A note on supersingular abelian varieties

2014/12/22 by Chia‐Fu Yu, Chia-Fu Yu, Yu, Chia-Fu
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT

paper · pdf · doi:10.48550/arxiv.1412.7107

14 pages. Revise Introduction; add Section 2; erratum to [Math. Res. Let., 2010]= arXiv:0905.0019

openalex publication_date 2014/12/22 · arxiv created 2017/06/10 · arxiv updated 2017/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we show that any supersingular abelian variety is isogenous to a superspecial abelian variety without increasing field extensions. The proof uses minimal isogenies and the Galois descent. We then construct a superspecial abelian variety which not directly defined over a finite field. This answers negatively to a question of the author [J. Pure Appl. Alg., 2013] concerning of endomorphism algebras occurring in Shimura curves. Endomorphism algebras of supersingular elliptic curves over an arbitrary field are also investigated. We correct a main result of the author's paper [Math. Res. Let., 2010].

Citations

• On finiteness of endomorphism rings of abelian varieties

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