Skip to main content

Iterated integrals of modular forms and noncommutative modular symbols

  • Chapter
Algebraic Geometry and Number Theory

Part of the book series: Progress in Mathematics ((PM,volume 253))

Summary

The main goal of this paper is to study properties of the iterated integrals of modular forms in the upper half-plane, possibly multiplied by z s−1, along geodesics connecting two cusps. This setting generalizes simultaneously the theory of modular symbols and that of multiple zeta values.

To Volodya Drinfeld, cordially and friendly.

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

  1. A. Ash and A. Borel, Generalized modular symbols, in J.-P. Labesse and J. Schwermer, eds., Cohomology of Arithmetic Groups and Automorphic Forms (Luminy, 1989, Lecture Notes in Mathematics, Vol. 1447, Springer-Verlag, New York, 1990, 57–75.

    Chapter  Google Scholar 

  2. A. Ash and L. Rudolph, The modular symbol and continued fractions in higher dimensions, Invent. Math., 55 (1979), 241–250.

    Article  MATH  MathSciNet  Google Scholar 

  3. K.-T. Chen, Iterated path integrals, Bull. Amer. Math. Soc., 83 (1977), 831–879.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Deligne, Multizeta Values, notes d’exposés, Institute for Advanced Study, Princeton, NJ, 2001.

    Google Scholar 

  5. P. Deligne and A. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. École Norm. Sup (4), 38-1 (2005), 1–56.

    MATH  MathSciNet  Google Scholar 

  6. V.G. Drinfeld, Two theorems on modular curves, Functional Anal. Appl., 7-2 (1973), 155–156.

    Article  MathSciNet  Google Scholar 

  7. V. G. Drinfeld, On quasi-triangular quasi-Hopf algebras and some groups closely associated with Gal \( \overline Q \)/Q) Algebra Anal., 2–4 (1990); Leningrad Math. J., 2–4 (1991), 829–860.

    Google Scholar 

  8. R. Elkik, Le théorème de Manin-Drinfeld, Astérisque, Vol. 183, Société Mathématique de France, Paris, 1990, 59–67.

    Google Scholar 

  9. A. Goncharov, Polylogarithms in arithmetic and geometry, in Proceedings of the International Congress of Mathematicians (ICM’ 94, Zurich, August 3–11, 1994, Vol. 1, Birkhäuser, Basel, 1995, 374–387.

    Google Scholar 

  10. A. Goncharov, Multiple ζ-values, Galois groups and geometry of modular varieties, in Proceedings of the Third European Congress of Mathematicians, Progress in Mathematics, Vol. 201, Birkhäuser Boston, Cambridge, MA, 2001, 361–392.

    Google Scholar 

  11. A. Goncharov, The double logarithm and Manin’s complex for modular curves, Math. Res. Lett., 4 (1997), 617–636.

    MATH  MathSciNet  Google Scholar 

  12. A. Goncharov, Multiple polylogarithms, cyclotomy, and modular complexes, Math. Res. Lett., 5 (1998), 497–516.

    MATH  MathSciNet  Google Scholar 

  13. A. Goncharov, Multiple polylogarithms and mixed Tate motives, math.AG/0103059, 2001.

    Google Scholar 

  14. A. Goncharov, Periods and mixed motives, math.AG/0202154, 2002.

    Google Scholar 

  15. A. Goncharov and Yu. Manin, Multiple zeta-motives and moduli spaces \( \overline {\rm M} _{0,n} \) 0,n , Composito Math., 140-1 (2004), 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Hain, Iterated integrals and algebraic cycles: Examples and prospects, in Contemporary Trends in Algebraic Geometry and Algebraic Topology (Tianjin, 2000), Nankai Tracts in Mathematics, Vol. 5, World Scientific, River Edge, NJ, 2002, 55–118.

    Google Scholar 

  17. M. Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys., 48 (1999), 35–72.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Kontsevich and D. Zagier, Periods, in B. Engquist and W. Schmid, eds., Mathematics Unlimited: 2001 and Beyond, Springer-Verlag, Berlin, 2001, 771–808.

    Google Scholar 

  19. Yu. Manin, Parabolic points and zeta-functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat., 36-1 (1972), 19–66 (in Russian); Math. USSR Izv., 6-1 (1972), 19–64 (in English); in Selected Papers, World Scientific, Singapore, 1996, 202–247.

    MATH  MathSciNet  Google Scholar 

  20. Yu. Manin, Periods of parabolic forms and p-adic Hecke series, Mat. Sb., 92-3 (1973), 378–401 (in Russian); Math. USSR Sb., 21-3 (1973), 371–393 (in English); in Selected Papers, World Scientific, Singapore, 1996, 268–290.

    MathSciNet  Google Scholar 

  21. Yu. Manin and M. Marcolli, Continued fractions, modular symbols, and noncommutative geometry, Selecta Math. (N.S.), 8 (2002), 475–521.

    Article  MATH  MathSciNet  Google Scholar 

  22. L. Merel, Quelques aspects arithmétiques et géométriques de la théorie des symboles modulaires, thèse de doctorat, Université Paris VI, Paris, 1993.

    Google Scholar 

  23. G. Racinet, Séries génératrices non-commutatives de polyzêtas et associateurs de Drinfeld, Thèse de doctorat, Universitde Picardie-Jules-Verne, Amiens, France, 2000.

    Google Scholar 

  24. G. Racinet, Doubles mélanges des polylogarithms multiples aux racines de l’unité, Publ. Math. IHÉS, 95 (2002), 185–231.

    MATH  MathSciNet  Google Scholar 

  25. G. Racinet, Summary of Algebraic Relations between Multiple Zeta Values, lecture notes, Max-Planck Institut für Mathematik, Bonn, 17 August, 2004.

    Google Scholar 

  26. V. Shokurov, Modular symbols of arbitrary weight, Functional Anal. Appl., 10-1 (1976), 85–86.

    Article  MATH  Google Scholar 

  27. V. Shokurov, The study of the homology of Kuga varieties, Math. USSR Izv., 16-2 (1981), 399–418.

    Article  MATH  MathSciNet  Google Scholar 

  28. V. Shokurov, Shimura integrals of cusp forms, Math. USSR Izv., 16-3 (1981), 603–646.

    Article  MATH  Google Scholar 

  29. T. Terasoma, Mixed Tate motives and multiple zeta values, Invent. Math., 149 (2002), 339–369.

    Article  MATH  MathSciNet  Google Scholar 

  30. D. Zagier, Hecke operators and periods of modular forms, in S. Gelbart, R. Howe, and P. Sarnak, eds., Festschrift in honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Israel Mathematics Conference Proceedings, Vol. 3, Weizmann Science Press, Jerusalem, 1990, 321–336.

    Google Scholar 

  31. D. Zagier, Values of zeta functions and their applications, in Proceedings of the First European Congress of Mathematics, Part II, Progress in Mathematics, Vol. 120, Birkhäuser, Basel, 1994, 497–512.

    Google Scholar 

  32. D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math., 104 (1991), 449–465.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Boston

About this chapter

Cite this chapter

Manin, Y.I. (2006). Iterated integrals of modular forms and noncommutative modular symbols. In: Ginzburg, V. (eds) Algebraic Geometry and Number Theory. Progress in Mathematics, vol 253. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4532-8_10

Download citation

Publish with us

Policies and ethics