The Manin constant and the modular degree

TL;DR

This paper proves new divisibility properties of the Manin constant for elliptic curves over rationals, combining automorphic and arithmetic geometry methods to advance understanding of the Manin conjecture.

Contribution

It establishes that the Manin constant divides the degree of the modular parametrization under broad conditions, improving the conjecture's status for many elliptic curves.

Findings

Proves $c mid ext{deg}(\

Establishes containment of the differential form in the global sections of the relative dualizing sheaf.

Analyzes $p$-adic bounds and representations at primes 2 and 3 to handle special cases.

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 N\'{e}ron 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 $c \mid \mathrm{deg}(\phi)$ 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 $\mathrm{deg}(\phi)$, 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 $\mathrm{GL}_2(\mathbb{Q}_2)$ and exhibit new cases in which $X_0(N)_{\mathbb{Z}}$ has rational singularities.