# On the consistency of the definition of the conductor for automorphic forms

Let $$\pi$$ be an irreducible admissible representation of $$\mathrm{GL}_2(F)$$, where $$F$$ is local non-archimedean. The local conductor associated to $$\pi$$ can be defined in two usual manners:

By its associated L-function

Godement and Jacquet associate to it the automorphic L-function $$L(s, \pi)$$. This L-function satisfies a functional equation of the form $$L(1-s, \tilde{\pi}) = \varepsilon(s, \pi) L(s, \pi)$$

This $$\varepsilon$$-factor is essentially of the form $$c^{s - \frac{1}{2}}$$, and we define the appearing real number $$c(\pi)=c$$ to be the conductor of $$\pi$$.

By some "depth" property

The other way to define the conductor is to follow the work of Casselman for $$\mathrm{GL}_2$$, and more generaly JPSS for $$\mathrm{GL}_n$$. We consider the decreasing sequence of compact open congruence subgroups, for $$r \geqslant 0$$: $$K_{0, \mathfrak{p}}\left(\mathfrak{p}^r\right) = \left\{ g \in \mathrm{GL}_2\left(\mathcal{O}_{\mathfrak{p}}\right) \ : \ g \equiv \left( \begin{array}{cc} \star & \star \\ 0 & \star \end{array} \right) \mod \mathfrak{p}^r \right\} \subseteq \mathrm{GL}_2(F).$$

The conductor of an irreducible admissible infinite-dimensional representation $$\pi_\mathfrak{p}$$ of $$\tilde{G_\mathfrak{p}}$$ with trivial central character (to ease notations) is then defined by: $$c(\pi) = N\mathfrak{p}^{f(\pi_{\mathfrak{p}})} ,$$

where: $$f\left(\pi_{\mathfrak{p}}\right) = \min \left\{r \geqslant 0 \ : \ \pi_{\mathfrak{p}}^{K_{0, \mathfrak{p}}\left(\mathfrak{p}^r\right)} \neq 0\right\}$$

Are they consistent?

Here is my question: is it obvious that those two definitions are the same?

• Have you looked at Ralf Schmidt's "Some remarks on local newforms for GL(2)"? Jan 18, 2017 at 17:55

These definitions are consistent, though it's not immediate.

The conductor quantifies the extent to which $$\pi$$ is ramified. As an aside, I prefer to write $$c(\pi)$$ for the conductor exponent of $$\pi$$, which is a nonnegative integer, so that $$\mathfrak{p}^{c(\pi)}$$ is the conductor of $$\pi$$, and $$q^{c(\pi)}$$ is the absolute conductor of $$\pi$$, where $$q = N(\mathfrak{p}) = \# \mathcal{O}_F / \mathfrak{p}$$. This isn't standard terminology though, as these are all unfortunately called the same thing by different people.

The conductor exponent is tied to a distinguished vector in $$\pi$$, called the newform of $$\pi$$. (It is also called the Whittaker newform, the essential Whittaker function, the newvector, or the essential vector; the terminology here hasn't reached a consensus, but newform is best in my opinion because for $$\mathrm{GL}_2$$, it is the local component of a classical newform.) There are two definitions of the newform of $$\pi$$, which are equivalent. I'll state them for $$\mathrm{GL}_n$$ instead of $$\mathrm{GL}_2$$. Throughout, $$n \geq 2$$ and $$F$$ is a nonarchimedean local field.

