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Transactions of the American Mathematical Society

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

 

 

Curves of genus $ 2$ with split Jacobian


Author: Robert M. Kuhn
Journal: Trans. Amer. Math. Soc. 307 (1988), 41-49
MSC: Primary 14H40; Secondary 11G10
DOI: https://doi.org/10.1090/S0002-9947-1988-0936803-3
MathSciNet review: 936803
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Abstract: We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If $ X$ is a curve of genus $ 2$, and $ f:X \to E$ a map from $ X$ to an elliptic curve, then $ X$ has split jacobian. It is not true that a complement to $ E$ in the jacobian of $ X$ is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve $ E' $ and algebraic $ f:X \to E' $, and we give an algorithm for finding that curve. The construction works in any characteristic other than two. Applications of the algorithm are given to give explicit examples in characteristics 0 and $ 3$.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0936803-3
Article copyright: © Copyright 1988 American Mathematical Society