# Higher Ext-groups in the triple product case

Li Cai  Academy for Multidisciplinary Studies
Beijing National Center for Applied Mathematics
Capital Normal University
Beijing, 10048, People’s Republic of China<EMAIL_ADDRESS>and  Yangyu Fan
Academy for Multidisciplinary Studies
Beijing National Center for Applied Mathematics
Capital Normal University
Beijing, 10048, People’s Republic of China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this short note, we compute higher extension groups for all irreducible
representations and deduce the multiplicity formula for finite length
representations in triple product case.

###### Contents

  1. 1 Introduction
  2. 2 The proof

##  1\. Introduction

Let $F$ be a $p$-adic field. Let $L/F$ be a cubic étale extension and $D/F$ be
a quaternion algebra. Let $G={\mathrm{Res}}_{L/F}D^{\times}$ and
$H=D^{\times}$. Note that the intersection of the center $Z_{G}$ of $G$ with
$H$ is $Z_{H}={\mathbb{G}}_{m}$. Denote by
${\mathrm{Rep}}(F^{\times}\backslash G(F))$ the category of smooth
$F^{\times}\backslash G(F)$-representations.

In this short note, we prove that the higher Ext groups vanish

${\mathrm{Ext}}^{i}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0,\quad
i>0.$

for any generic $\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$. Combining
with results from the local trace formula approach, we obtain a multiplicity
formula for any irreducible $\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$.

To state the result, we introduce more notations. For any irreducible
$\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$, consider its geometric
multiplicity

$m_{\mathrm{geo}}(\pi)=\sum_{T\in{\mathcal{T}}}|W(H,T)|^{-1}\int_{F^{\times}\backslash
T(F)}c_{\pi}(t)D^{H}(t)dt$

where

  * •

the support ${\mathcal{T}}={\mathcal{T}}(G,H)$ is a set of tori in $H$. If $D$
is split, ${\mathcal{T}}$ consists of $F^{\times}$ and the nonsplit maximal
tori in $H$. If $D$ is non-split, ${\mathcal{T}}$ consists of the nonsplit
maximal tori in $H$.

  * •

$W(H,T)=N_{H}(T)/Z_{H}(T)$.

  * •

$c_{\pi}$ is the regularized character on the semi-simple locus of $G(F)$ (See
[12, Definition 2.5]).

  * •

$D^{H}(t)=|\det(1-{\mathrm{ad}}(t))|_{{\mathfrak{h}}/{\mathfrak{h}}_{t}}|$ is
the Weyl discriminant on the semi-simple locus of $H(F)$.

  * •

the measure $dt$ is normalized so that the volume
${\mathrm{Vol}}(F^{\times}\backslash T(F),dt)=1$.

Via the local trace formula approach, Wan proves the following multiplicity
formula for tempered representations:

######  Theorem 1.1 (Wan [11], C.2 and C.3).

For any irreducible tempered $\pi\in{\mathrm{Rep}}(F^{\times}\backslash
G(F))$,

$m(\pi):=\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}(\pi,{\mathbb{C}})=m_{\mathrm{geo}}(\pi).$

######  Remark 1.2.

In fact, the above multiplicity formula holds for generic representations
combining the result of Prasad [8, 9] (See [11, Remark C. 2.3]).

In general, it is conjectured in [10, Conjecture 7.1] (for the Gan-Gross-
Prasad models) and [12, Conjecture 7.6] (for all spherical pairs) that the
multiplicity formula should hold for all
$\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$ of finite length with the
multiplicity $m(\pi)$ replaced by the Euler-Poincaré number

${\mathrm{EP}}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}}):=\sum_{i\geq
0}(-1)^{i}\dim_{\mathbb{C}}{\mathrm{Ext}}^{i}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}}).$

Philosophically, such multiplicity formula can be viewed as a kind of Riemann-
Roch theorem (See [10, Remark 7.2]).

######  Example 1.3.

Assume $D$ is split and $L=E\oplus F$ where $E/F$ is a quadratic field
extension. Let
$\pi=\pi_{1}\boxtimes\pi_{2}\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$ with

  * •

$\pi_{1}=I_{B(E)}^{{\mathrm{GL}}_{2}(E)}\chi_{1}\boxtimes\chi_{2}\in{\mathrm{Rep}}(F^{\times}\backslash{\mathrm{GL}}_{2}(E))$
being the normalized parabolic induction for characters $\chi_{1},\chi_{2}:\
E^{\times}\to{\mathbb{C}}^{\times}$ such that
$\chi_{1}\chi_{2}|_{F^{\times}}=1$;

  * •

$\pi_{2}={\mathbb{C}}\in{\mathrm{Rep}}(F^{\times}\backslash{\mathrm{GL}}_{2}(F))$
being the trivial representation.

Then

${\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})={\mathrm{Hom}}_{F^{\times}\backslash{\mathrm{GL}}_{2}(F)}(\pi_{1},{\mathbb{C}}).$

Set $\chi^{\prime}:=\chi_{1}{\overline{\chi_{2}}}$ where $-$ denotes the
Galois conjugation with respect to $E/F$. It is known that (see e.g. [6,
Theorem 5.2]) $m(\pi)\leq 1$ and the equality holds if and only if
$\chi_{1}|_{F^{\times}}=\chi_{2}|_{F^{\times}}=1$ or $\chi^{\prime}=1$.

On the other hand, by a property of the regularized characters of parabolic
inductions [12, Proposition 2.7]

$m_{\mathrm{geo}}(\pi)=\frac{1}{2}\int_{F^{\times}\backslash
E^{\times}}c_{\pi}(t)D^{H}(t)dt=\frac{1}{2}\int_{F^{\times}\backslash
E^{\times}}(\chi^{\prime}(t)+\chi^{\prime}(\bar{t}))dt.$

In particular, $m_{\mathrm{geo}}(\pi)\leq 1$ and the equality holds if and
only if $\chi^{\prime}=1$.

Therefore, in the case $\chi^{\prime}\not=1$ and
$\chi_{1}|_{F^{\times}}=\chi_{2}|_{F^{\times}}=1$,

$m(\pi)\not=m_{\mathrm{geo}}(\pi).$

This is compatible with Theorem 1.1 since $\pi_{2}$ (hence $\pi$) is neither
generic nor tempered.

The following is the main result of this paper.

######  Theorem 1.4.

For any irreducible $\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$, generic
if $D$ is split,

${\mathrm{Ext}}^{i}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0,\quad
i\geq 1.$

Moreover, for any $\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$ of finite
length, the multiplicity formula holds

$\mathrm{EP}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=m_{\mathrm{geo}}(\pi).$

######  Remark 1.5.

Previously,

  * •

The vanishing result is known for the Whittaker model: if $G$ is a connected
quasi-split reductive group over $F$, $B=TN$ a Borel subgroup of $G$, $\psi$ a
generic character on $N(F)$, then for any irreducible representation
$\pi\in{\mathrm{Rep}}(G(F))$ and any $i\geq 1$, by [10, Proposition 2.8]

${\mathrm{Ext}}_{N(F)}^{i}(\pi,\psi^{-1})\cong{\mathrm{Ext}}_{G(F)}^{i}\left(i_{N(F)}^{G(F)}\psi,\pi^{\vee}\right)\cong{\mathrm{Ext}}_{1}^{i}(\pi_{N,\psi},{\mathbb{C}})=0.$

For the Gan-Gan-Prasad models of general linear groups, the vanishing result
is due to Chan-Savin [5].

  * •

The only known cases of [12, Conjecture 7.6] are the group case [10,
Proposition 2.1(4)], the Whittaker models [12, Section 8.1], and the Gan-
Gross-Prasad models for general linear groups ([10, Theorem 4.2] and note that
in this case the support ${\mathcal{T}}={\mathcal{T}}(G,H)$ of
$m_{\mathrm{geo}}$ is $\\{1\\}$).

We explain the proof of Theorem 1.4. In fact, it is known that
${\mathrm{Ext}}^{i}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$ for any
$\pi$ when $i\geq 2$ (see [10, Proposition 2.9], also Proposition 2.1(1)
below) and ${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=0$ if $\pi$ is supercuspidal (see [3, Theorem 5.4.1],
also Proposition 2.1(1) below). The proof of the vanishing result is reduced
to showing that ${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=0$ for non-supercuspidal $\pi$. For this, we apply
standard tools: the geometric lemma of Bernstein-Zelevinsky and the Schneider-
Stuhler duality ([7, Theorem 2], see also Theorem 2.2 below). Once the
vanishing results are available, the multiplicity formula for finite length
representations is deduced from the tempered version (Theorem 1.1) by noting
both sides are additive and constant in an unramified twisting family.

In fact, we compute ${\mathrm{Ext}}^{i}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})$ for all irreducible
$\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$ and all $i$. The results for
$L=E\oplus F$ together with the Schneider-Stuhler duality implies the
following complete classification of irreducible
${\mathrm{GL}}_{2}(F)$-subrepresentations of irreducible
${\mathrm{GL}}_{2}(E)$-representations, which is analogous to [7, Proposition
9.1] for the case $L=F\oplus F\oplus F$ (in fact, there is an extra central
character condition in [7, Proposition 9.1]:
$\pi=\pi_{1}\boxtimes\pi_{2}\boxtimes\pi_{3}$ with $\pi_{3}$ having trivial
central character. This condition is dropped here).

######  Proposition 1.6.

Assume $L=E\oplus F$ and $D$ is split. For
$\pi=\pi_{1}\boxtimes\pi_{2}\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$
irreducible, $\pi_{2}^{\vee}$ is a ${\mathrm{GL}}_{2}(F)$-subrepresentation of
$\pi_{1}$ if and only if

  * •

$\pi_{1}=\xi\circ\det$ and $\pi_{2}=\xi^{-1}|_{F^{\times}}\circ\det$; or

  * •

$\pi_{2}^{\vee}$ is supercuspidal, and appears as a quotient of
$\pi_{1}|_{{\mathrm{GL}}_{2}(F)}$; or

  * •

$\pi_{2}={\mathrm{St}}_{F}\otimes\xi$,
$\pi_{1}=I_{B(E)}^{{\mathrm{GL}}_{2}(E)}\chi_{1}\boxtimes\chi_{2}$ such that
$\xi\chi_{1}|_{F^{\times}}=\xi\chi_{2}|_{F^{\times}}=1$ and
$\xi_{E}\chi^{\prime}\neq 1$. Here $\xi_{E}=\xi\circ N_{E/F}$,
$\chi^{\prime}=\chi_{1}{\overline{\chi_{2}}}$ and ${\mathrm{St}}_{F}$ is the
Steinberg representation for ${\mathrm{GL}}_{2}(F)$.

Acknowledgement The debt this work owes to [10] and [7] is clear. We thank
Professor Ye Tian for his consistent encouragement. We want to thank Professor
Kei-Yuen Chan for helpful discussions.

##  2\. The proof

For any reductive group $\Gamma$ over $F$ and any center character $\omega:\
Z_{\Gamma}(F)\to{\mathbb{C}}^{\times}$, let ${\mathrm{Rep}}(\Gamma(F),\omega)$
denote the full subcategory of ${\mathrm{Rep}}(\Gamma(F))$ consisting of
objects on which $Z_{\Gamma}(F)$ acts by $\omega$. For any objects
$\pi,\pi^{\prime}\in{\mathrm{Rep}}(\Gamma(F),\omega)\subset{\mathrm{Rep}}(\Gamma(F))$,
let ${\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(\Gamma(F),\omega)}(\pi,\pi^{\prime})$
(resp. ${\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi,\pi^{\prime})$) be the $i$-th
extension group in the category ${\mathrm{Rep}}(\Gamma(F),\omega)$ (resp.
${\mathrm{Rep}}(\Gamma(F))$). Let $\Gamma^{\prime}(F)$ be a closed subgroup of
the $p$-adic group $\Gamma(F)$ with modulus character
$\delta_{\Gamma^{\prime}}$. Note that when $\Gamma^{\prime}$ is reductive,
$\delta_{\Gamma^{\prime}}$ is trivial. Denote by
$I_{\Gamma^{\prime}(F)}^{\Gamma(F)}$ (resp.
$i_{\Gamma^{\prime}(F)}^{\Gamma(F)}$) the normalized (resp. compact)
induction. Note that for any $\sigma\in{\mathrm{Rep}}(\Gamma^{\prime}(F))$,
$(i_{\Gamma^{\prime}(F)}^{\Gamma(F)}\sigma)^{\vee}=I_{\Gamma^{\prime}(F)}^{\Gamma(F)}\sigma^{\vee}$
where $-^{\vee}$ stands for smooth dual in the proper category. Moreover if
$\Gamma^{\prime}=MN$ is a parabolic subgroup with Levi factor $M$, denote by
$J_{N}$ the normalized Jacquet functor for $\Gamma^{\prime}$.

