The Manin-Stevens constant in the semistable case

TL;DR

This paper proves Stevens' conjecture for semistable elliptic curves over Q, focusing on the challenging 2-primary case when the conductor is even, and extends results to X_0(n) parametrizations.

Contribution

It establishes Stevens' conjecture in the semistable case, especially addressing the complex 2-primary analysis and linking it to divisibility and oldforms.

Findings

Proves Stevens' conjecture for semistable elliptic curves.

Relates the conjecture to divisibility between degree and a congruence number.

Extends results to X_0(n) parametrizations and new cases of the Manin conjecture.

Abstract

Stevens conjectured that for every optimal parametrization $\phi\colon X_1(n) \rightarrow E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, the pullback of some N\'eron 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 $\mathrm{deg}\, \phi$ 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.