Skip to main content

Drinfeld modules and elliptic sheaves

  • Chapter
  • First Online:
Vector Bundles on Curves — New Directions

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1649))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • [An1] G. Anderson: t-Motives, Duke Math. J. 53 (1986), 457–502

    Article  MathSciNet  MATH  Google Scholar 

  • [An2] G. Anderson: Notes on t-Motives, Lectures given at the Institute for Advanced Studies, Princeton 1987.

    Google Scholar 

  • [An3] G. Anderson: A two-dimensional analogue of Stickelberger’s theorem, The Arithmetic of Function Fields (D. Goss, D. Hayes, and M. rosen, eds.), Proc. Workshop Ohio State Univ., June 17–26, 1991, de Gruyter, Berlin and New York, 1992, pp. 51–73

    Google Scholar 

  • [An4] G. Anderson: Rank one elliptic A-modules and A-harmonic series. Duke Math. J. 73 (1994), 491–542

    Article  MathSciNet  MATH  Google Scholar 

  • [Ar-Co-1] E. Arbarello, C. De Concini: On a set of equations characterizing Riemann matrices, Ann. of Math. (2) 120 (1984), 119–140

    Article  MathSciNet  MATH  Google Scholar 

  • [Ar-Co-2] E. Arbarello, C. De Concini: Geometrical aspects of the Kadomtsev-Petviashvili equation, Lecture notes in Mathematics 1451, Springer Verlag 1990, p. 95–137.

    Google Scholar 

  • [Art] M. Artin: Versal deformations and algebraic stacks, Inv. Math. 27, (1974), 165–189

    Article  MathSciNet  MATH  Google Scholar 

  • [Ba] H. Bass: Algebraic K-theory, W.A. Benjamin, Inc., New York 1968

    MATH  Google Scholar 

  • [Bo-Gu-Re] S. Bosch, U. Güntzer, R. Remmert. Non-Archimedean Analysis. Berlin-Heidelberg-New York: Springer 1984

    Book  MATH  Google Scholar 

  • [Br-Ti] F. Bruhat, J. Tits: Groupes réductifs sur un corps local. I, Données radicielles valués, Publ. Math. IHES 41, (1972), 5–251.

    Article  MathSciNet  Google Scholar 

  • [Bour] N. Bourbaki: Élements de mathematique, Algèbre commutative, chap. 7: Diviseurs Paris, Hermann 1965.

    MATH  Google Scholar 

  • [Bo-Ca] J.-F. Boutot, H. Carayol: Uniformisation p-adique des courbes de Shimura, Asterisque 196–197, (1991), 45–149

    MathSciNet  MATH  Google Scholar 

  • [Bu-Cha] J.L. Burchnall, T.W. Chaundy: Commutative ordinary differential operators, Proc. London Math. Soc. Ser. 2, 21, (1923), 420–440; Proc. Royal Soc. London Ser. A, 118 (1928), 557–583

    Article  MathSciNet  MATH  Google Scholar 

  • [Cal] H. Carayol: Non-abelian Lubin-Tate theory, in: Clozel, L., Milne, J.S. (ed), Automorphic forms, Shimura varieties and L-functions vol. II, Persp. in Math. 11, 15–40, Acad. Press Boston (1990)

    Google Scholar 

  • [Ca2] H. Carayol: Varietés de Drinfeld compactes, (d’après Laumon, Rapoport et Stuhler). Sém. Bourbaki, 44 ème année, 1991–92 Nr. 756

    Google Scholar 

  • [Carl] L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), 137–168

    Article  MathSciNet  MATH  Google Scholar 

  • [Ch] I.V. Cherednik: Uniformization of algebraic curves by discrete subgroups of PGL 2 (k w ) Math. USSR Sbornik, 29, (1976), 55–78

    Article  Google Scholar 

  • [De-Hu] P. Deligne, D. Husemöller: Survey of Drinfeld modules, Contemp. Math. 67 (1987), 25–91

    Article  MathSciNet  MATH  Google Scholar 

  • [De-Mu] P. Deligne, D. Mumford: The irreducibility of the space of curves of given genus, Publ. Math. I.H.E.S., Nr. 36, (1969), 75–110

    Article  MathSciNet  MATH  Google Scholar 

  • [De-Ra] P. Deligne, M. Rapoport: Les schémas de modules de courbes elliptiques. In: Modular functions of one variable II, Antwerpen conference 1972, Springer lecture notes in mathematics 349, (1973), 143–316

    Google Scholar 

  • [Dr1] V.G. Drinfeld: Elliptic modules, Mat. Sb. 94, (1974), 594–627; [English transl. in Math. USSR-Sb. 23 (1976), 561–592]

    MathSciNet  Google Scholar 

  • [Dr2] V.G. Drinfeld: Elliptic modules. II, Mat. Sb. 102, (1974) [English transl. in Math. USSR-Sb. 31 (1977), 159–170

    Google Scholar 

  • [Dr3] V.G. Drinfeld: Commutative subrings of some noncommutative rings, Funct. Anal. 11 (1977), 11–14

    MathSciNet  Google Scholar 

  • [Dr4] V.G. Drinfeld: Coverings of p-adic symmetric regions. Funct. Anal. Appl. 10 (1976), 107–115.

