Ranks of Jacobians in towers of function fields

2010/02/17 by Ulmer, Douglas, Zarhin, Yuri G.
#11G10 #11G35 #14G05 #14G25 #14K15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1002.3318

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/pr). In many cases we are able to compute the rank at every layer of the tower.

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