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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: [email protected], [email protected]
Ivan Horozov
Affiliation:
Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany, e-mail: [email protected], [email protected]
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Abstract

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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