aa r X i v : . [ m a t h . N T ] O c t HEIGHTS, RANKS AND REGULATORS OF ABELIAN VARIETIES
FABIEN PAZUKI
Abstract.
We give a lower bound on the Faltings height of an abelian variety over a numberfield by the sum of its injectivity diameter and the norm of its bad reduction primes. Itleads to an unconditional explicit upper bound on the rank of Mordell-Weil groups. Assumingthe height conjecture of Lang and Silverman, we then obtain a Northcott property for theregulator on the set of polarized simple abelian varieties defined over a fixed number field K , of dimension g and rank m K bounded from above and with dense K -rational points. Weremove the simplicity assumption in the principally polarized case by giving a refined versionof the Lang-Silverman conjecture. Keywords:
Heights, abelian varieties, regulators, Mordell-Weil.
Mathematics Subject Classification:
Hauteurs, rangs et régulateurs des variétés abéliennes . Résumé.
On minore la hauteur de Faltings d’une variété abélienne sur un corps de nom-bres par la somme de son diamètre d’injectivité et de la norme de ses premiers de mauvaiseréduction. Cela entraîne une majoration explicite inconditionnelle du rang des groupes deMordell-Weil. On obtient alors comme conséquence d’une conjecture de Lang et Silvermanune propriété de Northcott pour le régulateur sur l’ensemble des variétés abéliennes simples,polarisées et définies sur un corps de nombres, à dimension et rang bornés et dont les points ra-tionnels sont denses. On montre comment se passer de l’hypothèse de simplicité dans le cas depolarisation principale en proposant une version raffinée de la conjecture de Lang-Silverman.
Mots-Clefs:
Hauteurs, variétés abéliennes, régulateurs, Mordell-Weil.———1.
Introduction
Let K be a number field of degree d over Q and let M K stand for the set of all places of K .We denote by M ∞ K the set of archimedean places. For any place v ∈ M K , we denote by K v thecompletion of K with respect to the valuation | . | v . One fixes | p | v = p − as a normalization forany finite place v above a rational prime p . The local degree will be denoted by d v = [ K v : Q v ] .Let A be an abelian variety of dimension g defined over the number field K . The set ofrational points of A over K is finitely generated by the Mordell-Weil Theorem, and the aimof this article is to study links between the rank of the Mordell-Weil group, the regulator ofthe Mordell-Weil lattice and the Faltings height of the abelian variety A . Many thanks to Pascal Autissier, Marc Hindry, Robin de Jong, Qing Liu, Gaël Rémond and Martin Widmerfor exchanging ideas concerning this article. The referees gave good advice leading to several improvements,may they be warmly thanked. The author is supported by the DNRF Niels Bohr Professorship of LarsHesselholt and ANR-14-CE25-0015 Gardio.
F. Pazuki
We first give an inequality between the Faltings height h F + ( A/K ) of Definition 2.2 and thenorm of the bad reduction primes of A , interesting in itself and useful for the following results. Theorem 1.1.
Let g ≥ be an integer. Let K be a number field of degree d . There exist twoquantities c = c ( g ) > , c = c ( g ) ∈ R , such that for any abelian variety A of dimension g ,defined over K , one has h F + ( A/K ) ≥ c d log N A/K + c where N A/K is the product of the norms of the primes of K of bad reduction for A . Theexplicit values c = (12 g ) − g g g and c = − /c are valid. We believe this inequality will be useful in different contexts. We believe furthermore thatsome steps in the proof presented here could be useful as well (explicit Bertini, reduction tothe jacobian case by quotient with explicit bounds on their heights, etc). See Proposition3.2 for a detailed description of the argument. A similar statement (for the semi-stable case)was obtained in [HiPa16] independently. They show the result over function fields and withdifferent arguments. The main difference is the fact that the quantities c and c here don’tdepend on the base field K , but only on the dimension g . Their strategy of proof seems towork over number fields as well, based on rigid uniformization of abelian varieties. Remarkthat their lower bound (at least in the case of function fields) is given in terms of Tamagawanumbers. Another difference is that the quantities c and c here are given explicitly, andare not too extreme in the case of jacobian varieties, where c = 1 / g +1 and c = 0 , seeProposition 3.2 and Proposition 3.6 below. Unfortunately, it is unlikely that the explicitexpressions we obtain could be improved in a significant way using the strategy we followhere. The main reason is the use of Theorem 1.3 page 760 of [Rém10] where one alreadyhas a tower with three levels of exponents, plus the fact that the control given in [CaTa12]on the genus of the curve constructed by the Bertini argument is (more than) exponential:the combination of these inequalities is forcing our c and our c to behave like a four levelsexponential function in the dimension of A , and the aforementioned results both seem difficultto improve on.Whenever a polarization is given on A , one can obtain a richer lower bound. Let ( A, L ) be a polarized abelian variety of dimension g defined over the number field K . We givean inequality between the Faltings height h F + ( A/K ) of Definition 2.2, the norm of the badreduction primes of A over K and the injectivity diameter of ( A ( C ) , L ) . As a direct corollaryof Theorem 1.1 and of the Matrix Lemma (see Theorem 3.1 below) we obtain: Corollary 1.2.
Let g ≥ be an integer. Let K be a number field of degree d . There exist threeexplicit quantities c = c ( g ) > , c = c ( g ) > , c = c ( g ) ∈ R , such that for any abelianvariety A of dimension g , defined over K , for any ample line bundle L carrying a polarizationon A , one has h F + ( A/K ) ≥ c d log N A/K + c d X v ∈ M ∞ K d v ρ ( A v , L v ) − + c where N A/K is the product of the norms of the primes of K of bad reduction for A and ρ ( A v , L v ) is the injectivity diameter of A v ( C ) polarized by L v . The explicit values c = (12 g ) − g g g / , c = 1 / and c = − /c are valid. eights, ranks and regulators of abelian varieties A isthe semi-stable jacobian of a curve, one can take c = 1 / , c = 1 / and c = − g/ .One obtains the following result as another corollary of Theorem 1.1 and preexisting boundson the Mordell-Weil rank. Corollary 1.3.
Let A be an abelian variety of dimension g defined over a number field K ofdegree d and discriminant ∆ K . Let m K be the Mordell-Weil rank of A ( K ) . There exists aquantity c = c ( d, g ) > such that m K ≤ c max { , h F + ( A/K ) , log | ∆ K |} , and the explicit value c = (12 g ) g g g d is valid. Let us add another remark, namely if A = J C is the jacobian of a curve C of genus g ≥ (not necessarily semi-stable) over a number field K of degree d and discriminant ∆ K , one hasthe explicit m K ≤ g d g g max { , h F + ( J C /K ) } + gd g log | ∆ K | + g d g log 16 , as given in the proof of Corollary 1.3. The case of elliptic curves is given in Lemma 4.7 of[Paz14]. Corollary 1.3 will be used in the proof of Lemma 5.3.We then focus on the regulator of A ( K ) . We show that it verifies a Northcott property forsimple abelian varieties under a conjecture of Lang and Silverman, as proposed in [Paz16a]. Theorem 1.4.
