Congruences for Modular Forms mod 2 and QuaternionicS-ideal Classes

We prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin–Lehner eigenvalues. The proofs involve the notion of quaternionicS-ideal classes and the distribution of Atkin–Lehner signs among newforms.

CONGRUENCES FOR MODULAR FORMS OF NON-POSITIVE WEIGHT

In this paper, we consider modular forms f(z) whose q-series expansions ∑b(n)qn have coefficients in a localized ring of algebraic integers [Formula: see text]. Extending results of Serre and Ono, we show that if f has non-positive weight, a congruence of the form b(ℓn + a) ≡ 0 (mod ν), where ν is a place over ℓ in [Formula: see text], can hold for only finitely many primes ℓ ≥ 5. To obtain this, we establish an effective bound on ℓ in terms of the weight and the structure of f(z).