Canonical Heights on Varieties with Morphisms Compositio Mathematica, Tome 89, No 2 (1993), P

COMPOSITIO MATHEMATICA GREGORY S. CALL JOSEPH H. SILVERMAN Canonical heights on varieties with morphisms Compositio Mathematica, tome 89, no 2 (1993), p. 163-205 <<a href="/goto?url=http://www.numdam.org/item?id=CM_1993__89_2_163_0" target="_blank">http://www.numdam.org/item?id=CM_1993__89_2_163_0</a>> © Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (<a href="/goto?url=http://http://www.compositio.nl/" target="_blank">http: </a><a href="/goto?url=http://http://www.compositio.nl/" target="_blank">//http://www.compositio.nl/</a>) implique l’accord avec les conditions gé- nérales d’utilisation (<a href="/goto?url=http://www.numdam.org/conditions" target="_blank">http://www.numdam.org/conditions</a>). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce ﬁchier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques <a href="/goto?url=http://www.numdam.org/" target="_blank">http://www.numdam.org/ </a>1993. Mathematica 89: 163 163-205, Compositio ©1993 Kluwer Publishers. Printed in the Netherlands. Academic Canonical on varieties with heights morphisms GREGORY Mathematics S. CALL* Amherst Amherst, College, MA USA 01002, Department, and JOSEPHH. SILVERMAN** Mathematics Brown RI Providence, USA 02912, Department, University, Received 13 in final form 16 October 1992 1992; May accepted Let Aan be abelian defined over afield KDnumber have and let be aaavariety divisor on A. Néron and Tate the existence of symmetric canonical proven on characterized and satisfies the that is height hA,D A(k) by properties hA,D Weil for the divisor Dfor all height A,D([m]P) m2hA,D(P) Silverman that on certain K3 surfaces SP ~ A(K). Similarly, [19] proved with anon-trivial S ~ Sthere are two canonical ~: automorphism height for functions characterized the that are Weil hs by properties they heights +certain divisors Eand satisfy S(~P) = (7 for all PE43) 1 S(P) S(K) . In we will this these to aconstruct canonical paper on an generalize examples aVV - Vand aheight divisor arbitrary variety ~: morphism eigenvalue strictly greater possessing which is an with class q for 0 eigenclass than 1. Wewill also anumberof results about these canonical prove which should be useful for heights compu- arithmetic and numerical applications tations. Wenow describe the contents of this in more detail. paper Let Vbe adefined over anumber field let aV ~ V<ul style="display: flex;"><li style="flex:1">be </li><li style="flex:1">K, </li></ul>variety ~: and that there is adivisor class q E main result ~ R such morphism, suppose Div(V) that 0*,q some 03B1 > 1. Our first aqfor that says (Theorem 1.1) there is acanonical function height characterized the two that is aWeil function for by properties V,~,~ height the divisor class and ~*Research byNSFROA-DMS-8913113andan supported by Amherst Trustee supported Faculty Fellowship. ** Research and NSFDMS-8913113 aSloan Foundation Fellowship. 164 of the that is canonical we show then As an if 17 height, points ample, application contains which are only finitely many [13] 0-periodic, generalizing V(K) Narkiewicz and Lewis results of [10]. has shown howto intoasumoflocal the canonical on Néron [8]) height (see decompAo,sDe(P) = 03A3 hl,D anabelian v), where variety nVÂA,D(P, heights, is over the distinct of the sum canonical sum We likewise show that the places v K(P). constructed in Theorem 1.1 can be as aheight decomposed V,~,~ Here Eis divisor <ul style="display: flex;"><li style="flex:1">in </li><li style="flex:1">class </li></ul>the divisor and q, any Ewith the additional is aWeil local if f is any function for the divisor height property constant +aE then there is athat function ~*E satisfying div( f ), aso that in Theorem proven The existence of the canonical local and the fact that the canonical is 2.1, is height V,~,~ is the sumof the local height heights V,~,~ in Theorem 2.3. two Weil given for adivisor differ abounded and so amount, Any heights given by in the difference of the canonical Weil particular height any given V,~,~ is aon and For bounded constant V, q, 0height by important hV,~ depending hV,~. it is to have an bound for this constant. manyapplications explicit Such aboundwas and Zimmer