Towards the André-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points. (English) Zbl 1422.11140
Summary: We prove in this paper the Ax-Lindemann-Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of A. Silverberg [Compos. Math. 68, No. 3, 241–249 (1988; Zbl 0683.14002)] in a different approach. Then combining these results we prove the André-Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of \(\mathcal{A}_6^n\) and under the generalized Riemann hypothesis for all mixed Shimura varieties of abelian type.
Keywords:
André-Oort conjecture; Ax-Lindemann theorem; Galois orbits; special points; mixed Shimura varietiesCitations:
Zbl 0683.14002References:
| [1] | [1] Y. André, Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part, Compos. Math. 82 (1992), no. 1, 1-24. AndréY.Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed partCompos. Math.8219921124 · Zbl 0770.14003 |
| [2] | [2] Y. André, Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. reine angew. Math. 505 (1998), 203-208. AndréY.Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaireJ. reine angew. Math.5051998203208 · Zbl 0918.14010 |
| [3] | [3] Y. André, Shimura varieties, subvarieties, and CM points, Six lectures at the University of Hsinchu, August-September 2001. AndréY.Shimura varieties, subvarieties, and CM pointsSix lectures at the University of Hsinchu, August-September 2001 |
| [4] | [4] A. Ash, D. Mumford, D. Rapoport and Y. Tai, Smooth compactifications of locally symmetric varieties, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge 2010. AshA.MumfordD.RapoportD.TaiY.Smooth compactifications of locally symmetric varieties2nd ed.Cambridge Math. Lib.Cambridge University PressCambridge2010 · Zbl 0334.14007 |
| [5] | [5] J. Ax, On Schanuel’s conjectures, Ann. of Math. (2) 93 (1971), 252-268. 10.2307/1970774AxJ.On Schanuel’s conjecturesAnn. of Math. (2)931971252268 · Zbl 0232.10026 |
| [6] | [6] J. Ax, Some topics in differential algebraic geometry I: Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195-1204. 10.2307/2373569AxJ.Some topics in differential algebraic geometry I: Analytic subgroups of algebraic groupsAmer. J. Math.94197211951204 · Zbl 0258.14014 |
| [7] | [7] W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), no. 3, 442-528. 10.2307/1970457BailyW.BorelA.Compactification of arithmetic quotients of bounded symmetric domainsAnn. of Math. (2)8419663442528 · Zbl 0154.08602 |
| [8] | [8] D. Bertrand, Special points and Poincaré bi-extensions (with an appendix by B. Edixhoven), preprint (2011), http://arxiv.org/abs/1104.5178. <element-citation publication-type=”other“> BertrandD.Special points and Poincaré bi-extensions (with an appendix by B. Edixhoven)Preprint2011 <ext-link ext-link-type=”uri“ xlink.href=”>http://arxiv.org/abs/1104.5178 |
| [9] | [9] D. Bertrand, Unlikely intersections in Poincaré biextensions over elliptic schemes, Notre Dame J. Form. Log. 54 (2013), no. 3-4, 365-375. 10.1215/00294527-2143907BertrandD.Unlikely intersections in Poincaré biextensions over elliptic schemesNotre Dame J. Form. Log.5420133-4365375 · Zbl 1304.11053 |
| [10] | [10] D. Bertrand and B. Edixhoven, Pink’s conjecture, Poincaré bi-extensions and generalized Jacobians, in preparation. BertrandD.EdixhovenB.Pink’s conjecture, Poincaré bi-extensions and generalized JacobiansPreprintin preparation |
