Uniformizable families of $t$-motives
HTML articles powered by AMS MathViewer
- by Gebhard Böckle and Urs Hartl PDF
- Trans. Amer. Math. Soc. 359 (2007), 3933-3972 Request permission
Abstract:
Abelian $t$-modules and the dual notion of $t$-motives were introduced by Anderson as a generalization of Drinfeld modules. For such Anderson defined and studied the important concept of uniformizability. It is an interesting question and the main objective of the present article to see how uniformizability behaves in families. Since uniformizability is an analytic notion, we have to work with families over a rigid analytic base. We provide many basic results, and in fact a large part of this article concentrates on laying foundations for studying the above question. Building on these, we obtain a generalization of a uniformizability criterion of Anderson and, among other things, we establish that the locus of uniformizability is Berkovich open.References
- Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
- M. Demazure, A. Grothendieck: Schémas en Groupes I, II, III, LNM 151, 152, 153, Springer-Verlag, Berlin-Heidelberg 1970.
- Greg W. Anderson, $t$-motives, Duke Math. J. 53 (1986), no. 2, 457–502. MR 850546, DOI 10.1215/S0012-7094-86-05328-7
- Yves André, Period mappings and differential equations. From $\Bbb C$ to $\Bbb C_p$, MSJ Memoirs, vol. 12, Mathematical Society of Japan, Tokyo, 2003. Tôhoku-Hokkaidô lectures in arithmetic geometry; With appendices by F. Kato and N. Tsuzuki. MR 1978691
- Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709, DOI 10.1090/surv/033
- Vladimir G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 5–161 (1994). MR 1259429
- Gebhard Böckle, Global $L$-functions over function fields, Math. Ann. 323 (2002), no. 4, 737–795. MR 1924278, DOI 10.1007/s002080200325
- G. Böckle: An Eichler-Shimura isomorphism over function fields between Drinfeld modular forms and cohomology classes of crystals, preprint 2001, available under: http://www.exp-math.uni-essen.de/$\sim$boeckle .
- G. Böckle, R. Pink: A cohomological theory of crystals over function fields, in preparation.
- S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961, DOI 10.1007/978-3-642-52229-1
- V. G. Drinfel′d, Moduli varieties of $F$-sheaves, Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 23–41 (Russian). MR 902291
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- Siegfried Bosch and Werner Lütkebohmert, Formal and rigid geometry. I. Rigid spaces, Math. Ann. 295 (1993), no. 2, 291–317. MR 1202394, DOI 10.1007/BF01444889
- F. Gardeyn: $t$-Motives and Galois Representations, Dissertation Universiteit Gent, Oct. 2001.
- Francis Gardeyn, A Galois criterion for good reduction of $\tau$-sheaves, J. Number Theory 97 (2002), no. 2, 447–471. MR 1942970, DOI 10.1016/S0022-314X(02)00014-8
- F. Gardeyn: New criteria for uniformization of t -motives. Preprint 2001.
- David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR 1423131, DOI 10.1007/978-3-642-61480-4
- Urs Hartl, Uniformizing the stacks of abelian sheaves, Number fields and function fields—two parallel worlds, Progr. Math., vol. 239, Birkhäuser Boston, Boston, MA, 2005, pp. 167–222. MR 2176591, DOI 10.1007/0-8176-4447-4_{9}
- A. J. de Jong, Étale fundamental groups of non-Archimedean analytic spaces, Compositio Math. 97 (1995), no. 1-2, 89–118. Special issue in honour of Frans Oort. MR 1355119
- Johan de Jong and Marius van der Put, Étale cohomology of rigid analytic spaces, Doc. Math. 1 (1996), No. 01, 1–56. MR 1386046
- Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569, DOI 10.1515/9781400881710
- Reinhardt Kiehl, Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie, Invent. Math. 2 (1967), 191–214 (German). MR 210948, DOI 10.1007/BF01425513
- Werner Lütkebohmert, Vektorraumbündel über nichtarchimedischen holomorphen Räumen, Math. Z. 152 (1977), no. 2, 127–143. MR 430331, DOI 10.1007/BF01214185
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
- M. van der Put, Cohomology on affinoid spaces, Compositio Math. 45 (1982), no. 2, 165–198. MR 651980
- M. van der Put and P. Schneider, Points and topologies in rigid geometry, Math. Ann. 302 (1995), no. 1, 81–103. MR 1329448, DOI 10.1007/BF01444488
- Peter Schneider, Points of rigid analytic varieties, J. Reine Angew. Math. 434 (1993), 127–157. MR 1195693, DOI 10.1515/crll.1993.434.127
- P. Schneider and U. Stuhler, The cohomology of $p$-adic symmetric spaces, Invent. Math. 105 (1991), no. 1, 47–122. MR 1109620, DOI 10.1007/BF01232257
- Y. Taguchi and D. Wan, $L$-functions of $\phi$-sheaves and Drinfeld modules, J. Amer. Math. Soc. 9 (1996), no. 3, 755–781. MR 1327162, DOI 10.1090/S0894-0347-96-00199-3
Additional Information
- Gebhard Böckle
- Affiliation: Institut für Experimentelle Mathematik, Universität Duisburg-Essen, Campus Essen, Ellernstr. 29, D–45326 Essen, Germany
- ORCID: 0000-0003-1758-1537
- Email: boeckle@iem.uni-due.de
- Urs Hartl
- Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, D – 79104 Freiburg, Germany
- Received by editor(s): November 15, 2004
- Received by editor(s) in revised form: July 21, 2005
- Published electronically: February 23, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 3933-3972
- MSC (2000): Primary 11G09; Secondary 14G22
- DOI: https://doi.org/10.1090/S0002-9947-07-04136-0
- MathSciNet review: 2302519