for Weierstrass given byDem’janenko [4] curves, [25] families of elliptic for Mumford families Manin and Zarhin by [12] of abelian and Silverman and Tate for families of varieties, by [15] arbitrary abelian varieties. V - T Wefollowthe in andconsider aapproach [15] family of varieties with aV ~ V over Tand adivisor 0: class 7y satisfying map Then on almost all there is acanonical ~*~ = andwe 03B1~. fibers ’V, height Vt,~t~t, can ask to boundthe difference between this aWeil and height given in terms t. In Theorem 3.1 we show that there of the height hV,~ parameter are constants so that ci, C2 165 asimilar between This We also the difference estimate for 3.2) give (Theorem the canonical local and aWeil local height AV,E . height Vt,Et,~t given result for abelian varieties. generalizes Lang’s Suppose Weil [8] anow that the base Tof the aV- T is alet be curve, hT family Tcorresponding and let P: T ~ V on to divisor of 1, the height section. The degree over be afiber Vof Yis afunction and generic variety global efield the section Pcorresponds to arational Pv K(T), V(K(T)), point Theorem 1.1 the acanonical which can be evaluated at gives height Vt,~V,~V There are then three and which hT we show in Pv. point be heights V,~V,~V,Vt,~t,~V, aresult of Silverman may compared. Generalizing [15], Theorem 4.1 that Silverman’s result has been and in various original [2], generalized Silverman strengthened [20] Call Green and Tate We ways by have not [6], Lang [8, 9], [22]. been able to of these results in our yet prove any stronger general situation. In the fifth section we take thecanonical local the of how one up question and might efficiently thecanoni- compute heights V,E,~, therebyeventually cal In the case that Vis an Tate curve, global height (unpub- that the hV,E,CP’ elliptic aseries for lished) gave completion rapidly convergent v) provided V,(O),[2](P, of Kat v is not and Silverman Kv closed, algebraically [18] give described amodification of Tate’s series which works for all v. We series for our canonical local the series of Tate heights generalizing and Â V,E,cp and Silverman how such discuss 5.1) (Theorem 5.3) practice. briefly (Proposition in series could be implemented The final section is devoted to aof canonical local for description heights non-archimedean in termsofintersection In the case ofabelian intersection places varieties it is knownthat the local theory. computed general every extends canbe heights using on the Néron model. We show in that if Vamodel V has theory alocal such that Ov over rational aextends to complete ring point that the Vsection and such V ~ to afinite 0: <ul style="display: flex;"><li style="flex:1">morphism </li><li style="flex:1">morphism </li></ul>V ~ then the canonical local is (D: acertain intersection V, height the given by question V. Weleave for future index on exist. To of whether such models study in this we aof and local summarize, paper develop theory global 166 on varieties with non-unit divisorial canonical heights possessing morphisms We describe how these in families and heights vary be used for algebraic eigenclasses. which may of canonical the computational purposes. give algorithms on abelian varieties has had of arithmetic The theory heights profound several field Likewise, throughout applications geometry. applications and K3 canonical for arithmetic many It is our questions open heights in that the likewise of canonical are described [19]. hope general theory in will in useful this the described paper prove studying heights arithmetic of varieties. properties 1. Global canonical heights In this section we fix the data: following Kafield with aset of absolute values proper satisfying since global complete formula. Wewill call such afield aaglobal heightfield, function on product ait is for such fields that one can define height pn( K). could be anumber field or a one variable function (For example, K field. ) V/K asmooth, variety. projective aaaV ~ Vdefined over K. 0~~: morphism R. divisor class q E Weil Pic(V) 0 ~ R to function ~. height V(K) corresponding hV,~ hV,~: for details about fields and 2, 3, 4, (See [8], global height height with Chapters is an functions on Wefurther assume that q we assume that for 0 eigenclass varieties.) than 1. In