Assume the Lang-Silverman Conjecture 4.1. The set of Q -isomorphism classesof simple abelian varieties A , equipped with an ample and symmetric line bundle L , definedover a fixed number field K , of dimension g and rank m K bounded from above, with A ( K ) Zariski dense in A and with Reg L ( A/K ) bounded from above is finite. In this special case one restricts to simple abelian varieties where the Zariski density of A ( K ) is equivalent to having positive Mordell-Weil rank. Using a stronger height conjectureone obtains the following general statement, without any simplicity assumption, on the modulispace of principally polarized abelian varieties. Theorem 1.5.
Assume the stronger Lang-Silverman Conjecture 5.1. The set of Q -isomorphismclasses of principally polarized abelian varieties A , defined over a fixed number field K , of di-mension g and rank m K bounded from above, with A ( K ) Zariski dense in A and regulatorbounded from above is finite. As explained in [Paz16a], if one restricts to g = 1 one can replace the Lang-Silvermanconjecture by the ABC conjecture in the statements of Theorem 1.4 and Theorem 1.5.We divide the work as follows. In section 2 we give the definitions of the regulator andof the Faltings height of an abelian variety. In section 3 we prove Theorem 1.1: it relies onthe core of the work, Proposition 3.2, which gives the semi-stable version. The final step isthen given in Proposition 3.6. In section 4 we use the conjecture of Lang and Silverman todeduce Theorem 1.4. In section 5 we discuss how a stronger conjecture of Lang and Silvermantype imply Theorem 1.5. We conclude in section 6 with a comparaison with number fields,extending the dictionnary of [Paz14]. Note that a version of Theorem 1.4 for elliptic curves without the requirement that the rank is boundedfrom above is given in [Paz14] with an incorrect proof, see [Paz16b].
F. Pazuki Definitions
Let S be a set. We will say that a function f : S → R verifies a Northcott property on S iffor any real number B , the set { P ∈ S | f ( P ) ≤ B } is finite.Notation: the function denoted log is the reciprocal of the classical exponential function,so log e = 1 (we do not use the notation ln ). We will denote by O K the ring of integers of K .If K ′ is a finite extension of a number field K , we denote by N K ′ /K the relative norm. If p ′ isa prime ideal in O K ′ above the prime ideal p in O K , then e p ′ / p is the ramification index and f p ′ / p stands for the residual degree.2.1. Regulators of abelian varieties.
Let
A/K be an abelian variety over a number field K polarized by an ample and symmetric line bundle L . Let m K be the Mordell-Weil rank of A ( K ) . Let b h A,L be the Néron-Tate height associated with the pair ( A, L ) . Let < ., . > be theassociated bilinear form, given by < P, Q > = 12 (cid:16)b h A,L ( P + Q ) − b h A,L ( P ) − b h A,L ( Q ) (cid:17) for any P, Q ∈ A ( Q ) . This pairing on A × A depends on the choice of L . Definition 2.1.
Let P , ..., P m K be a basis of the lattice A ( K ) /A ( K ) tors , where A ( K ) is theMordell-Weil group. The regulator of A/K is defined by
Reg L ( A/K ) = | det( < P i , P j > ≤ i,j ≤ m K ) | , where by convention an empty determinant is equal to .As for the height, the regulator of an abelian variety depends on the choice of an ampleand symmetric line bundle L on A .There is a more intrinsic way of defining a regulator, that doesn’t depend on the choiceof L . Start with the natural pairing on the product of A with its dual abelian variety ˇ A given by the Poincaré line bundle P : for any P ∈ A ( Q ) and any Q ∈ ˇ A ( Q ) , define < P, Q > = b h A × ˇ A, P ( P, Q ) . Next choose a base P , ..., P m K of A ( K ) modulo torsion and a base Q , ..., Q m K of ˇ A ( K ) modulo torsion. Then define Reg(
A/K ) = | det( < P i , Q j > ≤ i,j ≤ m K ) | . Let us recall how these two regulators are linked (see for instance [Hin07] page 172). Let Φ L : A → ˇ A be the isogeny given by Φ L ( P ) = t ∗ P L ⊗ L − . One has the formula b h A,L ( P ) = − < P, Φ L ( P ) > . Hence if u is the index of the subgroup Φ L ( Z P ⊕ ... ⊕ Z P m K ) in ˇ A ( K ) modulo torsion, onehas(1) Reg L ( A/K ) = u − m K Reg(
A/K ) . Let us remark that when L induces a principal polarization, the index u is equal to . ThusTheorem 1.5 is valid with both regulators. eights, ranks and regulators of abelian varieties The Faltings height.
Let A be an abelian variety defined over a number field K , ofdimension g ≥ . Recall that O K is the ring of integers of K and let π : A −→
Spec( O K ) bethe Néron model of A over Spec( O K ) . Let ε : Spec( O K ) −→ A be the zero section of π and let ω A / O K be the maximal exterior power (the determinant) of the sheaf of relative differentials ω A / O K := ε ⋆ Ω g A / O K ≃ π ⋆ Ω g A / O K . For any archimedean place v of K , let σ be an embedding of K in C associated to v . Theassociated line bundle ω A / O K ,σ = ω A / O K ⊗ O K ,σ C ≃ H ( A σ ( C ) , Ω g A σ ( C )) is equipped with a natural L -metric k . k v given by k s k v = i g (2 π ) g Z A σ ( C ) s ∧ s . The O K -module ω A / O K is of rank and together with the hermitian norms k . k v at infinity itdefines an hermitian line bundle ω A / O K = ( ω A / O K , ( k . k v ) v ∈ M ∞ K ) over O K . It has a well definedArakelov degree d deg( ω A / O K ) . Recall that for any hermitian line bundle L over Spec( O K ) thedegree of L in the sense of Arakelov is defined as d deg( L ) = log L /s O K ) − X v ∈ M ∞ K d v log k s k v , where s is any non zero section of L . The resulting number does not depend on the choice of s in view of the product formula on the number field K .The Arakelov degree of this metrized bundle will give a translate of the classical Faltingsheight. Definition 2.2.
