theory . Bull. Symbolic Logic (2001), 37–57.[20] Szpiro, L.: Propri´et´es num´eriques du faisceau dualisant relatif . In: S´eminaire sur les pinceauxde courbes de genre au moint deaux, Ast´erisque (1981), Soci´et´e Math. de France, Paris,44–78.[21] Wilms, R.: On arithmetic intersection numbers on self-products of curves. . Preprint,arXiv:1903.12159 (2019).[22] Yamaki, K.:
Survey on the geometric Bogomolov conjecture . In: Actes de la Conf´erence “Non-Archimedean Analytic Geometry: Theory and Practice”, Publ. Math. Besan¸con Alg`ebreTh´eorie Nr. 2017/1 (2017), 137–193.[23] Zhang, S.:
Admissible pairing on a curve . Invent. Math. (1993), no. 1, 171–193.[24] Zhang, S.:
Small points and adelic metrics . J. Algebraic Geom. (1995), no. 2, 281–300.[25] Zhang, S.: Heights and reductions of semi-stable varieties . Compositio Math. (1996),no. 1, 77–105.[26] Zhang, S.:
Gross–Schoen cycles and dualising sheaves . Invent. Math. (2010), no. 1, 1–73.6 ROBERT WILMS