We record some basic properties of Ext-groups which are frequently used in the
following.

######  Proposition 2.1.

We have the following results for the Ext-groups.

  1. (1)

For any irreducible $\pi\in{\mathrm{Rep}}(\Gamma(F))$, smooth
$\pi^{\prime}\in{\mathrm{Rep}}(\Gamma(F))$ and any $i>\mathrm{split\ rank\
of}\ Z_{\Gamma}$

${\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi,\pi^{\prime})\cong{\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi^{\prime},\pi)=0.$

If $\pi$ is supercuspidal with central character $\omega$, then $\pi$ is both
projective and injective in ${\mathrm{Rep}}(\Gamma(F),\omega)$. In particular,
for any $i\geq 1$

${\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(\Gamma(F),\omega)}(\pi,\pi^{\prime})\cong{\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(\Gamma(F),\omega)}(\pi^{\prime},\pi)=0.$

  2. (2)

For any $\pi\in{\mathrm{Rep}}(\Gamma(F),\omega)$,
$\sigma\in{\mathrm{Rep}}(\Gamma(F),\omega^{-1})$ and any $i\geq 0$

${\mathrm{Ext}}^{i}_{Z_{\Gamma}(F)\backslash\Gamma(F)}(\pi\otimes\sigma,{\mathbb{C}})\cong{\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(\Gamma(F),\omega)}(\pi,\sigma^{\vee})\cong{\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(\Gamma(F),\omega^{-1})}(\sigma,\pi^{\vee}).$

  3. (3)

The restriction functor
${\mathrm{Rep}}(\Gamma(F))\to{\mathrm{Rep}}(\Gamma^{\prime}(F))$ sends
projective objects to projective objects, the normalized induction functor
$I_{\Gamma^{\prime}(F)}^{\Gamma(F)}:\
{\mathrm{Rep}}(\Gamma^{\prime}(F))\to{\mathrm{Rep}}(\Gamma(F))$ sends
injective objects to injective objects. Moreover, for any
$\sigma\in{\mathrm{Rep}}(\Gamma^{\prime}(F))$,
$\pi\in{\mathrm{Rep}}(\Gamma(F))$ and any $i\geq 0$

${\mathrm{Ext}}^{i}_{\Gamma(F)}\left(\pi,I_{\Gamma^{\prime}(F)}^{\Gamma(F)}\sigma\right)\cong{\mathrm{Ext}}^{i}_{\Gamma^{\prime}(F)}\left(\pi,\sigma\otimes\delta_{\Gamma^{\prime}}^{1/2}\right).$

When $\Gamma^{\prime}=MN$ is a parabolic subgroup with Levi factor $M$, the
normalized Jacquet functor $J_{N}:\
{\mathrm{Rep}}(\Gamma(F))\to{\mathrm{Rep}}(M(F))$ sends projective objects to
projective objects. For any $\sigma\in{\mathrm{Rep}}(M(F))$ and any $i\geq 0$

${\mathrm{Ext}}^{i}_{\Gamma(F)}\left(\pi,I_{\Gamma^{\prime}(F)}^{\Gamma(F)}\sigma\right)\cong{\mathrm{Ext}}^{i}_{M(F)}(J_{N}(\pi),\sigma).$

###### Proof.

The property of supercuspidal representations can be found in [3, Theorem
5.4.1]. Other results are summarized in [10, Section 2]. ∎

For any $\pi\in{\mathrm{Rep}}(\Gamma(F),\omega)$ irreducible, let $d(\pi)$
(resp. $d^{\prime}(\pi)$) be the split rank of $Z_{M}$ (resp.
$Z_{M}\cap[\Gamma,\Gamma]$), where $M$ is any Levi subgroup carrying the
cuspidal support of $\pi$. The key ingredient for computing higher Ext groups
is the following Schneider-Stuhler duality theorem.

######  Theorem 2.2 (Theorem 1,2 in [7]).

Let $\pi\in{\mathrm{Rep}}(\Gamma(F),\omega)$ be any irreducible representation
and $D(\pi)$ be the Aubert-Zelevinsky involution of $\pi$. Then

  * •

for any $\pi^{\prime}\in{\mathrm{Rep}}(\Gamma(F))$,
${\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi,\pi^{\prime})=0$ for $i>d(\pi)$ and for
$0\leq i\leq d(\pi)$, there is a non-degenerate pairing

${\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi,\pi^{\prime})\times{\mathrm{Ext}}^{d(\pi)-i}_{\Gamma(F)}(\pi^{\prime},D(\pi))\to{\mathrm{Ext}}_{\Gamma(F)}^{d(\pi)}(\pi,D(\pi))\cong{\mathbb{C}},$

  * •

for any $\pi^{\prime}\in{\mathrm{Rep}}(\Gamma(F),\omega)$,
${\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(\Gamma(F),\omega)}(\pi,\pi^{\prime})=0$
for $i>d^{\prime}(\pi)$ and for $0\leq i\leq d^{\prime}(\pi)$, there is a non-
degenerate pairing

${\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(\Gamma(F),\omega)}(\pi,\pi^{\prime})\times{\mathrm{Ext}}^{d^{\prime}(\pi)-i}_{{\mathrm{Rep}}(\Gamma(F),\omega)}(\pi^{\prime},D(\pi))\to{\mathrm{Ext}}_{{\mathrm{Rep}}(\Gamma(F),\omega)}^{d^{\prime}(\pi)}(\pi,D(\pi))\cong{\mathbb{C}}.$

######  Remark 2.3.

In general, the non-degeneracy means that if
$\pi^{\prime}=\varinjlim_{n}\pi_{n}^{\prime}$ is the inductive limit of finite
generated $\Gamma(F)$-submodules $\pi_{n}^{\prime}$, then

${\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi,\pi^{\prime})=\varinjlim_{n}{\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi,\pi_{n}^{\prime}),\quad{\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi^{\prime},D(\pi))=\varprojlim_{n}{\mathrm{Ext}}^{i}_{\Gamma(F)}(\pi_{n}^{\prime},D(\pi))$

is the inductive limit (resp. projective limit) of finite dimensional
${\mathbb{C}}$-spaces and the pairing is direct limit of perfect pairings on
these finite dimensional spaces.

In the triple product case, the result of Aizenbud-Sayag [1] guarantees all
the Ext-groups below are finite dimensional and the non-degeneracy has the
usual meaning.

In the following, we shall concentrate on the triple product case.

The case $D$ non-split is straightforward.

######  Proposition 2.4.

Assume $D$ is non-split. Then for any
$\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$ irreducible,

${\mathrm{Ext}}^{i}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0,\quad
i>0.$

###### Proof.

Note that $Z_{H}\backslash H$ is anisotropic. Then by Proposition 2.1 (1),
${\mathbb{C}}$ is injective and the statement follows. ∎

To deal with the $D$ split case, we need to consider the following Waldspurger
toric case.

######  Lemma 2.5.

Assume $D$ is split. Let $K\subset D$ be an étale quadratic $F$-algebra and
embed $F^{\times}$ into $D^{\times}(F)\times K^{\times}$ diagonally. Then for
any generic irreducible
$\pi\in{\mathrm{Rep}}(F^{\times}\backslash(D^{\times}(F)\times
K^{\times})\big{)}$,

${\mathrm{Ext}}^{i}_{F^{\times}\backslash
K^{\times}}(\pi,{\mathbb{C}})=0,\quad i\geq 1.$

###### Proof.

Assume $\pi=\sigma\boxtimes\xi$ with $\sigma\in{\mathrm{Rep}}(D^{\times}(F))$
and $\xi\in{\mathrm{Rep}}(K^{\times})$. If $K/F$ is a field extension, then
${\mathrm{Ext}}^{i}_{F^{\times}\backslash
K^{\times}}(\sigma\boxtimes\xi,{\mathbb{C}})=0$ for $i\geq 1$ by Proposition
2.1 (1).

If $K=F\oplus F$, ${\mathrm{Ext}}^{i}_{F^{\times}\backslash
K^{\times}}(\sigma\boxtimes\xi,{\mathbb{C}})=0$ for $i\geq 2$ by Proposition
2.1 (1). For $i=1$, one can apply Theorem 2.2 to
${\mathbb{C}}\in{\mathrm{Rep}}(F^{\times}\backslash K^{\times})$, where
$d({\mathbb{C}})=1$ and $D({\mathbb{C}})={\mathbb{C}}$. One has

$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
K^{\times}}(\sigma\boxtimes\xi,{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{F^{\times}\backslash
K^{\times}}({\mathbb{C}},\sigma\boxtimes\xi)=\dim_{\mathbb{C}}{\mathrm{Hom}}_{K^{\times}}(\xi^{-1},\sigma).$

Identify $K^{\times}$ with the split torus $T(F)$ and let
${\mathcal{K}}(\sigma)$ be the Kirillov model of $\sigma$. Write
$\xi=\xi_{1}\boxtimes\xi_{2}$. Then by [2, Proposition 4.7.2],

${\mathrm{Hom}}_{T(F)}(\xi^{-1},{\mathcal{K}}(\sigma)|_{T})=\\{\phi\in{\mathcal{K}}(\sigma)\
|\ \phi(ax)=\xi_{1}^{-1}(a)\phi(x)\text{ for any }a\in F^{\times}\\}=0.$

∎

Assume $D$ is split. By Proposition 2.1(1),
${\mathrm{Ext}}^{i}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$ for
$i\geq 2$. In the following, we compute
${\mathrm{Ext}}^{1}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})$ case by
case. Let $T\subset B\subset{\mathrm{GL}}_{2}$ be the diagonal torus and the
group of upper triangular matrices respectively, ${\mathrm{St}}_{F}$ be the
Steinberg representation for ${\mathrm{GL}}_{2}(F)$ and $|\cdot|_{F}$ be the
norm character on $F^{\times}$.

######  Proposition 2.6.

Assume $D$ is split and $L/F$ is a cubic field extension. Then for
$\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$ irreducible,
${\mathrm{Ext}}^{1}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$.

