Modularity of $\mathrm{PGL}_{2}({\mathbb{F}}_{p})$-representations over totally real fields

If $\chi:G_{K}\to k^{\times}$ is a character then its Teichmüller lift $X:G_{K}\to\overline{{\mathbb{Q}}}_{p}^{\times}$ is associated, by class field theory, to a finite order Hecke character $\Xi:{\mathbb{A}}_{K}^{\times}\to{\mathbb{C}}^{\times}$. If $\pi$ is a cuspidal automorphic representation which is regular algebraic of weight 0 and $\overline{r_{\iota}(\pi)}$ is conjugate to $\rho$, then $\pi\otimes(\Xi\circ\det)$ is also cuspidal and regular algebraic of weight 0 and $\overline{r_{\iota}(\pi\otimes(\Xi\circ\det))}$ is conjugate to $\rho\otimes\chi$.

It is clear from the definition that if $\rho$ is automorphic then so is $\sigma$. Conversely, if $\sigma$ is automorphic then there is a cuspidal, regular algebraic automorphic representation $\pi$ of $\mathrm{GL}_{2}(\mathbb{A}_{K})$ and isomorphism $\iota:\overline{{\mathbb{Q}}}_{p}\to{\mathbb{C}}$ such that $\mathrm{Proj}(\overline{r_{\iota}(\pi}))=\mathrm{Proj}(\rho)$. It follows that there exists a character $\chi:G_{K}\to\overline{{\mathbb{F}}}_{p}^{\times}$ such that $\rho$ is conjugate to $\overline{r_{\iota}(\pi)}\otimes\chi$. The automorphy of $\rho$ follows from the first part of the lemma. ∎

This follows from [Sno09, Theorem 7.6.1], on noting that the condition (A2) there can be replaced by the more general condition that ${\overline{\rho}}$ is non-exceptional (indeed, the condition (A2) is used to invoke [Kis09, Proposition 3.2.5], which is proved under this more general condition). To verify the existence of a potentially crystalline lift of ${\overline{\rho}}|_{G_{K_{v}}}$ for each $v|p$ (or in the terminology of loc. cit., the compatibility of ${\overline{\rho}}|_{G_{K_{v}}}$ with type $A$ or $B$) we apply [Sno09, Proposition 7.8.1] (when ${\overline{\rho}}|_{G_{K_{v}}}$ is irreducible) or [BLGG12, Lemma 6.1.6] (when ${\overline{\rho}}|_{G_{K_{v}}}$ is reducible). ∎

This follows from [KT17, Theorem 9.3]. ∎

Let $\psi:G_{K}\to\overline{{\mathbb{Z}}}_{p}^{\times}$ be the character such that $\psi\epsilon$ is the Teichmüller lift of $(\det{\overline{\rho}})\overline{\epsilon}$, and let $\rho:G_{K}\to\mathrm{GL}_{2}(\overline{{\mathbb{Z}}}_{p})$ be the lift of $\rho$ whose existence is asserted by Theorem 2.2. Then Theorem 2.3 implies the automorphy of $\rho|_{G_{L}}$, and the automorphy of $\rho$ itself and hence of ${\overline{\rho}}$ follows by cyclic descent, using the results of Langlands [Lan80]. ∎

Many of the results we quote here are stated in the case of $K={\mathbb{Q}}$ but hold more generally for totally real fields with minor modification. We will apply them in the more general setting without further comment.

If ${\overline{\rho}}$ is dihedral then this is a consequence of results of Hecke (see [Ser87, §5.1]). If $k={\mathbb{F}}_{3}$, it is a consequence of the Langlands–Tunnell theorem [Tun81] (see the discussion following Theorem 5.1 in [Wil95, Chapter 5]). We may thus assume for the remainder of the proof that $\lvert k\rvert>3$. We may also assume that for any abelian extension $L/K$, the restriction ${\overline{\rho}}|_{G_{L(\zeta_{p})}}$ is absolutely irreducible (as otherwise ${\overline{\rho}}$ would be dihedral).

Next suppose that $k={\mathbb{F}}_{5}$. We note that ${\overline{\rho}}$ is not exceptional, by [KT17, Lemma 3.1]. Let $L/K$ be the totally real cyclic extension cut out by $(\det{\overline{\rho}})\overline{\epsilon}$. By [SBT97, Theorem 1.2], there is an elliptic curve $E$ over $L$ such that ${\overline{\rho}}_{E,5}\cong{\overline{\rho}}|_{G_{L}}$ and ${\overline{\rho}}_{E,3}(G_{L})$ contains $\mathrm{SL}_{2}({\mathbb{F}}_{3})$. By the $k={\mathbb{F}}_{3}$ case of the theorem and by Theorem 2.3, we see that $E$ is automorphic, hence so is ${\overline{\rho}}|_{G_{L}}$. The automorphy of ${\overline{\rho}}$ then follows from Proposition 2.4. The $k={\mathbb{F}}_{7}$ case is similar, using [Man04, Proposition 3.1] instead of [SBT97, Theorem 1.2].

