A constraint for twist equivalence of cusp forms on GL$(n)$

TL;DR

This paper establishes a divisibility constraint on characters relating two cuspidal automorphic representations of GL(n), generalizing a question by Kaisa Matomäki and providing bounds involving conductors over number fields.

Contribution

It generalizes a question of Kaisa Matomäki by proving a divisibility condition on conductors for characters relating cuspidal automorphic representations of GL(n).

Findings

For two cuspidal automorphic representations, the conductor of the character satisfies Q^n divides N_1 N_2.

If local components are discrete series at ramified places, then Q^n divides the least common multiple of N_1 and N_2.

The result extends previous bounds and applies over general number fields.

Abstract

This Note answers, and generalizes, a question of Kaisa Matom\"aki. We show that give two cuspidal automorphic representations $\pi_1$ and $\pi_2$ of $GL_n$ over a number field $F$ of respective conductors $N_1,$ $N_2,$ every character $\chi$ such that $\pi_1\otimes\chi\simeq\pi_2$ of conductor $Q,$ satisfies the bound: $Q^n\mid N_1N_2.$ If at every finite place $v,$ $\pi_{1,v}$ is a discrete series whenever it is ramified, then $Q^n$ divides the least common multiple $[N_1, N_2].$

Equations

$$Q^n\mid N_1{N}_2.$$

$$\displaystyle U_{k}(D_{v})=\big{\{}x\in D^{\times}:\ \operatorname{ord}_{F_{v}}(\operatorname{Nrd}(x-1))\geq k\big{\}},\quad\forall\ k\geq 1;$$

$$\displaystyle l(\pi_{v}^{\prime})=\min_{k\leq 0}\big{\{}k:\ \pi^{\prime}_{v}\mid{U_{k}(D_{v})}=\operatorname{Id}\big{\}}.$$

$$A({\pi }_v^{\prime })=l({\pi }_v^{\prime })+d_v-1.$$

$$\mathrm{Nrd}(U_k(D_v))=U_k(D_v)\cap F_v^{\times }=U_{\lceil k/d_v\rceil }(F_v),$$

$$\displaystyle\operatorname{A}(\chi_{v})\leq\bigg{\lceil}\frac{l(\chi_{v}\circ\operatorname{Nrd})}{d_{v}}\bigg{\rceil},$$

$$l({\chi }_v\circ \mathrm{Nrd})=d_v\cdot (A({\chi }_v)-1)+1.$$

$$A({\pi }^{\prime }\otimes {\chi }_v)=l({\pi }^{\prime }\otimes {\chi }_v)+d_v-1.$$

$$\mathrm{JL}:\mathrm{Irr}(GL(d,F_v))⟶\mathrm{Irr}(D_v^{\times })$$

$$n\cdot\operatorname{A}(\chi_{v})\leq\max\big{\{}\operatorname{A}(\pi_{v}\otimes\chi_{v}),\operatorname{A}(\pi_{v})\big{\}},$$

$$l({\chi }_v\circ \mathrm{Nrd})\ge l({\pi }_v^{\prime }).$$

$$\displaystyle m_{j}n_{j}\cdot\operatorname{A}(\chi_{v})\leq\max\big{\{}\operatorname{A}(Q(\Delta_{j})\otimes\chi_{v}),\operatorname{A}(Q(\Delta_{j}))\big{\}},$$

$$\displaystyle\sum_{j=1}^{r}m_{j}n_{j}\cdot\operatorname{A}(\chi_{v})\leq\sum_{j=1}^{r}\max\big{\{}\operatorname{A}(Q(\Delta_{j}^{\prime})),\operatorname{A}(Q(\Delta_{j}))\big{\}}.$$

$$n\operatorname{A}(\chi_{v})\leq\sum_{j=1}^{r}\max\big{\{}\operatorname{A}(Q(\Delta_{j}^{\prime})),\operatorname{A}(Q(\Delta_{j}))\big{\}},$$

$$2A({\chi }_v)\le A({\chi }_v{\chi }_{1,v})+A({\chi }_v{\chi }_{2,v})=A({\pi }_{2,v})\le \max \{A({\pi }_{1,v}),A({\pi }_{2,v})\}.$$

$${\sigma }_v={\oplus }_{j=1}^{r_v}{\sigma }_{v,j},$$

$$L_v(s,{\sigma }_v\otimes {\chi }_v)=\prod _{j=1}^{r_v}{L}_v(s,{\sigma }_{v,j}\otimes {\chi }_v).$$

$$L_v(s,{\sigma }_{v,j}\otimes {\chi }_v)=L_v(s,{\tau }_{k_j+k^{\prime },{\nu }_j+{\nu }^{\prime }})={\Gamma }_{\mathbb{C}}(s+{\nu }_j+{\nu }^{\prime }+|k_j+k^{\prime }|/2).$$

$$C_v({\sigma }_{v,j}\otimes {\chi }_v)=(1+|{\nu }_j+{\nu }^{\prime }+|k_j+k^{\prime }|/2|{)}^2.$$

$$L_v(s,{\sigma }_{v,j}\otimes {\chi }_v)={\Gamma }_{\mathbb{R}}(s+{\nu }_j+{\nu }^{\prime }+(1-{\sigma }_{v,j}(\mathbf{j}){\chi }_v(\mathbf{j}))/2),$$

$$C_v({\sigma }_{v,j}\otimes {\chi }_v)=1+|{\nu }_j+{\nu }^{\prime }+(1-{\sigma }_{v,j}(\mathbf{j}){\chi }_v(\mathbf{j}))/2|.$$

$$L_v(s,{\sigma }_{v,j}\otimes {\chi }_v)={\Gamma }_{\mathbb{C}}(s+{\nu }_j+{\nu }^{\prime }+k_j/2).$$

$$C_v({\sigma }_{v,j}\otimes {\chi }_v)=(1+|{\nu }_j+{\nu }^{\prime }+k_j/2|{)}^2.$$

$$C_v({\pi }_v\otimes {\chi }_v)=C_v({\sigma }_v\otimes {\chi }_v)=\prod _{j=1}^{r_v}{C}_v({\sigma }_{v,j}\otimes {\chi }_v),$$