The height of
A/K is defined as h F + ( A/K ) := 1[ K : Q ] d deg( ω A / O K ) . This non-negative real number doesn’t depend on any choice of polarization on A . When A/K is semi-stable, this height only depends on the Q -isomorphism class of A . It is justa translate of the classical Faltings height h F ( A/K ) , we have h F + ( A/K ) = h F ( A/K ) + g log(2 π ) . If A/K is not semi-stable, one may use Chai’s base change conductor as in theformula (14) in the sequel as a complementary definition. See [Fa83] Satz 1, page 356 and 357for its basic properties, and for a comparison with the theta height in [Paz12] (based on ideasof Bost and David). We prefer to use this translate because it gives cleaner inequalities (see thejacobian case in Proposition 3.2 for instance). We recall here four classical properties: firstly, if A = A × A is a product of abelian varieties, one has h F + ( A/K ) = h F + ( A /K )+ h F + ( A /K ) .Secondly, the dual abelian variety of A has the same height as A by a result of Raynaud.Thirdly, if K ′ /K is a number field extension, one has h F + ( A/K ′ ) ≤ h F + ( A/K ) . Finallyif A/K is semi-stable, one defines the stable height by h F + ( A/ Q ) := h F + ( A/K ) , which isinvariant by number field extension.At finite places we focus on the bad reduction locus with the following quantity. Definition 2.3.
Let A be an abelian variety over a number field K . Let A →
Spec( O K ) beits Néron model. Let p be a prime of O K . If the special fiber A p is an abelian variety, we say F. Pazuki that p is a prime of good reduction for A , otherwise the prime is of bad reduction. We define N A/K = Y p ⊂O K , p bad for A N K/ Q ( p ) . Regarding archimedean places, let us recall what the injectivity diameter is.
Definition 2.4.
Let A be a complex abelian variety. Let L be a polarization on A . Let T A be the tangent space of A , let Γ A be its period lattice and H L the associated Riemann formon T A . The injectivity diameter is the positive number ρ ( A, L ) = min γ ∈ Γ \{ } p H L ( γ, γ ) , i.e. thefirst minimum in the successive minima of the period lattice of A .3. A lower bound for the Faltings height
We start by recalling Masser’s Matrix Lemma in Bost version (later precised by Autissierand Gaudron-Rémond). We then give a lower bound for the Faltings height by the norm ofthe bad reduction primes in the semi-stable case, then we obtain the result in the general caseby base change, hence deriving a proof of Theorem 1.1 and Corollary 1.2. This implies anupper bound on the Mordell-Weil rank of abelian varieties over number fields in terms of theFaltings height.3.1.
Archimedean places.
Let us start by the Matrix Lemma given in Théorème 1.1 page345 of Gaudron and Rémond [GaRe14b] (see also Autissier’s [Aut13] for good explicit constantsif the polarization is principal ; the first version was given by Bost for principally polarizedabelian varieties, as stated in Autissier’s paper). We give it here with the height h F + ( A/K ) = h F ( A/K ) + g log(2 π ) . Theorem 3.1. (Matrix Lemma) Let K be a number field such that A is defined over K ,polarized by an ample line bundle L . For any archimedean place v of K , denote by ρ ( A v , L v ) the injectivity diameter of the complex polarized abelian variety ( A v , L v ) , then d X v ∈ M ∞ K d v ρ ( A v , L v ) − ≤ h F + ( A/ Q ) + 39 g. The Matrix Lemma is true for the stable height h F + ( A/ Q ) , and we always have h F + ( A/K ) ≥ h F + ( A/ Q ) . Here the polarization is not necessarily principal.3.2. Bad reduction places.
We compare the height and the size of the bad primes of A over the base field K . We first give a proof of the inequality in the semi-stable case andthen obtain the general result using base change properties given in the next paragraph. Thefollowing proposition gives the result in the semi-stable case. Let us first recall the case ofelliptic curves, studied in [Paz14], where the argument is direct and produces easy constants.Let A = E be an elliptic curve. One has the exact formula h F + ( E/K ) = 112 d log N K/ Q (∆ E ) − X v ∈ M ∞ K d v log (cid:16) | ∆( τ v ) | (2 Im τ v ) (cid:17) , where ∆ E is the minimal discriminant of the curve, τ v is a period in the fundamental domainsuch that E ( K v ) ≃ C / Z + τ v Z and ∆( τ v ) = q Q + ∞ n =1 (1 − q n ) is the modular discriminant, eights, ranks and regulators of abelian varieties q = exp(2 πiτ v ) . A direct analytic estimate using Im τ v ≥ √ / provides us with(2) h F + ( E/K ) ≥ d log N E/K . Let’s move on to higher dimension.
Proposition 3.2.
Let
A/K be a semi-stable abelian variety of dimension g and defined overa number field K of degree d . Then there exists quantities c = c ( g ) > and c = c ( g ) ∈ R such that h F + ( A/K ) ≥ c d log N A/K + c . The explicit values c = (12 g ) − g g g and c = − /c are valid. If A is the jacobian of acurve of genus g ≥ , then one can even take c = and c = 0 .Proof. The proof is divided into six steps: we start by the case of jacobians in Step 1. Thenfor general abelian varieties, we reduce to the case of principally polarized abelian varietiesin Step 2 by Zarhin’s trick. We make use of several projective heights (theta height, heightà la Philippon, . . . ) to work on the inequality in Step 3. Then we explain in Step 4 how tofind a curve of small height on a principally polarized abelian variety (by a Bertini Theorem)with the extra constraint that it is defined over a finite extension of K with controlled rami-fication, that will help us reduce the general case to the first case of jacobians. We show thatthe abelian variety we started with is a quotient of the jacobian of this curve (by classicalarguments) in Step 5 and we can finally conclude (via Néron-Ogg-Shafarevich) by putting ev-erything together in Step 6. As A/K is semi-stable, its Faltings height is invariant by numberfield extension, this will be used in the sequel.
