## Notre Dame Journal of Formal Logic

### Algebraization, Transcendence, and $D$-Group Schemes

Jean-Benoît Bost

#### Abstract

We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over ${\overline {\mathbb {Q}}}$. This conjecture, closely related to the Grothendieck period conjecture for cycles of codimension $1$, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of $D$-group schemes attached to abelian schemes over algebraic curves over ${\overline {\mathbb {Q}}}$. We also derive the Grothendieck period conjecture for cycles of codimension $1$ in abelian varieties over ${\overline {\mathbb {Q}}}$ from a classical transcendence theorem à la Schneider–Lang.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 54, Number 3-4 (2013), 377-434.

Dates
First available in Project Euclid: 9 August 2013

https://projecteuclid.org/euclid.ndjfl/1376053771

Digital Object Identifier
doi:10.1215/00294527-2143961

Mathematical Reviews number (MathSciNet)
MR3091663

Zentralblatt MATH identifier
1355.11074

#### Citation

Bost, Jean-Benoît. Algebraization, Transcendence, and $D$ -Group Schemes. Notre Dame J. Formal Logic 54 (2013), no. 3-4, 377--434. doi:10.1215/00294527-2143961. https://projecteuclid.org/euclid.ndjfl/1376053771

#### References

• [1] André, Y., Une introduction aux motifs: Motifs purs, motifs mixtes, périodes, vol. 17 of Panoramas et Synthèses, Société Mathématique de France, Paris, 2004.
• [2] Andreatta, F., and L. Barbieri-Viale, “Crystalline realizations of 1-motives,” Mathematische Annalen, vol. 331 (2005), pp. 111–72.
• [3] Andreatta, F., and A. Bertapelle, “Universal extension crystals of $1$-motives and applications,” Journal of Pure and Applied Algebra, vol. 215 (2011), pp. 1919–44.
• [4] Andreotti, A., “Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves,” Bulletin de la Société Mathématique de France, vol. 91 (1963), pp. 1–38.
• [5] Anonymous (attributed to J.-P. Serre), “Correspondence,” American Journal of Mathematics, vol. 78 (1956), p. 898.
• [6] Baker, A., and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, vol. 9 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2007.
• [7] Berthelot, P., L. Breen, and W. Messing, Théorie de Dieudonné cristalline, II, vol. 930 of Lecture Notes in Mathematics, Springer, Berlin, 1982.
• [8] Bertrand, D., “Endomorphismes de groupes algébriques: Applications arithmétiques,” pp. 1–45 in Diophantine Approximations and Transcendental Numbers (Luminy, 1982), vol. 31 of Progress in Mathematics, Birkhäuser, Boston, 1983.
• [9] Bertrand, D., and A. Pillay, “A Lindemann–Weierstrass theorem for semi-abelian varieties over function fields,” Journal of the American Mathematical Society, vol. 23 (2010), pp. 491–533.
• [10] Birkenhake, C., and H. Lange, Complex Abelian Varieties, 2nd ed., vol. 302 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 2004.
• [11] Bogomolov, F., and M. L. McQuillan “Rational curves on foliated varieties,” preprint, 2001.
• [12] Bombieri, E., “Algebraic values of meromorphic maps,” Inventiones Mathematicae, vol. 10 (1970), pp. 267–87.
• [13] Bost, J.-B., “Algebraic leaves of algebraic foliations over number fields,” Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 93 (2001), pp. 161–221.
• [14] Bost, J.-B., “Germs of analytic varieties in algebraic varieties: Canonical metrics and arithmetic algebraization theorems,” pp. 371–418 in Geometric Aspects of Dwork Theory, vol. II, edited by A. Adolphson et al., Walter de Gruyter, Berlin, 2004.
