Root numbers and Selmer groups for the Jacobian varieties of Fermat curves

2018/09/25 by Jie Shu, Shu, Jie
Arts and Humanities · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1809.09285

openalex publication_date 2018/09/25 · openalex created_date 2018/10/05 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime number. Let K be the p-th cyclotomic field and F its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves X^p+Y^p=δ where δ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. When p\nmid δ, we bound the Selmer groups of these Jacobian varieties. Moreover, if p is regular and all prime ideals of K dividing δ are inert in K/F, the Selmer groups are explicitly determined and we verify the p-parity conjectures of these Jacobian varieties. We also give an asymptotic lower bound for the number of Fermat Jabobians for which the p-parity conjecture holds.

Citations

• Root numbers of Jacobi-sum Hecke characters

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