A note on non-ordinary primes for some genus-zero arithmetic groups

Abstract

Suppose that OL is the ring of integers of a number field L, and suppose thatf(z)=∑n=1∞af(n)qn∈Sk(Γ0(N)+)∩OL[[q]] is a normalized Hecke eigenform for Γ0(N)+. We say that f is non-ordinary at p if there is a prime ideal p⊂OL above p for which af(p)≡0(modp). In the authors' previous paper with Ken Ono [10] it was proved that there are infinitely many Hecke eigenforms for SL2(Z) such that are non-ordinary at any given finite set of primes. In this paper, we extend this result to some genus 0 subgroups of SL2(R), namely, the normalizers Γ0(N)+ of the congruence subgroups Γ0(N). Our result also generalizes some of Choi and Kim's result in [2].