Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T00:06:40.176Z Has data issue: false hasContentIssue false
  • English
  • Français

Iterated Integrals and Higher Order Invariants

Published online by Cambridge University Press:  20 November 2018

Anton Deitmar  and
Ivan Horozov
Anton Deitmar
Affiliation:
Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany, e-mail: deitmar@uni-tuebingen.de, ivan.horozov@uni-tuebingen.de
Ivan Horozov
Affiliation:
Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany, e-mail: deitmar@uni-tuebingen.de, ivan.horozov@uni-tuebingen.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that higher order invariants of smooth functions can be written as linear combinations of full invariants times iterated integrals. The non-uniqueness of such a presentation is captured in the kernel of the ensuing map from the tensor product. This kernel is computed explicitly. As a consequence, higher order invariants form a free module of the algebra of full invariants.

Keywords

14F3511F1255D3558A10higher order formsiterated integrals
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 3 , 01 June 2013 , pp. 544 - 552
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Chen, K.-T., Iterated integrals and exponential homomorphisms. Proc. London Math. Soc. (3) 4(1954), 502–512. http://dx.doi.org/10.1112/plms/s3-4.1.502 Google Scholar
[2] Chen, K.-T., Algebras of iterated path integrals and fundamental groups. Trans. Amer. Math. Soc. 156(1971), 359–379. http://dx.doi.org/10.1090/S0002-9947-1971-0275312-1 Google Scholar
[3] Chen, K.-T., Iterated path integrals. Bull. Amer. Math. Soc. 83(1977), no. 5, 831–879. http://dx.doi.org/10.1090/S0002-9904-1977-14320-6 Google Scholar
[4] Chinta, G., Diamantis, N., and O’Sullivan, C., Second order modular forms. Acta Arith. 103(2002), no. 3, 209–223. http://dx.doi.org/10.4064/aa103-3-2 Google Scholar
[5] Deitmar, A., Higher order group cohomology and the Eichler-Shimura map. J. Reine Angew. Math. 629(2009), 221–235. http://dx.doi.org/10.1515/CRELLE.2009.032 Google Scholar
[6] Deitmar, A., Higher order invariants in the case of compact quotients. Cent. Eur. J. Math. 9(2011), no. 1, 85–101. http://dx.doi.org/10.2478/s11533-010-0081-9 Google Scholar
[7] Deitmar, A., Lewis-Zagier correspondence for higher order forms. Pacific J. Math. 249(2011), no. 1, 11–21. http://dx.doi.org/10.2140/pjm.2011.249.11 Google Scholar
[8] Deitmar, A. and Diamantis, N., Automorphic forms of higher order. J. Lond. Math. Soc (2). 80(2009), no. 1, 18–34. http://dx.doi.org/10.1112/jlms/jdp015 Google Scholar
[9] Deitmar, A., A new multiple Dirichlet series induced by a higher-order form. Acta Arith. 142(2010), no. 4, 303–309. http://dx.doi.org/10.4064/aa142-4-1 Google Scholar
[10] Diamantis, N. and Kleban, P., New percolation crossing formulas and second-order modular forms. Commun. Number Theory Phys. 3(2009), no. 4, 677–696.Google Scholar
[11] Diamantis, N., Knopp, M., Mason, G., and O’Sullivan, C., L-functions of second-order cusp forms. Ramanujan J. 12(2006), no. 3, 327–347. http://dx.doi.org/10.1007/s11139-006-0147-2 Google Scholar
[12] Diamantis, N. and O’Sullivan, C., The dimensions of spaces of holomorphic second-order automorphic forms and their cohomology. Trans. Amer. Math. Soc. 360(2008), no. 11, 5629–5666. http://dx.doi.org/10.1090/S0002-9947-08-04755-7 Google Scholar
[13] Diamantis, N. and Sim, D., The classification of higher-order cusp forms. J. Reine Angew. Math. 622(2008), 121–153. http://dx.doi.org/10.1515/CRELLE.2008.067 Google Scholar
[14] Diamantis, N. and Sreekantan, R., Iterated integrals and higher order automorphic forms. Comment. Math. Helv. 81(2006), no. 2, 481–494. http://dx.doi.org/10.4171/CMH/60 Google Scholar
[15] Hain, R. M., The geometry of the mixed Hodge structure on the fundamental group. In: Algebraic geometry, Bowdoin, 1985 (Brunswick, Main, 1985), Proc. Sympos. Pure Math., 46, American Mathematical Society, Providence, RI, 1987, pp. 247–282.Google Scholar
[16] Kontsevich, M., Vassiliev's knot invariants. In: I. M. Gel’fand Seminar, Adv. Soviet Math., 16, American Mathematical Society, Providence, RI, 1993, pp. 137–150.Google Scholar
[17] Peters, C. A. M. and Steenbrink, J. H. M., Mixed Hodge structures. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 52, Springer-Verlag, Berlin, 2008.Google Scholar
[18] Sreekantan, R., Higher order modular forms and mixed Hodge theory. Acta Arith. 139(2009), no. 4, 321–340. http://dx.doi.org/10.4064/aa139-4-2 Google Scholar