Isogeny graphs with maximal real multiplication
type: Item
instance of: scholarly article
title: Isogeny graphs with maximal real multiplication
zbMATH Open document ID: 1457.11082
author: Sorina Ionica
author: Emmanuel Thomé
published in: Journal of Number Theory
publication date: 2019-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1407.6672
review text: Throughout, let \(K\) be a primitive CM-field of degree \(4\) with maximal totally real subfield \(K_0\), and let \(\mathcal{O}_K\) (resp. \(\mathcal{O}_{K_0}\)) be the ring of integers of \(K\) (resp. \(K_0\)). Assume further that \(\mathcal{O}_{K_0}\) is principal and \(K_0\) has class number one. Also, let \(l\) be a prime number and \(\mathfrak{l}\) a prime ideal of \(\mathcal{O}_{K_0}\) of norm \(l\), that is generated by a totally positive algebraic integer. Finally, let \(\mathcal{O}\) be an order in \(K,\) and let \(A\) be a principally polarizable abelian surface over a perfect field \(F\) with complex multiplication by \(\mathcal{O}.\) Then one of the main results of the paper under review describes the structure of the \(\mathfrak{l}\)-isogeny graph of principally polarized abelian surfaces (involving \(A\)) for the case when \(F=\mathbb{C}, K \ne \mathbb{Q}(\zeta_5),\) and \(\mathcal{O}\) has locally maximal real multiplication at \(\mathfrak{l}\) (see Proposition 21, from which we deduce that the structure of a connected component of the \(\mathfrak{l}\)-isogeny graph is precisely that of an \(l\)-isogeny graph for elliptic curves, called volcano (see e.g., [\textit{M. Fouquet} and \textit{F. Morain}, Lect. Notes Comput. Sci. 2369, 276--291 (2002; Zbl 1058.11041)]). The description is two-fold and depends on whether \(\mathfrak{l}\) is prime to the conductor \(\mathfrak{f}\) of \(\mathcal{O}\) or not, and if \(\mathfrak{l}\) is prime to \(\mathfrak{f}\), then there are three types of graphs that can occur, depending on the ramification behavior of \(\mathfrak{l}\) in \(\mathcal{O}_K.\) (More precisely, if \(\mathfrak{l}\) splits in \(\mathcal{O}_K\) (ramifies, is inert, respectively), then up to isomorphism, there are exactly two (one, zero, respectively) horizontal \(\mathfrak{l}\)-isogenies starting from \(A\), and all others are descending.) For another interesting result of the paper under review, we also consider the case of the base field being a …
reviewed by: WonTae Hwang
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14K02
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 7118791
zbMATH Keywords: abelian variety
zbMATH Keywords: complex multiplication
zbMATH Keywords: real multiplication
zbMATH Keywords: Tate pairing
zbMATH Keywords: endomorphism ring
zbMATH Keywords: algorithm
describes a project that uses: AVIsogenies
MaRDI profile type: Publication
OpenAlex ID: W2970019229
arXiv ID: 1407.6672
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DOI: 10.1016/J.JNT.2019.06.019
description: scientific article