Step 1 . We start by proving the result for jacobians of curves. If A = J C is the jacobian ofa curve C , the argument may be presented as follows. By the arithmetic Noether’s formulaof [MB89] Théorème 2.5 page 496 one has for a curve C of genus g ≥ (with semi-stablejacobian J C ) over a number field K of degree d , d h F + ( J C /K ) = ( ω C · ω C ) + X p prime p ⊂O K δ p ( C ) log N K/ Q ( p ) + X σ : K֒ → C δ ( C σ ) + dg log(2 π ) , where the auto-intersection ( ω C · ω C ) is non-negative, δ ( C σ ) is the delta invariant of Faltingsof the complex curve C σ and δ p ( C ) is the number of singular points in the fiber C p . It is zeroif and only if p is a prime ideal in O K of good reduction for C . A remark is that the quantity(3) d X p prime δ p ( C ) log N K/ Q ( p ) is invariant by number field extension of the base field K . Indeed, if one proceeds with abase change from O K to O K ′ , each double point in the fiber over a prime p of C/K becomessingular in the fiber over primes p ′ | p of C/K ′ with thickness equal to the ramification index e p ′ / p , so the number of double points gets multiplied by e p ′ / p by passing from p to p ′ , see theproof of Lemma 1.12 in [DeMu69].One has ( ω C · ω C ) ≥ and δ ( C σ ) ≥ − g log 2 π by [Wil16], hence ( ω C · ω C ) + X σ : K֒ → C δ ( C σ ) ≥ d · c ( g ) F. Pazuki where one can take c ( g ) = − g log 2 π . (Note that using the second inequality of Proposition2.4.8 page 102 of [Java14] we would get ( ω C · ω C ) + P σ : K֒ → C δ ( C σ ) ≥ − dg .)It proves that the height of J C satisfies(4) h F + ( J C /K ) ≥ d X p prime δ p ( C ) log N K/ Q ( p ) + c ( g ) , for c ( g ) = g log(2 π ) − g log 2 π = 0 . This completes the statement for jacobians, becauseany bad prime for J C is also a bad prime for C , so we have δ p ( C ) ≥ for any bad prime of J C . We now aim for a way to reduce to the case of jacobians. Step 2 . We may assume, using Zarhin’s trick, that the abelian variety is principally po-larized. Indeed if ˇ A stands for the dual of A , the abelian variety Z ( A ) = A × ˇ A carries aprincipal polarization, h F + ( Z ( A ) /K ) = 8 h F + ( A/K ) and N Z ( A ) /K = N A/K . It will have a lit-tle cost on the value of the quantities c and c . Let us now fix a principal polarization L on A . Step 3 . We will use the theory of Mumford theta coordinates as in the article of [DaPh02]pages 646–652, provided we do a field extension K ′ /K that enables us to choose a Mumfordtheta structure of level . The choice K ′ = K [ A [16]] is valid, and Lemma 4.7 page 2078 of[GaRe14a] implies(5) [ K ′ : K ] ≤ g . We choose an embedding Θ : A → P g − given by the theta sections of L ⊗ and wedefine the theta height of ( A, L ) by h Θ ( A, L ) = h (Θ ( O A )) . We will in fact show the lowerbound for the theta height of A : by virtue of the following inequality given in [Paz12](6) | h Θ ( A, L ) − h F + ( A/ Q ) | ≤ · g log(4 g ) log( h Θ ( A, L ) + 2) , it will lead to the lower bound we seek for the Faltings height of A as explained in Step 6.By Proposition 3.9 of [DaPh02] page 665, one has for any algebraic subvariety V ⊂ A theinequality (where N = 16 g − ) | b h P N ( V ) − h P N ( V ) | ≤ c ( g, dim V, deg V, h Θ ( A, L )) , where h P N ( V ) is the height of the variety V as defined in [DaPh02] page 644, the height b h P N ( V ) is defined in [Phi91] in Proposition 9 and the quantity c ( g, dim V, deg V, h Θ ( A, L )) > can betaken to be (4 g +1 h Θ ( A, L )+3 g log 2) · (dim V +1) · deg V . Picking V = A , one gets b h P N ( A ) = 0 , dim A = g , deg L A = g ! (the polarization is principal) and(7) h P N ( A ) ≤ c ( g )( h Θ ( A, L ) + 1) , where c ( g ) > only depends on the dimension of A , and one can take c ( g ) = 4 g +1 ( g !)( g +1) . Hence giving a lower bound on the height h P N ( A ) will imply a lower bound on the thetaheight, which in turn will imply a lower bound on the Faltings height by (6).By Theorem 1.3 page 760 of [Rém10] and Proposition 1.1 page 760 of [Rém10] one has thatfor any curve C in P N of genus g and degree deg C there exists a quantity c ( g , deg C ) > such that(8) h Θ ( J C , L Θ ) ≤ c ( g , deg C )( h P N ( C ) + 1) , eights, ranks and regulators of abelian varieties L Θ is the polarization associated to the theta divisor on J C . As C is embedded into itsjacobian by a theta embedding, one has deg( C ) = g and one can even take c = (6 g ) g g . Step 4 . The next goal is now to find an algebraic curve C on A of genus g ≤ c ( g ) suchthat h P N ( C ) ≤ c ( g ) h P N ( A ) . The proposition is already proved for g = 1 , we may wellsuppose that g ≥ from now on. We will cut A by g − hyperplanes H , ..., H g − in generalposition of height h ( H i ) ≤ c ( g ) h P N ( A ) . Using Bertini’s Theorem given in Theorem II.8.18of [Har06] page 179, there exists a dense open subset U such that for any hyperplane H in U , the intersection A ∩ H is non-singular and connected. As Q is algebraically closed, onehas U ( Q ) = ∅ , so there exist hyperplanes H with coordinates in Q and A ∩ H a geometricallyconnected non-singular variety in P N . To be able to choose hyperplans H i with height h ( H i ) ≤ c ( g ) h P N ( A ) , we use the following argument: assume we have an infinite set S M of algebraicnumbers of height less that M , where M ≥ is a fixed real number. This set can be infinitebecause we don’t impose an upper bound the degree of these algebraic numbers. Consider theinfinite set of all lines in the dual projective space ˇ P N with coefficients in S M . As U is an opendense subset, its complement can’t contain all these lines, so there exists infinitely many linesintersecting U . Pick one of these lines. It provides us with the desired hyperplane H i in P N .Repeat the argument g − times to obtain a smooth curve C , geometrically connected on A ,of genus g . Furthermore, we would like to ensure that the resulting field extension used todefine C will remain as little ramified as possible. The choice of the set S M is then crucial,we will now take the time to explain how it is done.Classical existence theorems for infinite unramified extensions of a given number field oftencome from the application of the Golod-Shafarevich inequality (see [GoSha64]). A quadraticfield with at least 5 different