| [11] | [11] D. Bertrand, D. Masser, A. Pillay and U. Zannier, Relative Manin-Mumford for semi-abelian surfaces, preprint (2013), http://arxiv.org/abs/1307.1008. <element-citation publication-type=”other“> BertrandD.MasserD.PillayA.ZannierU.Relative Manin-Mumford for semi-abelian surfacesPreprint2013 <ext-link ext-link-type=”uri“ xlink.href=”>http://arxiv.org/abs/1307.1008 · Zbl 1408.14139 |
| [12] | [12] D. Bertrand and A. Pillay, A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields, J. Amer. Math. Soc. 23 (2010), no. 2, 491-533. BertrandD.PillayA.A Lindemann-Weierstrass theorem for semi-abelian varieties over function fieldsJ. Amer. Math. Soc.2320102491533 · Zbl 1276.12003 |
| [13] | [13] E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Math. Monogr. 4, Cambridge University Press, Cambridge 2006. BombieriE.GublerW.Heights in Diophantine geometryNew Math. Monogr. 4Cambridge University PressCambridge2006 · Zbl 1115.11034 |
| [14] | [14] A. Borel, Linear algebraic groups, Grad. Texts in Math. 126, Springer, New York 1991. BorelA.Linear algebraic groupsGrad. Texts in Math. 126SpringerNew York1991 · Zbl 0726.20030 |
| [15] | [15] A. Chambert-Loir, Relations de dépendance et intersections exceptionnelles, Séminaire Bourbaki. Volume 2010/2011. Exposés 1027-1042. Avec table par noms d’auteurs de 1948/49 à 2009/10, Astérisque 348, Société Mathématique de France, Paris (2012), 149-188. Chambert-LoirA.Relations de dépendance et intersections exceptionnellesSéminaire Bourbaki. Volume 2010/2011. Exposés 1027-1042. Avec table par noms d’auteurs de 1948/49 à 2009/10Astérisque 348Société Mathématique de FranceParis2012149188 · Zbl 1273.14068 |
| [16] | [16] L. Clozel and E. Ullmo, Équidistribution de sous-variétés spéciales, Ann. of Math. (2) 161 (2005), 1571-1588. 10.4007/annals.2005.161.1571ClozelL.UllmoE.Équidistribution de sous-variétés spécialesAnn. of Math. (2)161200515711588 · Zbl 1099.11031 |
| [17] | [17] L. Clozel and E. Ullmo, Équidistribution adélique des tores et équidistribution des points CM, Doc. Math. J. DMV Extra Vol. (2006), 233-260. ClozelL.UllmoE.Équidistribution adélique des tores et équidistribution des points CMDoc. Math. J. DMV Extra Vol.2006233260 · Zbl 1144.14018 |
| [18] | [18] C. Daw, Degrees of strongly special subvarieties and the Andre-Oort conjecture, J. reine angew. Math. (2014), 10.1515/crelle-2014-0062. DawC.Degrees of strongly special subvarieties and the Andre-Oort conjectureJ. reine angew. Math.201410.1515/crelle-2014-0062 · Zbl 1368.11056 · doi:10.1515/crelle-2014-0062 |
| [19] | [19] C. Daw and A. Yafaev, An unconditional proof of the André-Oort conjecture for Hilbert modular surfaces, Manuscripta Math. 135 (2011), no. 1-2, 263-271. 10.1007/s00229-011-0445-xDawC.YafaevA.An unconditional proof of the André-Oort conjecture for Hilbert modular surfacesManuscripta Math.13520111-2263271 · Zbl 1215.14017 |
| [20] | [20] P. Deligne, Travaux de Shimura, Lecture Notes in Math. 244, Springer, Berlin 1971. DeligneP.Travaux de ShimuraLecture Notes in Math. 244SpringerBerlin1971 |
| [21] | [21] P. Deligne, Variété de Shimura: interprétation modulaire et techniques de construction de modèles canoniques, Automorphic forms, Shimura varieties, and \(L\)-functions. Part 2, Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence (1979), 274-290. DeligneP.Variété de Shimura: interprétation modulaire et techniques de construction de modèles canoniquesAutomorphic forms, Shimura varieties, and \(L\)-functions. Part 2Proc. Sympos. Pure Math. 33American Mathematical SocietyProvidence1979274290 |
