The Manin constant in the semistable case

Abstract

The Manin constant c of an elliptic curve E over Q is the nonzero integer that scales the pullback of a Néron differential under a minimal parametrization φ : X0(N ) Q ։ E into the differential ω f determined by the normalized newform f associated to 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. For the proof, we settle a part of the Manin conjecture by establishing an integrality property of ω f necessary for it to hold. We reduce the latter 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. E may be more canonical within the same isogeny class: for instance, X 1 (11) Q and X 0 (11) Q are distinct isogenous elliptic curves. The multiplicity one theorem ensures that the φ-pullback of a Néron differential ω E is a nonzero multiple of the differential ω f ∈ H 0 ((X Γ ) Q , Ω 1 ) associated to the normalized newform f whose Hecke eigenvalues agree with the Frobenius traces of E:

On the 2‐part of the Birch and Swinnerton‐Dyer conjecture for quadratic twists of elliptic curves

“…We suppose that the Manin constant of E has to be odd, which will be fully discussed in Section 2. However, we can remove the Manin constant assumption when 4 ∤ C by the recent work of Česnavičius [3]. Moreover, the conjecture that the Manin constant is always ±1 has been proved by Cremona for all optimal elliptic curves of conductor less than 390000 (see [8]).…”

The Birch--Swinnerton-Dyer exact formula for quadratic twists of elliptic curves

“…Moreover, if we write φ ′ : E ′ → E for the isogeny dual to φ, then φ ′ (P ′ ) = 2P = 0. It follows that E ′ (Q q ) [4] contains an element which is not in the kernel of the isogeny φ ′ , and so we must have E ′ (Q q ) [2] necessarily has order 4, contradicting our assumption. This completes the proof.…”

“…Note that, if we assume ν E is odd, the theorem gives the lower bound t E (M ) − 1. In fact, it has been shown that ν E is odd for 4 ∤ C by the work of Mazur [13], Abbes-Ullmo [1], Agashe-Ribet-Stein [2] and Česnavičius [4]. In addition, Cremona [10] has shown numerically that ν E is 1 for C ≤ 60000.…”

A conjecture of Watkins for quadratic twists

“…We recall that Manin [15] conjectured that if E is a strong Weil curve in the sense that m E is minimal within the isogeny class of E, then c E = 1. See [16,3,2,6] and the references therein.…”