Lower bounds for the ranks of CM types

Abstract

In this note, we discuss lower bounds for the ranks of CM types. Let (K, S) be a simple CM type, [K:Q]=2d and r its rank. It is known that r ≤ 1 + d. Lower bounds for r were given by K. Ribet ["Division Fields of Abelian Varieties with Complex Multiplication," 2e ser. mem., No. 2, pp. 75-94, Soc. Math. France, 1980]. When K/Q is Galois, we obtain a lower bound for r in terms of character degrees. A nearly optimal lower bound for r is also given in the case where K is the pth cyclotomic field and S is a simple CM type coming from the Fermat curve of degree p. © 1989.