| [22] | [22] P. Deligne, J. Milne, A. Ogus and K. Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math. 900, Springer, Berlin 1982. DeligneP.MilneJ.OgusA.ShihK.Hodge cycles, motives, and Shimura varietiesLecture Notes in Math. 900SpringerBerlin1982 · Zbl 0465.00010 |
| [23] | [23] L. van den Dries, Tame Topology and o-minimal Structures, London Math. Soc. Lecture Note Series 248, Cambridge University Press, Cambridge 1998. van den DriesL.Tame Topology and o-minimal StructuresLondon Math. Soc. Lecture Note Series 248Cambridge University PressCambridge1998 · Zbl 0953.03045 |
| [24] | [24] B. Edixhoven, Special points on the product of two modular curves, Compos. Math. 114 (1998), no. 3, 315-328. EdixhovenB.Special points on the product of two modular curvesCompos. Math.11419983315328 · Zbl 0928.14019 |
| [25] | [25] B. Edixhoven, On the André-Oort conjecture for Hilbert modular surfaces, Moduli of abelian varieties (Texel Island 1999), Birkhäuser, Basel (2001), 133-155. EdixhovenB.On the André-Oort conjecture for Hilbert modular surfacesModuli of abelian varietiesTexel Island 1999BirkhäuserBasel2001133155 · Zbl 1029.14007 |
| [26] | [26] B. Edixhoven, B. Moonen and F. Oort, Open problems in algebraic geometry, Bull. Sci. Math. 125 (2001), 1-22. 10.1016/S0007-4497(00)01075-7EdixhovenB.MoonenB.OortF.Open problems in algebraic geometryBull. Sci. Math.1252001122 · Zbl 1009.11002 |
| [27] | [27] B. Edixhoven and A. Yafaev, Subvarieties of Shimura varieties, Ann. of Math. (2) 157 (2003), no. 2, 621-645. 10.4007/annals.2003.157.621EdixhovenB.YafaevA.Subvarieties of Shimura varietiesAnn. of Math. (2)15720032621645 · Zbl 1053.14023 |
| [28] | [28] Z. Gao, Comparison of Galois orbits of special points of Shimura varieties, preprint (2014), http://www.math.u-psud.fr/ gao/articles. <element-citation publication-type=”other“> GaoZ.Comparison of Galois orbits of special points of Shimura varietiesPreprint2014 <ext-link ext-link-type=”uri“ xlink.href=”>http://www.math.u-psud.fr/ gao/articles |
| [29] | [29] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Première partie), Publ. Math. Inst. Hautes Études Sci. 20 (1964), 101-355. GrothendieckA.DieudonnéJ.Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Première partie)Publ. Math. Inst. Hautes Études Sci.201964101355 · Zbl 0136.15901 |
| [30] | [30] P. Habegger, Special points on fibered powers of elliptic surfaces, J. reine angew. Math. 685 (2013), 143-179. HabeggerP.Special points on fibered powers of elliptic surfacesJ. reine angew. Math.6852013143179 · Zbl 1318.14023 |
| [31] | [31] P. Habegger and J. Pila, Some unlikely intersections beyond André-Oort, Compos. Math. 148 (2012), no. 1, 1-27. 10.1112/S0010437X11005604HabeggerP.PilaJ.Some unlikely intersections beyond André-OortCompos. Math.14820121127 · Zbl 1288.11062 |
| [32] | [32] R. Hain and S. Zucker, Unipotent variations of mixed Hodge structure, Invent. Math. 88 (1987), 83-124. 10.1007/BF01405093HainR.ZuckerS.Unipotent variations of mixed Hodge structureInvent. Math.88198783124 · Zbl 0622.14007 |
| [33] | [33] J. Hwang and W. To, Volumes of complex analytic subvarieties of Hermitian symmetric spaces, Amer. J. Math. 124 (2002), no. 6, 1221-1246. 10.1353/ajm.2002.0038HwangJ.ToW.Volumes of complex analytic subvarieties of Hermitian symmetric spacesAmer. J. Math.1242002612211246 · Zbl 1024.32013 |
| [34] | [34] M. Kashiwara, A study of variation of mixed Hodge structure, Publ. RIMS Kyoto Univ. 22 (1986), 991-1024. 10.2977/prims/1195177264KashiwaraM.A study of variation of mixed Hodge structurePubl. RIMS Kyoto Univ.2219869911024 · Zbl 0621.14007 |
