The Manin constant, the modular degree, and Fourier expansions at cusps

Abstract: Let f be a normalized newform of weight k for $\Gamma_0(N)$. It is a natural question to try to understand the size (in a $p$-adic sense) of the "denominators" of the Fourier expansions of f at a cusp of $X_0(N)$. The problem is easy if N is square-free but is delicate when N is highly square-full. I will talk about recent joint work with Kȩstutis Česnavičius and Michael Neururer where we solve this problem using representation-theoretic techniques. Roughly speaking, we reduce the problem to bounding $p-$adic valuations of local Whittaker newforms and then use a "basic identity" (a consequence of the Jacquet-Langlands local functional equation) to reduce to $p$-adic properties of local epsilon factors of representations of $\GL_2(\Q_p)$.

A key application of our result is to understand the Manin constant c of an elliptic curve E over the rationals. The integer c scales the differential determined by the normalized newform f associated to E into the pullback of a N\'{e}ron differential under a minimal modular parametrization. Manin conjectured that c equals 1 or -1 for optimal parametrizations. We prove that c divides the degree of the parametrization under a minor assumption at the primes 2 and 3. For this result, we establish a certain integrality property of $\omega_f$ that follows from the Manin conjecture. We expect that this integrability property we prove here will be necessary for any further progress towards Manin's conjecture.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.