Jordi Guàrdia

Journal de Théorie des Nombres de Bordeaux

JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUXJORDI GUÀRDIAArakelov computations in genus 3 curvesJournal de Théorie des Nombres de Bordeaux, tome 13, no 1 (2001),p. 157-165<http://www.numdam.org/item?id=JTNB_2001__13_1_157_0>© Université Bordeaux 1, 2001, tous droits réservés.L’accès aux archives de la revue « Journal de Théorie des Nombresde Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’uneinfraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/157-1Arakelov computations in genus 3 curvespar JORDI GUÀRDIARÉSUMÉ. Les invariants d’Arakelov des surfaces arithmétiquessont bien connus pour le genre 1 et 2 ([4], [2]). Dans cette note,nous étudions la hauteur modulaire et la self-intersectiond’Arakelov pour une famille de courbes de genre 3 possédant beau-coup d’automorphismes, à savoirCn : Y4 = X4 - (4n - 2) X2Z2 + Z4 .La théorie d’Arakelov fait intervenir à la fois des calculs arithméti-ques et des calculs analytiques. Nous exprimons les périodes de Cnen termes d’intégrales elliptiques. Les substitutions utilisées dansles intégrales fournissent une décomposition de la jacobienne deCn en produit de trois courbes elliptiques. En utilisant l’isogéniecorrespondante, nous déterminons le modèle stable de la surfacearithmétique définie par Cn .Une fois calculés les périodes et le modèle stable de Cn, noussommes en mesure de déterminer la hauteur modulaire et la selfintersection du modèle canonique. Nous donnons une bonne esti-mation de cette hauteur modulaire, traduite par son comporte-ment logarithmique. Nous donnons également une minorationde la self-intersection qui montre qu’elle peut être arbitrairementgrande.Nous présentons ici nos calculs presque sans démonstrations.Les détails peuvent être lus dans [5] .ABSTRACT. Arakelov invariants of arithmetic surfaces are wellknown for genus 1 and 2 ([4], [2]). In this note, we study themodular height and the Arakelov self-intersection for a family ofcurves of genus 3 with many automorphisms:Cn: Y4 = X4 - (4n - 2)X2 Z2 + Z4.Arakelov calculus involves both analytic and arithmetic computa-tions. We express the periods of the curve Cn in terms of ellip-tic integrals. The substitutions used in these integrals provide asplitting of the jacobian of Cn as a product of three elliptic curves.Using the corresponding isogeny, we determine the stable modelof the arithmetic surface given by Cn. Once we have the periodsand the stable model of Cn, we can study the modular height andManuscrit reçu Ie 26 octobre 1999.Partially supported by DGICYT-PD-96-0166.158the self-intersection of the canonical sheaf. We can give a goodestimate for the modular height, which reflects its logarithmic be-haviour. We provide a lower bound for the self-intersection, whichshows that it can be arbitrarily large.We present here all our calculations on the curves Cn, almostwithout proofs. Details can be found in [5].1. A family of curves of genus 3We will study the curves given by the equation:for a E 0, :i:l. These are non-singular curves of genus 3. We notethat each of the four values of the parameter a, -a,1/a, -1/a gives riseto the same curve. Equation (1) is particularly well-suited to study thegeometry of the curves (7~) , but to study their arithmetic we should usethe equationobtained from (1) through the change of parameter a = + Ineither equation one can observe that these curves have many symmetries.More specifically, we prove:Proposition 1. Let a ¥= 0, ::1:1, :1:1 ::I: B/2. The automorphisms of C(a) arethe restrictions of the following 16 projectivities of p2(C):The group Aut(C(a» is isomorphic to a semidirect product Y4. Theautomorphismesform a system of generators for Aut(C(a».Proposition 2. The curve isomorphic to the Fermat curve offourth degree, F4 = ly4 = X4 + Z4}. The group Aut(F4) has order 96,and is isomorphic to a serraidirect product (Z/4Z x ~53.1592. Geometry of the curves C(a)The determination of the period lattice of the curves C(,,) is possiblethanks to their many symmetries. First of all, we can find a basis for thespace of holomorphic