TAMAGAWA NUMBERS OF DIAGONAL CUBIC SURFACES NUMERICAL EVIDENCE
22 pages
English

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TAMAGAWA NUMBERS OF DIAGONAL CUBIC SURFACES NUMERICAL EVIDENCE

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22 pages
English
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Niveau: Supérieur, Doctorat, Bac+8
TAMAGAWA NUMBERS OF DIAGONAL CUBIC SURFACES, NUMERICAL EVIDENCE? Emmanuel Peyre and Yuri Tschinkel Abstract. — A refined version of Manin's conjecture about the asymptotics of points of bounded height on Fano varieties has been developped by Batyrev and the authors. We test numer- ically this refined conjecture for some diagonal cubic surfaces. 1. Introduction The aim of this paper is to test numerically a refined version of a conjecture of Manin con- cerning the asymptotic for the number of rational points of bounded height on Fano varieties (see [BM] or [FMT] for Manin's conjecture and [Pe1] or [BT3] for its refined versions). Let V be a smooth Fano variety over a number field F and ??1V its anticanonical line bundle. Let Pic(V ) be the Picard group and NS(V ) the Néron-Severi group of V . We denote by Val(F ) the set of all places of F and by Fv the v-adic completion of F . Let (?·?v)v?Val(F ) be an adelic metric on ??1V . By definition, this is a family of v-adically continuous metrics on ??1V ?Fv which for almost all valuations v are given by a smooth model of V (see [Pe2]). These data define a height H on the set of rational points V (F ) given by ?x ? V (F ), ?y ? ??1V (x), H(x) = ∏ v?Val(F ) ?y??1v .

  • let

  • lebesgue measure

  • line bundle yields

  • anticanonical line

  • brauer-manin obstruction

  • bundle

  • asymptotic

  • constant ?h


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TAMAGAWA NUMBERS OF DIAGONAL CUBIC SURFACES, NUMERICAL EVIDENCE
Emmanuel Peyre and Yuri Tschinkel
Abstract. — A rened version of Manin's conjecture about the asymptot ics of points of bounded height on Fano varieties has been developped by Batyrev and the authors. We test numer-ically this rened conjecture for some diagonal cubic surfaces.
1. Introduction
The aim of this paper is to test numerically a rened version of a conjecture of Manin con-cerning the asymptotic for the number of rational points of bounded height on Fano varieties (see [BM] or [FMT] for Manin's conjecture and [Pe1] or [BT3] for its rened versions). LetVbe a smooth Fano variety over a number eldFandωV1its anticanonical line bundle. LetPic(V)be the Picard group andNS(V)the Néron-Severi group ofV. We denote byVal(F)the set of all places ofFand byFvthev-adic completion ofF. Let(kkv)vVal(F) be an adelic metric onωV1. By denition, this is a family ofv-adically continuous metrics onωV1Fvwhich for almost all valuationsvare given by a smooth model ofV(see [Pe2]). These data dene a heightHon the set of rational pointsV(F)given by xV(F)yωV1(x)H(x) =Ykykv1vVal(F) For every open subsetUVand every real numberHwe have nUH(H) = #{xU(F)|H(x)6H}<The problem is to understand the asymptotic behavior ofnUH(H)asHgoes to innity. It is expected that at least for Del Pezzo surfaces the following asymptotic formula holds: nUH(H) =θH(V)H(logH)t1(1 +o(1)) asH→ ∞, over appropriate nite extensionsEFof the groundeld. Here the open setU is the complement to exceptional curves,θH(V)>0andtis the rank of the Picard group of
1991Mathematics Subject Classication 11D25; secondary 14G05, 14J25.. — primary Math. Comp.7087–3n,)0002(763,332o
2
EMMANUEL PEYRE and YURI TSCHINKEL
VoverEcounter-examples to this conjecture in every dimension have . We>3[BT2] (see [BT3] for a discussion of higher dimensional varieties). In this paper we focus on the constantθH(V)the one hand, there is a theoretical . On description (1)θH(V) =α(V)β(V)τH(V) whereτH(V)is a Tamagawa number associated to the metrized anticanonical line bundle [Pe1],α(V)number dened in terms of the cone of effective divisors [is a rational Pe1] and the integerβ(V)is a cohomological invariant, which rst appeared in asymptotic formulas in [BT1]. On the other hand, let us consider a diagonal cubic surfaceVP3Qgiven by ax3+by3+cz3+dt3= 0witha b c dZandabcd6= 0. Our counting problem can be formulated as follows: nd all quadruples of integers(x y z t)with g.c.d.(x y z t) = 1andmax{|x||y||z||t|}6H which satisfy the equation above. Quadruples differing by a sign are counted once. Aproof of an asymptotic of the type (1) for smooth cubic surfaces seems to be out of reach of available methods, but one can numerically search for solutions of bounded height. The cubics with coefcients(1112)and(1113)and heightH62000were treated by Heath-Brown in [HB made substantial progress Swinnerton-Dyer both cases weak approximation fails.]. In towards an interpretation of the constantτH(V)[SD particular, he suggested that the]. In adelic integral deningτH(V)should be taken over the closure of rational pointsV(F)V(AF), rather than the whole adelic space. Our goal is to compute the theoretical constantθH(V)explicitely for certain diagonal cubic surfaces with and without obstruction to weak approximation and to compare the result with numerical data (with heightH6100000). We observe a very good accordance. Insection2wedenetheTamagawanumber.Thisdenitionislsightlydifferentfrom the one in [Pe1 sections 3, 4 and 5 we explain In], but the numbers coincide conjecturally. how to compute it. There is a subtlety at the places of bad reduction, notable at3, overlooked previously. In section 6 we compute the Brauer-Manin obstruction to weak approximation. And in section 7 we present the numerical results. These computations were made using a program of Bernstein which is described in [Be].
2. Conjectural constant
Notations2.1. — IfVis a scheme over a ringAandBanA-algebra, we denote byVBthe productV ×SpecASpecBand byV(B)the set ofB-points, that isHomSpecA(SpecBV). For any eldE, we denote byEa xed algebraic closure and byVthe varietyE. IfFemdiaeslniitswhehttoserifptesenfopetiecalrbeumanishtyfitnediew,dle OF. We denote bydFthe absolute value of its discriminant. Ifpis a nite place ofF, then Opis the ring of integers inFpandFpits residue eld. In the sequel we will always assume thatVis a smooth projective geometrically integral variety over a number eldFsatisfying the following conditions:
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