prime factors generally admits such an extension. The followingresult is of a similar spirit. Let k = Q ( √− · . By Maire [Mai00], the quadraticfield k admits an infinite everywhere unramified extension k † , which is a tower of unramified2-extensions. Let K ′ k be the compositum of K ′ and k over Q and let K ′′∞ = k † K ′ k be thecompositum of k † and K ′ k over k . Then K ′′∞ /K ′ k is unramified (classical, see PropositionB.2.4 page 592 of [BoGu07] for instance). We want to find small algebraic numbers in thisinfinite extension.Let F ⊂ k † be a finite extension of k . By applying Minkowski’s convex body Theorem as inthe proof of Theorem B.2.14 page 595 of [BoGu07] , there exists a non-zero algebraic number α F in O F generating F over Q (this is important) and with logarithmic absolute Weil heightless than log | ∆ F/ Q | / [ F : Q ] . Now | ∆ F/ Q | / [ F : Q ] = | ∆ k/ Q | / because F/k is unramified and [ k : Q ] = 2 , and log | ∆ k/ Q | / is bounded from above by log 10 < . Varying F along thetower, we get infinitely many α F because each α F is primitive in F , hence they are pairwisedistincts. We gather all these α F to define the set S M ⊂ K ′′∞ , for M = 14 , and thus get acurve C defined over a finite extension K ′′ ⊂ K ′′∞ unramified over K ′ k and with c ( g ) = 14 .Note that [ K ′ k : Q ] ≤ [ k : Q ][ K ′ : Q ] = 2[ K ′ : Q ] , and [ K ′ k : Q ] = [ K ′ k : K ′ ][ K ′ : Q ] , hence(9) [ K ′ k : K ′ ] ≤ . Here is a picture to help the reader follow the construction. See also [VaWi13] for better bounds in some cases. In the end the curve is defined with a finite number of coefficients in S M and K ′ . F. Pazuki k † K ′′∞ k K Q K ′ K ′ kF K ′′ The control on the height of the intersection defining C and on the degree of the intersectionis provided by Proposition 2.3 page 765 of [Rém10] which gives in our situation, as deg A = g ! and h ( H i ) ≤ c ( g ) h P N ( A ) ,(10) h P N ( C ) ≤ h P N ( A ∩ H ∩ · · · ∩ H g − ) ≤ c ( g )( h P N ( A ) + 1) , where one can take c ( g ) = g ( g !) + 14 . We also need to control the genus g of the curve C .Using calculations on the successive Hilbert polynomials of A ∩ H ∩ ... ∩ H i , one can take theexplicit bound g ≤ g g ! g = c ( g ) , see [CaTa12] for the details . Step 5 . The conjunction of (8), (10), (7) and the fact that g ≤ c ( g ) imply that over thefinite extension K ′′ /K there exists a curve C on A such that(11) h Θ ( J C , L Θ ) ≤ c ( g )( h Θ ( A, L ) + 1) where one can take c ( g ) = (12 g ) g g g , and by the universal property of the jacobian, fromthe inclusion of C in A one has a homomorphism J C → A . Let us show that it is surjective (wefollow the classical arguments given for instance in Proposition 6.1 page 104 and Theorem 10.1page 116-117 of [Mil08]). Let A be the image of the map, it is an abelian variety. Suppose A = A , we will derive a contradiction. There exists another non-zero abelian subvariety A in A such that A × A → A is an isogeny. In particular, A ∩ A is finite, so C ∩ A is finite because C is in A , and non-empty because one can always assume O ∈ C . Let W = A × A . It is in particular a non-singular projective variety. Let π be the compositionof Id × [2] : A × A → A × A with A × A → A . Then if p denotes the projection on thesecond factor, p ( π − ( C )) = [2] − ( C ∩ A ) , so π − ( C ) is not geometrically connected. Butit must be by Corollary 7.9 page 244 of [Har06] (or by lemma 10.3 of [Mil08]). This is thedesired contradiction, hence A = A .This implies that there exists an abelian variety B such that J C is isogenous to A × B .Isogenous abelian varieties share the same bad reduction primes by the Néron-Ogg-Shafarevichcriterion, because they have the same Tate modules (see Theorem 1 page 493 of [SeTa68] andCorollary 2 page 22 of [Fa86]). Thus, if we denote d ′′ = [ K ′′ : Q ] , we get that(12) d ′′ X p ′′ bad for A δ p ′′ ( C ) log N K ′′ / Q ( p ′′ ) ≤ d ′′ X p ′′ ⊂ O K ′′ δ p ′′ ( C ) log N K ′′ / Q ( p ′′ ) . Let us also remark that one can embed the curve in P , then using a theorem of Castelnuovo for curves in P given in Theorem 6.4 page 351 of [Har06], one has g ≤ deg P ( C ) . eights, ranks and regulators of abelian varieties Step 6 . Let us show that we have reduced the proof to the case of jacobians of curves.Following the previous steps we get d ′′ X p ′′ ⊂ O K ′′ δ p ′′ ( C ) log N K ′′ / Q ( p ) ≤ ( i ) h F + ( J C /K ′′ ) ≪ ( ii ) h Θ ( J C , L Θ ) ≪ ( iii ) h Θ ( A, L ) ≪ ( iv ) h F + ( A/K ′′ ) , where the implied constants depend only on g and the successive inequalities are • (i) is the case of curves given by inequality (4), • (ii) is the comparison between the theta height and the Faltings height of [Paz12] asrecalled in (6), • (iii) is inequality (11), • (iv) is again (6).If the curve C was defined over K , we could use on the far left side the invariance property(3). In the general case we have nevertheless inequality (12), and we get from there(13) d ′′ X p ′′ ⊂O K ′′ bad for A δ p ′′ ( C ) log N K ′′ / Q ( p ′′ ) ≥ d X p ⊂O K bad for A (cid:16) X p ′′ | p f p ′′ / p [ K ′′ : K ] (cid:17) log N K/ Q ( p ) where f p ′′ / p is the residual degree. Indeed, if p ′′ is a bad prime for A , it is also a bad prime for Jac( C ) , hence also a bad prime for C (the converse statement is wrong), hence δ p ′′ ( C ) ≥ .Using [ K ′′ : K ] = X p ′′ | p e p ′′ / p f p ′′ / p ≤ (cid:16) max p ′′ | p e p ′′ / p (cid:17) X p ′′ | p f p ′′ / p , one gets in (13) d ′′ X p ′′ bad for A δ p ′′ ( C ) log N K ′′ / Q ( p ′′ ) ≥ d X p bad for A p ′′ | p e p ′′ / p log N K/ Q ( p ) ≥ · g d log N A/K where the last inequality holds because the ramification index is controlled by e p ′′ / p ≤ [ K ′ k : K ′ ][ K ′ : K ] ≤ · g as K ′′ /K ′ k is unramified and as one has (5) and (9).One concludes by h F + ( A/K ′′ ) = h F + ( A/K ) on the far right side because A/K is alreadysemi-stable. At each and every step an explicit constant is provided, an easy calculation leadsto c = (12 g ) − g g g and c = − /c for the general case. These values are not expected tobe optimal. (cid:3) Reducing to the semi-stable case.
We explain in this section how to use base changeproperties to derive the general case from the semi-stable case. Let us start by the followingdefinition.
Definition 3.3.
