# Fourier coefficients of a modular form of higher level at a cusp other than $i\infty$

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.

• 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. Mar 1, 2022 at 15:26

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.

• How do you propose to compute the function h? This seems rather difficult to me. Feb 24, 2022 at 16:56
• 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 Feb 24, 2022 at 17:15