On the modularity of elliptic curves over $\mathbf{Q}$: Wild $3$-adic exercises
By Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor
Abstract
We complete the proof that every elliptic curve over the rational numbers is modular.
Introduction
In this paper, building on work of Wiles ReferenceWi and of Taylor and Wiles ReferenceTW, we will prove the following two theorems (see §2.2).
Theorem A
If $E_{/\mathbf{Q}}$ is an elliptic curve, then $E$ is modular.
Theorem B
If ${\overline{\rho }}:\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \rightarrow {\operatorname {GL}}_2(\mathbf{F}_5)$ is an irreducible continuous representation with cyclotomic determinant, then ${\overline{\rho }}$ is modular.
We will first remind the reader of the content of these results and then briefly outline the method of proof.
If $N$ is a positive integer, then we let $\Gamma _1(N)$ denote the subgroup of $\operatorname {SL}_2(\mathbf{Z})$ consisting of matrices that modulo $N$ are of the form
The quotient of the upper half plane by $\Gamma _1(N)$, acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve $Y_1(N)_{/\mathbf{C}}$. This curve has a natural model $Y_1(N)_{/\mathbf{Q}}$, which for $N >3$ is a fine moduli scheme for elliptic curves with a point of exact order $N$. We will let $X_1(N)$ denote the smooth projective curve which contains $Y_1(N)$ as a dense Zariski open subset.
Recall that a cusp form of weight $k\geq 1$ and level $N\geq 1$ is a holomorphic function $f$ on the upper half complex plane $\mathfrak{H}$ such that
•
for all matrices$$\begin{equation*} \left( \begin{array}{cc} {a} & {b} \\{c} & {d} \end{array} \right) \in \Gamma _1(N) \end{equation*}$$
and all $z \in \mathfrak{H}$, we have $f((az+b)/(cz+d))=(cz+d)^k f(z)$;
•
and $|f(z)|^2 (\operatorname {Im}z)^k$ is bounded on $\mathfrak{H}$.
The space $S_k(N)$ of cusp forms of weight $k$ and level $N$ is a finite-dimensional complex vector space. If $f \in S_k(N)$, then it has an expansion
$$\begin{equation*} f(z) = \sum _{n=1}^\infty c_n(f) e^{2\pi i nz} \end{equation*}$$
with $c \equiv 0 \bmod N$ and $d \equiv p \bmod N$. The operators $T_p$ for $p \nmid N$ can be simultaneously diagonalised on the space $S_k(N)$ and a simultaneous eigenvector is called an eigenform. If $f$ is an eigenform, then the corresponding eigenvalues, $a_p(f)$, are algebraic integers and we have $c_p(f)=a_p(f)c_1(f)$.
Let $\lambda$ be a place of the algebraic closure of $\mathbf{Q}$ in $\mathbf{C}$ above a rational prime $\ell$ and let $\overline{\mathbf{Q}}_\lambda$ denote the algebraic closure of $\mathbf{Q}_\ell$ thought of as a $\overline{\mathbf{Q}}$ algebra via $\lambda$. If $f \in S_k(N)$ is an eigenform, then there is a unique continuous irreducible representation
such that for any prime $p\nmid Nl$,$\rho _{f,\lambda }$ is unramified at $p$ and $\operatorname {tr}\rho _{f,\lambda }(\operatorname {Frob}_p)=a_p(f)$. The existence of $\rho _{f,\lambda }$ is due to Shimura if $k=2$ReferenceSh2, to Deligne if $k>2$ReferenceDe and to Deligne and Serre if $k=1$ReferenceDS. Its irreducibility is due to Ribet if $k>1$ReferenceRi and to Deligne and Serre if $k=1$ReferenceDS. Moreover $\rho$ is odd in the sense that $\det \rho$ of complex conjugation is $-1$. Also, $\rho _{f,\lambda }$ is potentially semi-stable at $\ell$ in the sense of Fontaine. We can choose a conjugate of $\rho _{f,\lambda }$ which is valued in ${\operatorname {GL}}_2({\mathcal {O}}_{\overline{\mathbf{Q}}_\lambda })$, and reducing modulo the maximal ideal and semi-simplifying yields a continuous representation
which, up to isomorphism, does not depend on the choice of conjugate of $\rho _{f,\lambda }$.