Next suppose that $k={\mathbb{F}}_{4}$. We can twist $\rho$ to assume that it is valued in $\mathrm{SL}_{2}({\mathbb{F}}_{4})$. Then [SBT97, Theorem 3.4] shows that there is an abelian surface $A$ over $F$ with real multiplication by ${\mathcal{O}}_{{\mathbb{Q}}(\sqrt{5})}$ such that the $G_{K}$-representation on $A[2]\cong{\mathbb{F}}_{4}^{2}$ is isomorphic to ${\overline{\rho}}$ and such that the $G_{K}$-representation on $A[\sqrt{5}]\cong{\mathbb{F}}_{5}^{2}$ has image containing $\mathrm{SL}_{2}({\mathbb{F}}_{5})$. By the $k={\mathbb{F}}_{5}$ case of the theorem, Theorem 2.2, and Theorem 2.3, we see that $A$ is automorphic, hence so is ${\overline{\rho}}$.

Finally suppose that $k={\mathbb{F}}_{9}$. Let $L/K$ be the totally real cyclic extension cut out by $(\det{\overline{\rho}})\overline{\epsilon}$. Then the argument of [Ell05, §2.5] shows that there is a solvable totally real extension $M/K$ containing $L/K$ and an abelian surface $A$ over $M$ with real multiplication by ${\mathcal{O}}_{{\mathbb{Q}}(\sqrt{5})}$ such that the $G_{M}$-representation on $A[3]\cong{\mathbb{F}}_{9}^{2}$ is isomorphic to $\overline{\rho}|_{G_{M}}\otimes\overline{\epsilon}$ and such that the $G_{M}$-representation on $A[\sqrt{5}]\cong{\mathbb{F}}_{5}^{2}$ has image containing $\mathrm{SL}_{2}({\mathbb{F}}_{5})$. By the $k={\mathbb{F}}_{5}$ case of the theorem, Theorem 2.2, and Theorem 2.3, we see that $A$ is automorphic, hence so is ${\overline{\rho}}|_{G_{M}}$. The automorphy of ${\overline{\rho}}$ follows from Proposition 2.4. ∎

The obstruction to lifting a continuous homomorphism $\sigma:G_{K}\to\mathrm{PGL}_{2}(k)$ to a continuous homomorphism $\rho:G_{K}\to\mathrm{GL}_{2}(\overline{k})$ lies in $H^{2}(G_{K},\overline{k}^{\times})$. Tate proved that $H^{2}(G_{K},\overline{k}^{\times})=0$ (see [Ser77, §6.5]) so a lift always exists. ∎

Let $H$ denote the $2$-Sylow subgroup of $k^{\times}$, of order $2^{m}$, and let $H^{\prime}\leq k^{\times}$ denote its prime-to-2 complement. If $0\leq k\leq m$, we write $G_{k}=\mathrm{GL}_{2}(k)/(2^{m-k}H\times H^{\prime})$, which is an extension

We show by induction on $k\geq 0$ that we can find a solvable, $S$-split extension $L_{k}/K$ and a homomorphism $\rho_{k}:G_{L_{k}}\to G_{k}$ lifting $\sigma|_{G_{L_{k}}}$ and such that for each $v\in S$ and each place $w|v$ of $L_{k}$, $\rho_{k}|_{G_{L_{k,w}}}=\rho_{v}\text{ mod }2^{m-k}H\times H^{\prime}$. The case $k=0$ is the existence of $\sigma$. The case $k=m$ implies the statement of the lemma, since $\mathrm{GL}_{2}(k)=G_{m}\times H^{\prime}$. (Note [AT09, Ch. X, Theorem 5] implies that any collection of characters $\chi_{v}:G_{K_{v}}\to H^{\prime}$ can be globalised to a character $\chi:G_{K}\to H^{\prime}$.)