$$C_{\infty }(\pi \times \chi )=\prod _{v\in {\Sigma }_{\infty }}{C}_v({\pi }_v\otimes {\chi }_v)$$

$$C(\pi \times \chi )=N(\pi \times \chi )\cdot C_{\infty }(\pi \times \chi )$$

$${\sigma }_{i,v}={\oplus }_{j=1}^{r_v}{\sigma }_{i,v;j},1\le i\le 2,$$

$${\sigma }_{1,v;j}\otimes {\chi }_v={\sigma }_{2,v;j},1\le j\le d_v,v\in {\Sigma }_{\infty }.$$

$$C_v({\chi }_v)=(1+|{\nu }_{2,j}-{\nu }_{1,j}+|k_{2,j}-k_{1,j}|/2|{)}^2.$$

$$C_v({\chi }_v)=1+|{\nu }_{2,j}-{\nu }_{1,j}+(1-{\sigma }_{2,v;j}{\sigma }_{1,v;j}^{-1}(\mathbf{j}))/2|.$$

Full text

[TABLE]

Suppose further that for every finite place $v,$ when $\pi_{1,v}$ or $\pi_{2,v}$ is ramified either it is in the discrete series or $n=2$ with trivial central character. Then $Q^{n}\mid[N_{1},N_{2}],$ where $[N_{1},N_{2}]$ is the least common multiple of $N_{1}$ and $N_{2}.$

In fact, the proof will give a corresponding local statement at each place, and the case of ramified principal series implies the optimality of the bound.

Also, for $GL(2),$ it was shown by the first author ( [Ram00]), using multiplicity one for $SL(2),$ that if $\operatorname{Ad}(\pi)\simeq\operatorname{Ad}(\pi^{\prime}),$ where $\operatorname{Ad}$ is the Gelbart-Jacquet adjoint lifting from GL$(2)$ to GL$(3)$, then $\pi^{\prime}\simeq\pi\otimes\chi$ for some character $\chi.$ So Theorem A elucidates the fibre of $\pi\rightarrow\operatorname{Ad}(\pi).$

Our proof of Theorem A reduces it, by the factorizability of the conductors, to a local statement, and further by the Bernstein-Zelevinsky classification, to a statement about discrete series representations of GL$(n,F_{v})$, and then makes use of the local Jacquet-Langlands correspondence to a question about twists of (finite-dimensional) irreducible representations of its inner form, the (multiplicative group of the) division algebra $D$ of dimension $n^{2}$ over $F_{v}$ of invariant $1/n$, where we appeal to some known results of Koch and Zink. (One can also work in the framework of Moy and Prasad.)

In our result, $\chi$ need not have finite order. Thus it is natural to ask for such a bound for the full analytic conductor, which includes an archimedean analysis. This is done in the last section, see Theorem B therein.

D. Prasad has referred us to the preprint ([Cor19]), which also treats character twists, but goes in another direction.

We thank Kaisa Matomäki, Dipendra Prasad and Maksym Radziwill for their interest.

1. Representations of Central Division Algebras

Let $v$ be a non-archimedean place of $F.$ Let $\mathfrak{o}_{F_{v}}$ be the ring of integers of $F_{v}.$ Let $\varpi_{v}$ be a fixed uniformizer. Let $\operatorname{ord}_{F_{v}}$ be the valuation normalized such that $\operatorname{ord}_{F_{v}}(\varpi)=1.$ Denote by $U_{k}(F_{v})=1+\varpi^{k}\mathfrak{o}_{F_{v}}$ if $k\geq 1.$ Set $U_{0}(F_{v})=\mathfrak{o}_{F_{v}}^{\times}.$

Let $\chi_{v}$ be a quasi-character of $F_{v}^{\times}.$ Denote by $\operatorname{A}(\chi_{v})$ the conductor exponent, i.e., the least integer $k\geq 0$ such that $\chi_{v}\mid_{U_{k}(F_{v})}=\operatorname{Id}.$

Let $D_{v}$ be a division algebra over $F_{v}$ of dimension $[D_{v}:F_{v}]=d_{v}^{2}.$ Denote by $\operatorname{Nrd}$ the reduced norm on $D_{v}.$ Let $\operatorname{ord}_{F_{v}}$ be the normalized valuation on $F_{v}.$ Denote by

[TABLE]

and set $U_{0}(D_{v})=\ker(\operatorname{ord}_{F_{v}}\circ\operatorname{Nrd}).$ Let $\pi_{v}^{\prime}$ be an irreducible admissible representation of $D_{v}^{\times}.$ Define the level $l(\pi_{v}^{\prime})$ of $\pi_{v}^{\prime}$ as following:

[TABLE]

Let $\chi_{v}$ be a quasi-character of $F_{v}^{\times}.$ Set $\pi_{v}^{\prime}\otimes\chi_{v}=\pi_{v}^{\prime}\otimes(\chi_{v}\circ\operatorname{Nrd}).$ Then it is clear from the definition that $l(\pi_{v}^{\prime}\otimes\chi_{v})=l(\chi_{v}\circ\operatorname{Nrd}),$ assuming $l(\chi_{v}\circ\operatorname{Nrd})\geq l(\pi_{v}^{\prime}).$

1.1. Local Level-Conductor Formula

Let $\pi^{\prime}_{v}$ be an irreducible admissible representation of $D_{v}^{\times}.$ One can define (e.g. [KZ80]) the conductor exponent $\operatorname{A}(\pi_{v}^{\prime})$ of $\pi_{v}^{\prime}$ via the $\epsilon$-factor. Then one has the following level-conductor formula ( (4.3.4) in Sec. 4.3 of [KZ80]), which is well known for $n=2.$

Lemma 2**.**

Let $\pi^{\prime}_{v}$ be an irreducible admissible representation of $D_{v}^{\times}.$ Then

[TABLE]

Remark*.*

One can state (2) equivalently in terms of the Moy-Prasad depth ( [MP94] and [MP96]).