    Article  MathSciNet  Google Scholar 

  • [Dr5] V.G. Drinfeld: Varieties of modules of F-sheaves, Funct. Anal. and its Appl. 21, (1987), 107–122

    Article  MathSciNet  Google Scholar 

  • [Dr6] V.G. Drinfeld: The proof of Petersson’s conjecture for GL(2) over a global field of characteristic p. Funct. Anal. and its Appl. 22, (1988), 28–43

    Article  MathSciNet  Google Scholar 

  • [Dr7] V.G. Drinfeld: Cohomology of compactified manifolds of modules of F-sheaves of rank 2, Journal of Soviet math., vol. 46, No. 1, (1989), 1789–1821

    Article  MathSciNet  Google Scholar 

  • [Dr8] V.G. Drinfeld: Letter to H. Carayol from 12.1.80

    Google Scholar 

  • [Fa1] G. Faltings: F-isocrystals on open varieties. Results and conjectures. Grothendieck Festschrift, Progress in Math., Birkhäuser 1990

    Google Scholar 

  • [Fa2] G. Faltings: The trace formula and Drinfeld’s upper halfplane, Duke Math. J. 76, 1994, 467–481

    Article  MathSciNet  MATH  Google Scholar 

  • [Fl-Ka] Y. Flicker, D. Kazhdan: Geometric Ramanujan conjecture and Drinfeld reciprocity law, in: Number theory, trace formulas and discrete groups (K. Aubert, E. Bombieri, D. Goldfeld, (eds), Academic press, (1989), 201–218

    Google Scholar 

  • [Fr-Pu] J. Fresnel, M. van der Put: Géométrie Analytique Rigide et Applications. Boston: Birkhäuser (1981)

    MATH  Google Scholar 

  • [Gek1] E.-U. Gekeler: On the de Rham isomorphism for Drinfeld modules. J. reine angew. Math. 401, (1989), 188–208

    MathSciNet  MATH  Google Scholar 

  • [Gek2] E.-U. Gekeler: De Rham cohomology for Drinfeld modules, Sém. Théorie des Nombres, Paris 1988–1989, Birkhäuser Basel and Boston, (1990), 57–85

    MATH  Google Scholar 

  • [Gen1] A. Genestier: Ramification du revêtement de Drinfeld, Thesè, Université de Paris Sud, Orsay, 1992

    Google Scholar 

  • [Gen2] A. Genestier: Espaces symétriques de Drinfeld, prepublication, Université de Paris-Sud, Orsay, 1995

    Google Scholar 

  • [Go-Iw] O. Goldmann, N. Iwahori: The space of p-adic norms. Acta Math. 109, (1963), 137–177

    Article  MathSciNet  MATH  Google Scholar 

  • [Go1] D. Goss: The algebraist’s upper half-plane. Bull. AMS 2, (1980), 391–415

    Article  MathSciNet  MATH  Google Scholar 

  • [Go2] D. Goss: Drinfeld modules: Cohomology and special functions, Proceedings of Symposia in Pure Mathematics, vol. 55, 1994, part 2, p. 309–362

    Article  MathSciNet  MATH  Google Scholar 

  • [Gro-Hop1] B. Cross, M. Hopkins: Equivariant vector bundles on the Lubin-Tate moduli space, Contemp. Math. 158, (1994), 23–88

    Article  MathSciNet  MATH  Google Scholar 

  • [Gro-Hop2] B. Gross, M. Hopkins: The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. 30, (1994), 76–86

    Article  MathSciNet  MATH  Google Scholar 

  • [Groth] A. Grothendieck: Élements de Géometrie Algebrique (EGA), rédigés avec la collaboration de J. Dieudonné, Publ. Math. I.H.E.S., 4, 8, 11, 17, 20, 24, 28, 32, Bures-Sur-Yvette, 1960–1967

    Google Scholar 

  • [Hur] J. Hurtubise: Algebraic geometry and completely integrable Hamiltonian systems, Canadian Math. Soc., Conference Proceedings vol. 12, 1992, 85–104

    MathSciNet  MATH  Google Scholar 

  • [Ka-Ma] N. Katz. B. Mazur: Arithmetic moduli of elliptic curves, Annals of Math. Studies 108, Princeton University Press, 1985.