differentials on C(~) :Proposition 3. Let x = XIZ, y = Y/Z. The differential formesyield an orthogonal basis for We can also determine a basis for the singular homology of the curves.We regard the curve C~a~ as a 4-sheeted covering of the complex projectiveline, through the map:Figure 1 .We can explicitly describe the monodromy of this map. For instance, apath on P (C) crossing the segment [- 1 /a, 1 /a], when lifted to C(a) through1607r goes from a sheet n to the sheet n + 2 (mod 4). The action of theautomorphism a of C(~) is easily described in terms of the covering ~: itlifts points one sheet. With all these data, we can proceed to build closedpaths on G(a) , lifting closed paths in I~2(~), as seen in figure 1. We referto the path sketched there as G. We proceed analogously to build the pathF indicated in figure 2 (numbers indicate the sheet on which each part ofthe path lies).Figure 2Proposition 4. The homology classes of the pathsform a symplectic basis for H¡(C(a), 7~).Once we have bases for the spaces HO (C(a), and H¡(G(a)’ Z) we com-pute the period lattice of the curve C(a) - The Abelian integrals that wemust compute can be reduced to elliptic integrals, and can thus be expressedin terms of elliptic functions. We obtain:Proposition 5. A normalized period matrix for the curves C(a) is (IlZa),where. ... II....... · ""II..." .161and I is the complete eltiptic integralo f the first kind.For instance, for the Fermat curve of fourth degree, isomorphic to thecurve C1+V2’ we re-obtain its well-known period matrix:B 2 2 -’-/The fact that all the Abelian integrals on C(a) can be reduced to ellipticintegrals tells us that the jacobian of the curve should split as aproduct of elliptic curves. We can show the situation exactly. Let p =Proposition 6. Let E be the elliptic curveand let En be the elliptic curvewhich is a Weierstrass model of the curve- - . - -We have isomorphisms defined over Ko:The quotient maps induce an isogeny of degree 8defined over Ko. ’ ’3. Arithmetic of the curves CnFrom now on, we will assume that n E N is a natural number, and, fortechnical reasons only, that n 0 0,1 (mod 25). The isogeny described inthe previous section is the key to the study of reduction of the curves Cn.It is well known that the the reduction of a curve is intimately related withthat of its jacobian. Moreover, at the level of Abelian varieties, the type ofreduction is invariant under isogenies. Thus, studying the reduction of the162elliptic curves we can achieve the reduction of the curves Cn. Firstof all, we determine the locus of good reduction:Proposition 7. The curve Cn has good reduction outside the primes di-viding n (n -1 ) .Let Kn = where a = 3 18 - 6 ~ and 24 ) 3 - j ( E~ )(an - 27) = 0. Let On be the ring of integers of K~. The elliptic curvesE, En acquire good reduction at 2 over the field since their Deuringmodels are defined over this field. The curve En has multiplicative reduc-tion at odd primes dividing n (n -1 ) . From this, we conclude:Proposition 8. a) The curve Cn has a stable model Cnst and a minimalregular model over On.b) At the On which divide 2, J(Cn) has Abelian reduction overCc) At the odd primes p in On which divides n(n-1), J(Cn) has semi-Abelianreduction over On, and the tolsc part of its N6ron model has dimension 1.We now describe the bad fibres of the stable model Cn of the curves Cn.At the odd primes, Prop. 8 gives enough information to describe the badfibres.Theorem 9. a) The special fibre of at an odd prime p in On dividingn (n - 1 ) has two elliptic components, intersecting at two different points.b) The intersection pairing on this fibre is given byThe situation at the primes dividing 2 is much more complicated, andrequires a detailed study of the reduction of the automorphisms of Gn.Once this is done, we obtain that:Theorem 10. a) The special fibre of Gn at a prime p in On dividing 2consists of three elliptic components, which intersect a rational componentat three different points. These intersection points map to 4-torsion pointsin the elliptic quotients of 163b) The intersection pairing on this fibre