Lemma 3**.**

Let $\pi^{\prime}_{v}$ be an irreducible admissible representation of $D_{v}^{\times}.$ Let $\chi_{v}$ be a quasi-character of $F_{v}^{\times}$ such that $l(\chi_{v}\circ\operatorname{Nrd})\geq l(\pi_{v}^{\prime}).$ Then $\operatorname{A}(\pi_{v}^{\prime}\otimes\chi_{v})=d_{v}\cdot\operatorname{A}(\chi_{v}).$

Proof.

Let $x\in F_{v}^{\times}.$ Then $\operatorname{ord}_{F_{v}}(\operatorname{Nrd}(x))=\operatorname{ord}_{F_{v}}(x^{d_{v}})=d_{v}\cdot\operatorname{ord}_{F_{v}}(x).$ Take $k=l(\chi_{v}\circ\operatorname{Nrd}).$ Then we have

[TABLE]

where $\lceil\cdot\rceil$ is the ceiling function. Note that $\chi_{v}$ is trivial on $U_{\lceil k/d_{v}\rceil}(F_{v})$ if and only if $\chi_{v}\circ\operatorname{Nrd}$ is trivial on $U_{k}(D_{v}).$ Then by definition, we have

[TABLE]

implying that $d_{v}\cdot(\operatorname{A}(\chi_{v})-1)\leq l(\chi_{v}\circ\operatorname{Nrd})-1.$ On the other hand, by definition, $l(\chi_{v}\circ\operatorname{Nrd})$ is the minimal integer such that $\chi_{v}$ is trivial on $U_{\lceil k/d_{v}\rceil}(F_{v}).$ Hence

[TABLE]

Since $l(\chi_{v}\circ\operatorname{Nrd})\geq l(\pi_{v}^{\prime}),$ one has $l(\pi_{v}^{\prime}\otimes\chi_{v})=l(\chi_{v}\circ\operatorname{Nrd}).$ Note that $\pi_{v}^{\prime}\otimes\chi_{v}$ is also irreducible. Applying (2) to $\pi_{v}^{\prime}\otimes\chi_{v}$ we thus deduce

[TABLE]

Then Lemma 3 follows from (3) and (4). ∎

1.2. The Local Jacquet-Langlands Correspondence

We recall briefly the Jacquet-Langlands correspondence here. Let $\operatorname{Irr}(GL(d,F_{v}))$ (resp. $\operatorname{Irr}(D_{v}^{\times})$) be the set of equivalence classes of irreducible essentially square-integrable representations of $GL(d,F_{v})$ (resp. $D_{v}^{\times}$). Then there exists a canonical bijection

[TABLE]

satisfying some properties (e.g. [ABPS16], Thm 2.2). Let $\pi_{v}$ be an irreducible essentially square-integrable representation of $GL(d,F_{v}).$ Let $\operatorname{A}(\pi_{v})$ be the conductor exponent of $\pi_{v}.$ Denote by $\pi_{v}^{\prime}=\operatorname{JL}(\pi_{v}).$ Then one also has $\operatorname{A}(\pi_{v}^{\prime}\otimes\chi_{v})=\operatorname{A}(\pi_{v}\otimes\chi_{v}),$ for any quasi-character $\chi_{v}$ of $F_{v}^{\times}.$

Proposition 4**.**

Let notation be as above. Assume $\pi_{v}$ is irreducible admissible and essentially square-integrable of $\operatorname{GL}(n,F_{v})$. Then either

[TABLE]

where the equality holds if $\operatorname{A}(\pi_{v}\otimes\chi_{v})\neq\operatorname{A}(\pi_{v}),$ or $\operatorname{A}(\pi_{v})=n\cdot\operatorname{A}(\chi_{v}).$

Proof.

Suppose $n\cdot\operatorname{A}(\chi_{v})>\max\{\operatorname{A}(\pi_{v}\otimes\chi_{v}),\operatorname{A}(\pi_{v})\}.$ Let $\pi_{v}^{\prime}=\operatorname{JL}(\pi_{v}).$ Then $\operatorname{A}(\pi_{v})=\operatorname{A}(\pi_{v}^{\prime}).$ Hence it follows from formulas (3), (4) and the assumption $n\cdot\operatorname{A}(\chi_{v})>\operatorname{A}(\pi_{v})$ that $l(\chi_{v}\circ\operatorname{Nrd})>l(\pi_{v}^{\prime}).$ We then deduce from Lemma 3 that $\operatorname{A}(\pi_{v}^{\prime}\otimes\chi_{v})=n\cdot\operatorname{A}(\chi_{v}).$ Therefore, applying the Jacquet-Langlands correspondence we get $\operatorname{A}(\pi_{v}\otimes\chi_{v})=n\cdot\operatorname{A}(\chi_{v}),$ a contradiction! Thus (5) holds.

Now we suppose $\operatorname{A}(\pi_{v}\otimes\chi_{v})\neq\operatorname{A}(\pi_{v}).$ Since $\operatorname{A}(\chi_{v})=\operatorname{A}(\chi_{v}^{-1}),$ after twisting by $\chi_{v}^{-1}$ if needed, we may assume that $\operatorname{A}(\pi_{v}\otimes\chi_{v})>\operatorname{A}(\pi_{v}),$ which amounts to $\operatorname{A}(\pi_{v}^{\prime}\otimes\chi_{v})>\operatorname{A}(\pi_{v}^{\prime}).$ Then by (4) one has $l(\pi_{v}^{\prime}\otimes\chi_{v})>l(\pi_{v}^{\prime}).$ Hence

[TABLE]

Suppose further that equality in (5) does not hold. Then $n\cdot\operatorname{A}(\chi_{v})<\operatorname{A}(\pi_{v}\otimes\chi_{v})=\operatorname{A}(\pi_{v}^{\prime}\otimes\chi_{v}).$ Then by Lemma 3 we deduce that $l(\chi_{v}\circ\operatorname{Nrd})<l(\pi_{v}^{\prime}),$ contradicting (6). In all, if $\operatorname{A}(\pi_{v}\otimes\chi_{v})\neq\operatorname{A}(\pi_{v}),$ then n\cdot\operatorname{A}(\chi_{v})=\max\big{\{}\operatorname{A}(\pi_{v}\otimes\chi_{v}),\operatorname{A}(\pi_{v})\big{\}}.

