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Suppose we have a modular curve of genus $0$, whose rational function field is generated by the modular function $f$. We can view $f$ as the parameter for some pencil of elliptic curves over $\mathbb{C}$. Under certain conditions, it is possible to express $f$ as the inverse function of the ratio of two linearly independent solutions of a second-order linear differential equation. The prototypical example is the case of the period integrals of the Legendre elliptic curve $y^2=x(x-1)(x-\lambda)$, which satisfy the Fuchsian equation $$\lambda(1-\lambda)D^2y + (1-2\lambda)Dy - y/4=0,$$ where $D=d/d\lambda$. We can interpret this differential equation as measuring the variation of the periods of an elliptic curve, as the parameter $\lambda$ changes.

Now my question is : have other Picard-Fuchs equations been calculated for modular functions? In principle, there should be many such equations; the Picard-Fuchs equation for Klein's $j$ function, without the calculation, is given in (Harnad, McKay). I have seen the calculation for the $\lambda$ case carried out in a few books. But I have not seen such equations for the Hauptmodul associated to the other genus $0$ modular curves.

Any thoughts, comments, questions or references are much appreciated.

Please be kind, as I am only an undergraduate. (There seems to be much "tough love" here!)

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Just to clarify (as I don't know your background in algebraic geometry and cohomology), for any family of smooth projective varieties parameterized by a variety $V$ there is a Picard-Fuchs PDE over $V$ expressing variation of the periods of deRham cohomology of the fibers (in fixed degree, such as degree 1 in the elliptic curve cases) as the base parameter changes. So your question is to make explicit a structure that always exists, not finding an ad hoc definition in each case. I guess Klein et al. worked it all out. I don't know a undergrad-level reference on this viewpoint of Picard-Fuchs. –  Boyarsky Jun 15 '10 at 6:15
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I'd suggest looking at work of Verrill and Doran. There are some articles in the CRM proceedings edited by McKay and Sebbar. There is also a brief description in Zagier's part of 1-2-3 of Modular forms. –  S. Carnahan Jun 15 '10 at 6:19
    
Boyarksy : thank you for your clarification. My background is not very strong, and I'm mostly relying on intuition here. I will soon be attempting to get a more technical grasp on my own question. Scott : many thanks, I will check out the references you mention. –  Bruno Joyal Jun 15 '10 at 6:27
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2 Answers

Brief answer: yes, there are a lot of articles where the computations are done. I even don't know where to start. The work of Helena Verrill and Chuck Doran as well as Zagier's lectures in the 1-2-3 of modular forms (mentioned by Scott Carnahan) are a definite must. In my joint work with G. Almkvist and D. van Straten this question is addressed as well and you will find some relevant references there (for example, to Don's Apery-like article). A lot of modular cases with the corresponding Picard--Fuchs differential equations are discussed in my other joint paper with Heng Huat Chan http://dx.doi.org/10.1112/S0025579309000436">[Mathematika 56:1 (2010) 107--117] and in a recent preprint New analogues of Clausen's identities arising from the theory of modular forms (which I am not allowed to distribute publicly). You may also check Robert S. Maier's arXiv:math/0501425 and his other publications.

I have to add the books which cover in full the hypergeometric differential equations uniformized by modular functions: [J.M. Borwein and P.B. Borwein, Pi and the AGM; A study in analytic number theory and computational complexity (Wiley, New York, 1987)] and [M. Yoshida, Hypergeometric functions, my love. Modular interpretations of configuration spaces, Aspects of Mathematics E32 (Friedr. Vieweg & Sohn, Braunschweig, 1997)].

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@Bruno: Since you are in Montreal, you can contact John McKay towards your question. As for your request to be kind, I highly recommend you Yoshida's book---I use it for my lectures in hypergeometric functions and modular forms. (It's out of print however, but please e-contact me directly if you have problems with access to the book.) –  Wadim Zudilin Jun 15 '10 at 8:06
    
Thank you so much! –  Bruno Joyal Jun 15 '10 at 16:46
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See the following papers as well...

  1. Linear Ordinary Differential Equations Satisfied by Modular Forms by Xiaolong Ji and Yujie Ma

  2. Modular Functions and Their Differential Equations by Joshua L. Wiczer See also http://arxiv.org/PS_cache/math/pdf/0611/0611291v1.pdf

  3. MR2031441 (2005b:11049) Yang, Yifan . On differential equations satisfied by modular forms. Math. Z. 246 (2004), no. 1-2, 1--19.

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Great! Thank you. –  Bruno Joyal Jun 15 '10 at 16:47
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