Now suppose that $\rho :G_\mathbf{Q}\rightarrow {\operatorname {GL}}_2(\overline{\mathbf{Q}}_\ell )$ is a continuous representation which is unramified outside finitely many primes and for which the restriction of $\rho$ to a decomposition group at $\ell$ is potentially semi-stable in the sense of Fontaine. To $\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_\ell /\mathbf{Q}_\ell )}$ we can associate both a pair of Hodge-Tate numbers and a Weil-Deligne representation of the Weil group of $\mathbf{Q}_\ell$. We define the conductor $N(\rho )$ of $\rho$ to be the product over $p \neq \ell$ of the conductor of $\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)}$ and of the conductor of the Weil-Deligne representation associated to $\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_\ell /\mathbf{Q}_\ell )}$. We define the weight $k(\rho )$ of $\rho$ to be $1$ plus the absolute difference of the two Hodge-Tate numbers of $\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_\ell /\mathbf{Q}_\ell )}$. It is known by work of Carayol and others that the following two conditions are equivalent:
•
$\rho \sim \rho _{f,\lambda }$ for some eigenform $f$ and some place $\lambda |\ell$;
•
$\rho \sim \rho _{f,\lambda }$ for some eigenform $f$ of level $N(\rho )$ and weight $k(\rho )$ and some place $\lambda |\ell$.
When these equivalent conditions are met we call $\rho$modular. It is conjectured by Fontaine and Mazur that if $\rho :G_\mathbf{Q}\rightarrow {\operatorname {GL}}_2(\overline{\mathbf{Q}}_\ell )$ is a continuous irreducible representation which satisfies
•
$\rho$ is unramified outside finitely many primes,
•
$\rho |_{\operatorname {Gal}(\overline{\mathbf{Q}}_\ell /\mathbf{Q}_\ell )}$ is potentially semi-stable with its smaller Hodge-Tate number $0$,
•
and, in the case where both Hodge-Tate numbers are zero, $\rho$ is odd,
Next consider a continuous irreducible representation ${\overline{\rho }}:\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \rightarrow {\operatorname {GL}}_2(\overline{\mathbf{F}}_\ell )$. Serre ReferenceSe2 defines the conductor $N({\overline{\rho }})$ and weight $k({\overline{\rho }})$ of ${\overline{\rho }}$. We call ${\overline{\rho }}$modular if ${\overline{\rho }}\sim {\overline{\rho }}_{f,\lambda }$ for some eigenform $f$ and some place $\lambda |\ell$. We call ${\overline{\rho }}$strongly modular if moreover we may take $f$ to have weight $k({\overline{\rho }})$ and level $N({\overline{\rho }})$. It is known from work of Mazur, Ribet, Carayol, Gross, Coleman, Voloch and others that for $\ell \geq 3$,${\overline{\rho }}$ is strongly modular if and only if it is modular (see ReferenceDi1). Serre has conjectured that all odd, irreducible ${\overline{\rho }}$ are strongly modular ReferenceSe2.
Now consider an elliptic curve $E_{/\mathbf{Q}}$. Let $\rho _{E,\ell }$ (resp. ${\overline{\rho }}_{E,\ell }$) denote the representation of $\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ on the $\ell$-adic Tate module (resp. the $\ell$-torsion) of $E(\overline{\mathbf{Q}})$. Let $N(E)$ denote the conductor of $E$. It is known that the following conditions are equivalent:
(1)
The $L$-function$L(E,s)$ of $E$ equals the $L$-function$L(f,s)$ for some eigenform $f$.