Assume $\operatorname{A}(\pi_{v})=n\cdot\operatorname{A}(\chi_{v}).$ Suppose at the same time that $\operatorname{A}(\pi_{v}\otimes\chi_{v})>\operatorname{A}(\pi_{v})=n\cdot\operatorname{A}(\chi_{v}).$ Then we have (6), which implies, by Lemma 3, that $\operatorname{A}(\pi_{v}\otimes\chi_{v})=n\cdot\operatorname{A}(\chi_{v}),$ contradicting our assumption. Hence $\operatorname{A}(\pi_{v}\otimes\chi_{v})\leq\operatorname{A}(\pi_{v}).$ Therefore, the equality of (5) holds when $\operatorname{A}(\pi_{v})=n\cdot\operatorname{A}(\chi_{v}).$ ∎

Remark*.*

In this Note, we haven’t used the local Langlands correspondence. Proposition 4 can be rephrased in terms of Artin conductors of Galois representations. If the Galois representation is irreducible (i.e. $\pi_{v}$ is supercuspidal), it has been pointed to us that Proposition 4 admits a short proof purely in the theory of the Weil-Deligne group by using the following argument: the Artin conductor of V is the integral over $s$ from $-1$ to infinity of $\operatorname{codim}(V^{G^{s}})$, where $G^{s}$ is the $s$-th higher ramification group in the upper numbering filtration of the Galois group. So if V is irreducible, it is simply $\dim V$ times the infimum of the set of s where $G_{s}$ acts trivially on V, plus 1. Let $\sigma_{v}$ be the Weil-Deligne representation associated to $\pi_{v}$ via local Langlands correspondence. The stated inequality then is equivalent to the statement that if $G_{s}$ acts nontrivially on $\chi$, then it acts nontrivially on $\sigma_{v}$ or ($\chi\otimes\sigma_{v}$), which is clear.

2. Proof of Theorem A

Let us note, before commencing the proof of Theorem A, that it is not hard to check that $Q\leq(N_{1}N_{2})^{n}.$ By the factorizability of conductors, it suffices to see this at each finite place $v.$ If $\sigma_{i,v},$ $i=1,2,$ is the $n$-dimensional representation of the Weil-Deligne group at $F_{v},$ associated to $\pi_{i,v},$ by the local Langlands correspondence then $\sigma_{2,v}\simeq\sigma_{1,v}\otimes\chi_{v},$ which implies that $\chi_{v}\hookrightarrow\check{\sigma_{1,v}}\otimes\sigma_{2,v}$; thus $Q_{v}\leq(N_{1,v}N_{2,v})^{n},$ since the conductor of $\check{\sigma_{1,v}}\otimes\sigma_{2,v}$ is bounded above by $N_{1,v}^{n}N_{2,v}^{n}.$ However, we do not appeal to anything from $p$-adic local Langlands correspondence.

For a supercuspidal representation $\sigma$ of $\operatorname{GL}(k,F_{v}),$ we denote by $\sigma(l)$ the twist of $\sigma$ with the character $|\cdot|^{l},$ i.e., the representation $g\mapsto|\det(g)|^{l}\sigma(g).$ For $m\geq 0,$ set $\Delta(\sigma,m)=[\sigma_{j},\sigma_{j}(1),\cdots,\sigma_{j}(m_{j}-1)].$ Let $\sigma^{\prime}$ be another supercuspidal representation of $\operatorname{GL}(k,F_{v}),$ and $m^{\prime}\geq 0.$ We say $\Delta(\sigma,m)$ precedes $\Delta(\sigma^{\prime},m^{\prime})$ if $\sigma^{\prime}\cong\sigma(m).$

Let $\pi(\sigma,m)$ be the irreducible representation of $\operatorname{GL}(km,F_{v})$ induced from $\sigma\times\sigma(1)\times\cdots\times\sigma(m-1).$ Denote by $Q(\Delta(\sigma,m))$ the unique irreducible quotient of $\pi(\sigma,m).$ Then by the Bernstein-Zelevinsky classification (see [Zel80]), any smooth irreducible representation of $\operatorname{GL}(n,F_{v})$ is isomorphic to the unique irreducible quotient $Q(\Delta(\sigma_{1},m_{1}),\cdots,\Delta(\sigma_{r},m_{r}))$ of $Q(\Delta(\sigma_{1},m_{1}))\times\cdots\times Q(\Delta(\sigma_{r},m_{r})),$ for some supercuspidal representations $\sigma_{j},$ such that $\Delta(\sigma_{j_{1}},m_{j_{1}})$ does not precede $\Delta(\sigma_{j_{2}},m_{j_{2}})$ for $j_{1}<j_{2}.$ We now use this classification to prove our main theorem.

Proof of Theorem A.

Write $\pi_{i}=\otimes_{v}^{\prime}\pi_{i,v},$ $1\leq i\leq 2.$ Let $v$ be a nonarchimedean place. Then $\pi_{i,v}$ is a smooth irreducible representation of $\operatorname{GL}(n,F_{v}).$ Hence, by Bernstein-Zelevinsky classification, $\pi_{1,v}$ is isomorphic to a representation of the form $Q(\Delta_{1},\cdots,\Delta_{r})$ for a unique (up to permutation) collection of intervals $\Delta_{j}=\Delta(\sigma_{j},m_{j}),$ with $\sigma_{j}$ a supercuspidal representation of some $\operatorname{GL}(n_{j},F_{v}),$ $1\leq j\leq r,$ $\sum_{j}m_{j}n_{j}=n;$ such that $\Delta_{i}$ does not precede $\Delta_{j}$ for $i<j.$ Moreover, each $Q(\Delta_{j})$ is essentially square-integrable.

