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arXiv · 1002.3318

Ranks of Jacobians in towers of function fields

Douglas Ulmer·Yuri G. Zarhin
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Abstract

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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Douglas Ulmer, Yuri G. Zarhin. 2010-07-08. Ranks of Jacobians in towers of function fields. https://arxiv.org/abs/1002.3318

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