Let A be an abelian variety defined over a discrete valuation field K p andlet K ′ p ′ be a finite extension of K p where A has semi-stable reduction, with ramificationindex e p ′ / p , where p ′ is a prime above p , and ω A/K p the determinant of differentials. Let h p : A O K p × O K p O K ′ p ′ → A O K ′ p ′ be the canonical base change morphism. Let Lie( h p ) be theinduced injective morphism on differentials. Let c ( A, p , p ′ ) = 1 e p ′ / p length O K ′ p ′ coker(Lie(h p )) F. Pazuki be the base change conductor, where if Γ( ., . ) stands for global sections one has coker(Lie(h p )) = Γ(Spec( O K p ) , ω A/K p ) ⊗ O K ′ p ′ Γ(Spec( O K ′ p ′ ) , ω A/K ′ p ′ ) . This conductor was defined by Chai in [Cha00], see also the reference [HaNi12] pages 90-98.It verifies in particular the two following key properties.
Proposition 3.4.
Let A be an abelian variety defined over a discrete valuation field K p and let K ′ p ′ be a finite extension of K p where A has semi-stable reduction with base change conductor c ( A, p , p ′ ) . Then one has (1) c ( A, p , p ′ ) = 0 if and only if A/K p has semi-stable reduction, (2) if A is not semi-stable at p , then c ( A, p , p ′ ) ≥ /e p ′ / p .Proof. The proof goes along the same lines as Proposition 4.3 of [Paz14] which deals withelliptic curves. As it is relatively short, we give it here for abelian varieties. Let us start byassuming that
A/K p has semi-stable reduction. Denote by A O K p the identity component ofthe Néron model of A over K p , one then has A O K p ⊗ O K ′ p ′ ≃ A O K ′ p ′ by Corollaire 3.3 page348 of SGA 7.1 [SGA72], hence the differentials are the same, so c ( A, p , p ′ ) = 0 .Reciprocally, one still has a map Φ : A O K p ⊗ O K ′ p ′ → A O K ′ p ′ . As c ( A, p , p ′ ) = 0 , the Liealgebras are the same and as Φ is an isomorphism on the generic fibers, Φ is birational. Onthe special fiber, Φ has finite kernel and is thus surjective because the dimensions are equal,here again because c ( A, p , p ′ ) = 0 .We have that Φ is quasi-finite and birational. As A O K ′ p ′ is normal, by Zariski’s MainTheorem found in Corollary 4.6 page 152 of [Liu02] for instance, Φ is an open immersion. So Φ is surjective and is also an open immersion, hence an isomorphism. This implies that A/K p is semi-abelian, and proves part (1). Part (2) is easier, if A is not semi-stable then the lengthin the definition of c ( A, p , p ′ ) is a positive integer, hence bigger than . (cid:3) We need a lemma.
Lemma 3.5.
Let
U ns denote the set of unstable primes of A over K . Let K ′ be a numberfield extension of K over which A has semi-stable reduction everywhere. Then one has (14) h F + ( A/K ) − h F + ( A/K ′ ) ≥ K ′ : Q ] X p ∈ Uns log N K/ Q ( p ) . Proof.
For a field F , we denote by A O F the Néron model of A over the base Spec O F . As K ′ is a finite extension of K , we have(15) h F + ( A/K ) − h F + ( A/K ′ ) = 1[ K : Q ] deg( ω A O K ) − K ′ : Q ] deg( ω A O K ′ ) , i.e. the archimedean parts cancel out. Let φ : K → K ′ be the inclusion, we have a morphism ω A O K ′ → φ ∗ ω A O K , taking degrees (see also the proof of Lemme 1.23 page 35 of [Del85]) oneobtains [ K ′ : K ] deg( ω A O K ) = deg( φ ∗ ω A O K ) = deg( ω A O K ′ )+ X p ⊂O K X p ′ | p length O K ′ p ′ (coker φ ) log N K ′ / Q ( p ′ ) , eights, ranks and regulators of abelian varieties [ K ′ : K ] deg( ω A O K ) ≥ deg( ω A O K ′ ) + X p ∈ Uns X p ′ | p [ K ′ : K ] e p ′ / p log N K/ Q ( p ) , hence dividing by [ K ′ : Q ] = [ K ′ : K ][ K : Q ] , and using e p ′ / p ≤ [ K ′ : K ]1[ K : Q ] deg( ω A O K ) ≥ K ′ : Q ] deg( ω A O K ′ ) + 1[ K ′ : Q ] X p ∈ Uns log N K/ Q ( p ) , which gives the result by using (15). (cid:3) We are now ready to perform a base change on the inequality of Proposition 3.2, which willbe the completion of the proof of Theorem 1.1.
Proposition 3.6. (Final step in the proof of Theorem 1.1.) Let g ≥ be an integer and K a number field of degree d . There exists c ( g ) > and c ( g ) ∈ R such that for any abelianvariety (not necessarily semi-stable) A defined over K , with dimension g , one has h F + ( A/K ) ≥ c d log N A/K + c , and one can take c = c / g and c = c . If A is the jacobian of a curve, one can take c = 1 / g +1 and c = 0 .Proof. Let N stA/K be the product of the norms of primes where A has semi-stable bad reduction.Let N unsA/K be the product of the norms of primes where A has unstable bad reduction. Bydefinition one has N A/K = N stA/K N unsA/K . Let K ′ be a number field extension of K such that A acquires semi-stable reduction everywhere over K ′ . Using equality (14), one gets h F + ( A/K ) ≥ h F + ( A/K ′ ) + 1[ K ′ : Q ] log N unsA/K . As A/K ′ has semi-stable reduction everywhere, one obtains by Proposition 3.2 that h F + ( A/K ′ ) ≥ c ( g ) 1 d ′ log N stA/K ′ + c ( g ) . Recall (use Theorem 6.2 page 413 of [SiZa95]) that one may choose the explicit extension K ′ = K [ A [12]] , hence the degree d ′ = [ K ′ : Q ] is controlled by the degree d = [ K : Q ] and by the dimension of A ; for instance, apply Lemma 4.7 page 2078 of [GaRe14a] to obtain d ′ = [ K [ A [12]] : K ] ≤ g . Now one has d ′ log N stA/K ′ = 1 d ′ X p ′ ⊂O K ′ log N K ′ / Q ( p ′ ) ≥ d X p ⊂O K p ′ | p e p ′ / p log N K/ Q ( p ) ≥ g d log N stA/K because e p ′ / p ≤ [ K ′ : K ] and so gathering the estimates we obtain h F + ( A/K ) ≥ c d log N stA/K + c d log N unsA/K + c ≥ min { c , c } d log N A/K + c , where the quantities c , c , c only depend on g . (cid:3) F. Pazuki
Note that for g = 1 , Proposition 3.6 is an improvement on Proposition 4.4 page 57 of[Paz14], both in the result and in the presentation: an equality of prime norms is incorrect in loc. cit. because of possible ramification of stable primes of K ′ /K , but the proof fortunatelyled to a weaker inequality in the end, so the result stated in loc. cit. still holds, and anyhowthe new result given here is better.We obtain an easy proof of Corollary 1.2 as the sum of Theorem 3.1 and Proposition 3.6.Apply Proposition 3.6 to get h F + ( A/K ) ≥ c log N A/K + c and Theorem 3.1 to get h F + ( A/K ) ≥ h F + ( A/ Q ) ≥ d X v ∈ M ∞ K d v ρ ( A v , L v ) − − g, then sum these two inequalities.We can now derive Corollary 1.3. Proof. (of Corollary 1.3)
