Torsion points on Fermat quotients of the form yⁿ = x^d + 1

2019/10/31 by Vishal Arul, Arul, Vishal
Mathematics · #11L05 #14H40 #14H55 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11L05 #msc:14H40 #msc:14H55

paper · pdf · doi:10.48550/arxiv.1910.14251

36 pages; improved exposition, reorganized sections to separate geometry and arithmetic and to avoid faulty Lemma 2.12 of v1, redid Magma calculations to work over finite fields instead over C to avoid interval arithmetic

arxiv created 2020/05/03 · arxiv updated 2020/05/05

Abstract

We classify all geometric torsion points on the Fermat quotients yⁿ = x^d + 1 where n, d ≥ 2 are coprime. In addition, we classify all geometric torsion points on the generic superelliptic curve yⁿ = (x - a₁) ⋯ (x - a_d), extending a result of Poonen and Stoll, who considered the hyperelliptic n = 2 case.

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