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In the paper by L.Clozel in this book (a French text), there is this conjecture (conjecture 4.5 p139)

Conjecture: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote by $E$ it's definition field. Then there exists a degree $n$ motive $M$ over a finite extension of $E$ such that the euler factors of the two L-function coïncide up to a translation of $\frac{1-n}{2}$.

First question: How can we generalize this conjecture for the group $GSp_{4}$?

On the over hand, in this paper, M.Harris associate to $\pi_{f}$ the finite part of a cuspidal representation $\pi$ of $GSp_{4}$ a motive $M(\pi_{f})$ as follows $$M(\pi_{f})=H^{3}_{!}(Sh,\tilde{V}_{\rho})[\pi_{f}]$$ It is done p.5, and note by the way that he does not tell us that it "is a motive associated to $\pi$", I just believe that it is what he mean because of the name of the section 1 and notation..

Second question: Assume $\pi$ is an algebraic representation of $GSp_{4}$, does the motive $M(\pi_{f})$ satisfy the conjecture stated in Clozel's paper and why??

Last question: Does the compatible system of Galois representation of $M(\pi_{f})$ coïncide with the representation in the Theorem 1 p.3 of this paper by Weissauer?

Bonus question: What about the more general case of $GSp_{2n}$ (including $Gl_{2}$!) ????

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The formula you quote from Harris defines a Galois representation, not a motive. We expect that there is a motive whose etale realisation is Harris' space, but that is not immediate.

The problems are: firstly, the Siegel threefold is not proper; secondly, the projector cutting out M(pi_f) might not be an idempotent in the Chow ring.

The first problem was solved by Wildeshaus using his theory of "interior motives". This gives a Grothendieck motive attached to pi, but sadly not a Chow motive, because of the second problem above. It is not known if Wildeshaus' Grothendieck motive can be upgraded to a Chow motive.

(Sorry for lack of nice formatting, typing on my phone.)

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  • $\begingroup$ Hi David, Thank's for your answer. Just about my last question, do you know if Harris's space as something in common with the Galois representation in Weissauer's paper? $\endgroup$
    – Marsault Chabat
    Commented Jul 14, 2022 at 13:01
  • $\begingroup$ For non-endoscopic, non CAP representations they are exactly the same. In the bad cases Harris' space might be a proper subspace of Weissauer's. $\endgroup$
    – David Loeffler
    Commented Jul 14, 2022 at 14:49
  • $\begingroup$ Thanks again David, that's helped a lot! $\endgroup$
    – Marsault Chabat
    Commented Jul 14, 2022 at 14:58

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