Ranks of Jacobians in towers of function fields

Details Ranks of Jacobians in towers of function fields Ranks of Jacobians in towers of function fields

2010-02-17

arxiv.org/abs/1002.3318v2

Mathematics

Number Theory

9 pages. To appear in Mathematical Research Letters

Scientific paper

Let $k$ be a field of characteristic zero and let $K=k(t)$ be the rational function field over $k$. In this paper we combine a formula of Ulmer for ranks of certain Jacobians over $K$ with strong upper bounds on endomorphisms of Jacobians due to Zarhin to give many examples of higher dimensional, absolutely simple Jacobians over $k(t)$ with bounded rank in towers $k(t^{1/p^r})$. In many cases we are able to compute the rank at every layer of the tower.

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