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What is the relationship between modular forms (and modular functions) over $\Gamma \subseteq SL_2(\mathbb Z)$, modular curves $Y(\Gamma)$, its singular cohomology $H^k(Y(\Gamma), \mathbb Z)$, and the group cohomology $H^k(\Gamma, M)$ for a suitable $M$?

Details

From what I understand (e.g. this question), a modular form of weight $k$ is equivalent to a symmetric $k$-tensor on the modular curve $Y(\Gamma)$: $$f(\frac{az+b}{cz+d}) = (cz+d)^{2k} f(z) \iff f(g \cdot z) (\text{d}(g \cdot z))^k = f(z) \text{d} z^k \text{ where } g = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ where value at cusps are controlled by growth conditions on the modular forms. But then what do the differential forms and the de Rham cohomology on $Y(\Gamma)$ mean?

Meanwhile, I remember from Akshay Venkatesh's Fields medal lecture that modular forms can be understood as elements of a group cohomology over the arithmetic group $\Gamma \subseteq SL_2(\mathbb{Z})$, although I can't remember what module it was over.

Uzu Lim
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  • The weight 2 case is detailed chapter 2 there – reuns Oct 16 '20 at 08:19
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    The result you're after is called the Eichler-Shimura isomorphism: for $k\ge 2$, there is a Hecke-equivarient isomorphism $H^1(\Gamma, \mathrm{Sym}^{k-2}(\mathbb C)) \cong M_k(\Gamma)\oplus\overline{S_k(\Gamma)}$. See, for example, section 6.2 (e.g. Prop 6.2.3, Thm 6.4.1) in these notes. – Mathmo123 Oct 18 '20 at 15:02
  • @Mathmo123 Thanks a lot! That's a delightful theorem. They constructed a map that sends a pair of modular form and cusp form into the first group cohomology using period integrals (?) along a path acted on by a group. – Uzu Lim Oct 18 '20 at 21:40
  • Do you know what the higher cohomologies $H^n(\Gamma, Sym^{k-2}(\mathbb C))$ would mean? – Uzu Lim Oct 18 '20 at 21:41
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    In the modular forms case, $H^n$ vanishes for $n\ge 3$ and $H^0$ and $H^2$ either vanish or consist of constants, so there isn't much interesting to say. However, for more general classes of modular forms (e.g. with larger groups or over larger fields), the higher cohomologies do become relevant. – Mathmo123 Oct 19 '20 at 05:38
  • Thanks for the valuable insight. Does the higher cohomology analogue include Eichler-Shimura-Harder, which says something about Bianchi modular forms? – Uzu Lim Oct 19 '20 at 20:36
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    I think both $H^1$ and $H^2$ are relevant for Bianchi forms. – Mathmo123 Oct 21 '20 at 19:27
  • @Mathmo123 A follow-up on your comment: does this isomorphism extend over $\mathbf{Z}$? That is, do we also have an isomorphism $H^1(\Gamma, \mathbf{Z}) \simeq M_2(\Gamma, \mathbf{Z}) \oplus \overline{S_2(\Gamma, \mathbf{Z})}$? (Here $M_2(\Gamma, \mathbf{Z})$ is the space of weight 2 modular forms with integer Fourier coefficients.) – Adithya Chakravarthy Nov 14 '22 at 22:33

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