• [15] Bost, J.-B., “Evaluation maps, slopes, and algebraicity criteria,” pp. 537–62 in International Congress of Mathematicians (Madrid, 2006), vol. II, European Mathematical Society, Zürich, 2006.
• [16] Bost, J.-B., and A. Chambert-Loir, “Analytic curves in algebraic varieties over number fields,” pp. 69–124 in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, vol. I, vol. 269 of Progress in Mathematics, Birkhäuser, Boston, 2009.
• [17] Bost, J.-B., and K. Künnemann, “Hermitian vector bundles and extension groups on arithmetic schemes, II: The arithmetic Atiyah extension, Astérisque, vol. 327 (2009), pp. 361–424.
• [18] Bouscaren, E., ed., Model Theory and Algebraic Geometry: An Introduction to E. Hrushovski’s Proof of the Geometric Mordell–Lang Conjecture, vol. 1696of Lecture Notes in Mathematics, Springer, Berlin, 1998.
• [19] Brylinski, J.-L., `$1$-motifs’ et formes automorphes (théorie arithmétique des domaines de Siegel),” pp. 43–106 in Conference on Automorphic Theory (Dijon, France, 1981), vol. 15 of Publications Mathématiques de l‘Université de Paris VII, Université de Paris VII, Paris, 1983.
• [20] Buium, A., Differential Function Fields and Moduli of Algebraic Varieties, vol. 1226 of Lecture Notes in Mathematics, Springer, Berlin, 1986.
• [21] Buium, A., Differential Algebraic Groups of Finite Dimension, vol. 1506 of Lecture Notes in Mathematics, Springer, Berlin, 1992.
• [22] Buium, A., “Intersections in jet spaces and a conjecture of S. Lang,” Annals of Mathematics (2), vol. 136 (1992), pp. 557–67.
• [23] Buium, A., “Effective bound for the geometric Lang conjecture,” Duke Mathematical Journal, vol. 71 (1993), pp. 475–99.
• [24] Buium, A., “Geometry of differential polynomial functions, I: Algebraic groups,” American Journal of Mathematics, vol. 115 (1993), pp. 1385–1444.
• [25] Buium, A., Differential Algebra and Diophantine Geometry, Actualités Mathématiques, Hermann, Paris, 1994.
• [26] Buium, A., and J. F. Voloch, “Integral points of abelian varieties over function fields of characteristic zero,” Mathematische Annalen, vol. 297 (1993), pp. 303–7.
• [27] Cartan, H., and J.-P. Serre, “Un théorème de finitude concernant les variétés analytiques compactes,” Comptes Rendus Académie des Sciences. Paris, vol. 237 (1953), pp. 128–30.
• [28] Chambert-Loir, A., “Théorèmes d’algébricité en géométrie diophantienne (d’après J.-B. Bost, Y. André, D. and G. Chudnovsky),” Astérisque, vol. 282 (2002), pp. 175–209, Séminaire Bourbaki, vol. 2000/2001, no. 886.
• [29] Chow, W.-L., “On compact complex analytic varieties,” American Journal of Mathematics, vol. 71 (1949), pp. 893–914.
• [30] Chow, W.-L., “Formal functions on homogeneous spaces,” Inventiones Mathematicae, vol. 86 (1986), pp. 115–30.
• [31] Chudnovsky, D. V., and G. V. Chudnovsky, “Applications of Padé approximations to the Grothendieck conjecture on linear differential equations,” pp. 52–100 in Number Theory (New York 1983–84), vol. 1135 of Lectures Notes in Mathematics, Springer, Berlin, 1985.
• [32] Chudnovsky, D. V., and G. V. Chudnovsky, “Padé approximations and Diophantine geometry,” Proceedings of the National Academy of Sciences of the USA, vol. 82(1985), pp. 2212–16.
• [33] Coleman, R. F., “The universal vectorial bi-extension and $p$-adic heights,” Inventiones Mathematicae, vol. 103 (1991), pp. 631–50.
• [34] Conforto, F., “Sopra le trasformazioni in sè della varietà di Jacobi relativa ad una curva di genere effettivo diverso dal genere virtuale, in ispecie nel caso di genere effettivo nullo,” Annali di Matematica Pura ed Applicata (4), vol. 27 (1948), pp. 273–91.