1. Recall that for a spherical (that is, unramified principal series) representation $$\pi'$$ of $$\mathrm{GL}_{n-1}(F)$$, there exists a spherical vector $$W'^{\circ}$$ in the Whittaker model $$\mathcal{W}(\pi',\overline{\psi})$$ of $$\pi'$$, so that $$W'^{\circ}(1_{n-1}) = 1$$ and $$W'^{\circ}(gk) = W(g)$$ for all $$g \in \mathrm{GL}_{n-1}(F)$$ and $$k \in K_{n-1} = \mathrm{GL}_{n-1}(\mathcal{O}_F)$$; that such a vector exists and is unique is classical. Now let $$\pi$$ be a (possibly ramified) generic irreducible admissible representation of $$\mathrm{GL}_n(F)$$. Then there exists a unique Whittaker function $$W^{\circ}$$ in the Whittaker model $$\mathcal{W}(\pi,\psi)$$ of $$\pi$$ satisfying $$W^{\circ} \left(g \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}\right) = W^{\circ}(g), \qquad W^{\circ}(1_n) = 1$$ for every $$g \in \mathrm{GL}_n(F)$$ and $$k \in K_{n-1}$$ such that for every spherical representation $$\pi'$$ of $$\mathrm{GL}_{n-1}(F)$$ with associated spherical vector $$W'^{\circ}$$, the Eulerian integral $$\Psi(s,W^{\circ},W'^{\circ}) = \int\limits_{N_{n-1}(F) \backslash \mathrm{GL}_{n-1}(F)} W^{\circ} \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} W'^{\circ}(g) |\det g|^{s - 1/2} \, dg$$ is equal to the Rankin-Selberg $$L$$-function $$L(s, \pi \times \pi')$$. $$W^{\circ}$$ is called the newform of $$\pi$$. (In general, $$\Psi(s,W,W')/L(s,\pi \times \pi')$$ is a polynomial in $$q^{-s}$$; the point is that there exists a distinguished choice of $$W \in \mathcal{W}(\pi,\psi)$$, $$W' \in \mathcal{W}(\pi',\overline{\psi})$$ for which this polynomial is equal to $$1$$.)
2. For a nonnegative integer $$m$$, let $$K_1(\mathfrak{p}^m)$$ denote the congruence subgroup of $$K_n = \mathrm{GL}_n(\mathcal{O}_F)$$ given by $$\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in K_n : c \in \mathrm{Mat}_{1 \times (n-1)}(\mathfrak{p}^m), \\ d - 1 \in \mathfrak{p}^m \right\}.$$ (Note that necessarily $$a \in K_{n-1}$$ and $$b \in \mathrm{Mat}_{n-1 \times 1}(\mathcal{O}_F)$$.) Let $$\pi$$ be a (possibly ramified) generic irreducible admissible representation of $$\mathrm{GL}_n(F)$$. Then there exists a minimal $$m$$ for which the vector subspace $$\pi^{K_1(\mathfrak{p}^m)} = \left\{v \in \pi : \pi(k) \cdot v = v \quad \text{for all k \in K_1(\mathfrak{p}^m)}\right\}$$ of $$\pi$$ is not equal to $$\{0\}$$. We denote by $$c(\pi)$$ this minimal $$m$$. Then $$\pi^{K_1(\mathfrak{p}^{c(\pi)})}$$ is one dimensional, so that there exists a unique Whittaker function $$W^{\circ}$$ in the Whittaker model $$\mathcal{W}(\pi,\psi)$$ of $$\pi$$ satisfying $$W^{\circ} (gk) = W^{\circ}(g), \qquad W^{\circ}(1_n) = 1$$ for every $$g \in \mathrm{GL}_n(F)$$ and $$k \in K_1(\mathfrak{p}^{c(\pi)})$$.

The original proof of the existence and uniqueness of the newform $$W^{\circ}$$ of $$\pi$$ is via (1), in the paper Conducteur des répresentations du group linéaire by Jacquet, Piatetski-Shapiro, and Shalika. However, it was noticed by Matringe several years ago that the proof was in fact incomplete. He gave a correct proof, as did Jacquet.