(2)
The $L$-function$L(E,s)$ of $E$ equals the $L$-function$L(f,s)$ for some eigenform $f$ of weight $2$ and level $N(E)$.
(3)
For some prime $\ell$, the representation $\rho _{E,\ell }$ is modular.
(4)
For all primes $\ell$, the representation $\rho _{E,\ell }$ is modular.
(5)
There is a non-constant holomorphic map $X_1(N)(\mathbf{C}) \rightarrow E(\mathbf{C})$ for some positive integer $N$.
(6)
There is a non-constant morphism $X_1(N(E)) \rightarrow E$ which is defined over $\mathbf{Q}$.
The implications (2) $\Rightarrow$ (1), (4) $\Rightarrow$ (3) and (6) $\Rightarrow$ (5) are tautological. The implication (1) $\Rightarrow$ (4) follows from the characterisation of $L(E,s)$ in terms of $\rho _{E,\ell }$. The implication (3) $\Rightarrow$ (2) follows from a theorem of Carayol ReferenceCa1. The implication (2) $\Rightarrow$ (6) follows from a construction of Shimura ReferenceSh2 and a theorem of Faltings ReferenceFa. The implication (5) $\Rightarrow$ (3) seems to have been first noticed by Mazur ReferenceMaz. When these equivalent conditions are satisfied we call $E$modular.
It has become a standard conjecture that all elliptic curves over $\mathbf{Q}$ are modular, although at the time this conjecture was first suggested the equivalence of the conditions above may not have been clear. Taniyama made a suggestion along the lines (1) as one of a series of problems collected at the Tokyo-Nikko conference in September 1955. However his formulation did not make clear whether $f$ should be a modular form or some more general automorphic form. He also suggested that constructions as in (5) and (6) might help attack this problem at least for some elliptic curves. In private conversations with a number of mathematicians (including Weil) in the early 1960’s, Shimura suggested that assertions along the lines of (5) and (6) might be true (see ReferenceSh3 and the commentary on [1967a] in ReferenceWe2). The first time such a suggestion appears in print is Weil’s comment in ReferenceWe1 that assertions along the lines of (5) and (6) follow from the main result of that paper, a construction of Shimura and from certain “reasonable suppositions” and “natural assumptions”. That assertion (1) is true for CM elliptic curves follows at once from work of Hecke and Deuring. Shimura ReferenceSh1 went on to check assertion (5) for these curves.
Our approach to Theorem A is an extension of the methods of Wiles ReferenceWi and of Taylor and Wiles ReferenceTW. We divide the proof into three cases.
(1)
If ${\overline{\rho }}_{E,5}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {5}))}$ is irreducible, we show that $\rho _{E,5}$ is modular.
(2)
If ${\overline{\rho }}_{E,5}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {5}))}$ is reducible, but ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {-3}))}$ is absolutely irreducible, we show that $\rho _{E,3}$ is modular.
(3)
If ${\overline{\rho }}_{E,5}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {5}))}$ is reducible and ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}/\mathbf{Q}(\sqrt {-3}))}$ is absolutely reducible, then we show that $E$ is isogenous to an elliptic curve with $j$-invariant$0$,$(11/2)^3$, or $-5(29)^3/2^5$ and so (from tables of modular elliptic curves of low conductor) is modular.
In each of cases (1) and (2) there are two steps. First we prove that ${\overline{\rho }}_{E,\ell }$ is modular and then that $\rho _{E,\ell }$ is modular. In case (1) this first step is our Theorem B and in case (2) it is a celebrated theorem of Langlands and Tunnell ReferenceL, ReferenceT. In fact, in both cases $E$ obtains semi-stable reduction over a tame extension of $\mathbf{Q}_\ell$ and the deduction of the modularity of $\rho _{E,\ell }$ from that of ${\overline{\rho }}_{E,\ell }$ was carried out in ReferenceCDT by an extension of the methods of ReferenceWi and ReferenceTW. In the third case we have to analyse the rational points on some modular curves of small level. This we did, with Elkies’ help, in ReferenceCDT.
