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Here's a well-known lemma about modular curves:

Let $\pi_1, \pi_2$ be the two degeneracy maps $Y_1(Np) \to Y_1(N)$, for $p \nmid N$, corresponding to $z \mapsto z$ and $z \mapsto pz$. Then as correspondences $Y_1(Np) \rightrightarrows Y_1(N)$, we have $$ \pi_1 \circ U_p = T_p \circ \pi_1 - \langle p \rangle \pi_2,$$ $$ \pi_2 \circ U_p = p \pi_1. $$

I like this lemma a lot, because it implies that if $f$ is a weight 2 newform of level $N$, then the characteristic polynomial of $U_p$ on the space of old forms at level $Np$ spanned by $\pi_1^*(f)$ and $\pi_2^*(f)$ is exactly the local $L$-factor of $f$ at $p$.

What's the analogue of this for genus 2 Siegel modular forms?

(It's shown in Richard Taylor's thesis that for a Siegel eigenform $f$ of level $\Gamma$ prime to $p$, the space of oldforms at level $\Gamma \cap \Gamma_0(p)$ -- i.e. things in $\Gamma$ whose lower left $2\times 2$ submatrix is divisible by $p$ -- corresponding to $f$ is 4-dimensional, and the characteristic polynomial of the Hecke operator $U_p$ on this space is the degree 4 spin L-factor of $f$ at $p$. But Taylor's proof goes via local representation theory and doesn't seem to give an explicit basis of this space of oldforms in terms pullbacks via degeneracy maps.

Is there any more explicit, "classical" proof of Taylor's result that can be understood in terms of commutation relations between Hecke operators and degeneracy maps, as above?)

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