###### Proof.

By Proposition 2.1 (1,3)(the modulus character of $H$ is trivial), when $\pi$
is supercuspidal,

${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})={\mathrm{Ext}}^{1}_{F^{\times}\backslash
G(F)}(\pi,I_{H(F)}^{G(F)}{\mathbb{C}})=0.$

If $\pi$ is one-dimensional, then by Theorem 2.2 for
${\mathbb{C}}\in{\mathrm{Rep}}(F^{\times}\backslash H(F))$, where
$D({\mathbb{C}})={\mathrm{St}}_{F}$ and $d({\mathbb{C}})={\mathbb{C}}$,

$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}({\mathrm{St}}_{F},\pi)=0.$

Note that when $\pi$ is a special series, then $\pi\hookrightarrow
I_{B(L)}^{{\mathrm{GL}}_{2}(L)}\chi_{1}\boxtimes\chi_{2}$ with
$\chi_{1}\chi_{2}^{-1}=|\cdot|_{L}$. Thus it suffices to show that for
$\pi=I_{B(L)}^{{\mathrm{GL}}_{2}(L)}\chi\in{\mathrm{Rep}}(F^{\times}\backslash
G(F))$ where $\chi=\chi_{1}\boxtimes\chi_{2}$ with
$\chi_{1}\chi_{2}^{-1}\neq|\cdot|_{L}^{-1}$,

${\mathrm{Ext}}^{1}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0.$

By the geometric lemma (see [9, Section 6.1]), there exists an exact sequence
of $F^{\times}\backslash H(F)$-representations

$0\to i_{F^{\times}}^{H(F)}{\mathbb{C}}\to\pi\to
I_{B(F)}^{H(F)}\chi\delta_{B}\to 0$

where $\delta_{B}$ is the modulus character of $B(F)$. The short exact
sequence induces a long exact sequence

$\displaystyle 0\to$ $\displaystyle{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(I_{B(F)}^{H(F)}\chi\delta_{B},{\mathbb{C}})\to{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})\to{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(i_{F^{\times}}^{H(F)}{\mathbb{C}},{\mathbb{C}})\to$
$\displaystyle{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(I_{B(F)}^{H(F)}\chi\delta_{B},{\mathbb{C}})\to{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})\to{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(i_{F^{\times}}^{H(F)}{\mathbb{C}},{\mathbb{C}})\to 0.$

By Proposition 2.1(2,3),

${\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(i_{F^{\times}}^{H(F)}{\mathbb{C}},{\mathbb{C}})={\mathbb{C}},\quad{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(i_{F^{\times}}^{H(F)}{\mathbb{C}},{\mathbb{C}})=0.$

Note that

  * •

if $\pi$ is reducible, $I_{B(F)}^{H(F)}\chi\delta_{B}$ is irreducible;

  * •

if $\pi$ is irreducible, we may assume $I_{B(F)}^{H(F)}\chi\delta_{B}$ is
irreducible by replacing $\chi$ by $\chi^{w}$, the twisting of $\chi$ by
$w=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$, if necessary.

Thus

${\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(I_{B(F)}^{H(F)}\chi\delta_{B},{\mathbb{C}})=0.$

Moreover by Theorem 2.2 for
${\mathbb{C}}\in{\mathrm{Rep}}(F^{\times}\backslash H(F))$, where
$D({\mathbb{C}})={\mathrm{St}}_{F}$ and $d({\mathbb{C}})=1$,

$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(I_{B(F)}^{H(F)}\chi\delta_{B},{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}({\mathrm{St}}_{F},I_{B(F)}^{H(F)}\chi\delta_{B})=0.$

Consequently, ${\mathrm{Hom}}_{H(F)}(\pi,{\mathbb{C}})={\mathbb{C}}$ and
${\mathrm{Ext}}^{1}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$. ∎

######  Proposition 2.7.

Assume $D$ is split and $L=E\oplus F$. For
$\pi=\pi_{1}\boxtimes\pi_{2}\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$
irreducible, $\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})\leq 1$ with the equality holds if and only if

  * •

$\pi_{2}=\xi\circ\det$,
$\pi_{1}=I_{B(E)}^{{\mathrm{GL}}_{2}(E)}\chi_{1}\boxtimes\chi_{2}$ such that
$\chi_{1}|_{F^{\times}}\xi=\chi_{2}|_{F^{\times}}\xi=1$ and
$\xi_{E}\chi^{\prime}\neq 1$ (See Proposition 1.6 for the notations $\xi_{E}$
and $\chi^{\prime}$); or

  * •

$\pi_{1}=\xi\circ\det$ and
$\pi_{2}={\mathrm{St}}_{F}\otimes\xi^{-1}|_{F^{\times}}$.

###### Proof.

Denote by $\omega^{-1}$ the central character of $\pi_{2}$. Then by
Proposition 2.1 (2)

${\mathrm{Ext}}^{i}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})\cong{\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(H(F),\omega)}(\pi_{1},\pi_{2}^{\vee})$

and if $\pi_{1}$ or $\pi_{2}$ is supercuspidal,
${\mathrm{Ext}}^{1}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$.

When $\pi_{1}$ is one-dimensional, by Theorem 2.2 for
${\mathbb{C}}\in{\mathrm{Rep}}(F^{\times}\backslash H(F))$, where
$D({\mathbb{C}})={\mathrm{St}}_{F}$ and $d({\mathbb{C}})=1$,

$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}({\mathrm{St}}_{F},\pi)\leq
1$

with the equality holds if and only if $\pi_{1}=\xi\circ\det$ and
$\pi_{2}={\mathrm{St}}_{F}\otimes\xi^{-1}|_{F^{\times}}$.

Note that when $\pi_{1}$ is a special series, $\pi\hookrightarrow
I_{B(E)}^{{\mathrm{GL}}_{2}(E)}\chi_{1}\boxtimes\chi_{2}$ with
$\chi_{1}\chi_{2}^{-1}=|\cdot|_{E}$. Thus it suffices to consider
$\pi_{1}=I_{B(E)}^{{\mathrm{GL}}_{2}(E)}\chi$ for
$\chi=\chi_{1}\boxtimes\chi_{2}$ where
$\chi_{1}\chi_{2}^{-1}\neq|\cdot|_{E}^{-1}$. By the geometric lemma (See [9,
Section 4.1]), there is an exact sequence of $H(F)$-representations

$0\to i_{E^{\times}}^{H(F)}\chi^{\prime}\to\pi_{1}\to
I_{B(F)}^{H(F)}\chi\delta_{B}^{1/2}\to 0.$

By Proposition 2.1(2,3) and Lemma 2.5,

${\mathrm{Hom}}_{H(F)}(i_{E^{\times}}^{{\mathrm{GL}}_{2}(F)}\chi^{\prime},\pi_{2}^{\vee})\cong{\mathrm{Hom}}_{E^{\times}}(\pi_{2}\otimes\chi^{\prime},{\mathbb{C}});$
${\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(i_{E^{\times}}^{{\mathrm{GL}}_{2}(F)}\chi^{\prime},\pi_{2}^{\vee})\cong{\mathrm{Ext}}^{1}_{F^{\times}\backslash
E^{\times}}(\pi_{2}\otimes\chi^{\prime},{\mathbb{C}})=0;$
${\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(H(F),\omega)}(I_{B(F)}^{{\mathrm{GL}}_{2}(F)}\chi\delta_{B}^{1/2},\pi_{2}^{\vee})\cong{\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(T(F),\omega^{-1})}(J_{N}(\pi_{2}),\delta_{B}^{-1/2}\chi^{-1})\cong{\mathrm{Ext}}^{i}_{F^{\times}\backslash
T(F)}(J_{N}(\pi_{2})\otimes\delta_{B}^{1/2}\chi,{\mathbb{C}}).$

Then by Theorem 2.2 for ${\mathbb{C}}\in{\mathrm{Rep}}(F^{\times}\backslash
T(F))$, where $D({\mathbb{C}})={\mathbb{C}}$ and $d({\mathbb{C}})=1$,

$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(I_{B(F)}^{{\mathrm{GL}}_{2}(F)}\chi\delta_{B}^{1/2},\pi_{2}^{\vee})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}(I_{B(F)}^{{\mathrm{GL}}_{2}(F)}\chi\delta_{B}^{1/2},\pi_{2}^{\vee})\leq
1$

with the equality holds if and only if $\delta_{B}^{-1/2}\chi^{-1}|_{T(F)}$ is
a Jordan factor of $J_{N}(\pi_{2})$. By the long exact sequence

$\displaystyle 0\to$
$\displaystyle{\mathrm{Hom}}_{H(F)}(I_{B(F)}^{H(F)}\chi\delta_{B}^{1/2},\pi_{2}^{\vee})\to{\mathrm{Hom}}_{H(F)}(\pi_{1},\pi_{2}^{\vee})\to{\mathrm{Hom}}_{H(F)}(i_{E^{\times}}^{H(F)}\chi^{\prime},\pi_{2}^{\vee})\to$
$\displaystyle{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(I_{B(F)}^{H(F)}\chi\delta_{B}^{1/2},\pi_{2}^{\vee})\to{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(\pi_{1},\pi_{2}^{\vee})\to{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(i_{E^{\times}}^{H(F)}\chi^{\prime},\pi_{2}^{\vee})\to
0,$

one has that

  1. (i)

${\mathrm{EP}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{E^{\times}}(\pi_{2}\otimes\chi^{\prime},{\mathbb{C}})\leq
1$

  2. (ii)

unless $\delta_{B}^{-1/2}\chi^{-1}|_{T(F)}$ is a Jordan factor of
$J_{N}(\pi_{2})$, ${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=0,$

  3. (iii)

if ${\mathrm{Hom}}_{E^{\times}}(\pi_{2}\otimes\chi^{\prime},{\mathbb{C}})=0$,

$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(I_{B(F)}^{H(F)}\chi\delta_{B}^{1/2},\pi_{2}^{\vee}).$

If $\delta_{B}^{-1/2}\chi^{-1}|_{T(F)}$ is a Jordan factor of $J_{N}(\pi_{2})$
and $\pi_{1}=I_{B(E)}^{{\mathrm{GL}}_{2}(E)}\chi$ sits in the exact sequence

$0\to{\mathrm{St}}_{E}\otimes\mu\to\pi_{1}\to\mu\to 0,$

one has $\pi_{2}=I_{B(F)}^{{\mathrm{GL}}_{2}(F)}\chi^{-1}\delta_{B}^{-1/2}$ is
irreducible. Then by Saito-Tunnell, ${\mathrm{EP}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=1$ and hence

$\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}({\mathrm{St}}_{E}\otimes\mu,\pi_{2}^{\vee})=\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}({\mathrm{St}}_{E}\otimes\mu,\pi_{2}^{\vee})+1.$

Note that

$\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}({\mathrm{St}}_{E}\otimes\mu,\pi_{2}^{\vee})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}({\mathrm{St}}_{E}\otimes\mu\otimes\pi_{2},{\mathbb{C}})\leq 1.$

one deduce ${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}({\mathrm{St}}_{E}\otimes\mu\otimes\pi_{2},{\mathbb{C}})=0$ and hence
${\mathrm{Ext}}^{1}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$ from the
induced long exact sequence.

Assume $\pi_{1}=I_{B(E)}^{{\mathrm{GL}}_{2}(E)}\chi$ is irreducible. If
$\chi_{1}|_{F^{\times}}\neq\chi_{2}|_{F^{\times}}$, then up to replacing
$\chi$ by $\chi^{w}$, one can make $\delta_{B}^{-1/2}\chi^{-1}|_{T(F)}$
different from the Jordan Holder factors of $J_{N}(\pi_{2})$. Then
${\mathrm{Ext}}^{1}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$ by (ii).

Now assume moreover $\xi^{-1}=\chi_{1}|_{F^{\times}}=\chi_{2}|_{F^{\times}}$.
Then $\delta_{B}^{-1/2}\chi^{-1}|_{T(F)}$ is a Jordan factor of
$J_{N}(\pi_{2})$ if and only if $\pi_{2}=\xi\circ\det$ or
${\mathrm{St}}_{F}\otimes\xi$. In the case
$\pi_{2}={\mathrm{St}}_{F}\otimes\xi$,

  * •

if ${\mathrm{EP}}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=1$, then
${\mathrm{Ext}}^{1}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$ since
$\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}(\pi,{\mathbb{C}})\leq 1$;

  * •

if ${\mathrm{EP}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{E^{\times}}(\pi_{2}\otimes\chi^{\prime},{\mathbb{C}})=0$,
then by Theorem 2.2 for
${\mathrm{St}}_{F}\in{\mathrm{Rep}}(F^{\times}\backslash H(F))$, where
$D({\mathrm{St}}_{F})={\mathbb{C}}$ and $d({\mathrm{St}}_{F})=1$,

$\displaystyle\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})$
$\displaystyle=\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(I_{B(F)}^{H(F)}\chi\delta_{B}^{1/2}\otimes\xi,{\mathrm{St}}_{F})$
$\displaystyle=\dim_{\mathbb{C}}{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}({\mathbb{C}},I_{B(F)}^{{\mathrm{GL}}_{2}(F)}\delta_{B}^{1/2})=0.$

In the case $\pi_{2}=\xi\circ\det$, we shall give the criterion for
${\mathrm{Ext}}_{F^{\times}\backslash
H(F)}^{1}(\pi,{\mathbb{C}})={\mathbb{C}}$ via studying
${\mathrm{EP}}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})$. By the above
(i), ${\mathrm{EP}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{E^{\times}}(\pi_{2}\otimes\chi^{\prime},{\mathbb{C}})\leq
1$, where the equality holds if and only if $\xi_{E}\chi^{\prime}=1$.

