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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Torsion points on abelian étale coverings of $ {\bf P}\sp 1-\{0,1,\infty\}$


Author: Robert F. Coleman
Journal: Trans. Amer. Math. Soc. 311 (1989), 185-208
MSC: Primary 11G30; Secondary 14H30, 14H40
MathSciNet review: 974774
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Abstract: Let $ X \to {{\mathbf{P}}^1}$ be an Abelian covering of degree $ m$ over $ {\mathbf{Q}}({\mu _m})$ unbranched outside 0, $ 1$ and $ \infty$. If the genus of $ X$ is greater than $ 1$ embed $ X$ in its Jacobian $ J$ in such a way that one of the points above 0, $ 1$ or $ \infty$ is mapped to the origin. We study the set of torsion points of $ J$ which lie on $ X$. In particular, we prove that this set is defined over an extension of $ {\mathbf{Q}}$ unramified outside $ 6m$. We also obtain information about the orders of these torsion points.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0974774-5
PII: S 0002-9947(1989)0974774-5
Article copyright: © Copyright 1989 American Mathematical Society