We will use as a pivot the quantity N A/K . Applying Theorem 5.1 of[Rém10] page 775, there exists quantities c = c ( K, g ) > and c = c ( K, g ) ≥ such that m K ≤ c ( K, g ) log N A/K + c ( K, g ) . The quantities depend on the degree and the discrimi-nant of the base field here. This last inequality doesn’t require semi-stability of A . ApplyingProposition 3.6 of the present text one obtains log N A/K ≤ c ( K, g ) max { h F + ( A/K ) , } , alsovalid in general. Use the explicit quantities (valid in general) of Theorem 5.1 of [Rém10] page775, it leads to m K ≤ g d g log N A/K + gd g (log | ∆ K | + g d log 16) , and combine withProposition 3.6. It proves the corollary. In the case of jacobians combine with Proposition 3.2which gives h F + ( J C /K ) ≥ d log N J C /K in the semi-stable case and Proposition 3.6 whichgives h F + ( J C /K ) ≥ d g log N J C /K for the general case. (cid:3) Lang-Silverman conjecture and regulators
We give here a conjecture of Lang and Silverman ([Si84b] page 396 or [Paz12] ). Through-out this section, we will use the notation End( A ) · P = A to say that the set End( A ) · P isZariski dense in A . Conjecture 4.1. (Lang-Silverman) Let g ≥ be an integer. For any number field K , thereexists a positive quantity c = c ( K, g ) such that for any abelian variety A/K of dimension g and any ample symmetric line bundle L on A , for any point P ∈ A ( K ) , one has: (cid:16) End( A ) · P = A (cid:17) ⇒ (cid:16)b h A,L ( P ) ≥ c max n h F + ( A/K ) , o(cid:17) , where b h A,L ( . ) is the Néron-Tate height associated to the line bundle L and h F + ( A/K ) is the(relative) Faltings height of the abelian variety A/K . Remark 4.2.
We only require the condition
End( A ) · P Zariski dense, not necessarily Z · P Zariski dense. Let us consider the following situation: let A be a simple abelian varietyover K and let A = A × A . Choose P = ([ n ] P , P ) ∈ A ( K ) . If P is non-torsion, then Z · P is a strict abelian subvariety (of degree growing with n ), whereas End( A ) · P = A . As This first version of the conjecture is known to be wrong, consider for instance the point ( P, on a variety A × A where P is non-torsion and let the height of A tend to infinity. However the philosophy of theconjecture is clearly the same as the original statement, a generic point can’t have too small height. This stronger version is also known to be wrong, see Remark 4.2 and section 5 of the present article for aclarification. eights, ranks and regulators of abelian varieties b h A,L ( P ) = ( n + 1) b h A ,L ( P ) for the product polarization L , and as h F + ( A/K ) =2 h F + ( A /K ) , the point P verifies the expected lower bound if the point P does. Proposition 4.3.
Assume the Lang-Silverman Conjecture 4.1. Let K be a number field and g, m ≥ be integers. There exists a quantity c = c ( K, g, m ) > such that for any simpleabelian variety A defined over K of dimension g , of rank m over K , polarized by an ampleand symmetric line bundle L , Reg L ( A/K ) ≥ (cid:16) c max { h F + ( A/K ) , } (cid:17) m . Proof.
Let us denote h = max { h F + ( A/K ) , } and for any i ∈ { , ..., m } , the Minkowski i th-minimum λ i = λ i ( A ( K ) /A ( K ) tors ) . Apply Minkowski’s successive minima inequality to theMordell-Weil lattice, λ · · · λ m ≤ m m/ (Reg L ( A/K )) / . Now, as A is simple, any non-torsion point verifies End( A ) · P = A , so using m times theinequality of Conjecture 4.1 one gets(16) Reg L ( A/K ) ≥ c m h m m m , which gives the result. (cid:3) We thus obtain Theorem 1.5 as a corollary of Proposition 4.3. Indeed if the rank is nonzero, as soon as the regulator, the rank and the dimension are bounded from above, the heightwill be bounded from above, hence the claimed finiteness. This may be expressed in otherwords by: the Lang-Silverman conjecture implies that the regulator
Reg L ( A/K ) verifies aNorthcott property on the set of polarized simple abelian varieties (modulo isomorphisms) ofdimension g defined over a fixed number field K with A ( K ) Zariski dense and Mordell-Weilrank bounded from above.
Remark 4.4.
Back to inequality (16), in view of Corollary 1.3, we have h ≫ m . Anyimprovement of the form h ≫ m ε (for a fixed ε > ) would lead to a stronger Northcottproperty, without assuming that the rank is bounded from above. See also the addendum[Paz16b]. 5. A stronger lower bound conjecture
We would like to refine the conjecture of Lang and Silverman to take care of the exceptionalpoints in Conjecture 4.1: what can be said about the points P verifying End( A ) · P ( A ? Conjecture 5.1. (Lang-Silverman, new strong version) Let g ≥ be an integer. For anynumber field K , there exists two positive quantities c = c ( K, g ) and c = c ( K, g ) suchthat for any abelian variety A/K of dimension g and any ample symmetric line bundle L on A , for any point P ∈ A ( K ) , one has: Such an attempt has been proposed in Conjecture 1.8 of [Paz12], but it unfortunately fails because ofsituations similar to the one described in Remark 4.2 where certain points fall in the first case but should fallin the second instead. This was communicated to the author by the referee of another project, may he bewarmly thanked here. We fix the problem by changing the condition given there as Z · P = A by the weaker End( A ) · P = A . We also add a dependance in deg L ( A ) in the attempt to control the degree of B thanks to aremark of Gaël Rémond. F. Pazuki • either there exists an abelian subvariety B ⊂ A , B = A , of degree deg L ( B ) ≤ c deg L ( A ) and such that the order of the point P modulo B is bounded from aboveby c , • or one has End( A ) · P is Zariski dense and b h A,L ( P ) ≥ c max n h F + ( A/K ) , o , where b h A,L ( . ) is the Néron-Tate height associated to the line bundle L and h F + ( A/K ) is the (relative) Faltings height of the abelian variety A/K . This is a strong statement. It implies the strong torsion conjecture for example. Indeed,any torsion point P ∈ A ( K ) tors falls into the first case because its canonical height is zero.Hence the order of P is bounded from above solely in terms of K and g and of the cardinalityof the torsion subgroup of a strict abelian subvariety B . An easy induction on the dimensionof A gives a bound on the order of P solely in terms of K and g , hence on the exponent ofthe torsion group as well, hence on the cardinal of the torsion group A ( K ) tors as well.This strong form of the conjecture is motivated by Théorème 1.4 page 511 of [Da93] andThéorèmes 1.8 and 1.13 of [Paz13]. Remark that in both of these works, the abelian varietiesconsidered are principally polarized, hence the dependance in the degree of A is only throughthe dimension g .Let us see now how this statement can help in understanding the link between the Mordell-Weil group A ( K ) and the abelian subvarieties of A . The following quantity will play a keyrole in this paragraph. Definition 5.2.