If ${\mathrm{EP}}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=1$, as the
second term in the long exact sequence has multiplicity $\leq 1$, we must have
${\mathrm{Ext}}_{F^{\times}\backslash H(F)}^{1}(\pi,{\mathbb{C}})=0$.

If ${\mathrm{EP}}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}})=0$, then as
the last term in the last exact sequence vanishes, we must have

$\dim_{\mathbb{C}}{\mathrm{Hom}}_{E^{\times}}(\pi_{2}\otimes\chi^{\prime},{\mathbb{C}})=0,\quad\dim_{\mathbb{C}}{\mathrm{Ext}}_{F^{\times}\backslash
H(F)}^{1}(\pi,{\mathbb{C}})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}(I_{B(F)}^{H(F)}\chi\delta_{B}^{1/2},\pi_{2}^{\vee}).$

It is then easy to see that
$\dim_{\mathbb{C}}{\mathrm{Ext}}_{F^{\times}\backslash
H(F)}^{1}(\pi,{\mathbb{C}})=1$ if and only if $\chi|_{F^{\times}}\xi=1$ and
$\xi_{E}\chi^{\prime}\not=1$.

∎

One can deduce Proposition 1.6 immediately from Proposition 2.7.

###### Proof of Proposition 1.6.

Denote the central character of $\pi_{2}$ by $\omega^{-1}$. By Theorem 2.2,

  * •

when $\pi_{2}$ is supercuspidal, by Proposition 2.1 (1),

${\mathrm{Hom}}_{H(F)}(\pi_{2}^{\vee},\pi_{1})\cong{\mathrm{Hom}}_{H(F)}(\pi_{1},\pi^{\vee}_{2});$

  * •

when $\pi_{2}$ is non-supercuspidal, $d(\pi_{2})=1$ and

$\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}(\pi^{\vee}_{2},\pi_{1})=\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(\pi_{1},D(\pi^{\vee}_{2}))=\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi_{1}\otimes D(\pi_{2}),{\mathbb{C}}).$

Since $D(\xi\circ\det)={\mathrm{St}}_{F}\otimes\xi$ and
$D({\mathrm{St}}_{F}\otimes\xi)=\xi\circ\det$, the statement follows from
Proposition 2.7 immediately. ∎

######  Proposition 2.8.

Assume $D$ is split and $L=F\oplus F\oplus F$. For
$\pi=\pi_{1}\boxtimes\pi_{2}\boxtimes\pi_{3}\in{\mathrm{Rep}}(F^{\times}\backslash
G(F))$ with $\pi_{i}\in{\mathrm{Rep}}({\mathrm{GL}}_{2}(F))$ irreducible,
$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})\leq 1$ with the equality holds if and only if up to
reordering and twisting,

  * •

$\pi_{1}={\mathrm{St}}_{F}$ and $\pi_{2}=\pi_{3}={\mathbb{C}}$, or

  * •

$\pi_{1}=\pi_{2}^{\vee}$ are principal series and $\pi_{3}={\mathbb{C}}$.

###### Proof.

Denote the central character of $\pi_{3}$ by $\omega^{-1}$. Then by
Proposition 2.1 (i) (ii)

${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})\cong{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(\pi_{1}\otimes\pi_{2},\pi_{3}^{\vee})$

and if $\pi_{3}$ is supercuspidal, ${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=0$. In the following, we assume none of $\pi_{i}$ is
supercuspidal.

If $\pi_{3}$ is one dimensional, which we may assume to be ${\mathbb{C}}$ by
twisting, then

${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})\cong{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega_{1})}(\pi_{1},\pi_{2}^{\vee})$

where $\omega_{1}$ is the central character of $\pi_{1}$. Then by Theorem 2.2
for $\pi_{2}^{\vee}$,

$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})\leq 1$

with the equality holds if and only if up to reordering and twisting,

  * •

$\pi_{1}={\mathrm{St}}_{F}$ and $\pi_{2}={\mathbb{C}}$,

  * •

$\pi_{1}=\pi_{2}^{\vee}$ are principal series.

If $\pi_{1}$, $\pi_{2}$ are special series, which we assume to be
${\mathrm{St}}_{F}$ by twisting, and $\pi_{3}$ is generic, then by [8, Lemma
5.4], there exists an exact sequence of ${\mathrm{GL}}_{2}(F)$-representations

$0\to{\mathrm{St}}_{F}\to
i_{T(F)}^{{\mathrm{GL}}_{2}(F)}{\mathbb{C}}\to\pi_{1}\otimes\pi_{2}\to 0,$

which induces the long exact sequence

$\displaystyle 0\to$ $\displaystyle{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(\pi_{1}\otimes\pi_{2},\pi_{3}^{\vee})\to{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(i_{T(F)}^{H(F)}{\mathbb{C}},\pi_{3}^{\vee})\to{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}({\mathrm{St}}_{F},\pi^{\vee}_{3})\to$
$\displaystyle{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi_{1}\otimes\pi_{2},\pi_{3}^{\vee})\to{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(i_{T(F)}^{H(F)}{\mathbb{C}},\pi_{3}^{\vee})\to{\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F))}({\mathrm{St}}_{F},\pi^{\vee}_{3})\to 0.$

By Lemma 2.5,

${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(i_{T(F)}^{H(F)}{\mathbb{C}},\pi_{3}^{\vee})=0,\quad{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(i_{T(F)}^{H(F)}{\mathbb{C}},\pi_{3}^{\vee})={\mathbb{C}}.$

Thus

  * •

when $\pi_{3}={\mathrm{St}}_{F}$, ${\mathrm{Hom}}_{H(F)}(\pi,{\mathbb{C}})=0$
by [8, Theorem 1.2] and hence ${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=0$,

  * •

when $\pi_{3}\neq{\mathrm{St}}_{F}$, ${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi|_{H},{\mathbb{C}})={\mathrm{Hom}}_{F^{\times}\backslash
H(F)}({\mathrm{St}}_{F},\pi^{\vee}_{3})=0$.

If $\pi_{1}=I_{B(F)}^{{\mathrm{GL}}_{2}(F)}\xi_{1}$,
$\pi_{2}=I_{B(F)}^{{\mathrm{GL}}_{2}(F)}\xi_{2}$ and $\pi_{3}$ is generic.
Then by the geometric lemma (see [8, Section 5]), there is an exact sequence
of ${\mathrm{GL}}_{2}(F)$-representations

$0\to
i_{T(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}^{w})\to\pi_{1}\otimes\pi_{2}\to
I_{B(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}\delta^{1/2})\to 0,$

which leads to a long exact sequence

$\displaystyle 0\to$
$\displaystyle{\mathrm{Hom}}_{H(F)}(I_{B(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}\delta_{B}^{1/2}),\pi_{3}^{\vee})\to{\mathrm{Hom}}_{H(F)}(\pi_{1}\otimes\pi_{2},\pi_{3}^{\vee})\to$
$\displaystyle{\mathrm{Hom}}_{H(F)}(i_{T(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}^{w}),\pi_{3}^{\vee})\to{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(I_{B(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}\delta_{B}^{1/2}),\pi_{3}^{\vee})\to$
$\displaystyle{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(\pi_{1}\otimes\pi_{2},\pi_{3}^{\vee})\to{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(i_{T(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}^{w}),\pi_{3}^{\vee})\to
0.$

By Proposition 2.1 (2)(3),

${\mathrm{Ext}}^{i}_{{\mathrm{Rep}}(H(F),\omega)}(i_{T(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}^{w}),\pi_{3}^{\vee})={\mathrm{Ext}}^{i}_{F^{\times}\backslash
T(F)}(\pi_{3}\otimes\xi_{1}\xi_{2}^{w},{\mathbb{C}}).$

Moreover by Theorem 2.2 for
$\pi_{3}^{\vee}\in{\mathrm{Rep}}(F^{\times}\backslash H(F),\omega)$, where
$d(\pi_{3}^{\vee})=1$,

$\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(I_{B(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}\delta_{B}^{1/2}),\pi_{3}^{\vee})=\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}(I_{B(F)}^{{\mathrm{GL}}_{2}(F)}(\xi_{1}\xi_{2}\delta_{B}^{1/2}),\pi_{3}^{\vee}).$

Then by Lemma 2.5, one has

$\displaystyle\dim_{\mathbb{C}}{\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(\pi_{1}\otimes\pi_{2},\pi_{3}^{\vee})$
$\displaystyle=\dim_{\mathbb{C}}{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})-\dim_{\mathbb{C}}{\mathrm{Hom}}_{F^{\times}\backslash
T(F)}(\pi_{3}\otimes\xi_{1}\xi_{2}^{w},{\mathbb{C}})$
$\displaystyle=\dim_{\mathbb{C}}{\mathrm{Hom}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})-1\geq 0.$

Since $\dim_{\mathbb{C}}{\mathrm{Hom}}_{H(F)}(\pi|_{H},{\mathbb{C}})\leq 1$,
one has ${\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})={\mathrm{Ext}}^{1}_{{\mathrm{Rep}}(H(F),\omega)}(\pi_{1}\otimes\pi_{2},\pi_{3}^{\vee})=0.$

Interchanging the roles of $\pi_{i}$, the statement follows. ∎

Immediately, we deduce the following corollary, which slightly generalizes [7,
Proposition 9.1] (by dropping the central character condition).

######  Corollary 2.9.

Let $\pi=\boxtimes_{i=1}^{3}\pi_{i}\in{\mathrm{Rep}}(F^{\times}\backslash
G(F))$ be an irreducible smooth representation. Then $\pi_{3}^{\vee}$ is a
${\mathrm{GL}}_{2}(F)$-subrepresentation of $\pi_{1}\otimes\pi_{2}$ if and
only if

  * •

$\pi_{3}^{\vee}$ is supercuspidal, and appears as a quotient of
$\pi_{1}\otimes\pi_{2}$;

  * •

$\pi_{3}^{\vee}={\mathrm{St}}_{F}\otimes\xi$ and
$\pi_{1}=\pi_{2}^{\vee}\otimes\xi$ is a principal series;

  * •

$\pi_{1}$ or $\pi_{2}$ is one-dimensional and
$\pi_{3}^{\vee}\cong\pi_{1}\otimes\pi_{2}$.

