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Transactions of the American Mathematical Society

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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
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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

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

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