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I am looking for a dictionary that relates the level of a classical genus 2 Siegel modular form and the local components of the corresponding automorphic representation of $Gsp_4(\mathbb{A}_{\mathbb{Q}})$ (Relative to the Sally-Tadic classification of the non-supercuspidals). The notes of J. Booher compile the relevant information for classical elliptic modular forms (http://stanford.edu/~jbooher/expos/adelic_mod_forms.pdf) and I would love to have a similar breakdown for Siegel if such a treatise exists.

For instance: If I start with a Siegel modular form $F$ of level $N$, and $p \vert N$, what representations (of type I-XI in S-T) should I expect to see and when?

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    $\begingroup$ Relevant: www2.math.ou.edu/~rschmidt/papers/rims2.pdf $\endgroup$ – user31415 Sep 2 '16 at 0:13
  • $\begingroup$ Incidentally, there's a trivial but dangerously misleading typo in the concluding Theorem 5.6 of Booher's notes: in the third bullet point, "if the conductor of $\pi_{f, p}$ is $p^r$" should say "if the conductor of $\omega_{p}$ is $p^r$". $\endgroup$ – David Loeffler Sep 2 '16 at 12:39
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Look at the tables of Ralf Schmidt and Brooks Roberts:

Tables for Representations of GSp(4)

Section 2 also tells you which representations you should find in Saito-Kurokawa versus non-Saito-Kurokawa spaces.

Section 5 tells you about what representations can occur for prime level for various congruence subgroups. They also do higher level for paramodular groups (see also their book). I saw a similar table in a talk by Ralf 2 days ago for higher level Siegel congruence subgroups--I don't think it's public yet, but I expect it will appear in a future paper.

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