2019/11/21 by Kęstutis Česnavičius, Cesnavicius, Kestutis, Michael Neururer +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1911.09446
openalex publication_date 2019/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Manin constant c of an elliptic curve E over ℚ is the nonzero integer that scales the differential ωf determined by the normalized newform f associated to E into the pullback of a Néron differential under a minimal parametrization ϕ\colon X0(N)ℚ \twoheadrightarrow E. Manin conjectured that c = ± 1 for optimal parametrizations, and we prove that in general c | deg(ϕ) under a minor assumption at 2 and 3 that is not needed for cube-free N or for parametrizations by X1(N)ℚ. Since c is supported at the additive reduction primes, which need not divide deg(ϕ), this improves the status of the Manin conjecture for many E. Our core result that gives this divisibility is the containment ωf ∈ H0(X0(N), Ω), which we establish by combining automorphic methods with techniques from arithmetic geometry; here the modular curve X0(N) is considered over ℤ and Ω is its relative dualizing sheaf over ℤ. We reduce this containment to p-adic bounds on denominators of the Fourier expansions of f at all the cusps of X0(N)ℂ and then use the recent basic identity for the p-adic Whittaker newform to establish stronger bounds in the more general setup of newforms of weight k on X0(N). To overcome obstacles at 2 and 3, we analyze nondihedral supercuspidal representations of GL2(ℚ2) and exhibit new cases in which X0(N)ℤ has rational singularities.