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I've been trying to learn a bit about modular forms, and mainly using the Sherman-Diamond textbook. Now, looking at modular forms of higher level where we might have more than one cusp of the compactified modular curve, I was wondering how I might get an expansion of a given modular form around a given cusp, similarly to how we get them around $\tau = i\infty$, or equivalently $q = e^{2\pi i \tau} = 0$?

Specifically, are the expansions around different cusps related to the one around infinity? I was thinking that there must be a way to permute cusps and so to relate these expansions.

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  • $\begingroup$ I'm not sure what you have in mind for how one gets coefficients at the $\infty$ cusp. If it's "perform an integral with the modular form", then it's possible to do the same thing at other cusps. Specifically, is $\sigma_a$ is a map sending the cusp $a$ to $\infty$, then you can study the Fourier expansion of $f |_{\sigma_a^{-1}}$ (the slash operator) pretending it's a typical Fourier expansion at $\infty$. See for instance section 2.7 of Iwaniec's Topics in Classical Aut Forms. Expansions at different cusps are related by Atkin-Lehner operators. See papers of Atkin-Lehner and Atkin-Li. $\endgroup$
    – davidlowryduda
    Mar 1, 2022 at 15:26

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For $f \in M_{2k}(\Gamma_0(N))$ you can say that $\displaystyle\frac{f(z)}{j'(z)^k}$ is meromorphic on $X_0(N)$ so that $$\frac{f(z)}{j'(z)^k}=h(j(z),j(Nz))\qquad \text{with }\quad h\in \Bbb{C}(x,y)$$

Once you know the coefficients of this rational function $h$ then for any $\alpha\in SL_2(\Bbb{Z})$, $$(c_\alpha z+d_\alpha)^{-2k} f(\alpha z) = h(j(z),j(N \alpha z)) (c_\alpha z+d_\alpha)^{-2k}j'(\alpha z)^k$$ $$= h(j(z),j(\frac{Az+B}{D})) j'(z)^k$$

Where the Fourier expansion at $i\infty$ of the RHS is found from that of $j$ (which is known from its expression in term of $E_4,E_6$).

If $f\in M_{2k}(\Gamma(N))$ then it is the same idea except that $\displaystyle\frac{f(z)}{j'(z)^k}$ will be a rational function in $j$ and all the $j(\frac{az+b}{d})$, $ad=N,b\in 0\ldots d-1$.

If $f$ has an odd weight then you can do the same for $f^2$ and deduce from it the Fourier expansion of $f$ at each cusp.

So morally the problem gets relatively easy once you algebraized the (even weight) modular forms as some $k$-tensor of $1$-forms with no pole.


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  • $\begingroup$ How do you propose to compute the function h? This seems rather difficult to me. $\endgroup$ Feb 24, 2022 at 16:56
  • $\begingroup$ Yes clearly I was stuck at this part. I think that knowing enough Fourier coefficients at each cusp is mostly equivalent to finding $h$, as from those coefficients we know the first few coefficients of $Tr_{\Bbb{C}(X_0(N))/\Bbb{C}(j)}(f^4/E_4^k j^k j_N^l)$ whence their polynomial expression in $j$. @DavidLoeffler $\endgroup$
    – reuns
    Feb 24, 2022 at 17:15

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