2019/11/30 by Wojciech Wawrów · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Explicit formulae #Finite field #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Hyperelliptic curve #Jacobian matrix and determinant #Torsion (gastropod) #Upper and lower bounds #math.AG #math.NT #msc:14G10 #msc:14H40 #msc:14H45
paper · pdf · doi:10.5802/jtnb.1158
published in Journal de Théorie des Nombres de Bordeaux 33(1), 223-235 (Institut de Mathématiques de Bordeaux) · 11 pages
openalex created_date 2019/11/22 · openalex publication_date 2021/05/21 · arxiv created 2021/06/09 · arxiv updated 2021/06/10 · openalex updated_date 2026/08/05
We prove a result describing the structure of a specific subgroup of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> -torsion of the Jacobian of a general superelliptic curve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , generalizing the structure theorem for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> -torsion of a hyperelliptic curve. We study existence of torsion on curves of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:math> over finite fields of characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> . We apply those results to bound from below the Mordell–Weil ranks of Jacobians of certain superelliptic curves over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> .