other words, eigenvalue greater It follows from of functions that functoriality height [8] Weil and the choice of The the on the V, 0, map Ov(1) variety depends first Our PEfunction but it is bounded of height independently V (K). which hV,~, result the that there exists aWeil function associated for says to ~ Ov(1) entirely disappears. THEOREM 1.1. aWith notation as there exists above, uniquefunction 167 the two conditions : satisfying following in Theorem 1.1 the canonical Wecall the function described V,~,~ height If one or more divisor associated to the and the on Vclass ~ q, 0. morphism it is will sometimes omit of the elements of the we clear, 0) triple (V, from notation. the 1. Aan let [n] : A - Abe the Let be abelian variety, multiplica- divisor. Example tion-by-n map is, for some n and let q E be a2, Pic(A) symmetric As is one then has the relation well-known, = q.) [-1]*~ [n]*= (That n2TJ, acanonical so one obtains height satisfying ~(nP) = h~ hA,~,[n] andTate. Néron This is the classical canonical constructed height by n2~(P). of Theorem 1.1 is an for Our easy (See, [8, Chapter 5].) argument proof more example, aextension of Tate’s to slightly general setting. XC<ul style="display: flex;"><li style="flex:1">P2 </li><li style="flex:1">P2be </li></ul>asmooth K3surface The two the intersec- Example2. Let given by SaS~P2 are tion of aand (2, 2)-form (1, 1)-form. projections 0-2: S~ S. Let double so each an involution covers, on S, o-1, gives say and be sections of Ç1, e2 F- let 0 Pic(S) hyperplane type (1, 0) (0, 1) respectively, +2and define divisor classes on S the formulas B13, by Then one can check that Further 0’2 0 03C31. let 0 = obtain two canonical and It is worth so we heights noting hS,~,~+ hS,~-1,~-. in of the effective cone each lie on the that and ~ R, ~+ ~- Pic(S) is useful for boundary +is This but that their sum the arithmetic of S. observation this ~+ ~- ample. studying aFor more details concerning example, including stated of the facts we have and formulas that can be used to proof explicit the canonical see and height, compute [19] [3]. IPn ~ Pn be a3. Let of d : 2. Then any Example 0: degree so we can morphism ~*~ 7L divisor class q E satisfies Theorem Pic(Pn) ~ d~, but never apply 1.1. had been observed earlier Notice Tate, (This just by published.) that as in the case of abelian one can to be varieties, take q ample, which means that 1.1.1 below is Corollary applicable. athe canonical we can result Using height, easily prove strong rationality 168 for Pis called We recall that apoints. point pre-periodic for ~ pre-periodic if its orbit Pis if some iterate is is finite. ~i(P) (Equivalently, pre-periodic aperiodic.) bound for Lewis The corollary gives strong rationality following pre-periodic and It results of Narkiewicz who generalizes [13] [10], prove points. in K-rational Lewis’ the cases that there are only finitely many pre-periodic points An and VPn in uses basic Vrespectively. functions, few lines. proof, particular, pro- reduces but our use of the canonical of Weil to height height perties the ajust proof COROLLARY 1.1.1. Let V/K ~VIK be aover a0: morphism defined divisor class numberfield and that there is an suppose K, ample qsatisfying Let PE(1). V (K). P0. is pre-periodicfor 4J contains only if ifand only finitely many (a) hV,7J,f/J(P) More gen- 0. pre-periodic points for (b) V (K) erally, is aset bounded so in it contains of points defined height, particular onlyfinitely many degree. over all extensions ofKofa bounded If Pis for then takes on 0, Proof. (a) many only finitely pre-periodic hV,~(~n(P)) distinct and so values, if then 0, Conversely, hV,~,~(P) {P, Hence the set <ul style="display: flex;"><li style="flex:1">is </li><li style="flex:1">is </li></ul>aset of bounded so it finite. is ...} O(P), ~2(P), height, Therefore this is where we use the fact that is P(Note ~ample.) 0pre- periodic. If Pis for then from so ~, (b) pre-periodic is bounded. (a), V,~,~(P) = hV,~(P) = form +This shows that the a0(l) V,~,~(P) pre-periodic points set of bounded The rest of is then since aset of

Canonical Heights on Varieties with Morphisms Compositio Mathematica, Tome 89, No 2 (1993), P

Details

Download

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

Support