It thus only remained to prove Theorem B. Let ${\overline{\rho }}$ be as in that theorem. Twisting by a quadratic character, we may assume that ${\overline{\rho }}|_{\operatorname {Gal}( \overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ falls into one of the following cases (see §2.2):
(1)
${\overline{\rho }}$ is unramified at $3$.
(2)
${\overline{\rho }}(I_3)$ has order $5$.
(3)
${\overline{\rho }}(I_3)$ has order $4$.
(4)
${\overline{\rho }}(I_3)$ has order $12$ and ${\overline{\rho }}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ has conductor $27$.
(5)
${\overline{\rho }}(I_3)$ has order $3$.
(6)
${\overline{\rho }}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ is induced from a character $\chi : \operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3(\sqrt {-3})) \to \mathbf{F}_{25}^\times$ such that $\chi (-1)=-1$ and$$\begin{equation*} \chi (\sqrt {-3})=\chi (1+3\sqrt {-3})-\chi (1-3\sqrt {-3}), \end{equation*}$$
where we use the Artin map (normalised to take uniformisers to arithmetic Frobenius) to identify $\chi$ with a character of $\mathbf{Q}_3(\sqrt {-3})^\times$.
We will refer to these as the $f=1,3,9,27, 81$ and $243$ cases respectively.
Using the technique of Minkowski and Klein (i.e. the observation that the moduli space of elliptic curves with full level $5$ structure has genus $0$; see for example ReferenceKl), Hilbert irreducibility and some local computations of Manoharmayum ReferenceMan, we find an elliptic curve $E_{/\mathbf{Q}}$ with the following properties (see §2.2):
${\overline{\rho }}_{E,3}$ is surjective onto ${\operatorname {GL}}_2(\mathbf{F}_3)$,
•
and
(1)
in the $f=1$ case, either ${\overline{\rho }}_{E,3}|_{I_3} \otimes \mathbf{F}_9 \sim \omega _2\oplus \omega _2^3$ or$$\begin{equation*} {\overline{\rho }}_{E,3}|_{I_3} \sim \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \end{equation*}$$
and is peu ramifié;
(2)
in the $f=3$ case,$$\begin{equation*} {\overline{\rho }}_{E,3}|_{I_3} \sim \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right); \end{equation*}$$
(3)
in the $f=9$ case, ${\overline{\rho }}_{E,3}|_{I_3} \otimes \mathbf{F}_9 \sim \omega _2\oplus \omega _2^3$;
(4)
in the $f=27$ case,$$\begin{equation*} {\overline{\rho }}_{E,3}|_{I_3} \sim \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \end{equation*}$$
and is très ramifié;
(5)
in the $f=81$ case,$$\begin{equation*} {\overline{\rho }}_{E,3}|_{I_3} \sim \left( \begin{array}{cc} {1} & {*} \\{0} & {\omega } \end{array} \right) \end{equation*}$$
and is très ramifié;
(6)
in the $f=243$ case,$$\begin{equation*} {\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)} \sim \left( \begin{array}{cc} {\omega } & {*} \\{0} & {1} \end{array} \right) \end{equation*}$$
is non-split over $\overline{\mathbf{Q}}_3^{\ker {\overline{\rho }}}$ and is très ramifié.
(We are using the terms très ramifié and peu ramifié in the sense of Serre ReferenceSe2. We are also letting $\omega$ denote the $\bmod 3$ cyclotomic character and $\omega _2$ the second fundamental character $I_3 \rightarrow \mathbf{F}_{9}^\times$, i.e.
We will often use the equality $\omega =\omega ^{-1}$ without further remark.) We emphasise that for a general elliptic curve over $\mathbf{Q}$ with ${\overline{\rho }}_{E,5} \cong {\overline{\rho }}$, the representation ${\overline{\rho }}_{E,3}$ does not have the above form, rather we are placing a significant restriction on $E$.
