What is the conductor of an automorphic representation for $\Gamma_0(q)$ in $GSp(4)$?

Question

Let $\pi$ be a generic cuspidal automorphic representation on $GSp(4)$, with level $\Gamma_0(q)$ (the group of symplectic matrices with lower left block divisible by $q$), i.e. $$\Gamma_0(q) = \left\{ \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in Sp_4(\mathbb{Z}) \ : \ C \equiv 0 \mod q \right\}$$

I would like to understand how this notion of level $q$ interacts with the epsilon factor in the functional equations of the attached L-functions, i.e. the classical notion of conductor.

There are two natural representations of the Langlands dual group of $(G)Sp_4$ into a general linear group, viz. the standard $\rho_1 : GSp_4(\mathbb{C}) \to GL_4(\mathbb{C})$ and the spinor $\rho_2 : SO_5(\mathbb{C}) \to GL_5(\mathbb{C})$. These two representations conjecturally induce a compatible map between associated automorphic representations, and allow to define associated L-functions by pulling back the corresponding notions already defined in the $GL_n$ setting. We therefore set $L(s, \pi, \rho_1)$ (resp. $L(s,\pi, \rho_2)$) to be the corresponding L-functions.

These L-functions have been studied precisely in various works (e.g. Piatetskii-Shapiro or Takloo-Bighash), and have been related to local newforms of Roberts and Schmidt, providing explicit formulas in some cases. The global L-function, completed by suitable Archimedean factors, has a functional equation of the form $$\Lambda(1-s, \tilde\pi) = \varepsilon_\pi N_\pi^{s-\tfrac12} \Lambda(s, \pi)$$ for a certain integer $N_\pi$. We call it the arithmetic conductor of $\pi$.

My question is: do we know $N_\pi$ in terms of the level $q$? (I emphasize level since $N_\pi$ is in fact known, by Roberts-Schmidt local newform theory, in terms of the paramodular level; but here I am using a kind of different level, mimicking $GL_n$ congruence subgroups).

Answer 1

To make the question well-posed, I'm going to suppose that we fix $\pi$ and take $q$ to be the smallest integer such that $\pi$ has nonzero invariants under $\Gamma_0(q)$. Then $q$ gives you some information about $\pi$, but I don't think you can reliably expect it to nail down the arithmetic conductor exactly; it is rather a miracle that the paramodular invariants do encode the conductor in all cases, you can't just expect a random family of subgroups always to do this.

The question is, of course, a local one: for $\pi$ a representation of $\operatorname{GSp}_4(\mathbb{Q}_\ell)$ having invariants under the Siegel congruence subgroup $Si(\ell^r) \subseteq \operatorname{GSp}_4(\mathbb{Z}_\ell)$, but not under $Si(\ell^{r-1})$, what can the conductor of $\pi$ be?

For $r = 1$ this can be completely answered using the results of Schmidt's paper "Iwahori-spherical representations" (see the tables in the Roberts–Schmidt book for a handy summary). In particular, if we suppose $\pi$ is generic, then $\pi$ has invariants under $Si(\ell)$, but not $Si(1) = \operatorname{GSp}_4(\mathbb{Z}_\ell)$, if and only if $\pi$ is a representation of type IIa, IIIa or VIa with unramified inducing data; and its conductor is $\ell$ if the type is IIa, but $\ell^2$ if the type is IIIa or VIa. This shows that the minimal $r$ at which $\pi$ has nontrivial $Si(\ell^r)$-invariants is not enough to encode the conductor.

(PS. Your question seems to implicitly suggest that the "arithmetic conductors" $N_\pi$ appearing in the functional equation of the degree 4 $L$-factor and the degree 5 $L$-factor are the same. Are you sure about this? I am not sure it is true.)