• [35] Conforto, F., “Sulla nozione di corpi equivalenti e di corpi coincidenti nella teoria delle funzioni quasi abeliane,” Rendiconti del Seminario Matematico della Università di Padova, vol. 18(1949), pp. 292–310.
• [36] Deligne, P., “Théorie de Hodge, II,” Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 40 (1971), pp. 5–57.
• [37] Demailly, J.-P., “Formules de Jensen en plusieurs variables et applications arithmétiques,” Bulletin de la Société Mathématique de France, vol. 110 (1982), pp. 75–102.
• [38] Ehresmann, C., “Les connexions infinitésimales dans un espace fibré différentiable,” pp. 29–55 in Colloque de Topologie (éspaces fibres), Bruxelles, 1950, Masson, Paris, 1951.
• [39] Faltings, G., “Algebraisation of some formal vector bundles,” Annals of Mathematics (2), vol. 110 (1979), pp. 501–14.
• [40] Faltings, G., “Some theorems about formal functions,” Publications of the Research Institute for Mathematical Sciences, vol. 16 (1980), pp. 721–37.
• [41] Faltings, G., “Formale Geometrie und homogene Räume,” Inventiones Mathematicae, vol. 64 (1981), pp. 123–65.
• [42] Gasbarri, C., “Analytic subvarieties with many rational points,” Mathematische Annalen, vol. 346 (2010), pp. 199–243.
• [43] Gillet, H., “Differential algebra: A scheme theory approach,” pp. 95–123 in Differential Algebra and Related Topics (Newark, NJ, 2000), World Sci. Publ., River Edge, NJ, 2002.
• [44] Grothendieck, A., Éléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I, vol. 11 of Publications Mathématiques. Institut de Hautes Études Scientifiques, 1961.
• [45] Grothendieck, A., Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962.
• [46] Grothendieck, A., “On the de Rham cohomology of algebraic varieties,” Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 29 (1966), pp. 95–103.
• [47] Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. Séminaire de Géométrie Algébrique du Bois-Marie 1962 (SGA 2), vol. 2 of Advanced Studies in Pure Mathematics, North-Holland, Amsterdam; Masson et Cie, Paris, 1968.
• [48] Gunning, R. C., Introduction to Holomorphic Functions of Several Variables, Vol. II: Local Theory, Wadsworth and Brooks/Cole Mathematics Series, Wadsworth and Brooks/Cole, Monterey, CA, 1990.
• [49] Hartshorne, R., “Cohomological dimension of algebraic varieties,” Annals of Mathematics (2), vol. 88 (1968), pp. 403–50.
• [50] Hartshorne, R., Ample Subvarieties of Algebraic Varieties, vol. 156 of Lecture Notes in Mathematics, Springer, Berlin, 1970.
• [51] Hartshorne, R., “On the De Rham cohomology of algebraic varieties,” Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 45 (1975), pp. 5–99.
• [52] Herblot, M., “Algebraic points on meromorphic curves,” preprint, arXiv:1204.6336v1 [math. NT].
• [53] Hironaka, H., and H. Matsumura, “Formal functions and formal embeddings,” Journal of the Mathematical Society of Japan, vol. 20 (1968), pp. 52–82.
• [54] Hodge, W. V. D., The Theory and Applications of Harmonic Integrals, Cambridge University Press, Cambridge, England: Macmillan, New York, 1941.
• [55] Hrushovski, E., “The Mordell-Lang conjecture for function fields,” Journal of the American Mathematical Society, vol. 9 (1996), pp. 667–690.
• [56] Hrushovski, E., and B. Zilber, “Zariski geometries,” Journal of the American Mathematical Society, vol. 9 (1996), pp. 1–56.
• [57] Illusie, L., “Crystalline cohomology,” pp. 43–70 in Motives (Seattle, 1991), vol. 55 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, 1994.
• [58] Illusie, L., “Grothendieck’s existence theorem in formal geometry,” pp. 179–233 in Fundamental Algebraic Geometry, vol. 123 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2005.