In each case our strategy is to prove that $\rho _{E,3}$ is modular and so deduce that ${\overline{\rho }}\sim {\overline{\rho }}_{E,5}$ is modular. Again we use the Langlands-Tunnell theorem to see that ${\overline{\rho }}_{E,3}$ is modular and then an analogue of the arguments of ReferenceWi and ReferenceTW to conclude that $\rho _{E,3}$ is modular. This was carried out in ReferenceDi2 in the cases $f=1$ and $f=3$, and in ReferenceCDT in the case $f=9$. (In these cases the particular form of ${\overline{\rho }}_{E,3}|_{I_3}$ is not important.) This leaves the cases $f=27$,$81$ and $243$, which are complicated by the fact that $E$ now only obtains good reduction over a wild extension of $\mathbf{Q}_3$. In these cases our argument relies essentially on the particular form we have obtained for ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ (depending on ${\overline{\rho }}_{E,5}|_{I_3}$). We do not believe that our methods for deducing the modularity of $\rho _{E,3}$ from that of ${\overline{\rho }}_{E,3}$ would work without this key simplification. It seems to be a piece of undeserved good fortune that for each possibility for ${\overline{\rho }}|_{I_3}$ we can find a choice for ${\overline{\rho }}_{E,3}|_{\operatorname {Gal}(\overline{\mathbf{Q}}_3/\mathbf{Q}_3)}$ for which our methods work.
Following Wiles, to deduce the modularity of $\rho _{E,3}$ from that of ${\overline{\rho }}_{E,3}$, we consider certain universal deformations of ${\overline{\rho }}_{E,3}$ and identify them with certain modular deformations which we realise over certain Hecke algebras. The key problem is to find the right local condition to impose on these deformations at the prime $3$. As in ReferenceCDT we require that the deformations lie in the closure of the characteristic zero points which are potentially Barsotti-Tate (i.e. come from a $3$-divisible group over the ring of integers of a finite extension of $\mathbf{Q}_3$) and for which the associated representation of the Weil group (see for example Appendix B of ReferenceCDT) is of some specified form. That one can find suitable conditions on the representation of the Weil group at $3$ for the arguments of ReferenceTW to work seems to be a rare phenomenon in the wild case. It is here we make essential use of the fact that we need only treat our specific pairs $({\overline{\rho }}_{E,5},{\overline{\rho }}_{E,3})$.
Our arguments follow closely the arguments of ReferenceCDT. There are two main new features. Firstly, in the $f=243$ case, we are forced to specify the restriction of our representation of the Weil group completely, rather than simply its restriction to the inertia group as we have done in the past. Secondly, in the key computation of the local deformation rings, we now make use of a new description (due to Breuil) of finite flat group schemes over the ring of integers of any $p$-adic field in terms of certain (semi-)linear algebra data (see ReferenceBr2 and the summary ReferenceBr1). This description enables us to make these computations. As the persistent reader will soon discover they are lengthy and delicate, particularly in the case $f=243$. It seems miraculous to us that these long computations with finite flat group schemes in §7, §8 and §9 give answers completely in accord with predictions made from much shorter computations with the local Langlands correspondence and the modular representation theory of ${\operatorname {GL}}_2(\mathbf{Q}_3)$ in §3. We see no direct connection, but cannot help thinking that some such connection should exist.
Notation
In this paper $\ell$ denotes a rational prime. In §1.1, §4.1, §4.2 and §4.3 it is arbitrary. In the rest of §1 and in §5 we only assume it is odd. In the rest of the paper we only consider $\ell =3$.