| [35] | [35] B. Klingler, E. Ullmo and A. Yafaev, The hyperbolic Ax-Lindemann-Weierstrass conjecture, preprint (2013), http://arxiv.org/abs/1307.3965. <element-citation publication-type=”other“> KlinglerB.UllmoE.YafaevA.The hyperbolic Ax-Lindemann-Weierstrass conjecturePreprint2013 <ext-link ext-link-type=”uri“ xlink.href=”>http://arxiv.org/abs/1307.3965 · Zbl 1372.14016 |
| [36] | [36] B. Klingler and A. Yafaev, The André-Oort conjecture, Ann. of Math. (2) 180 (2014), no. 3, 867-925. KlinglerB.YafaevA.The André-Oort conjectureAnn. of Math. (2)18020143867925 · Zbl 1377.11073 |
| [37] | [37] J. Kollár, Shafarevich maps and automorphic forms, Princeton University Press, Princeton, 1995. KollárJ.Shafarevich maps and automorphic formsPrinceton University PressPrinceton1995 · Zbl 0871.14015 |
| [38] | [38] J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, Automorphic forms, Shimura varieties, and \(L\)-functions. Vol. I (Ann Arbor 1988), Perspect. Math. 10, Academic Press, Boston (1990), 283-414. MilneJ.Canonical models of (mixed) Shimura varieties and automorphic vector bundlesAutomorphic forms, Shimura varieties, and \(L\)-functions. Vol. IAnn Arbor 1988Perspect. Math. 10Academic PressBoston1990283414 · Zbl 0704.14016 |
| [39] | [39] J. Milne, Introduction to Shimura varieties, Harmonic analysis, the trace formula, and Shimura varieties (Toronto 2003), Clay Math. Proc. 4, American Mathematical Society, Providence (2005), 265-378. MilneJ.Introduction to Shimura varietiesHarmonic analysis, the trace formula, and Shimura varietiesToronto 2003Clay Math. Proc. 4American Mathematical SocietyProvidence2005265378 · Zbl 1148.14011 |
| [40] | [40] B. Moonen, Linearity properties of Shimura varieties, I, J. Algebraic Geom. 7 (1988), no. 3, 539-467. MoonenB.Linearity properties of Shimura varieties, IJ. Algebraic Geom.719883539467 · Zbl 0956.14016 |
| [41] | [41] D. Mumford, The red book of varieties and schemes, 2nd expanded ed., Lecture Notes in Math. 1358, Springer, Berlin 1999. MumfordD.The red book of varieties and schemes2nd expanded ed.Lecture Notes in Math. 1358SpringerBerlin1999 · Zbl 0945.14001 |
| [42] | [42] M. Orr, La conjecture d’André-Pink: Orbites de Hecke et sous-variétés faiblement spéciales, PhD thesis, Université Paris-Sud, 2013. OrrM.La conjecture d’André-Pink: Orbites de Hecke et sous-variétés faiblement spécialesPhD thesisUniversité Paris-Sud2013 |
| [43] | [43] M. Orr, Families of abelian varieties with many isogenous fibres, J. reine angew. Math. (2013), 10.1515/crelle-2013-0058. OrrM.Families of abelian varieties with many isogenous fibresJ. reine angew. Math.201310.1515/crelle-2013-0058 · Zbl 1349.14143 · doi:10.1515/crelle-2013-0058 |
| [44] | [44] C. Peters and J. Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3) 52, Springer, Berlin 2008. PetersC.SteenbrinkJ.Mixed Hodge structuresErgeb. Math. Grenzgeb. (3) 52SpringerBerlin2008 · Zbl 1138.14002 |
| [45] | [45] Y. Peterzil and S. Starchenko, Around Pila-Zannier: The semi-abelian case, preprint (2009), http://math.haifa.ac.il/kobi/Manin-Mumford1.pdf. <element-citation publication-type=”other“> PeterzilY.StarchenkoS.Around Pila-Zannier: The semi-abelian casePreprint2009 <ext-link ext-link-type=”uri“ xlink.href=”>http://math.haifa.ac.il/kobi/Manin-Mumford1.pdf |
| [46] | [46] Y. Peterzil and S. Starchenko, Definability of restricted theta functions and families of abelian varieties, Duke J. Math. 162 (2013), 731-765. 10.1215/00127094-2080018PeterzilY.StarchenkoS.Definability of restricted theta functions and families of abelian varietiesDuke J. Math.1622013731765 · Zbl 1284.03215 |
