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The Manin constant

Let $ A$ be a quotient of $ J_0(N)$ .

Definition 8.6.1 (Manin constant)   The Manin constant of $ A$ is the index in its saturation of $ \H ^1(\mathcal{A},\Omega_{\mathcal{A}/\mathbb{Z}})$ in $ S_2(\Gamma_0(N);\mathbb{Z})$ under the appropriate natural identifications (see [ARS06]).

Problem 8.6.2   Find (and implement) an algorithm to compute the prime divisors of the Manin constant of $ A$ .

This is closely related to understanding $ q$ -expansions at cusps other than $ \infty$ .

Problem 8.6.3   Give a modular form $ f=\sum a_n q^n\in S_2(\Gamma_0(N))$ compute the $ q^{1/h}$ -expansion of $ f$ at all the cusps.

Remark 8.6.4 (From John Cremona:)   Prob 8.6.3 was dealt with in Delaunay's thesis.

Problem 8.6.5   Implement in SAGE Delaunay's algorithm to compute the $ q^{1/h}$ -expansion of a modular form at all cusps.

Problem 8.6.6   Give a modular form $ f=\sum a_n q^n\in S_k(\Gamma_1(N))$ compute the $ q^{1/h}$ -expansion of $ f$ at all the cusps.



William Stein 2006-10-20