In the last section of Jacquet, Piatetski-Shapiro, and Shalika's paper, they show that (1) implies (2) via the functional equation for $$L(s,\pi \times \pi')$$. They make use of the fact that the epsilon factor $$\epsilon(s,\pi \times \pi',\psi)$$ is equal to $$\epsilon(1/2, \pi \times \pi',\psi) q^{-(n - 1) c(\pi) (s - 1/2)}$$ for some nonnegative integer $$c(\pi)$$, and it is precisely this nonnegative integer $$c(\pi)$$ that satisfies $$\pi^{K_1(\mathfrak{p}^{c(\pi)})} \ni W^{\circ}$$ and $$\pi^{K_1(\mathfrak{p}^m)} = \{0\}$$ for all $$m < c(\pi)$$.

I don't know if the fact that (2) implies (1) has appeared anywhere in print, but I do know how to prove it. There is a paper by Miyauchi that shows that the (Whittaker) newform $$W^{\circ}$$ given by (2) is such that $$\Psi(s,W^{\circ}) = \int_{F^{\times}} W^{\circ} \begin{pmatrix} x & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & & \cdots & 1 \end{pmatrix} |x|^{s - \frac{n - 1}{2}} \, d^{\times} x$$ is equal to the $$L$$-function $$L(s,\pi)$$, and the same method of proof (using Hecke operators) shows that $$\Psi(s,W^{\circ},W'^{\circ}) = L(s, \pi \times \pi')$$ for all spherical representations $$\pi'$$ of $$\mathrm{GL}_{n-1}(F)$$. Now you can work backwards to show that $$\epsilon(s,\pi \times \pi',\psi)$$ is equal to $$\epsilon(1/2, \pi \times \pi',\psi) q^{-(n - 1) c(\pi) (s - 1/2)}$$, as in Martin Dickson's answer.

• Hi! I have looked at the references by Jacquet--Piatetski-Shapiro--Shalika and by Matringe, and it looked to me that in both cases they claim to also prove that (2) implies (1). [JPS] Section 5 Théorème (iii) ; [M] Last sentence of the paragraph after the main Theorem of the introduction. Or am I misreading this? Sep 13, 2022 at 14:58
• @Aurel In both cases that you mention, they are showing that (1) implies (2): they have previously proven the existence of a test vector $W^{\circ}$ for the $\mathrm{GL}_n \times \mathrm{GL}_{n - 1}$ Rankin-Selberg when the second representation is spherical, and then they use this to prove that there exists a unique $K_1(\mathfrak{p}^{c(\pi)})$-fixed vector in $\pi$ and that there is no $K_1(\mathfrak{p}^m)$-fixed vector for $m < c(\pi)$. Sep 13, 2022 at 17:41

Yes this is true, but I don't think it's completely obvious. The following argument is taken from Roberts--Schmidt Local newforms for $GSp_4$'', which uses the same argument as the $GL_2$ case.

Jacquet--Langlands proves the local functional equation $$Z(1-s, \pi(w_0) W) = \gamma(s, W) Z(s, W)$$ where $W$ is in the Whittaker model, $Z(s, W)$ is the usual zeta integral, and $w_0 = \begin{bmatrix} & -1 \\ 1 \end{bmatrix}$ is the long Weyl element. For any positive integer $n$, define $w_n = \begin{bmatrix} & -1 \\ \varpi^n \end{bmatrix}$; we can write the above as $$Z(1-s, \pi(w_n) W) = q^{n(s-1/2)} \gamma(s, W) Z(s, W).$$

Now take $\pi$ of conductor $\mathfrak{p}^n$ (in the depth sense), and $W$ a generator for the space of $K(\mathfrak{p}^n)$-fixed vectors. It is known (presumably implicit in J--L) that one can choose $W$ s.t. $Z(s, W) = L(s, \pi)$. Also, one easily sees that $\pi(w_n)$ normalises $K(\mathfrak{p}^n)$ and hence acts on its fixed vectors, and so the LHS is also the corresponding local $L$-function, times the eigenvalue of $w_n$. Now dividing by the local $L$-functions to get the definition of the $\epsilon$-factor, one now sees that the conductor in the $\epsilon$-factor is what it should be.