Then it follows from $\pi_{2,v}\simeq\pi_{1,v}\otimes\chi_{v}$ and uniqueness of the quotient, that $\pi_{2,v}$ is isomorphic to a representation of the form $Q(\Delta_{1}^{\prime},\cdots,\Delta_{r}^{\prime}),$ where $\Delta_{j}^{\prime}=\Delta(\sigma_{j}\otimes\chi_{v},m_{j}),$ $1\leq j\leq r.$ Then by Proposition 4 we have, for each $1\leq i\leq r,$ that

[TABLE]

namely, m_{j}n_{i}\cdot\operatorname{A}(\chi_{v})\leq\max\big{\{}\operatorname{A}(Q(\Delta_{j}^{\prime})),\operatorname{A}(Q(\Delta_{j}))\big{\}}. Summing through all $1\leq j\leq r$ one then obtain

[TABLE]

Note that $\sum_{j=1}^{r}m_{j}n_{j}=n.$ Hence,

[TABLE]

from which we deduce that $Q_{v}^{n}$ divides $N_{1,v}N_{2,v}.$ Thus (1) follows.

Suppose further for every finite place $v,$ $\pi_{1,v}$ is a discrete series whenever $\pi_{1,v}$ is ramified. Let $v$ be a nonarchimedean place as above. Then $r=1.$ So we have by (7) that n\operatorname{A}(\chi_{v})\leq\max\big{\{}\operatorname{A}(Q(\Delta_{1}^{\prime})),\operatorname{A}(Q(\Delta_{1}))\big{\}}=\max\{\operatorname{A}(\pi_{1,v}),\operatorname{A}(\pi_{2,v})\}, from which we deduce that $Q_{v}^{n}$ divides $[N_{1,v},N_{2,v}].$ Now Theorem A follows from the above analysis and the following claim:

Claim 5**.**

Let notation be as before. Assume $n=2$ and $\pi_{1}$ has trivial central character. Then $Q^{2}\mid[N_{1},N_{2}].$

∎

Proof of Claim 5.

Let notation be as in the proceeding proof. From the above proof we know that $2\operatorname{A}(\chi_{v})\leq\max\{\operatorname{A}(\pi_{1,v}),\operatorname{A}(\pi_{2,v})\}$ if $r=1.$ Hence, it suffices to consider the case when $r=2,$ in which case $\pi_{i,v}$ is a principal series, $1\leq i\leq 2.$ We may write $\pi_{1,v}=\chi_{1,v}\boxplus\chi_{2,v}$ and thus $\pi_{2,v}\simeq\chi_{v}\chi_{1,v}\boxplus\chi_{v}\chi_{2,v}.$

On the other hand, since $\pi_{1}$ has trivial central character, $\operatorname{A}(\chi_{1,v})=\operatorname{A}(\chi_{2,v}).$ Suppose $\operatorname{A}(\chi_{v})\leq\operatorname{A}(\chi_{1,v}).$ Then $2\operatorname{A}(\chi_{v})\leq\operatorname{A}(\chi_{1,v})+\operatorname{A}(\chi_{2,v})=\operatorname{A}(\pi_{1,v}),$ which is no more than $\max\{\operatorname{A}(\pi_{1,v}),\operatorname{A}(\pi_{2,v})\}.$ Therefore, we may suppose $\operatorname{A}(\chi_{v})>\operatorname{A}(\chi_{1,v}).$ By Proposition 4 we deduce that $\operatorname{A}(\chi_{v})\leq\max\{\operatorname{A}(\chi_{j,v}),\operatorname{A}(\chi_{v}\chi_{j,v})\},$ for $1\leq j\leq 2;$ moreover, since $\operatorname{A}(\chi_{1,v})=\operatorname{A}(\chi_{2,v})<\operatorname{A}(\chi_{v})\leq\max\{\operatorname{A}(\chi_{j,v}),\operatorname{A}(\chi_{v}\chi_{j,v})\},$ $j=1,2,$ then from the condition when the equality holds therein, we obtain that $\operatorname{A}(\chi_{v})=\operatorname{A}(\chi_{v}\chi_{j,v}),$ for $j=1,2.$ Therefore, we deduce that

[TABLE]

Hence Claim 5 follows. ∎

This completes the proof of Theorem A. Moreover, towards the extreme cases of Theorem A, we have the following:

Let notation be as before. Let $\pi_{1}=\operatorname{Ind}(\chi_{1},\chi_{2})$ be an induced representation of $GL(2,\mathbb{A}_{\mathbb{Q}}).$ Let $p_{1},p_{2}$ be two distinct primes. Suppose $\chi_{i}$ has arithmetic conductor $p_{i},$ $1\leq i\leq 2.$ Let $\chi=\chi_{1}^{-1}\chi_{2}^{-1},$ and $\pi_{2}=\pi_{1}\otimes\chi.$ Then both $\pi_{1}$ and $\pi_{2}$ have arithmetic conductor $p_{1}p_{2},$ namely, $N_{1}=N_{2}=p_{1}p_{2}.$ Also, $\chi$ has arithmetic conductor $Q=p_{1}p_{2}.$ And we thus have $Q^{2}=N_{1}N_{2}.$

• 2.

Let $\pi_{1}$ be a cuspidal representation on $GL(n,\mathbb{A}_{F})$ such that for every finite place $v,$ $\pi_{1,v}$ is a discrete series whenever $\pi_{1,v}$ is ramified. Suppose further that $n\mid\operatorname{A}(\pi_{1,v}),$ for each finite place $v.$ Let $\chi$ be an idele class character of Artin conductor exponent $\operatorname{A}(\chi_{v})=\operatorname{A}(\pi_{1,v})/n,$ for any finite place $v.$ Let $\pi_{2}=\pi_{1}\otimes\chi.$ Then it follows from Proposition 4 that $n\operatorname{A}(\chi_{v})=\max\{\operatorname{A}(\pi_{1,v}),\operatorname{A}(\pi_{2,v})\},$ for all finite place $v.$ Thus $Q^{n}=[N_{1},N_{2}]$ in this case.