| [47] | [47] J. Pila, O-minimality and the André-Oort conjecture for ℂn{\mathbb{C}^n}, Ann. of Math. (2) 173 (2011), 1779-1840. PilaJ.O-minimality and the André-Oort conjecture for ℂn{\mathbb{C}^n}Ann. of Math. (2)173201117791840 · Zbl 1243.14022 |
| [48] | [48] J. Pila, Special point problems with elliptic modular surfaces, Mathematika 60 (2014), no. 1, 1-31. 10.1112/S0025579313000168PilaJ.Special point problems with elliptic modular surfacesMathematika6020141131 · Zbl 1372.11063 |
| [49] | [49] J. Pila and J. Tsimerman, The André-Oort conjecture for the moduli space of Abelian surfaces, Compos. Math. 149 (2013), 204-216. 10.1112/S0010437X12000589PilaJ.TsimermanJ.The André-Oort conjecture for the moduli space of Abelian surfacesCompos. Math.1492013204216 · Zbl 1304.11055 |
| [50] | [50] J. Pila and J. Tsimerman, Ax-Lindemann for 𝒜g{\mathcal{A}_g}, Ann. of Math. (2) 179 (2014), no. 2, 659-681. PilaJ.TsimermanJ.Ax-Lindemann for 𝒜g{\mathcal{A}_g}Ann. of Math. (2)17920142659681 · Zbl 1305.14020 |
| [51] | [51] J. Pila and A. Wilkie, The rational points of a definable set, Duke J. Math. 133 (2006), 591-616. 10.1215/S0012-7094-06-13336-7PilaJ.WilkieA.The rational points of a definable setDuke J. Math.1332006591616 · Zbl 1217.11066 |
| [52] | [52] J. Pila and U. Zannier, Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), 149-162. PilaJ.ZannierU.Rational points in periodic analytic sets and the Manin-Mumford conjectureAtti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.192008149162 · Zbl 1164.11029 |
| [53] | [53] R. Pink, Arithmetical compactification of mixed Shimura varieties, Bonner Math. Schriften 209, University of Bonn, Bonn 1989. PinkR.Arithmetical compactification of mixed Shimura varietiesBonner Math. Schriften 209University of BonnBonn1989 · Zbl 0748.14007 |
| [54] | [54] R. Pink, A combination of the conjectures of Mordell-Lang and André-Oort, Geometric methods in algebra and number theory, Progr. Math. 253, Birkhäuser, Basel (2005), 251-282. PinkR.A combination of the conjectures of Mordell-Lang and André-OortGeometric methods in algebra and number theoryProgr. Math. 253BirkhäuserBasel2005251282 · Zbl 1200.11041 |
| [55] | [55] V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Pure Appl. Math. 139, Academic Press, Boston 1994. PlatonovV.RapinchukA.Algebraic groups and number theoryPure Appl. Math. 139Academic PressBoston1994 · Zbl 0806.11002 |
| [56] | [56] G. Rémond, Autour de la conjecture de Zilber-Pink, J. Théor. Nombres Bordeaux 21 (2009), no. 2, 405-414. 10.5802/jtnb.677RémondG.Autour de la conjecture de Zilber-PinkJ. Théor. Nombres Bordeaux2120092405414 · Zbl 1196.11083 |
| [57] | [57] A. Silverberg, Torsion points on abelian varieties of CM-type, Compos. Math. 68 (1988), 241-249. SilverbergA.Torsion points on abelian varieties of CM-typeCompos. Math.681988241249 · Zbl 0683.14002 |
| [58] | [58] J. Steenbrink and S. Zucker, Variation of mixed Hodge structure I, Invent. Math. 80 (1985), 489-542. 10.1007/BF01388729SteenbrinkJ.ZuckerS.Variation of mixed Hodge structure IInvent. Math.801985489542 · Zbl 0626.14007 |
| [59] | [59] J. Tsimerman, Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points, J. Amer. Math. Soc. 25 (2012), 1091-1117. 10.1090/S0894-0347-2012-00739-5TsimermanJ.Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special pointsJ. Amer. Math. Soc.25201210911117 · Zbl 1362.11057 |
