The relative class number one problem for function fields, II

2022/06/05 by Kiran S. Kedlaya, Kedlaya, Kiran S.
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2206.02084

openalex publication_date 2022/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish that any finite extension of function fields of genus greater than 1 whose relative class group is trivial is Galois and cyclic. This depends on a result from a preceding paper which establishes a finite list of possible Weil polynomials for both fields. Given this list, we analyze most cases by computing options for the splittings of low-degree places in the extension, then consider the effect of these options on the Weil polynomials of certain isogeny factors of the Jacobian of the Galois closure. In one case, we use instead an analysis based on principal polarizations, modeled on an argument of Howe.

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