3. Comparison of Analytic Conductors

Let $\pi$ be unitary cuspidal representations on $GL(n,\mathbb{A}_{F}),$ where $F$ is a global field. Denote by $\Sigma$ the set of places of $F.$ Then $\pi=\otimes^{\prime}_{v}\pi_{v}.$ Let $L(s,\pi)$ be the principal $L$-function associated to $\pi.$ Then $L(s,\pi)=\prod_{v\in\Sigma}L_{v}(s,\pi_{v}),$ where $\operatorname{Re}(s)\gg 0.$ Let $\Sigma_{\infty}$ be the set of archimedean places of $F.$ To define analytic conductor of $\pi,$ we need to recall the definition of each $L_{v}(s,\pi_{v})$ for $v\in\Sigma_{\infty}.$

In this section,we fix a place $v\in\Sigma_{\infty},$ denote by $\sigma_{v}$ the $n$-dimensional Weil-Deligne representation corresponding to $\pi_{v}$ under the local Langlands correspondence. Then $L_{v}(s,\pi_{v})=L_{v}(s,\sigma_{v}).$ Moreover, let $\chi_{v}$ be a quasi-character of $F_{v}^{\times},$ we have $L_{v}(s,\pi_{v}\times\chi_{v})=L_{v}(s,\sigma_{v}\otimes\chi_{v}).$ Hence, it suffices to recall the definition of $L_{v}(s,\sigma_{v}\otimes\chi_{v}),$ which is a product of Gamma functions. Write

[TABLE]

where each $\sigma_{v,j}$ is an irreducible representation of the Weil group $\mathcal{W}_{F,v}.$ Hence,

[TABLE]

To define archimedean conductor, we shall describe each $L_{v}(s,\sigma_{v,j}\otimes\chi_{v})$ explicitly. Since our approach is using Langlands classification ( [Del73]), we will separate the cases when $F_{v}\simeq\mathbb{R}$ and $F_{v}\simeq\mathbb{C}.$

• Case 1:

Assume that $F_{v}\simeq\mathbb{C}.$ One has $\mathcal{W}_{F,v}\simeq\mathbb{C}^{\times}.$ So all irreducible representations are one dimensional. We may write any such characters as $\tau_{k,\nu}(z)=(z/|z|)^{k}|z|_{\mathbb{C}}^{\nu}=(z/|z|)^{k}|z|^{2\nu},$ for $k\in\mathbb{Z}$ and $\nu\in\mathbb{C}.$ The local $L$-function associated to this character is $L_{v}(s,\tau_{k,\nu})=\Gamma_{\mathbb{C}}(s+\nu+|k|/2),$ where $\Gamma_{\mathbb{C}}(s):=2(2\pi)^{-s}\Gamma(s).$

Let $\sigma_{v,j}=\tau_{k_{j},\nu_{j}},$ $k_{j}\in\mathbb{Z}$ and $\nu_{j}\in\mathbb{C},$ $1\leq j\leq r.$ Also, one can write $\chi_{v}$ as $\tau_{k^{\prime},\nu^{\prime}}$ for some $k^{\prime}\in\mathbb{Z}$ and $\nu^{\prime}\in\mathbb{C}.$ Since $\tau_{k,\nu}\otimes\tau_{k^{\prime},\nu^{\prime}}=\tau_{k+k^{\prime},\nu+\nu^{\prime}},$ we then have

[TABLE]

Define the archimedean conductor of $\sigma_{v,j}\otimes\chi_{v}$ in this case to be

[TABLE]

• Case 2:

Assume that $F_{v}\simeq\mathbb{R}.$ One has $\mathcal{W}_{F,v}\simeq\mathbb{C}^{\times}\sqcup\bm{j}\mathbb{C}^{\times},$ where $\bm{j}^{2}=-1$ and $\bm{j}z\bm{j}^{-1}=\bar{z}$ for any $z\in\mathbb{C}^{\times}.$ Hence each irreducible representation $\sigma_{v,j}$ of $\mathcal{W}_{F,v}$ is of dimension 1 or 2.

• (a)

If $\dim\sigma_{v,j}=1,$ then its restriction to $\mathbb{C}^{\times}$ is of the form $\tau_{0,\nu_{j}}$ for some $\nu_{j}\in\mathbb{C}$ ( (3.2) of [K94]). Also, we can write $\chi_{v}=\tau_{0,\nu^{\prime}}$ for some $\nu^{\prime}\in\mathbb{C}.$ In this case, we have

[TABLE]

where $\Gamma_{\mathbb{R}}(s):=\pi^{-s/2}\Gamma(s/2).$ Define the archimedean conductor of $\sigma_{v,j}\otimes\chi_{v}$ in this case to be

[TABLE]

• (b)

If $\dim\sigma_{v,j}=2,$ we may assume that $\sigma_{v,j}$ is induced from $\mathbb{C}^{\times}$ to $GL(2,\mathbb{R})$ by $\tau_{k_{j},\nu_{j}},$ where $k_{j}\in\mathbb{N}_{\geq 1}$ and $\nu_{j}\in\mathbb{C}.$ Then $\sigma_{v,j}\otimes\chi_{v}$ is induced from $\mathbb{C}^{\times}$ by $\tau_{k_{j},\nu_{j}+\nu^{\prime}}.$ The $L$-factor is defined to be

[TABLE]

Define the archimedean conductor of $\sigma_{v,j}\otimes\chi_{v}$ in this case to be

[TABLE]

Definition 6**.**

Let notation be as above. Define the archimedean conductor of $\pi_{v}\otimes\chi_{v}$ to be

[TABLE]

where each $C_{v}(\sigma_{v,j}\otimes\chi_{v})$ is defined via (9), (10) or (11). Let $N(\pi\times\chi)$ be the arithmetic conductor of $\pi\times\chi.$ We set

[TABLE]

to be the archimedean conductor of $\pi\times\chi.$ And let

[TABLE]

be the analytic conductor of $\pi\times\chi.$

Recall that $\pi_{1}$ and $\pi_{2}$ are unitary cuspidal representations on $GL(n,\mathbb{A}_{F}).$ Denote by $\sigma_{i,v}$ the Weil-Deligne representation associated to $\pi_{i,v},$ $1\leq i\leq 2.$ Assume $\pi_{1}\otimes\chi\simeq\pi_{2}.$ Then $\sigma_{1,v}\otimes\chi_{v}\simeq\sigma_{2,v},$ for all $v\in\Sigma_{\infty}.$ Let

