A note on non-ordinary primes

Abstract

Suppose that O L is the ring of integers of a number field L, and suppose that(note: q := e 2πiz ) is a normalized Hecke eigenform for SL 2 (Z). We say that f is non-ordinary at a prime p if there is a prime ideal p ⊂ O L above p for which a f (p) ≡ 0 (mod p).For any finite set of primes S, we prove that there are normalized Hecke eigenforms which are non-ordinary for each p ∈ S. The proof is elementary and follows from a generalization of work of Choie, Kohnen and the third author [1].

Ramanujan Congruences for Fractional Partition Functions

“…It remains an open question [5] whether a generic normalized Hecke eigenform is non-ordinary at an infinite number of primes. Jin, Ma, and Ono [6] proved that if S is a finite set of primes, there are infinitely many normalized Hecke eigenforms on SL 2 (Z) that are ℓ-non-ordinary for each ℓ ∈ S. Their key result (Theorem 2.5 of [6]) identifies a sufficient condition to guarantee the ℓ-nonordinarity of all cusp forms with Fourier coefficients in O L . Combining Theorem 2.5 and Proposition 2.1 of [6], we obtain the following result, which is crucial in the formulation of ℓ being good for α.…”

“…where O L is the ring of integers of a number field L, is ℓ-non-ordinary if a(ℓ) ≡ 0 (mod ℓO L ). The notion of ℓ-non-ordinarity is a slight strengthing of the notion of non-ordinarity at ℓ studied by Jin, Ma, Ono [6], and others. If f is a normalized Hecke eigenform, ℓ-non-ordinarity is equivalent to f being annihilated by the Hecke operator T ℓ modulo ℓ, that is, f | T ℓ ≡ 0 (mod ℓO L ).…”

“…If f is a normalized Hecke eigenform, ℓ-non-ordinarity is equivalent to f being annihilated by the Hecke operator T ℓ modulo ℓ, that is, f | T ℓ ≡ 0 (mod ℓO L ). The second condition (1.6) for ℓ being good for α is a technical criterion that guarantees the ℓ-non-ordinarity of every normalized eigenform in S 12k via a result of Jin, Ma, and Ono [6] (see Section 2.2). The third condition (1.7) ensures that the ℓ-non-ordinarity of the normalized eigenforms extends to ℓ-nonordinarity for suitable linear combinations of these eigenforms.…”

A Note on Congruences for Weakly Holomorphic Modular Forms

“…Proof. For completeness, we present the proof, which is given in [3]. Every weakly holomorphic modular form f of weight 2 is of the form P (j(z))E 14 (z)∆(z) −1 , for some polynomial P (x).…”

“…Jin, Ma, and Ono generalized this result in [3]. This allowed them to show that given any finite set of primes S, there are infinitely many level 1 normalized Hecke eigenforms f such that every p ∈ S is non-ordinary for f .…”