###### Proof.

Denote the central character of $\pi_{3}$ by $\omega^{-1}$. By Theorem 2.2 and
Proposition 2.1 (2),

  * •

when $\pi_{3}$ is supercuspidal,

${\mathrm{Hom}}_{H(F)}(\pi_{3}^{\vee},\pi_{1}\otimes\pi_{2})={\mathrm{Hom}}_{H(F)}(\pi_{1}\otimes\pi_{2},\pi^{\vee}_{2}),$

  * •

when $\pi_{3}$ is non-supercuspidal,

${\mathrm{Hom}}_{H(F)}(\pi^{\vee}_{3},\pi_{1}\otimes\pi_{2})={\mathrm{Ext}}^{1}_{F^{\times}\backslash
H(F)}(\pi_{1}\otimes\pi_{2}\otimes D(\pi_{3}),{\mathbb{C}}).$

Then the statement follows immediately from Proposition 2.8. ∎

###### Proof of Theorem 1.4.

The vanishing of higher Ext-groups for generic representations is proved in
Propositions 2.6, 2.7, 2.8. In particular, for
$\pi\in{\mathrm{Rep}}(F^{\times}\backslash G(F))$ tempered,

$m(\pi)={\mathrm{EP}}_{F^{\times}\backslash H(F)}(\pi,{\mathbb{C}}).$

Hence, by the multiplicity formula for tempered representations (Theorem 1.1),

${\mathrm{EP}}_{F^{\times}\backslash
H(F)}(\pi,{\mathbb{C}})=m(\pi)=m_{\mathrm{geo}}(\pi).$

For any Levi subgroup $M$ of $G$, denote by $\mathrm{Tmp}(F^{\times}\backslash
M(F))$ the set of tempered representations on $F^{\times}\backslash M(F)$ and
${\widehat{F^{\times}\backslash M(F)}}^{\mathrm{un}}$ the set of unramified
characters on $F^{\times}\backslash M(F)$. Consider the subcategory
${\mathrm{Rep}}^{0}(F^{\times}\backslash G(F))$ of
${\mathrm{Rep}}(F^{\times}\backslash G(F))$ consisting of
$I_{P}^{G}(\sigma\otimes\chi)$ where $P=MN$ is a parabolic subgroup of $G$,
$\sigma\in\mathrm{Tmp}(F^{\times}\backslash M(F))$ and
$\chi\in{\widehat{F^{\times}\backslash M(F)}}^{\mathrm{un}}$. We have the
following two lemmas.

######  Lemma 2.10.

The Grothendieck group of the abelian sub-category of finite length
representations in ${\mathrm{Rep}}(F^{\times}\backslash G(F))$ can be
generated by ${\mathrm{Rep}}^{0}(F^{\times}\backslash G(F))$.

###### Proof.

For any character $\omega:\ F^{\times}\to{\mathbb{C}}^{\times}$, set
${\mathrm{Rep}}^{0}(D^{\times}(F),\omega):={\mathrm{Rep}}(D^{\times}(F),\omega)\cap{\mathrm{Rep}}^{0}(D^{\times}(F))$.
Then by the classification of irreducible $D^{\times}(F)$-representations, for
any irreducible $\pi\in{\mathrm{Rep}}(D^{\times}(F),\omega)$, there exists
$\pi_{0}\in{\mathrm{Rep}}^{0}(D^{\times}(F),\omega)$ such that the
semisimplification of $\pi_{0}$ is the direct sum of $\pi$ and another
irreducible representation in ${\mathrm{Rep}}^{0}(D^{\times},\omega)$. The
lemma follows from this fact. ∎

######  Lemma 2.11.

Fix $\sigma\in\mathrm{Tmp}(F^{\times}\backslash M(F))$ for some Levi $M$ of
$G$ and consider the unramified twisting family
$\pi_{\chi}=I_{P}^{G}\sigma\otimes\chi\in{\mathrm{Rep}}^{0}(F^{\times}\backslash
G(F))$ with $\chi\in{\widehat{F^{\times}\backslash M(F)}}^{\mathrm{un}}$. For
any $T\in{\mathcal{T}}$, the function

$\chi\mapsto c_{\pi_{\chi}}|_{T}$

is constant. In particular, the geometric multiplicity is constant for an
unramified twisting family in the sense that the function

$\chi\mapsto m_{\mathrm{geo}}(\pi_{\chi})$

is constant.

###### Proof.

We have the following fact on the regularized character of a parabolic
induction [12, Proposition 2.7]. Let $\pi=I_{P}^{G}\sigma$ be a finite length
$G(F)$-representation induced from $P=MN$. Then for any semi-simple $x\in
G(F)$

$D^{G}(x)^{1/2}c_{\pi}(x)=\sum_{y\in{\mathfrak{X}}_{M}(x)}D^{M}(y)^{1/2}c_{\sigma}(y)$

where ${\mathfrak{X}}_{M}(x)$ is the set of representatives for
$M(F)$-conjugacy classes of elements in $M(F)$ that are $G(F)$-conjugated to
$x$.

Let $T\in{\mathcal{T}}$. If ${\mathfrak{X}}_{M}(t)$ is empty for any $t\in
T(F)$, $c_{\pi_{\chi}}|_{T}=0$ for any $\chi$. Assume there exists $t\in T(F)$
such that ${\mathfrak{X}}_{M}(t)$ is nonempty, or equivalently, there is an
embedding of $T$ into $M$. This happens exactly when

  * •

$M=G$;

  * •

$L=E\oplus F$, $T=E^{\times}$ and $M=A_{E}\times D^{\times}$ where $A_{E}$ is
the diagonal torus of ${\mathrm{GL}}_{2,E}$. In this case,

${\mathfrak{X}}_{M}(t)=\left\\{\left(\begin{pmatrix}t&0\\\
0&\bar{t}\end{pmatrix},t\right),\quad\left(\begin{pmatrix}\bar{t}&0\\\
0&t\end{pmatrix},t\right)\right\\}.$

To show the constancy of $c_{\pi_{\chi}}|_{T}$ for any unramified character
$\chi$ on $M(F)$, it is enough to prove that

$F^{\times}\backslash T(F)\subset\left(F^{\times}\backslash
M\right)^{0}=\bigcap_{\mu\in\mathrm{Rat}\left(F^{\times}\backslash
M\right)}\ker|\mu|$

where $\mathrm{Rat}\left(F^{\times}\backslash M\right)$ is the group of
rational characters on $M$. As $F^{\times}\backslash T$ is anisotropic,
$\mathrm{Rat}(F^{\times}\backslash T)=0$ so that for any
$\mu\in\mathrm{Rat}\left(F^{\times}\backslash M\right)$, $\mu(t)=1$ for any
$t\in T$. ∎

Now, as the both sides of the multiplicity formula is additive, by (1), we
only need to consider representations in
${\mathrm{Rep}}^{0}(F^{\times}\backslash G(F))$, that is, unramified twists of
tempered representations. By (2), the geometric multiplicity is constant for
an unramified twisting family. Meanwhile, it is known that the Euler-Poincaré
number is constant for an unramified twisting family (see [1, Theorem E(4)] or
[4, Proposition 3.18] for a more general situation). Therefore, the
multiplicity formula for tempered representations implies the formula for any
irreducible representation.

∎

######  Remark 2.12.

In fact, the constancy of the geometric multiplicity holds in general. It can
be proved similarly as the above special case Lemma 2.11 (Note that in
general, any torus $T$ in the support ${\mathcal{T}}(G,H)$ satisfies
$T/Z_{G,H}$ is anisotropic. See [12, Definition 4.3].)

## References

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††thanks: This project was initiated at the workshop WIN Women in Numbers in
November 2008. The authors would like to thank the Banff International
Research Station for hosting the workshop and the National Security Agency,
the Fields Institute, the Pacific Institute for the Mathematical Sciences,
Microsoft Research, and University of Calgary for their financial support.
Author Pries was partially supported by NSF grant 07-01303. Author Im was
partially supported by the Korea Science and Engineering Foundation (KOSEF)
grant (No. R01-2007-000-10660-0) funded by the Korea government (MOST). The
authors would also like to thank the referee for helpful comments

# Semi-direct Galois covers of the affine line

Linda Gruendken  Department of Mathematics, University of Pennsylvania, David
Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104-6395
<EMAIL_ADDRESS>Laura Hall-Seelig  Department of Mathematics and
Statistics, Lederle Graduate Research Tower, University of Massachusetts,
Amherst, MA 01003-9305<EMAIL_ADDRESS>Bo-Hae Im  Department of
Mathematics, Chung-Ang University, 221, Heukseok-dong, Dongjak-gu, Seoul
156-756
South Korea<EMAIL_ADDRESS>Ekin Ozman  Department of Mathematics, University
of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706
<EMAIL_ADDRESS>Rachel Pries  Department of Mathematics, Colorado State
University, Fort Collins, CO 80523-1874, USA<EMAIL_ADDRESS>Katherine Stevenson  Department of Mathematics, California State University,
18111 Nordhoff St, Northridge, CA 91330-8313<EMAIL_ADDRESS>
###### Abstract.

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a
semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ where
$b$ is a positive integer and $\ell$ is a prime distinct from $p$. In this
paper, we study Galois covers $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over
$\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ which
admits a covering map of this form and we give an explicit formula for this
genus in terms of $\ell$ and $p$. The minimal genus occurs when $b$ equals the
order $a$ of $\ell$ modulo $b$ and we also prove that the number of curves $Z$
of this minimal genus which admit such a covering map is at most $(p-1)/a$
when $p$ is odd.

###### 1991 Mathematics Subject Classification:

14H30, 14E20, 14F40

##  1\. Introduction

Let $k$ be an algebraically closed field of characteristic $p>0$. In sharp
contrast with the situation in characteristic $0$, there exist Galois covers
$\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over infinity. By Abhyankar’s
Conjecture [2], proved by Raynaud and Harbater [9], [4], a finite group $G$
occurs as the Galois group of such a cover $\psi$ if and only if $G$ is
quasi-$p$, i.e., $G$ is generated by $p$-groups. This result classifies all
the finite quotients of the fundamental group $\pi_{1}(\mathbb{A}^{1}_{k})$.
It does not, however, determine the profinite group structure of
$\pi_{1}(\mathbb{A}^{1}_{k})$ because this fundamental group is an infinitely
generated profinite group.

There are many open questions about Galois covers
$\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over infinity. For example, given
a finite quasi-$p$ group $G$, what is the smallest integer $g$ for which there
exists a cover $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over infinity with
$Z$ of genus $g$? As another example, suppose $G$ and $H$ are two finite
quasi-$p$ groups such that $H$ is a quotient of $G$. Given an unramified
Galois cover $\phi$ of $\mathbb{A}^{1}_{k}$ with group $H$, under what
situations can one dominate $\phi$ with an unramified Galois cover $\psi$ of
$\mathbb{A}^{1}_{k}$ with Galois group $G$? Answering these questions will
give progress towards understanding how the finite quotients of
$\pi_{1}(\mathbb{A}^{1}_{k})$ fit together in an inverse system. These
questions are more tractible for quasi-$p$ groups that are $p$-groups since
the maximal pro-$p$ quotient $\pi_{1}^{p}(\mathbb{A}^{1}_{k})$ is free (of
infinite rank) [10].

In this paper, we study Galois covers $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified
only over $\infty$ whose Galois group is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$, where
$\ell$ is a prime distinct from $p$. Such a cover $\psi$ must be a composition
$\psi=\phi\circ\omega$ where $\omega:Z\to Y$ is unramified and
$\phi:Y\to\mathbb{P}^{1}_{k}$ is an Artin-Schreier cover ramified only over
$\infty$. The cover $\phi$ has an affine equation $y^{p}-y=f(x)$ for some
$f(x)\in k[x]$ with degree $s$ prime-to-$p$. The $\ell$-torsion
$\mathrm{Jac}\,(Y)[\ell]$ of the Jacobian of $Y$ is isomorphic to
$(\mathbb{Z}/\ell\mathbb{Z})^{2g_{Y}}$. When $f(x)=x^{s}$, we determine how an
automorphism $\tau$ of $Y$ of order $p$ acts on $\mathrm{Jac}\,(Y)[\ell]$.
This allows us to construct a Galois cover
$\psi_{a}:Z_{a}\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$ which
dominates $\phi$, such that the Galois group of $\psi_{a}$ is
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ where
$a$ is the order of $\ell$ modulo $p$ (Section 3). We prove that the genus of
$Z_{a}$ is minimal among all natural numbers that occur as the genus of a
curve $Z$ which admits a covering map $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified
only over $\infty$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. We
also prove that the number of curves $Z$ of this minimal genus which admit
such a covering map is at most $(p-1)/a$ when $p$ is odd (Section 4).

##  2\. Quasi-$p$ semi-direct products

We recall which groups occur as Galois groups of covers of
$\mathbb{P}^{1}_{k}$ ramified only over $\infty$.

######  Definition 2.1.

A finite group is a quasi $p$-group if it is generated by all of its Sylow
$p$-subgroups.

It is well-known that there are other equivalent formulations of the quasi-$p$
property, such as the next result.