[TABLE]

where each $\sigma_{i,v;j}$ is an irreducible representation of $\mathcal{W}_{F,v}.$ Hence,

[TABLE]

• Case 1:

Assume that $F_{v}\simeq\mathbb{C}.$ One can write $\sigma_{i,v;j}=\tau_{k_{i,j},\nu_{i,j}},$ $k_{i,j}\in\mathbb{Z}$ and $\nu_{i,j}\in\mathbb{C},$ $1\leq j\leq r.$ Also, one can write $\chi_{v}$ as $\tau_{k^{\prime},\nu^{\prime}}$ for some $k^{\prime}\in\mathbb{Z}$ and $\nu^{\prime}\in\mathbb{C}.$ Then (12) yields $k_{1,j}+k^{\prime}=k_{2,j}$ and $\nu_{1,j}+\nu^{\prime}=\nu_{2,j}.$ Hence

[TABLE]

• Case 2:

Assume that $F_{v}\simeq\mathbb{R}.$ If $\dim\sigma_{i,v;j}=1,$ then its restriction to $\mathbb{C}^{\times}$ is of the form $\tau_{0,\nu_{i,j}}$ for some $\nu_{i,j}\in\mathbb{C}.$ Also, we can write $\chi_{v}=\tau_{0,\nu^{\prime}}$ for some $\nu^{\prime}\in\mathbb{C}.$ In this case, we have $\nu_{1,j}+\nu^{\prime}=\nu_{2,j}.$ Hence

[TABLE]

If $\dim\sigma_{i,v;j}=2,$ we may assume that $\sigma_{i,v;j}$ is induced from $\mathbb{C}^{\times}$ to $GL(2,\mathbb{R})$ by $\tau_{k_{i,j},\nu_{i,j}},$ where $k_{i,j}\in\mathbb{N}_{\geq 1}$ and $\nu_{i,j}\in\mathbb{C}.$ Then $\sigma_{1,v;j}\otimes\chi_{v}$ is induced from $\mathbb{C}^{\times}$ by $\tau_{k_{1,j},\nu_{1,j}+\nu^{\prime}}.$ Hence $k_{1,j}=k_{2,j}$ and $\nu_{1,j}+\nu^{\prime}=\nu_{2,j},$ implying

[TABLE]

Lemma 7**.**

Assume $F_{v}\simeq\mathbb{R}.$ Then

[TABLE]

Moreover, the above bound is sharp.

Proof.

Since $\dim\sigma_{1,v;j}=1$ or $2,$ we shall discuss the two cases separately.

• Case I.

Suppose $\dim\sigma_{1,v;j}=1.$ Then $\dim\sigma_{2,v;j}=1.$ Hence by (14) we have

[TABLE]

• (a).

Assume $\sigma_{2,v;j}(\bm{j})\sigma_{1,v;j}(\bm{j})=1.$ If $\sigma_{2,v;j}(\bm{j})=\sigma_{1,v;j}(\bm{j})=1,$ then $C_{v}(\chi_{v})=1+|\nu_{2,j}-\nu_{1,j}|\leq 1+|\nu_{2,j}|+|\nu_{1,j}|=1+|\nu_{2,j}+(1-\sigma_{2,v;j}(\bm{j}))/2|+|\nu_{1,j}+(1-\sigma_{1,v;j}(\bm{j}))/2|.$ Thus $C_{v}(\chi_{v})\leq C_{v}(\sigma_{1,v;j})C_{v}(\sigma_{2,v;j}).$

Suppose $\sigma_{2,v;j}(\bm{j})=\sigma_{1,v;j}(\bm{j})=-1.$ By Corollary 2.5 of [JS81], $|\operatorname{Re}(\nu_{1,j})|\leq 1/2$ and $|\operatorname{Re}(\nu_{2,j})|\leq 1/2.$ Thus $|\nu_{i,j}|\leq|\nu_{i,j}+1|=|\nu_{i,j}+(1-\sigma_{i,v;j}(\bm{j}))/2|,$ $1\leq i\leq 2.$ Therefore, $C_{v}(\chi_{v})=1+|\nu_{2,j}-\nu_{1,j}|\leq 1+|\nu_{2,j}|+|\nu_{1,j}|\leq 1+|\nu_{2,j}+(1-\sigma_{2,v;j}(\bm{j}))/2|+|\nu_{1,j}+(1-\sigma_{1,v;j}(\bm{j}))/2|.$ Hence, again, we have $C_{v}(\chi_{v})\leq C_{v}(\sigma_{1,v;j})C_{v}(\sigma_{2,v;j}).$

• (b).

Assume $\sigma_{2,v;j}(\bm{j})\sigma_{1,v;j}(\bm{j})=-1.$ Then $C_{v}(\chi_{v})=1+|\nu_{2,j}-\nu_{1,j}+1|.$ If $\sigma_{2,v;j}(\bm{j})=-1,$ then $C_{v}(\chi_{v})\leq 1+|\nu_{2,j}+1|+|\nu_{1,j}|\leq(1+|\nu_{1,j}|)(1+|\nu_{2,j}+1|)=C_{v}(\sigma_{1,v;j})C_{v}(\sigma_{2,v;j}).$ If $\sigma_{2,v;j}(\bm{j})=1,$ then $C_{v}(\chi_{v})=1+|\nu_{2,j}-\nu_{1,j}+1|\leq 1+|\nu_{1,j}+1|+|\nu_{2,j}|+2\leq 3(1+|\nu_{2,j}|)(1+|\nu_{1,j}+1|),$ namely, we have $C_{v}(\chi_{v})=3C_{v}(\sigma_{1,v;j})C_{v}(\sigma_{2,v;j}).$

In all, if $\dim\sigma_{1,v;j}=1,$ then we deduce that

[TABLE]

• Case II.

