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The Manin constant in the semistable case

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  13 August 2018

Kęstutis Česnavičius
Kęstutis Česnavičius*
Affiliation:
Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email kestutis@math.u-psud.fr
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Abstract

For an optimal modular parametrization $J_{0}(n){\twoheadrightarrow}E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^{0}(E,\unicode[STIX]{x1D6FA}^{1})$. Multiple authors generalized his conjecture to higher-dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic étale and de Rham cohomologies of abelian varieties over $p$-adic fields and exhibit a new exactness result for Néron models.

Keywords

elliptic curveHecke algebraManin constantmodular curveNéron modelnewform

MSC classification

Primary: 11G05: Elliptic curves over global fields 11G10: Abelian varieties of dimension $> 1$
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 11F11: Holomorphic modular forms of integral weight 14G35: Modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 9 , September 2018 , pp. 1889 - 1920
Copyright
© The Author 2018 

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