The Manin constant and the modular degree

Abstract

The Manin constant $c$ of an elliptic curve $E$ over $\mathbb{Q}$ is the nonzero integer that scales the differential $\omega\_f$ determined by the normalized newform $f$ associated to $E$ into the pullback of a Néron differential under a minimal parametrization $\phi\colon X\_0(N)\mathbb{Q} \twoheadrightarrow E$. Manin conjectured that $c = \pm 1$ for optimal parametrizations, and we prove that in general $c \mid \deg(\phi)$ under a minor assumption at $2$ and $3$ that is not needed for cube-free $N$ or for parametrizations by $X\_1(N)\mathbb{Q}$. Since $c$ is supported at the additive reduction primes, which need not divide $\deg(\phi)$, this improves the status of the Manin conjecture for many $E$. Our core result that gives this divisibility is the containment $\omega\_f \in H^0(X\_0(N), \Omega)$, which we establish by combining automorphic methods with techniques from arithmetic geometry; here the modular curve $X\_0(N)$ is considered over $\mathbb{Z}$ and $\Omega$ is its relative dualizing sheaf over $\mathbb{Z}$. We reduce this containment to $p$-adic bounds on denominators of the Fourier expansions of $f$ at all the cusps of $X\_0(N)\_\mathbb{C}$ 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 $X\_0(N)$. To overcome obstacles at $2$ and $3$, we analyze nondihedral supercuspidal representations of $\operatorname{GL}\_2(\mathbb{Q}2)$ and exhibit new cases in which $X\_0(N)\mathbb{Z}$ has rational singularities.