Suppose $\dim\sigma_{1,v;j}=2.$ Then $\dim\sigma_{2,v;j}=2.$ Hence by (15) we have

[TABLE]

Let $\sigma_{i,v;j}$ be induced from $\mathbb{C}^{\times}$ by $\tau_{k_{i,j},\nu_{i,j}},$ where $k_{i,j}\in\mathbb{N}_{\geq 1}$ and $\nu_{i,j}\in\mathbb{C},$ $1\leq i\leq 2.$ Then $\sigma_{1,v;j}\otimes\chi_{v}$ is induced from $\mathbb{C}^{\times}$ by $\tau_{k_{1,j},\nu_{1,j}+\nu^{\prime}}.$ Hence $k_{1,j}=k_{2,j}$ and $\nu_{1,j}+\nu^{\prime}=\nu_{2,j}.$ Then by triangle inequality we have

[TABLE]

Recall that in this case $C_{v}(\sigma_{i,v;j})=(1+|\nu_{i,j}+k_{i,j}/2|)^{2},$ $1\leq i\leq 2.$ So

[TABLE]

Let $\sigma_{i,v}=\oplus_{j=1}^{r_{v}}\sigma_{i,v;j}$ be the decomposition of $\sigma_{i,v}$ into irreducible representations. Let $r_{l}$ be the number of $\sigma_{i,v;j}$’s such that $\dim\sigma_{i,v;j}=l,$ $1\leq l\leq 2.$ Then $r_{1}+2r_{2}=n.$ It the follows from (17) and (18) that

[TABLE]

Thus (16) follows. Moreover, from the above proof, it is clear that the equality in (16) holds if $r_{1}=n$ (so $r_{2}=0$) and $\nu_{1,j}=-1,$ $\nu_{2,j}=0,$ for all $1\leq j\leq r_{v}.$ ∎

Lemma 8**.**

Assume $F_{v}\simeq\mathbb{C}.$ Then

[TABLE]

Moreover, the above bound is sharp.

Proof.

Since $F_{v}\simeq\mathbb{C},$ we can write $\sigma_{i,v;j}=\tau_{k_{i,j},\nu_{i,j}},$ $k_{i,j}\in\mathbb{Z}$ and $\nu_{i,j}\in\mathbb{C},$ $1\leq j\leq r.$ Write $\chi_{v}$ as $\tau_{k^{\prime},\nu^{\prime}}$ for some $k^{\prime}\in\mathbb{Z}$ and $\nu^{\prime}\in\mathbb{C}.$ Then (12) yields $k_{1,j}+k^{\prime}=k_{2,j}$ and $\nu_{1,j}+\nu^{\prime}=\nu_{2,j}.$ So by (13), $C_{v}(\chi_{v})=(1+|\nu_{2,j}-\nu_{1,j}+|k_{2,j}-k_{1,j}|/2|)^{2}.$ By triangle inequality we have

[TABLE]

On the other hand, we have the following inequality:

Claim 9**.**

Let $1\leq i\leq 2.$ Then $|\nu_{i,j}|+|k_{i,j}|/2\leq 1+3|\nu_{i,j}+|k_{i,j}|/2|.$

Therefore, by Claim 9, one obtains the upper bound for $C_{v}(\chi_{v}):$

[TABLE]

Hence, by (9) we have $C_{v}(\chi_{v})\leq 9\cdot C_{v}(\sigma_{1,v;j})C(\sigma_{2,v;j}).$ Invoking with the decomposition $\sigma_{i,v}=\oplus_{j=1}^{r_{v}}\sigma_{i,v;j}$ we thus get

[TABLE]

Therefore, (19) follows. ∎

Proof of Claim 9.

• (a).

Suppose $k_{i,j}=0.$ Then $|\nu_{i,j}|+|k_{i,j}|/2=|\nu_{i,j}|.$ Hence $|\nu_{i,j}|+|k_{i,j}|/2\leq 1+3|\nu_{i,j}+|k_{i,j}|/2|.$ holds trivially.

• (b).

Suppose $|k_{i,j}|=1.$ Then by triangle inequality, $|\nu_{i,j}|+|k_{i,j}|/2=|\nu_{i,j}|+1/2\leq|\nu_{i,j}+1/2|+1\leq 1+3|\nu_{i,j}+|k_{i,j}|/2|.$

• (c).

Suppose $|k_{i,j}|\geq 2.$ Note that by Corollary 2.5 of [JS81], $|\operatorname{Re}(\nu_{i,j})|\leq 1/2.$ Then $|\operatorname{Re}(\nu_{i,j})|\leq|\operatorname{Re}(\nu_{i,j})+|k_{i,j}|/2|,$ as $|k_{i,j}|/2\geq 1.$ Also, $2|\operatorname{Re}(\nu_{i,j})+|k_{i,j}|/2|\geq|k_{i,j}|/2.$ So $|\nu_{i,j}|+|k_{i,j}|/2\leq|\nu_{i,j}+|k_{i,j}|/2|+|k_{i,j}|/2\leq|\nu_{i,j}+|k_{i,j}|/2|+2|\operatorname{Re}(\nu_{i,j})+|k_{i,j}|/2|\leq 1+3|\nu_{i,j}+|k_{i,j}|/2|.$

In all, Claim 9 holds. ∎

Remark*.*

Our proofs of Lemma 7 and Lemma 8 would imply an explicit version of Lemma A.2 in [HB19] (in the case when $n=n^{\prime}$). Also, the original proof of Lemma A.2 there is not quite complete as the inequality chain right above (A. 13) (see P. 14 of [HB19]) is not correct for $k=1.$

Theorem B**.**

Let $\pi_{1}$ and $\pi_{2}$ be unitary cuspidal representations on $GL(n,\mathbb{A}_{F})$ such that $\pi_{1}\otimes\chi\simeq\pi_{2},$ for a character $\chi.$ Let $C_{1}$ (resp. $C_{2}$) be the analytic conductor of $\pi_{1}$ (resp. $\pi_{2}$). Let $\chi$ be a Hecke character on $F$. Denote by $C$ the analytic conductor of $\chi.$ Then

[TABLE]

where $[F:\mathbb{Q}]$ is the degree of $F/\mathbb{Q}.$

Proof.

Let $r_{1}$ (resp. $r_{2}$) be the number of real (complex) places of $F.$ Then $r_{1}+2r_{2}=[F:\mathbb{Q}].$ By Lemma 7 and Lemma 8, one has

[TABLE]

Then (21) follows from Theorem A. ∎