2017/11/24 by Avinash Kulkarni, Kulkarni, Avinash
Computer Science · Mathematics · #11E72 (Secondary) #11G30 (Primary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1711.08843
openalex publication_date 2017/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a field of characteristic 0, let C/k be a uniquely trigonal genus 4 curve, and let P ∈ C(k) be a simply ramified point of the uniquely trigonal morphism. We construct an assignment of an orbit of an algebraic group of type E8 acting on a specific variety to each element of JC(k)/2. The algebraic group and variety are independent of the choice of (C,P). We also construct a similar identification for uniquely trigonal genus 4 curves C with P ∈ C(k) a totally ramified point of the trigonal morphism. Our assignments are analogous to the assignment of a genus 3 curve with a rational point (C,P) to an orbit of an algebraic group of type E7 exhibited by Jack Thorne. Our assignment is also analogous to one constructed by Bhargava and Gross, who use it determine average ranks of hyperelliptic Jacobians.