Lower bounds for the ranks of CM types
type: Item
instance of: scholarly article
title: Lower bounds for the ranks of CM types
zbMATH Open document ID: 0723.11062
DOI: 10.1016/0022-314X(89)90025-5
author: Liem Mai
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: Let (K,S) be a simple CM type and r its type. If \((K:{\mathbb{Q}})=2d\) then an upper bound for r is \(1+d\). K. Ribet has given a lower bound for r, in 1980, namely \(r\geq 2+\log_ 2d\). The author proves that if K/\({\mathbb{Q}}\) is a Galois extension, with Galois group G then \(r\geq 1+\sum 'd_{\pi}\), where the sum ranges over odd irreducible representations \(\pi\) with \(\pi\) (\(\tau\))\(\neq 0\), where \(\tau =\sum_{s\in G}s\in {\mathbb{Z}}[G]\). In the special case when K is the pth cyclotomic field and S is a simple CM type coming from the Fermat curve of degree p, he gives sharper lower bounds.
Mathematics Subject Classification ID: 11R99
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 14K22
zbMATH DE Number: 4191822
zbMATH Keywords: ranks
zbMATH Keywords: CM type
zbMATH Keywords: Fermat curve
reviewed by: Jannis A. Antoniadis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4191822