The Manin-Stevens constant in the semistable case

Abstract

Stevens conjectured that for every optimal parametrization $ϕ\colon X_1(n) \rightarrow E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, the pullback of some Néron differential on $E$ is the differential associated to the normalized new eigenform that corresponds to the isogeny class of $E$. We prove this conjecture under the assumption that $E$ is semistable, the key novelty lying in the $2$-primary analysis when $n$ is even. For this analysis, we first relate the general case of the conjecture to a divisibility relation between $\mathrm{deg}\, ϕ$ and a certain congruence number and then reduce the semistable case to a question of exhibiting enough suitably constrained oldforms. Our methods also apply to parametrizations by $X_0(n)$ and prove new cases of the Manin conjecture.