• [59] Jannsen, U., Mixed Motives and Algebraic $K$-Theory, with appendices by S. Bloch and C. Schoen, vol. 1400 of Lecture Notes in Mathematics, Springer, Berlin, 1990.
• [60] Kebekus, S., L. Solá Conde, and M. Toma, “Rationally connected foliations after Bogomolov and McQuillan,” Journal of Algebraic Geometry, vol. 16 (2007), pp. 65–81.
• [61] Kodaira, K., and D. C. Spencer, “Divisor class groups on algebraic varieties,” Proceedings of the National Academy of Sciences of the USA, vol. 39(1953), pp. 872–77.
• [62] Kodaira, K., and D. C. Spencer, “Groups of complex line bundles over compact Kähler varieties,” Proceedings of the National Academy of Sciences of the USA, vol. 39(1953), pp. 868–72.
• [63] Kohn, J. J., P. A. Griffiths, H. Goldschmidt, E. Bombieri, B. Cenkl, P. Garabedian, and L. Nirenberg, “Donald C. Spencer (1912–2001),” Notices of the American Mathematical Society, vol. 51 (2004), pp. 17–29.
• [64] Kowalski, P., and A. Pillay, “Quantifier elimination for algebraic $D$-groups,” Transactions of the American Mathematical Society, vol. 358 (2006), pp. 167–81.
• [65] Lang, S., “Transcendental points on group varieties,” Topology, vol. 1 (1962), pp. 313–18.
• [66] Lang, S., “Algebraic values of meromorphic functions,” Topology, vol. 3 (1965), pp. 183–91.
• [67] Lang, S., “Algebraic values of meromorphic functions, II,” Topology, vol. 5 (1966), pp. 363–70.
• [68] Lang, S., Introduction to Transcendental Numbers, Addison-Wesley Series in Mathematics, Addison-Wesley, Reading, MA; London, Ontario, 1966.
• [69] Le Potier, J., “Fibrés de Higgs et systèmes locaux,” Astérisque, vol. 201-203 (1991), pp. 221–68, Séminaire Bourbaki 1990/91, no. 737.
• [70] Malgrange, B., “Differential algebraic groups,” pp. 292–312 in Algebraic Approach to Differential Equations, World Scientific, Hackensack, NJ, 2010.
• [71] Manin, Ju. I., “Algebraic curves over fields with differentiation” (in Russian), Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, vol. 22 (1958), pp. 737–56.
• [72] Manin, Ju. I., “The Hasse–Witt matrix of an algebraic curve,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 25 (1961), pp. 153–72.
• [73] Manin, Ju. I., “Rational points on algebraic curves over function fields,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 27 (1963), pp. 1395–440.
• [74] Marker, D., “Manin kernels,” pp. 1–21 in Connections Between Model Theory and Algebraic and Analytic Geometry, vol. 6 of Quaderni di Matematica, Dept. Math., Seconda Univ. Napoli, Caserta, 2000.
• [75] Mazur, B., and W. Messing, Universal Extensions and One Dimensional Crystalline Cohomology, vol. 370 of Lecture Notes in Mathematics, Springer, Berlin, 1974.
• [76] Messing, W., “The universal extension of an abelian variety by a vector group,” pp. 359–72 in Symposia Mathematica, Vol. XI (Rome, 1972), Academic Press, London, 1973.
• [77] Mumford, D., Algebraic Geometry, I: Complex Projective Varieties, vol. 221of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 1976.
• [78] Oda, T., “The first de Rham cohomology group and Dieudonné modules,” Annales Scientifiques de l’École Normale Supérieure (4), vol. 2(1969), pp. 63–135.
• [79] Pillay, A., “Model theory and Diophantine geometry,” Bulletin of the American Mathematical Society (N.S.), vol. 34 (1997), pp. 405–22.
• [80] Pillay, A., “Some foundational questions concerning differential algebraic groups,” Pacific Journal of Mathematics, vol. 179 (1997), pp. 179–200.
