2016/04/07 by Kęstutis Česnavičius, Cesnavicius, Kestutis · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1604.02165
Stevens conjectured that for every optimal parametrization ϕ\colon X1(n) → E of an elliptic curve E over 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 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 X0(n) and prove new cases of the Manin conjecture.