is given by4. Arakelov invariants for arithmetic surfacesHaving a great deal of geometric and arithmetic information about acurve, one can proceed to study it from the Arakelov viewpoint. We willnow do so with the curves Cn. We will study the two basic Arakelovinvariants for curves: the modular height and the self-intersection. Webriefly recall their definitions (details can be found in [4]).Given a curve X of genus g, defined over a number field K, consider anextension where X acquires stable reduction, and let ;rest ---+ T =be its stable model. The relative dualizing sheaf úJxest/T can beprovided with an admissible metric, giving rise to an Arakelov line sheafWxestlT-Definition 11. The modular height of the curve X isThe Arakelov self-intersection of X ismhere (.,.) Ar denotes the Arakelov intersection pairing d4])-For genus g = 1, it is known that e(X) = 0 and h(X) can be expressedas a function h(X) = in terms of the periods of the curve, itsdiscriminant and its locus of bad reduction. For genus g = 2, Bost - Mestre- Moret-Bailly ([3])give explicit formulas for both invariants, where againthe periods and the reduction of the curve play an important role. Butthere is not much more information about these two invariants in general.Ullmo ([8]) has proved that e(X) &#x3E; 0 for g &#x3E; 1, and Abbes and Ullmo ([l])have found upper and lower bounds for the self-intersection of modularcurves Xo(N), N squarefree.1645. Modular height of the curves CnTheorem 12. Let Tn = The height of the curveCn is given bySketch of the proof The modular height of a curve coincides with thatof its jacobian as an Abelian variety. A result of Raynaud ([7]) controlsthe behaviour of the modular height of Abelian varieties through isogeniesA ~ B of degree a prime power p’: we have that h(A) = h(B) + k logp,with k E Z, Ikl  e/2. Moreover, the modular height of a product ofAbelian varieties is the sum of the heights of the factors. We thus obtain:Now we use the explicit formula for the height of the elliptic curves given in([4]), to determine the terms h(E), h(En). Note that Tn is the fundamentalperiod of the elliptic curve En. 06. A lower bound for e(Cn)Lemma 13 ([6]). Let X - S = Spec(OK) be a stable arithmetic surfaceof 2. Let pl, ... , pt be the primers in OK where the fibres of Xare reducible. We have thatWe can apply this result to the curves Cn, since we have a precise de-scription of the primes where their stable model has reducible fibres: theyare the primes which divide n(n - 1). We thus obtain:Proposition 14.As the degree ~.I~n : K] is always less than or equal to 1152, this resultimplies that the Arakelov self-intersection of a curve can be arbitrarilylarge.165References[1] A. ABBES, E. ULLMO, Auto-intersection du dualisant relatif des courbes modulaires X0(N).J. reine angew. Math. 484 (1997), 1-70.[2] J.-B. BOST, Fonctions de Green-Anakelov, fonctions théta et courbes de genre 2. C.R. Acad.Sci. Paris Série I 305 (1987), 643-646.[3] J.-B. BosT, J.-F. MESTRE, L. MORET-BAILLY, Sur le calcul explicite des "classes de Chern"des surfaces arithmétiques de genre 2. Astérisque 183 (1990), 69-105.[4] G. FALTINGS, Calculus on arithmetic surfaces. Ann. of Math. 119 (1984), 387-424.[5] J. GUÀRDIA, Geometria aritmética en una famlia de corbes de genere 3. Thesis, Universitatde Barcelona, 1998.[6] A. MORIWAKI, Lower bound of self-intersection of dualizing sheaves on arithmetic surfaceswith reducible fibres. Compositio Mathematica 105 (1997), 125-140.[7] M. RAYNAUD, Hauteurs et isogénies. Séminaire sur les pinceaux arithmétiques: la conjecturede Mordell, exp. VII. Astérisque 127 (1985), 199-234.[8] E. ULLMO, Positivité et discrétion des points algébriques des courbes. Annals of Math. 147(1998), 167-179.Jordi GUARDIADept. algebra i GeometriaUniversitat de BarcelonaGran Via de les Corts Catalanes, 585E-08007 BarcelonaEspanaE-rrtail : 

Arakelov computations in genus $3$ curves

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Arakelov computations in genus $3$ curves

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Arakelov computations in genus $3$ curves