If $F$ is a field we let $\overline{F}$ denote a separable closure, $F^{\operatorname {ab}}$ the maximal subextension of $\overline{F}$ which is abelian over $F$ and $G_F$ the Galois group $\operatorname {Gal}(\overline{F}/F)$. If $F_0$ is a $p$-adic field (i.e. a finite extension of $\mathbf{Q}_p$) and $F'/F_0$ a (possibly infinite) Galois extension, then we let $I_{F'/F_0}$ denote the inertia subgroup of $\operatorname {Gal}(F'/F_0)$. We also let $I_{F_0}$ denote $I_{\overline{F}_0 /F_0}$,$\operatorname {Frob}_{F_0}\in G_{F_0}/I_{F_0}$ denote the arithmetic Frobenius element and $W_{F_0}$ denote the Weil group of $F_0$, i.e. the dense subgroup of $G_{F_0}$ consisting of elements which map to an integer power of $\operatorname {Frob}_{F_0}$. We will normalise the Artin map of local class field theory so that uniformisers and arithmetic Frobenius elements correspond. (We apologise for this convention, which now seems to us a bad choice. However we feel it is important to stay consistent with ReferenceCDT.) We let ${\mathcal {O}}_{F_0}$ denote the ring of integers of $F_0$,$\wp _{F_0}$ the maximal ideal of ${\mathcal {O}}_{F_0}$ and $k_{F_0}$ the residue field ${\mathcal {O}}_{F_0}/\wp _{F_0}$. We write simply $G_p$ for $G_{\mathbf{Q}_p}$,$I_p$ for $I_{\mathbf{Q}_p}$ and $\operatorname {Frob}_p$ for $\operatorname {Frob}_{\mathbf{Q}_p}$. We also let $\mathbf{Q}_{p^n}$ denote the unique unramified degree $n$ extension of $\mathbf{Q}_p$ in $\overline{\mathbf{Q}}_p$. If $k$ is any perfect field of characteristic $p$ we also use $\operatorname {Frob}_p$ to denote the $p^{th}$-power automorphism of $k$ and its canonical lift to the Witt vectors $W(k)$.
We write $\epsilon$ for the $\ell$-adic cyclotomic character and sometimes $\omega$ for the reduction of $\epsilon$ modulo $\ell$. We write $\omega _2$ for the second fundamental character $I_\ell \rightarrow \mathbf{F}_{\ell ^2}^\times$, i.e.
We also use $\omega$ and $\omega _2$ to denote the Teichmuller lifts of $\omega$ and $\omega _2$.
We let ${\mathbf{1}}$ denote the trivial character of a group. We will denote by $V^\vee$ the dual of a vector space $V$.
If $g:A \rightarrow B$ is a homomorphism of rings and if $X_{/\operatorname {Spec}A}$ is an $A$-scheme, then we sometimes write ${^gX}$ for the pullback of $X$ by $\operatorname {Spec}g$. We adopt this notation so that ${}^g({}^hX)={}^{gh}X$. Similarly if $\theta :X \rightarrow Y$ is a morphism of schemes over $A$ we will sometimes write ${^g\theta }$ for the pullback of $\theta$ by $\operatorname {Spec}g$.
By finite flat group scheme we always mean commutative finite flat group scheme. If $F_0$ is a field of characteristic $0$ with fixed algebraic closure $\overline{F}_0$ we use without comment the canonical identification of finite flat $F_0$-group schemes with finite discrete $\operatorname {Gal}(\overline{F}_0/F_0)$-modules, and we will say that such objects correspond. If $R$ is a Dedekind domain with field of fractions $F$ of characteristic $0$, then by a model of a finite flat $F$-group scheme $G$ we mean a finite locally free $R$-group scheme ${\mathcal {G}}$ and an isomorphism $i: G \stackrel {\sim }{\rightarrow }{\mathcal {G}}\times F'$. As in Proposition 2.2.2 of ReferenceRa the isomorphism classes of models for $G$ form a lattice ($({\mathcal {G}},i) \geq ({\mathcal {G}}',i')$ if there exists a map of finite flat group schemes ${\mathcal {G}}\rightarrow {\mathcal {G}}'$ compatible with $i$ and $i'$) and we can talk about the inf and sup of two such models. If $R$ is also local we call the model $({\mathcal {G}},i)$ local-local if its special fibre is local-local. When the ring $R$ is understood we sometimes simply refer to $({\mathcal {G}},i)$, or even just ${\mathcal {G}}$, as an integral model of $G$.