For the induction step, suppose the induction hypothesis holds for a fixed value of $k$. We consider the obstruction to lifting $\rho_{k}$ to a homomorphism $\rho_{k+1}:G_{L_{k}}\to G_{k+1}$. This defines an element of $H^{2}(G_{L_{k}},{\mathbb{Z}}/2{\mathbb{Z}})$ which is locally trivial at the places of $L_{k}$ lying above $S$. We can therefore find an extension of the form $L_{k+1}=L_{k}\cdot E_{k+1}$, where $E_{k+1}/K$ is a solvable $S$-split extension, such that the image of this obstruction class in $H^{2}(G_{L_{k+1}},{\mathbb{Z}}/2{\mathbb{Z}})$ vanishes and so there is a homomorphism $\rho^{\prime}_{k+1}:G_{L_{k+1}}\to G_{k+1}$ lifting $\rho_{k}|_{G_{L_{k+1}}}$.

If $v\in S$ and $w|v$ is a place of $L_{k+1}$ then there is a character $\chi_{w}:G_{L_{k+1,w}}\to{\mathbb{Z}}/2{\mathbb{Z}}$ such that $\rho^{\prime}_{k+1}|_{G_{L_{k+1,w}}}=(\rho_{v}\text{ mod }2^{m-(k+1)}H\times H^{\prime})\cdot\chi_{w}$. We can certainly find a character $\chi:G_{L_{k+1}}\to{\mathbb{Z}}/2{\mathbb{Z}}$ such that $\chi|_{G_{L_{k+1,w}}}=\chi_{w}$ for each such place $w$. The induction step is complete on taking $\rho_{k+1}=\rho^{\prime}_{k+1}\cdot\chi$.

It remains to explain why we can choose $K$ to be CM if $L$ is. Since the extensions $E_{k}$ in the proof are required only to satisfy some local conditions, which are vacuous if $K$ is CM, we can choose the fields $E_{k}$ to be of the form $KE_{k}’$ where $E_{k}’$ is a totally real extension, in which case the field $L$ constructed in the proof is seen to be CM. ∎

We consider the short exact sequence of groups

where the last group is the subgroup of $(g,\alpha)\in\mathrm{PGL}_{2}(k)\times k^{\times}$ such that $\Delta(g)=\alpha\text{ mod }(k^{\times})^{2}$. By hypothesis the pair $(\sigma,\chi)$ defines a homomorphism $\Sigma:G_{K}\to\mathrm{PGL}_{2}(k)\times_{\Delta}k^{\times}$ such that for every place $v$ of $K$, $\Sigma|_{G_{K_{v}}}$ lifts to $\mathrm{GL}_{2}(k)$ (see Remark 2.9). The subgroup of locally trivial elements of $H^{2}(G_{K},\{\pm 1\})$ is trivial, by class field theory, so $\Sigma$ lifts to a homomorphism $\rho:G_{K}\to\mathrm{GL}_{2}(k)$, as required. ∎

By Lemma 2.7, we can lift $\sigma$ to a representation ${\overline{\rho}}:G_{K}\to\mathrm{GL}_{2}(\overline{k})$. Then ${\overline{\rho}}|_{G_{L}}$ is automorphic and we can apply Proposition 2.4 to conclude that ${\overline{\rho}}$ is automorphic, hence that $\sigma$ is automorphic. ∎

When $k={\mathbb{F}}_{2}$ or ${\mathbb{F}}_{4}$, the map $\mathrm{SL}_{2}(k)\to\mathrm{PGL}_{2}(k)$ is an isomorphism, so $\sigma$ trivially lifts to a $\mathrm{GL}_{2}(k)$ representation and we can apply Theorem 2.5. The case when $|k|=3$ follows from [Tun81]. In the other cases, we can assume that $\sigma|_{G_{K(\zeta_{p})}}$ is absolutely irreducible (as otherwise $\sigma$ lifts to a dihedral representation). Let $S_{\infty}$ be the set of infinite places of $K$ and choose a finite set $S^{\prime}$ of finite places of $K$ at which $\sigma$ is unramified such that $\operatorname{Gal}(\overline{K}^{\ker(\sigma|_{G_{K(\zeta_{p})}})}/K)$ is generated by $\{\operatorname{Frob}_{v}\}_{v\in S^{\prime}}$. We can apply Lemma 2.8, see also Remark 2.9, with $S=S_{\infty}\cup S^{\prime}$ to find a solvable, totally real extension $L/K$ such that $\sigma$ lifts to a representation ${\overline{\rho}}:G_{L}\to\mathrm{GL}_{2}(k)$ such that ${\overline{\rho}}|_{G_{L(\zeta_{p})}}$ is absolutely irreducible and ${\overline{\rho}}$ is not exceptional if $p=5$. Then Theorem 2.5 implies the automorphy of ${\overline{\rho}}$ and Proposition 2.11 implies the automorphy of $\sigma$, as desired. ∎