Let A be an abelian variety over a number field K . Let m K denote theMordell-Weil rank of A ( K ) . Define m = sup { rank( B ( K )) | B strict abelian subvariety of A } . We will call the relative quantity m K − m the Zariski rank of the Mordell-Weil group A ( K ) .Note that m K − m > is equivalent to A ( K ) being Zariski dense in A . This Zariski rankcould be compared with the following quantity for a number field K . If r K is the rank of unitsin K , let r denote the maximal rank of units in a strict subfield of K . As already noticed in[Paz14] in the easier case of elliptic curves, the Zariski rank m K − m plays the same role (atleast when one gives lower bounds on the regulator in both contexts) as the relative rank ofunits r K − r for number fields.The next lemma studies the size of the successive minima of the Mordell-Weil lattice modulotorsion, where the square of the norm is implicitly given by the Néron-Tate height. We believethis version could lead in the future to some improvements in Theorem 1.5. Lemma 5.3.
Assume Conjecture 5.1. Let ( A, L ) be a polarized abelian variety of dimension g defined over a number field K . For any i ∈ { , ..., m K } , let λ i be the i -th successive minimaof the lattice A ( K ) /A ( K ) tors . Then there is a quantity c = c ( K, g, deg L ( A )) > such that for any i , λ i ≥ c i ,if i > m , λ i ≥ c max { , h F + ( A/K ) } .Proof. Within the proof, we will use the symbol c ∗ for a positive quantity only depending on g ,on K and on deg L ( A ) . We allow the value of this quantity c ∗ to vary at some steps within theproof, as long as it depends only on g , on K and on deg L ( A ) and stays positive. If c ( K, g ) eights, ranks and regulators of abelian varieties c = max { , max ≤ i ≤ g c ( K, i ) } , thefield K being fixed.Let B denote the set of all abelian subvarieties B in A of degree bounded from above by c g deg L ( A ) : it contains the subvarieties appearing in the first case of Conjecture 5.1, andwe raise c to the power g to be able to use an induction on the dimension g towards theend. This is a finite set with cardinal bounded from above by a quantity depending only on g , on K (because c only depends on g and K ) and on deg L ( A ) . The reader interested inan explicit upper bound for the cardinal of this set can refer to Proposition 4.1 page 529 of[Rém00].Choose an integer i ∈ { , ..., m K } and define Z i = [ B ∈B rank( B ( K ))
Assume Conjecture 5.1. Let K be a number field, let g ≥ be an integer,let m ≥ be an integer. There exists a quantity c = c ( K, g, m ) > such that for anyprincipally polarized abelian variety A defined over K of dimension g , equipped with an ampleand symmetric line bundle L , with A ( K ) of rank m , Reg L ( A/K ) ≥ (cid:16) c max { h F + ( A/K ) , } (cid:17) m − m . F. Pazuki
Proof.
Let us denote h = max { h F + ( A/K ) , } and m = rank( A ( K )) , and for any i ∈{ , ..., m } , λ i = λ i ( A ( K ) /A ( K ) tors ) .The inequality is trivial for m = 0 . From now on, let us assume m = 0 . Apply Minkowski’ssuccessive minima inequality to the Mordell-Weil lattice, λ · · · λ m ≤ m m Reg L ( A/K ) . Now apply lemma 5.3 with deg L ( A ) = g ! to get(17) Reg L ( A/K ) ≥ c m − m h m − m λ · · · λ m m m , If m = 0 , the inequality is the one claimed. Let us suppose that m = 0 . Apply againLemma 5.3 to get(18) Reg L ( A/K ) ≥ m − m ( c ) m ( m !) ( c h ) m − m . Hence the claimed inequality, as ≤ m ≤ m . Finally, if the regulator is bounded from abovethen the height is bounded from above as soon as m = m , hence the claimed finiteness, as m > m is equivalent with the fact that A ( K ) is Zariski dense in A . (cid:3) Theorem 1.5 follows directly from Proposition 5.4, because the set of principally polar-ized abelian varieties defined over a fixed number field K , of fixed dimension g such that A ( K ) is Zariski dense in A and with regulator and rank bounded from above is also a set ofbounded height under Conjecture 5.1. Note that in view of (1), one can replace Reg L ( A/K ) by Reg(
A/K ) in Theorem 1.5 because the polarization is principal.6. Conclusion
We generalize here the last section of [Paz14] to abelian varieties, extending the dictionarygiven in [Hin07] as well.
Number field K Abelian variety
A/K zeta function ζ K ( s ) ↔ L ( A, s ) L functionlog of discriminant log | D K | ↔ h F + ( A ) Faltings heightregulator R K ↔ Reg(
A/K ) regulatorclass number h K ↔ | X ( A/K ) | Tate − Shafarevitch grouptorsion ( U K ) tors ↔ ( A × ˇ A )( K ) tors torsion of A and dual ˇ A degree d ↔ g dimensionmax subrank of units r ↔ m max rank of ab . subvar . relative unit ranks r K − r ↔ m K − m Zariski rank of A ( K )CM field r K = r ↔ m K = m A ( K ) non Z . densenon − CM field r K > r ↔ m K > m A ( K ) Zariski dense Remark 6.1.
One could prefer to put in link the property “ A ( K ) Zariski dense in A ” with“ K generated by units”. Let us remark that A ( K ) Zariski dense is equivalent to m K > m ,but on the number field side there exists some CM fields K that are generated by units, so K generated by units is not equivalent to r K > r . However, regarding the finiteness propertyobtained from giving an upper bound for the regulator, one may replace the property of being eights, ranks and regulators of abelian varieties A ( K ) on the abelian side. References [Aut13]
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Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5,2100 Copenhagen Ø, Denmark, and Institut de Mathématiques de Bordeaux, Université deBordeaux, 351, cours de la Libération, 33405 Talence, France.
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