GENERATION OF CYCLOTOMIC HECKE FIELDS BY MODULAR L-VALUES WITH CYCLOTOMIC TWISTS

Abstract

Let $p$ be an odd prime. We show that the compositum of the Hecke field of a normalized Hecke eigen cuspform for ${\rm GL}(2)$ over $\Bbb{Q}$ and a cyclotomic field of a $p$-power degree over $\Bbb{Q}$, namely the cyclotomic Hecke field, is generated by a single algebraic critical value of the corresponding $L$-function twisted by a Dirichlet character of sufficiently large $p$-power conductor when the level of cuspform is relatively prime to $p$. The same result holds when the level is divisible by $p$ if we assume further that the cuspform does not have an inner twist. This is a stronger version of a result of Luo-Ramakrishnan. As a consequence, we reformulate a result of H. Hida on the growth of Hecke fields in a Hida family, in terms of special $L$-values.