######  Lemma 2.2.

A finite group is a quasi $p$-group if and only if it has no nontrivial
quotient group whose order is relatively prime to $p$.

The importance of the quasi-$p$ property is that it characterizes which finite
groups occur as Galois groups of unramified covers of the affine line.

######  Theorem 2.3.

A finite group occurs as the Galois group of a Galois cover of the projective
line $\mathbb{P}^{1}_{k}$ ramified only over infinity if and only if it is a
quasi-$p$ group.

###### Proof.

This is a special case of Abhyankar’s Conjecture [2] which was jointly proved
by Harbater [4] and Raynaud [9]. ∎

We now restrict our attention to groups $G$ that are semi-direct products of
the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. The
semi-direct product action is determined by a homomorphism
$\iota:\mathbb{Z}/p\mathbb{Z}\to\mathrm{Aut}\,((\mathbb{Z}/\ell\mathbb{Z})^{b})$.

######  Lemma 2.4.

Suppose a quasi-$p$ group $G$ is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ for a
positive integer $b$.

  1. (1)

Then $G$ is not a direct product.

  2. (2)

Moreover, $b\geq\mathrm{ord}_{p}(\ell)$ where $\mathrm{ord}_{p}(\ell)$ is the
order of $\ell$ modulo $p$.

###### Proof.

Part (1) is true since $(\mathbb{Z}/\ell\mathbb{Z})^{b}$ cannot be a quotient
of the quasi-$p$ group $G$. For part (2), the structure of a semi-direct
product $({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$
depends on a homomorphism
$\iota:\mathbb{Z}/p\mathbb{Z}\to\mathrm{Aut}\,(\mathbb{Z}/\ell\mathbb{Z})^{b}$.
By part (1), $\iota$ is an inclusion. Thus
$\mathrm{Aut}\,(\mathbb{Z}/\ell\mathbb{Z})^{b}\simeq\mathrm{GL}_{b}({\mathbb{Z}}/\ell{\mathbb{Z}})$
has an element of order $p$. Now

$|\mathrm{GL}_{b}({\mathbb{Z}}/\ell{\mathbb{Z}})|=(\ell^{b}-1)(\ell^{b}-\ell)\cdots(\ell^{b}-\ell^{b-1}).$

Thus $\ell^{\beta}\equiv 1\bmod p$ for some positive integer $\beta\leq b$
which implies $b\geq\mathrm{ord}_{p}(\ell)$. ∎

######  Lemma 2.5.

If $a=\mathrm{ord}_{p}(\ell)$, then there exists a semi-direct product of the
form $({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$
which is quasi-$p$. It is unique up to isomorphism.

###### Proof.

If $a=\mathrm{ord}_{p}(\ell)$, then there is an element of order $p$ in
$\mathrm{Aut}\,((\mathbb{Z}/\ell\mathbb{Z})^{a})$ and so there is an injective
homomorphism
$\iota:\mathbb{Z}/p\mathbb{Z}\to\mathrm{Aut}\,((\mathbb{Z}/\ell\mathbb{Z})^{a})$.
Thus there exists a non-abelian semi-direct product $G$ of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. To
show that $G$ is quasi-$p$, suppose $N$ is a normal subgroup of $G$ whose
index is relatively prime to $p$. Then $N$ contains an element $\tau$ of order
$p$. By [3, 5.4, Thm. 9], since $G$ is not a direct product and
$(\mathbb{Z}/\ell\mathbb{Z})^{a}$ is normal in $G$, the subgroup
$\langle\tau\rangle$ is not normal in $G$. Thus $\langle\tau\rangle$ is a
proper subgroup of $N$. It follows that $\ell$ divides $|N|$ and so $N$
contains an element $h$ of order $\ell$ by Cauchy’s theorem. Recall that
$\mathrm{Aut}\,((\mathbb{Z}/\ell\mathbb{Z})^{\beta})$ contains no element of
order $p$ for any positive integer $\beta<a$. Thus the group generated by the
conjugates of $h$ under $\tau$ has order divisible by $\ell^{a}$. Thus $N=G$
and $G$ has no non-trivial quotient group whose order is relatively prime to
$p$. By Lemma 2.2, $G$ is quasi-$p$.

The uniqueness follows from [8, Lemma 6.6]. ∎

##  3\. Explicit construction of
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$-Galois
covers of $\mathbb{A}^{1}_{k}$

In this section, we give concrete examples of Galois covers
$\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$ with Galois group of
the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. To
compute the genus of the covering curve $Z$, we will need to determine the
higher ramification groups of $\psi$.

######  Definition 3.1.

Let $L/K$ be a Galois extension of function fields of curves with Galois group
$G$ and let $P,P^{\prime}$ be primes of $K$ and $L$ such that $P^{\prime}|P$.
Let $\nu_{P^{\prime}}$ and ${\mathcal{O}}_{P^{\prime}}$ be the corresponding
valuation function and valuation ring for $P^{\prime}$. For any integer
$i\geq-1$, the $i$th ramification group of $P^{\prime}|P$ is

$I_{i}(P^{\prime}|P)=\\{\sigma\in G\ |\ \nu_{P^{\prime}}(\sigma(z)-z)\geq
i+1,\forall z\in{\mathcal{O}}_{P^{\prime}}\\}.$

######  Lemma 3.2.

Suppose $f(x)\in k[x]$ is a polynomial of degree $s$ for a positive integer
$s$ prime to $p$. Let $\phi:Y\to\mathbb{P}^{1}_{k}$ be the cover of curves
corresponding to the field extension

$k(x)\hookrightarrow k(x)[y]/(y^{p}-y-f(x)).$

  1. (1)

Then $\phi:Y\to\mathbb{P}^{1}_{k}$ is a Galois cover with Galois group
$\mathbb{Z}/p\mathbb{Z}$ ramified only at the point $P_{\infty}$ over
$\infty$.

  2. (2)

The $i$th ramification group at $P_{\infty}$ satisfies

$I_{i}=\left\\{\begin{array}[]{ll}\mathbb{Z}/p\mathbb{Z}&\textrm{if }i\leq
s\\\ 0&\textrm{if }i>s.\end{array}\right.$

  3. (3)

The genus $g_{Y}$ of $Y$ is equal to

$g_{Y}=(p-1)(s-1)/2.$

###### Proof.

For part (1), note that the extension $k(x)\hookrightarrow
k(x)[y]/(y^{p}-y-f(x))$ is cyclic of degree $p$, with Galois group generated
by the automorphism $\tau:y\mapsto y+1$ of order $p$. Let $P$ be a finite
prime of $k(x)$ and let $\nu_{P}$ be the corresponding valuation. Then
$\nu_{P}(f(x))\geq 0$, hence $P$ is unramified by [12, Prop. III.7.8(b)]. For
the infinite prime $\infty$ with corresponding valuation $\upsilon_{\infty}$,
we have

$\nu_{{\infty}}(f(x)-(z^{p}-z))\leq 0$

for all $z\in k[x]$ thus $P_{\infty}$ is totally ramified by [12, Prop.
III.7.8(c)].

To prove part (2), we note that furthermore

$\upsilon_{P_{\infty}}(y^{p}-y)=\upsilon_{P_{\infty}}(f(x))=\upsilon_{P_{\infty}}(x^{s})=-sp,$

which implies that

$\upsilon_{P_{\infty}}(y)=-s.$

Now let $\widehat{\theta}$ be the completion of the valuation ring of
$k(x)[y]/(y^{p}-y-f(x))$ at $P_{\infty}$, and let $\pi_{\infty}$ be a
generator of the unique prime in $\widehat{\theta}$. Then write
$y=\pi_{\infty}^{-s}u$, where $u$ is a unit in $\widehat{\theta}\simeq
k[[\pi_{\infty}]]$. Since $k$ is algebraically closed,
$\sqrt[s]{u}\in\widehat{\theta}$, and so $\sqrt[s]{y}\in\widehat{\theta}$.
After possibly changing $\pi_{\infty}$, we can assume without loss of
generality that $\sqrt[s]{y}=\pi_{\infty}^{-1}$. Recalling that $\tau$ acts on
$y$ by $\tau(y)=y+1$, we have

$\displaystyle\tau(\pi_{\infty})$ $\displaystyle=$
$\displaystyle\tau\left(1/y\right)^{1/s}=\left(\pi_{\infty}^{s}/(1+\pi_{\infty}^{s})\right)^{1/s}$
$\displaystyle=$
$\displaystyle\pi_{\infty}(1-\pi_{\infty}^{s}+\pi_{\infty}^{2s}-+\ldots)^{\frac{1}{s}}$
$\displaystyle=$
$\displaystyle\pi_{\infty}-(1/s)\pi_{\infty}^{s+1}+a_{2s+1}\pi_{\infty}^{2s+1}-+\ldots.$

Thus $\upsilon_{P_{\infty}}(\tau(\pi_{\infty})-\pi_{\infty})=s+1$, which
completes the proof of part (2).

To find the genus $g_{Y}$ of $Y$ for part (3), we make use of the Riemann-
Hurwitz formula

$2g_{Y}-2=p(-2)+\sum_{i=0}^{\infty}\left(|I_{i}|-1\right),$

where $I_{i}$ denotes the $i$th ramification group at $P_{\infty}$, [5, Thms.
7.27 & 11.72]). From part (2), we then obtain that $g_{Y}=(p-1)(s-1)/2$. ∎

Recall the following facts about the $p$th cyclotomic polynomial
$\Phi_{p}(t):=t^{p-1}+\cdots+1$, which is the minimal polynomial over
${\mathbb{Q}}$ of a primitive $p$th root of unity $\zeta_{p}$. Now
${\mathbb{Q}}(\zeta_{p})$ is a Galois extension of $\mathbb{Q}$, unramified
over $\ell$ since $\ell\not=p$, and all primes over $\ell$ have the same
residue field degree. The irreducible factors of $\Phi_{p}(t)$ modulo $\ell$
are in one-to-one correspondence with the primes of ${\mathbb{Z}}[\zeta_{p}]$
over $\ell$, and each of their degrees is equal to the residue field degree of
the corresponding prime over $\ell$. The latter equals the order
$a=\mathrm{ord}_{p}(\ell)$ of $\ell$ modulo $p$ [3, Ch. 12.2, Exercise #20].

We shall soon explicitly construct a cover of $\mathbb{P}^{1}_{k}$ ramified
only over $\infty$ with Galois group
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. But
before we do so, we start with a specific example.

######  Example 3.3.

Let $p$ be an odd prime. Consider the Artin-Schreier cover
$\phi:Y_{2}\to\mathbb{P}^{1}_{k}$ corresponding to the field extension
$k(x)\hookrightarrow k(x)[y]/(y^{p}-y-x^{2})$. By Lemma 3.2(3), the genus of
$Y_{2}$ is $g_{Y}=(p-1)/2$.

Let $\mathrm{Jac}\,(Y)$ be the Jacobian of $Y$. The automorphism $\tau$ of $Y$
given by $\tau(y)=y+1$ defines an automorphism of $\mathrm{Jac}\,(Y)$ of order
$p$.

Now we describe the action of $\tau$ on the subgroup $\mathrm{Jac}\,(Y)[2]$ of
2-torsion points of $\mathrm{Jac}\,(Y)$ explicitly. Note that since
$2g_{Y}=(p-1)$, then $\mathrm{Jac}\,(Y)[2]$ is isomorphic to
$(\mathbb{Z}/2\mathbb{Z})^{p-1}$ by [7, pg. 64]. Thus we can represent $\tau$
as an element of $\mathrm{GL}_{p-1}(\mathbb{Z}/2\mathbb{Z})$.