    Google Scholar 

  • [Kaz] D. Kazhdan: An introduction to Drinfeld’s “shtuka”, Proc. Sym. Pure Math. 33, part 2, (1979), 347–356

    Article  MathSciNet  MATH  Google Scholar 

  • [Ko] N. Koblitz: p-adic numbers, p-adic analysis and Zeta functions, Sec. edition, Springer Verlag, 1984

    Google Scholar 

  • [Kri] I.M. Krichever: Methods of algebraic geometry in the theory of nonlinear equations, Russ. Math. Surveys 32 (1977), 185–214

    Article  MathSciNet  MATH  Google Scholar 

  • [Ku] A. Kurihara: Construction of p-adic unit balls and the Hirzebruch proportionality. Amer. J. Math. 102, (1980), 565–648

    Article  MathSciNet  MATH  Google Scholar 

  • [Laf] L. Lafforgue: D-stukas de Drinfeld, Université de Paris Sud, Orsay, 1994

    Google Scholar 

  • [Lau1] G. Laumon: Sur les modules de Krichever, Preprint

    Google Scholar 

  • [Lau2] G. Laumon: Cohomology of Drinfeld modular varieties, part I, Cambridge University Press, 1996, Part II, to appear

    Google Scholar 

  • [Lau-Mo] G. Laumon, L. Moret-Bailly: Champs algébriques; Prépublications de l’université de Paris Sud, 193

    Google Scholar 

  • [Lau-Ra-Stxxx] G. Laumon, M. Rapoport, U. Stuhler: D-elliptic sheaves and the Langlands correspondence; Inventiones mathematicae 113, 1993, 217–338

    Article  MathSciNet  MATH  Google Scholar 

  • [MacL] S. MacLane: Homology, Springer-Verlag, 1967

    Google Scholar 

  • [Mi] St. Milne: Etale Cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton 1980

    MATH  Google Scholar 

  • [Mu1] D. Mumford: An analytic construction of degenerating curves over complete local rings. Compositio Math. 24, (1972), 129–174

    MathSciNet  MATH  Google Scholar 

  • [Mu2] D. Mumford: An Algebro-geometric construction of commuting oeprators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations, (Proc. Internat. Sympos. Alg. Geometry, Kyoto, 1977), Kinokuniy Book Store, Tokyo, (1978), 115–153

    Google Scholar 

  • [Mu3] D. Mumford: Tata Lectures on Theta II, Birkhäuser-Verlag, Basel, Switzerland, and Cambridge, MA 1984

    MATH  Google Scholar 

  • [Mustxxx] G. Mustafin: Nonarchimedian uniformization, Math. USSR Sbornik 34 (1978), 187–214

    Article  MATH  Google Scholar 

  • [Pr-Se] A. Pressley, G. Segal: Loop Groups, Oxford University press 1986

    Google Scholar 

  • [Ra] M. Rapoport: On the bad reduction of Shimura varieties, in L. Clozel, J.S. Milne (ed), Persp. in Math. 11, Academic Press, Boston 1990, 253–321

    Google Scholar 

  • [Ra-Zi] M. Rapoport, Th. Zink: Period spaces for p-divisible groups, Annals of Mathematics Studies, Princeton University Press, 1996

    Google Scholar 

  • [Rei] I. Reiner, Maximal orders, Academic Press 1975

    Google Scholar 

  • [Sc-Stxxx] P. Schneider, U. Stuhler: The cohomology of p-adic symmetric spaces, Invent. math. 105, (1991), 47–122

    Article  MathSciNet  MATH  Google Scholar 

  • [Sch] I. Schur: Über vertauschbare lineare Differentialausdrücke, Sitzungsber. der Berliner Math. Gesell. 4 (1905), 2–8

    MATH  Google Scholar 

  • [Se-Wi] G.B. Segal, G. Wilson: Loop groups and equations of KdV type, Publ. Math. I.H.E.S. 61 (1985), 5–65

    Article  MathSciNet  MATH  Google Scholar 

  • [Sesh] C.S. Seshadri: Fibrés vectoriels sur les courbes algébriques (redigée par J.M. Drezet). Asterisque 96, (1982)

    Google Scholar 

  • [Shio] T. Shiota: Characterization of Jacobian varieties in terms of soliton equations, Inv. Math. 83 (1986), 333–382

    Article  MathSciNet  MATH  Google Scholar 

  • [Stxxx] U. Stuhler: p-adic homogeneous spaces and moduli problems, Math. Zeitschrift 192, (1986), 491–540

    Article  MathSciNet  MATH  Google Scholar 

  • [Pu-Vo] M. van der Put, H. Voskuil: Symmetric spaces associated to split algebraic groups over a local field, J. reine angew. Math. 433, (1992), 69–100.

    MathSciNet  MATH  Google Scholar 

  • [Va-Po-Ma] J.M. Vázquez, M. Porras, and F.J.P. Martin: The algebraic formalism of soliton equations over arbitrary base fields, preprint 1996, 1–34 (alggeom/9606009)

    Google Scholar 

  • [Ver] J.-L. Verdier: Equations differentielles algébriques, Séminaire de l’École Normale Supérieure 1979–82, Birkhäuser (1983), 215–236

    Google Scholar 

Download references

Authors

Editor information

M. S. Narasimhan

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag

About this chapter

Cite this chapter

Blum, A., Stuhler, U. (1997). Drinfeld modules and elliptic sheaves. In: Narasimhan, M.S. (eds) Vector Bundles on Curves — New Directions. Lecture Notes in Mathematics, vol 1649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094426

Download citation

  • DOI: https://doi.org/10.1007/BFb0094426

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62401-1

  • Online ISBN: 978-3-540-49701-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics