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I'm asking for a reference where I can find proof of isomorphism

$$H^{3}_{\text{cusp}}(Y(U),F_{\lambda})\simeq H^{3}_{\text{par}}(Y(U),F_{\lambda}),$$

where $Y(U)$ is the level $U$ shimura variety of $\mathbf{GSp(4)}$ and $F_{\lambda}$ the local system associated to the weight $\lambda$.

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  • $\begingroup$ Do you want such a result in all degrees of the cohomology? $\endgroup$
    – random123
    Commented Apr 23, 2023 at 10:50
  • $\begingroup$ oohh no only for degree 3, I'm going to edit thank you. $\endgroup$ Commented Apr 24, 2023 at 16:42
  • $\begingroup$ What are your definitions of parabolic and cuspidal cohomology? $\endgroup$ Commented Apr 24, 2023 at 17:30
  • $\begingroup$ I am trying to answer correctly: Parabolic is the image of $H_{c}(Y(U))\rightarrow H(Y(U))$ and Cuspidal is the $\frak{g},K$- cohomology of the space of cuspidal forms (which can be defined using the cohomology of the completion of $Y(U)$). $\endgroup$ Commented Apr 25, 2023 at 14:07
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    $\begingroup$ But actually, since yesterday (after @random123's comment) I tried to find it on my own and found it in a paper I've known for a long time now... Which makes me ashamed, but to my credit it is hidden inside two pages of calculation of cohomology spaces. You can find it on page 5-6 (293-294) of Taylor's "On the l-adic cohomology of Siegel threefold". $\endgroup$ Commented Apr 25, 2023 at 14:13

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As requested in the comments, I'm writing an answer with references and a short explanation. Because I don't deeply understand some step, I hope someone in the future will clarify this post.

You can find it on page 5-6 (293-294) of Taylor's "On triple Siegel l-adic cohomology" or on page 10 of Mokrane and Tilouine "Cohomology of Siegel manifolds with p-adic integral coefficients and applications".

The proof in the second reference I quote above is a bit clearer (to me, at least).

First of all, it is a known fact that it remains to show that $H^{k}_{\text{cusp}}=H^{k}_{(2)}$ where the right hand side denotes the $L^{ 2}$-cohomology. Now, according to Matsushima's formula, it remains to show that any representation appearing in $H_{(2)}$ is cupsidal and appears with the same multiplicity as in $H_{\text{cusp}}$. The Vogan-Zuckerman classification tells you that if $\pi_{\infty}$ appears in $H_{(2)}^{3}$ then it is tempered (this is why being with $\mathbf{GSp}_{4}$ is important but it is unclear for me, see bellow). Finally, a theorem of Wallach claims that for such $\pi_{\infty}$ the representations $\pi_{\infty}\pi_{f}$ are cuspidal and $\pi_{\infty}$ occur with the multiplicity desired.

I will simply conclude with a few words on the Vogan-Zuckerman classification. The paper quoted is D. Vogan, G. Zuckermann "Unit representations with non-zero cohomology". I'm having trouble finding where in this article is a result that allows us to prove that any $L^{2}$-cohomological representation of $\mathbf{GSp}_{4}$ is tempered. Maybe someone can comment or add an answer.

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