For $0\leq i\leq p-1$, let $P_{i}$ denote the closed point of $Y$ at which the
function $y-i$ vanishes. For each $i$, the divisors $P_{i}$ and
$D_{i}=P_{i}-P_{\infty}$ on $Y$ can be identified with elements of
$\mathrm{Jac}\,(Y)$. Let $O$ be the identity element of $\mathrm{Jac}\,(Y)$,
i.e., the linear equivalence class of principal divisors. Then the divisor
$2D_{i}$ is equivalent to $O$ since ${\rm div}(y-i)=2D_{i}$. Moreover since
${\rm div}(x)=D_{0}+D_{1}+\cdots+D_{p-1}$ is equivalent to $0$, we have
$D_{i}\in\mathrm{Jac}\,(Y)[2]$ with the only relation
$D_{p-1}=-(D_{0}+D_{1}+\cdots+D_{p-2})$. In particular, $D_{0},\ldots,D_{p-2}$
form a basis of $\mathrm{Jac}\,(Y)[2]$. With respect to this basis, the action
of $\tau$ can be represented by the $(p-1)\times(p-1)$-matrix

$\left(\begin{array}[]{ccccc}0&0&\ldots&0&-1\\\ 1&0&\ldots&0&-1\\\
0&1&\ldots&0&-1\\\ \vdots&\vdots&\ddots&0&-1\\\ 0&0&\ldots&1&-1\\\
\end{array}\right).$

The characteristic polynomial of $\tau$ is $\Phi_{p}(t)=1+t+\ldots
t^{p-1}\in(\mathbb{Z}/2\mathbb{Z})[t]$, which factors into irreducible
polynomials each of degree equaling the order of $2$ modulo $p$. In
particular, $\tau$ acts irreducibly on $\mathrm{Jac}\,(Y)[2]$ if and only if
$2$ is a primitive root modulo $p$, i.e., if and only if $p$ is an Artin
prime.

For example, if $p=3$, then $\tau$ acts irreducibly on $\mathrm{Jac}\,(Y)[2]$
with minimal polynomial $\Phi_{3}(t)=t^{2}+t+1$. If $p=7$, then $2$ has order
$3$ modulo $7$ and the factorization of $\Phi_{7}(t)$ into irreducible
polynomials is $\Phi_{7}(t)\equiv(x^{3}+x^{2}+1)(x^{3}+x+1)$ modulo $2$. Thus
the action of $\tau$ on $\mathrm{Jac}\,(Y)[2]$ can be represented by the
$6\times 6$-matrix

$\left(\begin{array}[]{cc}A_{1}&0\\\ 0&A_{2}\\\ \end{array}\right)$

where $A_{1}$ and $A_{2}$ are the irreducible $3$-dimensional companion
matrices of $x^{3}+x^{2}+1$ and $x^{3}+x+1$ respectively.

For the rest of the paper, let $\phi_{s}:Y_{s}\to\mathbb{P}^{1}_{k}$ be the
Artin-Schreier cover corresponding to the field extension

$k(x)\hookrightarrow k(x)[y]/(y^{p}-y-x^{s}).$

We show that $\phi_{s}$ can be dominated by a Galois cover of
$\mathbb{P}^{1}_{k}$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ for $a$
equal to the order of $\ell$ modulo $p$.

######  Proposition 3.4.

Let $s$ and $\ell$ be primes distinct from $p$. Let
$\phi_{s}:Y_{s}\to\mathbb{P}^{1}_{k}$ be the Artin-Schreier cover with affine
equation $y^{p}-y=x^{s}$. Let $a=\mathrm{ord}_{p}(\ell)$ be the order of
$\ell$ modulo $p$. Then there exists an unramified Galois cover
$\omega:Z_{a}\to Y_{s}$ with Galois group
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$ such that
$\psi_{a}=\phi_{s}\circ\omega:Z_{a}\rightarrow\mathbb{P}^{1}_{k}$ is a Galois
cover of $\mathbb{P}^{1}_{k}$ ramified only over $\infty$ whose Galois group
is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes\mathbb{Z}/p\mathbb{Z}$.

###### Proof.

By Lemma 3.2(1), $\phi_{s}:Y_{s}\rightarrow\mathbb{P}^{1}_{k}$ is a Galois
cover with Galois group $\mathbb{Z}/p\mathbb{Z}$ ramified only at the point
$P_{\infty}$ over $\infty$. The genus $g_{s}$ of $Y_{s}$ is $(p-1)(s-1)/2$.
Consider two commuting automorphisms of $Y_{s}$ defined by

$\tau:\begin{cases}x\mapsto x,\\\ y\mapsto
y+1,\end{cases}~{}~{}\sigma:\begin{cases}x\mapsto\zeta_{s}x,\text{ where
}\zeta_{s}\text{ is a primitive $s$th root of unity,}\\\ y\mapsto
y.\end{cases}$

Let $\mathrm{Jac}\,(Y_{s})$ be the Jacobian of $Y_{s}$. Then $\tau$ and
$\sigma$ define commuting automorphisms of $\mathrm{Jac}\,(Y_{s})$ of orders
$p$ and $s$ respectively. Therefore, $\mathrm{End}\,(\mathrm{Jac}\,(Y_{s}))$
contains a ring isomorphic to
$\mathbb{Z}[\zeta_{p},\zeta_{s}]\cong\mathbb{Z}[\zeta_{ps}]$, which is a
$\mathbb{Z}$-module of rank $\phi(ps)=(p-1)(s-1)=2g_{s}$. Then
${\mathbb{Q}}(\zeta_{ps})$ is contained in
$\mathrm{End}\,(\mathrm{Jac}\,(Y_{s}))\otimes{\mathbb{Q}}$. In other words,
$\mathrm{Jac}\,(Y_{s})$ has complex multiplication by
${\mathbb{Q}}(\zeta_{ps})$.

For a prime $\ell$ distinct from $p$, the automorphism $\tau$ induces an
action on the subgroup $\mathrm{Jac}\,(Y_{s})[\ell]$ of $\ell$-torsion points
of $\mathrm{Jac}\,(Y_{s})$. Recall that there is a bijection between
$\ell$-torsion points $D$ of $\mathrm{Jac}\,(Y_{s})$ and unramified
$(\mathbb{Z}/\ell\mathbb{Z})$-Galois covers $\omega_{D}:Z_{D}\to Y_{s}$ [6,
Prop. 4.11]. Also $D$ has order $\ell$ if and only if $Z_{D}$ is connected.
This induces a bijection between orbits of $\tau$ on the set of unramified
$(\mathbb{Z}/\ell\mathbb{Z})$-Galois covers $\omega_{D}:Z_{D}\to Y_{s}$ and on
the set of $\ell$-torsion points of $\mathrm{Jac}\,(Y_{s})$. For a point $D$
of order $\ell$ of $\mathrm{Jac}\,(Y_{s})$, consider the compositum
$\omega:Z\to Y_{s}$ of all of the conjugates
$\omega_{\tau^{j}(D)}:Z_{\tau^{i}(D)}\to Y_{s}$ for $0\leq j\leq p-1$:

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathbb{Z}}/\ell{\mathbb{Z}})}$$\textstyle{Z_{\tau(D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathbb{Z}}/\ell{\mathbb{Z}})}$$\textstyle{Z_{\tau^{2}(D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathbb{Z}}/\ell{\mathbb{Z}})}$$\textstyle{\ldots}$$\textstyle{Z_{\tau^{p-1}(D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\mathbb{Z}}/\ell{\mathbb{Z}})}$$\textstyle{Y_{s}}$

Then $Z$ is invariant under $\tau$ and so
$\phi_{s}\circ\omega:Z\to\mathbb{P}^{1}_{k}$ is Galois. Moreover,
$\phi_{s}\circ\omega$ is the Galois closure of
$\phi_{s}\circ\omega_{D}:Z_{D}\to\mathbb{P}^{1}_{k}$.

Suppose there is a non-trivial one-dimensional $\tau$-invariant subspace of
$\mathrm{Jac}\,(Y_{s})[\ell]$ with eigenvalue $1$; i.e. $\tau$ acts trivially
on this subgroup of order $\ell$. This yields a cover
$\psi_{s}\circ\omega_{1}:Z_{1}\to Y_{s}\to\mathbb{P}^{1}_{k}$. Since the
action of $\tau$ is trivial, $\psi_{s}\circ\omega_{1}$ is Galois, ramified
only over $\infty$, with abelian Galois group
$\mathbb{Z}/\ell\mathbb{Z}\times\mathbb{Z}/p\mathbb{Z}$. This contradicts
Lemma 2.4.

Since $\tau$ has order $p$, the minimal polynomial $m_{\tau}(t)$ of $\tau$
divides $t^{p}-1=(t-1)(t^{p-1}+\cdots+1)$ in $(\mathbb{Z}/\ell\mathbb{Z})[t]$.
From the preceding paragraph, there is no non-trivial one-dimensional
$\tau$-invariant subspace of $\mathrm{Jac}\,(Y_{s})[\ell]$ with eigenvalue
$1$. This implies that $m_{\tau}(t)$ divides the $p$th cyclotomic polynomial
$\Phi_{p}(t)=t^{p-1}+\cdots+1$ in $(\mathbb{Z}/\ell\mathbb{Z})[t]$. The
irreducible factors of $\Phi_{p}(t)$ in $(\mathbb{Z}/\ell\mathbb{Z})[t]$ all
have degree $a$. Thus the degree of $m_{\tau}(t)$ equals $a$.

Since $2g_{s}=(p-1)(s-1)$, we have
$\mathrm{Jac}\,(Y_{s})[\ell]\cong(\mathbb{Z}/\ell\mathbb{Z})^{(p-1)(s-1)}$, so
we can represent $\tau$ as an element of
$\mathrm{GL}_{(p-1)(s-1)}(\mathbb{Z}/\ell\mathbb{Z})$. We can choose a basis
of $\mathrm{Jac}\,(Y_{s})[\ell]$ such that $\tau$ is represented as an element
of $\mathrm{GL}_{(p-1)(s-1)}(\mathbb{Z}/\ell\mathbb{Z})$ in block form. The
first irreducible subrepresentation of $\tau$ has dimension $a$. Moreover,
since ${\mathbb{Q}}(\zeta_{ps})$ is a Galois extension of ${\mathbb{Q}}$, the
block form of $\tau$ consists entirely of irreducible blocks of the same size.
In particular, the number of irreducible blocks is $(p-1)(s-1)/a$. In other
words, $\tau$ can be represented by an element of
$\mathrm{GL}_{(s-1)(p-1)}(\mathbb{Z}/\ell\mathbb{Z})$ of the form

$\left(\begin{array}[]{ccc}A_{1}&&0\\\ &A_{2}&\\\ &\ddots&\\\
0&&A_{(p-1)(s-1)/a}\end{array}\right),$

where $A_{i}$ is an $a\times a$ matrix representing an $a$-dimensional
irreducible subrepresentation of $\tau$ on $\mathrm{Jac}\,(Y_{s})[\ell]$.

Using the bijection between orbits of $\mathrm{Jac}\,(Y_{s})[\ell]$ and orbits
of $(\mathbb{Z}/\ell\mathbb{Z})$-covers of $Y_{s}$ under $\tau$ and the above
observation for the action of $\tau$ on $\mathrm{Jac}\,(Y_{s})[\ell]$, there
exists an unramified $(\mathbb{Z}/\ell\mathbb{Z})^{a}$-Galois cover
$\omega:Z_{a}\to Y_{s}$ such that
$\psi_{a}=\phi_{s}\circ\omega:Z_{a}\rightarrow\mathbb{P}^{1}_{k}$ is a Galois
cover of $\mathbb{P}^{1}_{k}$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes\mathbb{Z}/p\mathbb{Z}$. Also
$\psi_{a}$ is ramified only over infinity since $\phi_{s}$ is ramified only
over $\infty$ and since $\omega$ is unramified. ∎

##  4\. Minimal genus of
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$-Galois
covers of $\mathbb{A}^{1}_{k}$

In this section, we find the minimal genus of a curve $Z$ that admits a
covering map $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$, with
Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. The
minimal genus depends only on $\ell$ and $p$. We consider the cases $p$ odd
and $p=2$ separately. We also prove that the number of curves $Z$ of this
minimal genus which admit such a covering map is at most $(p-1)/a$ when $p$ is
odd and at most $\ell+1$ when $p=2$. The following lemma will be useful.

