Supersingular elliptic curves, theta series and weight two modular forms

Abstract

Let p p be a prime, and let M \mathcal {M} denote the space of weight two modular forms on Γ 0 ( p ) \Gamma _{0}(p) all of whose Fourier coefficients are integral, except possibly for the constant term, which should be either integral or half-integral. We prove that M \mathcal {M} is spanned as a Z \mathbb {Z} -module by theta series attached to the unique quaternion algebra that is ramified at p p , at infinity, and at no other primes.