Periods of iterated integrals of holomorphic forms on a compact Riemann surface

Author:
Shu Yin Hwang Ma

Journal:
Trans. Amer. Math. Soc. **264** (1981), 295-300

MSC:
Primary 14H15; Secondary 14H20, 30F30, 32G20

DOI:
https://doi.org/10.1090/S0002-9947-1981-0603764-X

MathSciNet review:
603764

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Holomorphic forms are integrated iteratedly along paths in a compact Riemann surface of genus , thus inducing a homomorphism from the fundamental group to a proper multiplicative subgroup of the group of units in , where denotes the space of holomorphic forms on is the complex dual of , means the associated tensor algebra and 11'' means completion with respect to the natural grading. The associated homomorphisms from to reduces to the classical case when . We show that the images of are always cocompact in and are discrete for all if and only if the Jacobian variety of is isogenous to for some elliptic curve with complex multiplication.

**[1]**K. T. Chen,*Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula*, Ann. of Math. (2)**65**(1957), 163-178. MR**0085251 (19:12a)****[2]**-,*Algebras of iterated path integrals and fundamental groups*, Trans. Amer. Math. Soc.**156**(1971), 359-379. MR**0275312 (43:1069)****[3]**N. Koblitz,*Gamma function identities and elliptic differentials on Fermt curves*, Duke Math. J.**45**(1978), 87-99. MR**0476752 (57:16309)****[4]**A. N. Parsin,*A generalization of the Jacobian variety*, Amer. Math. Soc. Transl. (2)**84**(1969), 187-196.**[5]**H. E. Rauch,*Theta functions with applications to Riemann surfaces*, William & Wilkins, Baltimore, Md., 1974. MR**0352108 (50:4595)****[6]**H. E. Rauch and J. Lewittes,*The Riemann surface of Klein with**automorphisms*, Problems in Analysis (Papers dedicated to Solomon Bochner, 1969), Princeton Univ. Press, Princeton, N.J., 1970, pp. 297-308. MR**0346150 (49:10876)****[7]**R. Ree,*Lie elements and an algebra associated with shuffles*, Ann. of Math. (2)**68**(1958), 210-220. MR**0100011 (20:6447)****[8]**D. Sullivan,*Differential forms and the topology of manifolds*, Manifolds-Tokyo 1973 (Proc. Internat. Conf. on Manifolds and Related Topics in Topology), Univ. Tokyo Press, Tokyo, 1975, pp. 37-49. MR**0370611 (51:6838)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14H15,
14H20,
30F30,
32G20

Retrieve articles in all journals with MSC: 14H15, 14H20, 30F30, 32G20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0603764-X

Keywords:
Iterated integrals,
compact Riemann surfaces,
period matrices,
Jacobian varieties,
Lie group,
holomorphic forms

Article copyright:
© Copyright 1981
American Mathematical Society