The Manin constant and the modular degree
Kęstutis Česnavičius
Univ. Paris-Sud, Université Paris-Saclay, Orsay, FranceMichael Neururer
Technische Universität Darmstadt, GermanyAbhishek Saha
Queen Mary University of London, UK
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Abstract
The Manin constant of an elliptic curve over is the nonzero integer that scales the differential determined by the normalized newform associated to into the pullback of a Néron differential under a minimal parametrization . Manin conjectured that for optimal parametrizations, and we prove that in general under a minor assumption at and that is not needed for cube-free or for parametrizations by . Since is supported at the additive reduction primes, which need not divide , this improves the status of the Manin conjecture for many . Our core result that gives this divisibility is the containment , which we establish by combining automorphic methods with techniques from arithmetic geometry; here the modular curve is considered over and is its relative dualizing sheaf over . We reduce this containment to -adic bounds on denominators of the Fourier expansions of at all the cusps of and then use the recent basic identity for the -adic Whittaker newform to establish stronger bounds in the more general setup of newforms of weight on . To overcome obstacles at and , we analyze nondihedral supercuspidal representations of and exhibit new cases in which has rational singularities.
Cite this article
Kęstutis Česnavičius, Michael Neururer, Abhishek Saha, The Manin constant and the modular degree. J. Eur. Math. Soc. 26 (2024), no. 2, pp. 573–637
DOI 10.4171/JEMS/1367