We use Serre’s terminology peu ramifié and très ramifié; see ReferenceSe2.
1. Types
1.1. Types of local deformations
By an $\ell$-type we mean an equivalence class of two-dimensional representations
over $\overline{\mathbf{Q}}_\ell$ which have open kernel and which can be extended to a representation of $W_{\mathbf{Q}_\ell }$. By an extended $\ell$-type we shall simply mean an equivalence class of two-dimensional representations
over $\overline{\mathbf{Q}}_\ell$ with open kernel.
Suppose that $\tau$ is an $\ell$-type and that $K$ is a finite extension of $\mathbf{Q}_\ell$ in $\overline{\mathbf{Q}}_\ell$. Recall from ReferenceCDT that a continuous representation $\rho$ of $G_\ell$ on a two-dimensional $K$-vector space $M$ is said to be of type$\tau$ if
(1)
$\rho$ is Barsotti-Tate over $F$ for any finite extension $F$ of $\mathbf{Q}_\ell$ such that $\tau |_{I_F}$ is trivial;
(2)
the restriction of $WD(\rho )$ to $I_\ell$ is in $\tau$;
(3)
the character $\epsilon ^{-1} \det \rho$ has finite order prime to $\ell$.
(For the definition of “Barsotti-Tate” and of the representation $WD(\rho )$ associated to a potentially Barsotti-Tate representation, see §1.1 and Appendix B of ReferenceCDT.) Similarly if $\tau '$ is an extended $\ell$-type, then we say that $\rho$ is of extended type$\tau '$ if
(1)
$\rho$ is Barsotti-Tate over $F$ for any finite extension $F$ of $\mathbf{Q}_\ell$ such that $\tau '|_{I_F}$ is trivial;
(2)
$WD(\rho )$ is equivalent to $\tau '$;
(3)
the character $\epsilon ^{-1} \det \rho$ has finite order prime to $\ell$.
Note that no representation can have extended type $\tau '$ unless $\det \tau '$ is of the form $\chi _1\chi _2$ where $\chi _1$ has finite order prime to $\ell$ and where $\chi _2$ is unramified and takes an arithmetic Frobenius element to $\ell$; see Appendix B of ReferenceCDT. (Using Theorem 1.4 of ReferenceBr2, one can show that for $\ell$ odd one obtains equivalent definitions of “type $\tau$” and “extended type $\tau '$” if one weakens the first assumption to simply require that $\rho$ is potentially Barsotti-Tate.)
Now fix a finite extension $K$ of $\mathbf{Q}_\ell$ in $\overline{\mathbf{Q}}_\ell$ over which $\tau$ (resp. $\tau '$) is rational. Let ${\mathcal {O}}$ denote the integers of $K$ and let $k$ denote the residue field of ${\mathcal {O}}$. Let
be a continuous representation of $G_\ell$ on a two-dimensional $k$-vector space $V$ and suppose that $\operatorname {End}_{k[G_\ell ]}V = k$. One then has a universal deformation ring $R_{V,{\mathcal {O}}}$ for ${\overline{\rho }}$ (see, for instance, Appendix A of ReferenceCDT).
We say that a prime ideal $\mathfrak{p}$ of $R_{V,{\mathcal {O}}}$ is of type $\tau$ (resp. of extended type $\tau '$) if there exist a finite extension $K'$ of $K$ in $\overline{\mathbf{Q}}_\ell$ and an ${\mathcal {O}}$-algebra homomorphism $R_{V,{\mathcal {O}}} \to K'$ with kernel $\mathfrak{p}$ such that the pushforward of the universal deformation of $\rho$ over $R_{V,{\mathcal {O}}}$ to $K'$ is of type $\tau$ (resp. of extended type $\tau '$).
