The Manin constant and the modular degree

The Manin constant c of an elliptic curve E over Q is the nonzero integer that scales the differential cof determined by the normalized newform f associated to E into the pullback of a Neron differential under a minimal parametrization 0: X0(N)Q ->> E. Manin conjectured that c = +1 for optimal parametrizations, and we prove that in general c i deg(0) under a minor assumption at 2 and 3 that is not needed for cube-free N or for parametrizations by X1(N)Q. Since c is supported at the additive reduction primes, which need not divide deg(0), this improves the status of the Manin conjecture for many E. Our core result that gives this divisibility is the containment cof E H0(X0(N), S2), which we establish by combining automorphic methods with techniques from arithmetic geometry; here the modular curve X0(N) is considered over Z and S2 is its relative dualizing sheaf over Z. We reduce this containment to p-adic bounds on denominators of the Fourier expansions of f at all the cusps of X0(N)C and then use the recent basic identity for the p-adic Whittaker newform to establish stronger bounds in the more general setup of newforms of weight k on X0(N). To overcome obstacles at 2 and 3, we analyze nondihedral supercuspidal representations of GL2(Q2) and exhibit new cases in which X0(N)Z has rational singularities.