2017/10/14 by Alina Dudeanu, Dimitar Jetchev, Dudeanu, Alina +5
Computer Science · Mathematics · #11G10 #11G15 #14H42 #14K02 #14K25 #14Q15 #Abelian group #Algebra over a field #Algebraic Geometry and Number Theory #Botany #Combinatorics #Computer science #Cryptography and Residue Arithmetic #Cyclic group #Discrete logarithm #Discrete mathematics #FOS: Mathematics #Finite field #Genus #Isogeny #Logarithm #Mathematical analysis #Mathematics #Multiplication (music) #Number Theory (math.NT) #Polynomial #Polynomial and algebraic computation #Pure mathematics #Quotient #math.NT #msc:11G10 #msc:11G15 #msc:14H42 #msc:14K02 #msc:14K25 #msc:14Q15
paper · pdf · doi:10.48550/arxiv.1710.05147
36 pages
openalex publication_date 2017/10/14 · arxiv created 2020/09/30 · arxiv updated 2020/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study quotients of principally polarized abelian varieties with real multiplication by Galois-stable finite subgroups and describe when these quotients are principally polarizable. We use this characterization to provide an algorithm to compute explicit cyclic isogenies from kernel for abelian varieties with real multiplication over finite fields. Our algorithm is polynomial in the size of the finite field as well as the degree of the isogeny and is based on Mumford's theory of theta functions and theta embeddings. Recently, the algorithm has been successfully applied to obtain new results on the discrete logarithm problem in genus 2 as well as to study the discrete logarithm problem in genus 3.