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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Periods of iterated integrals of holomorphic forms on a compact Riemann surface
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by Shu Yin Hwang Ma PDF
Trans. Amer. Math. Soc. 264 (1981), 295-300 Request permission

Abstract:

Holomorphic forms are integrated iteratedly along paths in a compact Riemann surface $M$ of genus $g$, thus inducing a homomorphism from the fundamental group $\Gamma = {\pi _1}(M,{P_0})$ to a proper multiplicative subgroup $G$ of the group of units in $\widehat {T({\Omega ^{1 \ast }})}$, where ${\Omega ^1}$ denotes the space of holomorphic forms on $T$ is the complex dual of ${\Omega ^1}$, $T$ means the associated tensor algebra and 11$\hat { }$” means completion with respect to the natural grading. The associated homomorphisms from $\Gamma /{\Gamma ^{(n + 1)}}$ to $G/{G^{(n + 1)}}$ reduces to the classical case ${H_1}(M) \to {\Omega ^{1 \ast }}$ when $n = 1$. We show that the images of $\Gamma /{\Gamma ^{(n + 1)}}$ are always cocompact in $G/{G^{(n + 1)}}$ and are discrete for all $n \geqslant 2$ if and only if the Jacobian variety $J(M)$ of $M$ is isogenous to ${E^g}$ for some elliptic curve $E$ with complex multiplication.
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • 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