Lower bounds for the ranks of CM types
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.
scientific article; zbMATH DE number 4191822