• [81] Pillay, A., “Algebraic $D$-groups and differential Galois theory,” Pacific Journal of Mathematics, vol. 216 (2004), pp. 343–60.
• [82] Poincaré, H., “Sur les fonctions abéliennes,” Acta Mathematica, vol. 26 (1902), pp. 43–98.
• [83] Puiseux, V., “Recherches sur les fonctions algébriques,” Journal de Mathématiques pures et appliquées, vol. 15 (1850), pp. 365–480.
• [84] Puiseux, V., “Nouvelles recherches sur les fonctions algébriques,” Journal de Mathématiques pures et appliquées, vol. 16 (1851), pp. 228–240.
• [85] Raynaud, M., “Théorèmes de Lefschetz en cohomologie cohérente et en cohomologie étale,” vol. 103 of Bulletin de la Société Mathématique de France, Soc. Math. France, Paris, 1975.
• [86] Remmert, R., “Meromorphe Funktionen in kompakten komplexen Räumen,” Mathematische Annalen, vol. 132 (1956), pp. 277–88.
• [87] Riemann, B., “Theorie der Abel’schen Functionen,” Journal für die Reine und Angewandte Mathematik, vol. 54 (1857), pp. 115–55.
• [88] Rosenlicht, M., “Extensions of vector groups by abelian varieties,” American Journal of Mathematics, vol. 80 (1958), pp. 685–714.
• [89] Schneider, T., “Zur Theorie der Abelschen Funktionen und Integrale,” Journal für die Reine und Angewandte Mathematik, vol. 183 (1941), pp. 110–28.
• [90] Schneider, T., Einführung in die transzendenten Zahlen, Springer, Berlin, 1957.
• [91] Serre, J.-P., “Fonctions automorphes: Quelques majorations dans le cas où ${X}/{G}$ est compact,” Séminaire H. Cartan, vol. 6 (1953–1954), no. 2.
• [92] Serre, J.-P., “Géométrie algébrique et géométrie analytique,” Université de Grenoble. Annales de l’Institut Fourier, vol. 6 (1955–1956), pp. 1–42.
• [93] Serre, J.-P., Groupes algébriques et corps de classes, vol. 7 of Publications de l’institut de mathématique de l’université de Nancago, VII, Hermann, Paris, 1959.
• [94] Severi, F., Funzioni quasi abeliane, 2nd augmented edition, vol. 20 of Pontificiae Academiae Scientiarum Scripta Varia, Pontificia Academia Scientiarum, Vatican City, 1961.
• [95] Shafarevich, I. R., Basic Algebraic Geometry, translated from the Russian by K. A. Hirsch, revised printing of vol. 213 of Grundlehren der Mathematischen Wissenschaften, Springer Study Edition, Springer, Berlin, 1977.
• [96] Siegel, C. L., “Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten,” Nachrichten der Akademie der Wissenschaften in Göttingen II—Mathematisch-Physikalische Klasse, vol. 1955, pp. 71–77.
• [97] Simpson, C. T., “Moduli of representations of the fundamental group of a smooth projective variety, I,” Publications Mathématiques. Institut des Hautes Études Scientifiques, vol. 79 (1994), pp. 47–129.
• [98] Simpson, C. T., “Moduli of representations of the fundamental group of a smooth projective variety, II,” Publications Mathématiques. Institut des Hautes Études Scientifiques, vol. 80 (1994), pp. 5–79.
• [99] Thimm, W., “Über meromorphe Abbildungen von komplexen Mannigfaltigkeiten,” Mathematische Annalen, vol. 128 (1954), pp. 1–48.
• [100] Waldschmidt, M., Nombres transcendants et groupes algébriques, with appendices by D. Bertrand and J.-P. Serre, vol. 69–70 of Astérisque, Société Mathématique de France, Paris, 1979.
• [101] Wüstholz, G., Zum Periodenproblem, Inventiones Mathematicae, vol. 78 (1984), pp. 381–91.
• [102] Zariski, O., “Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields,” Memoirs of the American Mathematical Society, vol. 1951, no. 5.
• [103] Zariski, O., Algebraic Surfaces, with appendices by S. S. Abhyankar, J. Lipman, and D. Mumford, reprint of the 1971 edition, Classics in Mathematics, Springer, Berlin, 1995.