######  Lemma 4.1.

Let $G$ be a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ where
$\ell$ is a prime distinct from $p$. If $\psi:Z\to\mathbb{P}^{1}_{k}$ is a
Galois cover ramified only over $\infty$ with Galois group $G$, then the
subcover $\omega:Z\to Y$ with Galois group
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}$ is unramified.

###### Proof.

The quotient of $G$ by the normal subgroup
$N=({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}$ is ${\mathbb{Z}}/p{\mathbb{Z}}$. Thus
the cover $\psi$ is a composition $\psi=\phi\circ\omega$ where
$\phi:Y\to\mathbb{P}^{1}_{k}$ has Galois group ${\mathbb{Z}}/p{\mathbb{Z}}$
and is totally ramified at the unique point $P_{\infty}$ over $\infty$ and
where $\omega:Z\to Y$ has Galois group $N$ and is branched only over
$P_{\infty}$. Then $\omega$ is a prime-to-$p$ abelian cover of $Y$. Let $g$ be
the genus of $Y$. Then by [1, XIII, Cor. 2.12], the prime-to-$p$ fundamental
group of $Y-\\{P_{\infty}\\}$ is isomorphic to the prime-to-$p$ quotient
$\Gamma$ of the free group on generators
$\\{a_{1},b_{1},\ldots,a_{g},b_{g},c\\}$ subject to the relation
$\prod_{i=1}^{g}[a_{i},b_{i}]=c^{-1}$. The cover $\omega$ corresponds to a
surjection of $\Gamma$ onto $N$ where $c$ maps to the canonical generator of
inertia $\gamma$ of a point of $Z$ over $P_{\infty}$. Thus $N$ is generated by
elements $\\{\alpha_{1},\beta_{1},\ldots,\alpha_{g},\beta_{g},\gamma\\}$
subject to the relation $\prod_{i=1}^{g}[\alpha_{i},\beta_{i}]=\gamma^{-1}$.
Then $\gamma=1$ since $N$ is abelian and so $\omega$ is unramified. ∎

######  Theorem 4.2.

Let $p$ be an odd prime. Let $\ell$ be a prime distinct from $p$ and let $a$
be the order of $\ell$ modulo $p$. Then:

  1. (1)

There exists a Galois cover $\psi_{a}:Z_{a}\to\mathbb{P}^{1}_{k}$ ramified
only over $\infty$ whose Galois group is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ such
that $g_{Z_{a}}=1+\ell^{a}(p-3)/2$.

  2. (2)

The integer $g_{Z_{a}}$ is the minimal genus of a curve $Z$ which admits a
covering map $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$ with
Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ for any
positive integer $b$.

  3. (3)

There are at most $(p-1)/a$ isomorphism classes of curves $Z$ which admit a
Galois covering map as in part (1) with minimal genus $g_{Z_{a}}$.

###### Proof.

By the construction in Proposition 3.4, there exists a Galois cover
$\psi_{a}:Z_{a}\to\mathbb{P}^{1}_{k}$ ramified only over $\infty$ whose Galois
group is a semi-direct product of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. We
compute the genus of the curve $Z_{a}$. Recall that $\psi_{a}$ is a
composition $\psi=\phi_{2}\circ\omega$ where $\omega:Z\to Y_{2}$ is an
unramified $({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois cover and
$\phi_{2}:Y_{2}\to\mathbb{P}^{1}_{k}$ has Artin-Schreier equation
$y^{p}-y=x^{2}$. Then $Y_{2}$ has genus $g_{Y_{2}}=(p-1)/2$ by Lemma 3.2(3).
By the Riemann-Hurwitz formula,
$2g_{Z_{a}}-2=\ell^{a}(2g_{Y_{2}}-2)=\ell^{a}(p-3)$, i.e.,
$g_{Z_{a}}=1+\ell^{a}(p-3)/2$. This completes part (1).

For part (2), suppose $\psi:Z\to\mathbb{P}^{1}_{k}$ is a Galois cover ramified
only over $\infty$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$. If $g$
is the genus of $Z$, we will show that $g\geq g_{Z_{a}}$. As described in the
proof of Lemma 4.1, the cover $\psi$ is a composition $\psi=\phi\circ\omega$
where $\phi:Y\to\mathbb{P}^{1}_{k}$ has Galois group $\mathbb{Z}/p\mathbb{Z}$
and is ramified only over $\infty$ and where $\omega$ is unramified with group
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{b}$. By the Riemann-Hurwitz formula,
$2g-2=\ell^{b}(2g_{Y}-2)$.

By Artin-Schreier theory, $\phi$ is given by an equation $y^{p}-y=f(x)$ where
$f\in k[x]$ has degree $s$ for some integer $s$ relatively prime to $p$. Since
the genus $g_{Y}$ of $Y$ is $(p-1)(s-1)/2$ by Lemma 3.2 (3), we should make
$s$ as small as possible. The value $s=1$ is impossible since then $Y$ is a
projective line and there do not exist Galois covers of the projective line
ramified only over one point with Galois group $\mathbb{Z}/\ell\mathbb{Z}$.
Thus $s=2$ yields the smallest possible value for $g_{Y}$, namely $(p-1)/2$.
Recall that $b\geq a$ by Lemma 2.4. Thus $g\geq 1+\ell^{a}(p-3)/2=g_{Z_{a}}$.

For part (3), suppose $\psi:Z\to\mathbb{P}^{1}_{k}$ is a Galois cover ramified
only over $\infty$ with Galois group of the form
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}\rtimes{\mathbb{Z}}/p{\mathbb{Z}}$ and the
genus of $Z$ satisfies $g_{Z}=1+\ell^{a}(p-3)/2$. As in part (2), $\psi$
factors as $\phi\circ\omega$ where $\omega:Z\to Y$ is an unramified
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois cover, where
$\phi:Y\to\mathbb{P}^{1}_{k}$ is an Artin-Schreier cover ramified only over
$\infty$, and where $Y$ has genus $(p-1)/2$. By Lemma 3.2(3), $Y$ has an
affine equation $y^{p}-y=a_{2}x^{2}+a_{1}x+a_{0}$ for some $a_{0},a_{1}\in k$
and $a_{2}\in k^{*}$. Since $p$ is odd and $k$ is algebraically closed, it is
possible to complete the square and write
$a_{2}x^{2}+a_{1}x+a_{0}=x_{1}^{2}+\epsilon$ where
$x_{1}=\sqrt{a_{2}}x+a_{1}/2\sqrt{a_{2}}$. After modifying by an automorphism
of the projective line, specifically by the affine linear transformation
$x\mapsto x_{1}$, the equation for $Y$ can be rewritten as
$y^{p}-y=x_{1}^{2}+\epsilon$. Since $k$ is algebraically closed, there exists
$\delta\in k$ such that $\delta^{p}-\delta=\epsilon$. Let $y_{1}=y-\delta$.
After the change of variables $y\mapsto y_{1}$, the curve $Y$ is isomorphic to
the curve $Y_{2}$ with affine equation $y_{1}^{p}-y_{1}=x_{1}^{2}$. Thus there
is a unique possibility for the isomorphism class of the curve $Y$.

From the proof of Proposition 3.4, there is a bijection between
$\tau$-invariant connected unramified
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois covers of $Y_{2}$ and orbits of
$\tau$ on points $D$ of order $\ell$ on $\mathrm{Jac}\,(Y_{2})$. The action of
$\tau$ on $\mathrm{Jac}\,(Y_{2})[\ell]$ decomposes into $(p-1)/a$ irreducible
subrepresentations. Each of these is distinct, because the irreducible factors
of $\Phi_{p}(t)\in(\mathbb{Z}/\ell\mathbb{Z})[t]$ are distinct. Thus there are
$(p-1)/a$ choices for a $\tau$-invariant unramified
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois cover of $Y_{2}$. Thus there are
at most $(p-1)/a$ isomorphism classes of curves $Z$ which admit a Galois
covering map as in part (1) with minimal genus $g_{Z_{a}}$. ∎

We note that the set of curves which are unramified
$({\mathbb{Z}}/\ell{\mathbb{Z}})^{a}$-Galois covers of $Y_{2}$ may contain
fewer than $(p-1)/a$ isomorphism classes of curves.

######  Theorem 4.3.

Let $p=2$ and let $\ell$ be an odd prime. Then:

  1. (1)

There exists a Galois cover $\psi:Z\to\mathbb{P}^{1}_{k}$ ramified only over
$\infty$ with Galois group of the form
$\mathbb{Z}/\ell\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$.

  2. (2)

The minimal genus of a curve $Z$ which admits a covering map as in part (1) is
$g_{Z}=1$.

  3. (3)

There are at most $\ell+1$ isomorphism classes of curves $Z$ which admit a
Galois covering map as in part (1) with minimal genus $g_{Z}=1$.

###### Proof.

Note that the order of $\ell$ modulo $2$ is $a=1$. For part (1), Lemma 2.5
shows that there exists a semi-direct product of the form
$\mathbb{Z}/\ell\mathbb{Z}\rtimes\mathbb{Z}/2\mathbb{Z}$ which is quasi-$2$.
The result is then immediate from Theorem 2.3.

Suppose $\psi:Z\to\mathbb{P}^{1}_{k}$ is a Galois cover ramified only over
$\infty$ with Galois group as in part (1). As before, $\psi$ factors as a
composition $\phi\circ\omega$. where $\omega:Z\to Y$ has Galois group
$\mathbb{Z}/\ell\mathbb{Z}$ and $\phi:Y\to\mathbb{P}^{1}_{k}$ is an Artin-
Schreier extension with affine equation $y^{2}-y=f(x)$ for some $f(x)\in k[x]$
of odd degree $s$. By Lemma 4.1, $\omega$ is unramified. The minimal genus for
$Z$ will thus occur when $s$ is as small as possible. As before, $s=1$ is
impossible, and so $s=3$ is the smallest choice. In this case, by Lemma
3.2(3), $g_{Y}=1$, i.e., $Y$ is an elliptic curve. By the Riemann-Hurwitz
formula, the minimal genus for $Z$ is $g_{Z}=1+\ell(g_{Y}-1)=1$, which
completes part (2).

For part (3), since $k$ is algebraically closed, we can complete the cube of
$f(x)$ and make the corresponding change of variables, which is a scaling and
translation of $x$. So we can assume that $Y$ has affine equation
$y^{2}-y=x^{3}+a_{1}x+a_{0}$ for some $a_{0},a_{1}\in k$. Then it follows from
[11, Appendix A, Prop. 1.1c] that the $j$-invariant of $Y$ is $j(Y)=0$ and
that the discriminant is $\Delta(Y)=(-1)^{4}=1$. Since $k$ is algebraically
closed, by [11, Appendix A, Prop. 1.2b], all elliptic curves $Y$ with $j(Y)=0$
are isomorphic over $k$. Thus there is a unique choice for $Y$ up to
isomorphism. Without loss of generality, we may assume that $Y=Y_{3}$ has
affine equation $y^{2}-y=x^{3}$.

From the proof of Proposition 3.4, the action of $\tau$ on
$\mathrm{Jac}\,(Y_{3})[\ell]$ decomposes into the direct sum of two
$1$-dimensional subrepresentations. In other words, the action of $\tau$ is
diagonal with both eigenvalues equal to $-1$. The number of non-trivial
$\tau$-invariant subgroups of $\mathrm{Jac}\,(Y_{3})[\ell]$ is the number of
subgroups of order $\ell$ in $(\mathbb{Z}/\ell\mathbb{Z})^{2}$, which is
$\ell+1$. As in Theorem 4.2, this implies that there are at most $\ell+1$
isomorphism classes of curves $Z$ which admit a Galois covering map as in part
(1) with minimal genus $g_{Z}=1$. ∎

We note that the set of curves which are unramified
$\mathbb{Z}/\ell\mathbb{Z}$-Galois covers of $Y_{3}$ may contain fewer than
$\ell+1$ isomorphism classes of curves.

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