| [60] | [60] E. Ullmo, Autour de la conjecture d’André-Oort, preprint (2012), http://www.math.u-psud.fr/ ullmo/Prebublications/CoursCIRM2011.pdf; Notes de cours pour les états de la recherche sur la conjecture de Zilber-Pink (CIRM 2011). <element-citation publication-type=”other“> UllmoE.Autour de la conjecture d’André-OortPreprint2012 <ext-link ext-link-type=”uri“ xlink.href=”>http://www.math.u-psud.fr/ ullmo/Prebublications/CoursCIRM2011.pdfNotes de cours pour les états de la recherche sur la conjecture de Zilber-Pink (CIRM 2011) |
| [61] | [61] E. Ullmo, Applications du théorème d’Ax-Lindemann hyperbolique, Compos. Math. 150 (2014), 175-190. 10.1112/S0010437X13007446UllmoE.Applications du théorème d’Ax-Lindemann hyperboliqueCompos. Math.1502014175190 · Zbl 1326.14062 |
| [62] | [62] E. Ullmo and A. Yafaev, A characterisation of special subvarieties, Mathematika 57 (2011), no. 2, 263-273. 10.1112/S0025579311001628UllmoE.YafaevA.A characterisation of special subvarietiesMathematika5720112263273 · Zbl 1236.14029 |
| [63] | [63] E. Ullmo and A. Yafaev, Galois orbits and equidistribution of special subvarieties: Towards the André-Oort conjecture, Ann. of Math. (2) 180 (2014), no. 3, 823-865. 10.4007/annals.2014.180.3.1UllmoE.YafaevA.Galois orbits and equidistribution of special subvarieties: Towards the André-Oort conjectureAnn. of Math. (2)18020143823865 · Zbl 1328.11070 |
| [64] | [64] E. Ullmo and A. Yafaev, Hyperbolic Ax-Lindemann theorem in the cocompact case, Duke J. Math. 163 (2014), no. 2, 433-463. 10.1215/00127094-2410546UllmoE.YafaevA.Hyperbolic Ax-Lindemann theorem in the cocompact caseDuke J. Math.16320142433463 · Zbl 1375.14096 |
| [65] | [65] E. Ullmo and A. Yafaev, Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites Galoisiennes de points spéciaux, Bull. Soc. Math. France 143 (2015), 197-228. 10.24033/bsmf.2683UllmoE.YafaevA.Nombre de classes des tores de multiplication complexe et bornes inférieures pour orbites Galoisiennes de points spéciauxBull. Soc. Math. France1432015197228 · Zbl 1323.11043 |
| [66] | [66] C. Voisin, Hodge theory and complex algebraic geometry I, Cambridge Stud. Adv. Math. 76, Cambridge University Press, Cambridge 2002. VoisinC.Hodge theory and complex algebraic geometry ICambridge Stud. Adv. Math. 76Cambridge University PressCambridge2002 · Zbl 1005.14002 |
| [67] | [67] J. Wildeshaus, The canonical construction of mixed sheaves on mixed Shimura varieties, Realizations of polylogarithms, Lecture Notes in Math. 1650, Springer, Berlin (1997), 77-140. WildeshausJ.The canonical construction of mixed sheaves on mixed Shimura varietiesRealizations of polylogarithmsLecture Notes in Math. 1650SpringerBerlin199777140 · Zbl 0877.11001 |
| [68] | [68] A. Yafaev, On a result of Ben Moonen on the moduli space of principally polarised abelian varieties, Compos. Math. 141 (2005), no. 5, 1103-1108. 10.1112/S0010437X05000291YafaevA.On a result of Ben Moonen on the moduli space of principally polarised abelian varietiesCompos. Math.1412005511031108 · Zbl 1093.14034 |
| [69] | [69] A. Yafaev, A conjecture of Yves André’s, Duke J. Math. 132 (2006), no. 3, 393-407. 10.1215/S0012-7094-06-13231-3YafaevA.A conjecture of Yves André’sDuke J. Math.13220063393407 · Zbl 1097.11032 |
| [70] | [70] B. Zilber, Exponential sums equations and the Schanuel conjecture, J. Lond. Math. Soc. (2) 65 (2002), no. 1, 27-44. 10.1112/S0024610701002861ZilberB.Exponential sums equations and the Schanuel conjectureJ. Lond. Math. Soc. (2)65200212744 · Zbl 1030.11073 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