Let $\tau$ be an $\ell$-type and $\tau '$ an irreducible extended $\ell$-type. If there do not exist any prime ideals $\mathfrak{p}$ of type $\tau$ (resp. of extended type $\tau '$), we define $R_{V,{\mathcal {O}}}^D = 0$ (resp. $R_{V,{\mathcal {O}}}^{D'}=0$). Otherwise, define $R_{V,{\mathcal {O}}}^D$ (resp. $R_{V,{\mathcal {O}}}^{D'}$) to be the quotient of $R_{V,{\mathcal {O}}}$ by the intersection of all $\mathfrak{p}$ of type $\tau$ (resp. of extended type $\tau '$). We will sometimes write $R_{V,{\mathcal {O}}}^\tau$ (resp. $R_{V,{\mathcal {O}}}^{\tau '}$) for $R_{V,{\mathcal {O}}}^D$ (resp. $R_{V,{\mathcal {O}}}^{D'}$). We say that a deformation of ${\overline{\rho }}$ is weakly of type $\tau$ (resp. weakly of extended type $\tau '$) if the associated local ${\mathcal {O}}$-algebra map $R_{V,{\mathcal {O}}} \rightarrow R$ factors through the quotient $R_{V,{\mathcal {O}}}^D$ (resp. $R_{V,{\mathcal {O}}}^{D'}$). We say that $\tau$ (resp. $\tau '$) is weakly acceptable for ${\overline{\rho }}$ if either $R_{V,{\mathcal {O}}}^D = 0$ (resp. $R_{V,{\mathcal {O}}}^{D'} = 0$) or there is a surjective local ${\mathcal {O}}$-algebra map ${\mathcal {O}}\llbracket X\rrbracket \twoheadrightarrow R_{V,{\mathcal {O}}}^D$ (resp. ${\mathcal {O}}\llbracket X\rrbracket \twoheadrightarrow R_{V,{\mathcal {O}}}^{D'}$). We say that $\tau$ (resp. $\tau '$) is acceptable for ${\overline{\rho }}$ if $R_{V,{\mathcal {O}}}^D \ne 0$ (resp. $R_{V,{\mathcal {O}}}^{D'} \ne 0$) and if there is a surjective local ${\mathcal {O}}$-algebra map ${\mathcal {O}}\llbracket X\rrbracket \twoheadrightarrow R_{V,{\mathcal {O}}}^D$ (resp. ${\mathcal {O}}\llbracket X\rrbracket \twoheadrightarrow R_{V,{\mathcal {O}}}^{D'}$).
If $K'$ is a finite extension of $K$ in $\overline{\mathbf{Q}}_\ell$ with valuation ring ${\mathcal {O}}'$ and residue field $k'$, then ${\mathcal {O}}' \otimes _{{\mathcal {O}}} R_{V,{\mathcal {O}}}^D$ (resp. ${\mathcal {O}}' \otimes _{{\mathcal {O}}} R_{V,{\mathcal {O}}}^{D'}$) is naturally isomorphic to $R_{V\otimes _k k',{\mathcal {O}}'}^{D}$ (resp. $R_{V\otimes _k k', {\mathcal {O}}'}^{D'}$). Thus (weak) acceptability depends only on $\tau$ (resp. $\tau '$) and ${\overline{\rho }}$, and not on the choice of $K$. Moreover $\tau$ (resp. $\tau '$) is acceptable for ${\overline{\rho }}$ if and only if $\tau$ (resp. $\tau '$) is acceptable for ${\overline{\rho }}\otimes _k k'$.
Although it is of no importance for the sequel, we make the following conjecture, part of which we already conjectured as Conjecture 1.2.1 of ReferenceCDT.
Conjecture 1.1.1
Suppose that $\tau$ is an $\ell$-type and $\tau '$ an absolutely irreducible extended $\ell$-type. A deformation $\rho :G_\ell \rightarrow \operatorname {GL}(M)$ of ${\overline{\rho }}$ to the ring of integers ${\mathcal {O}}'$ of a finite extension $K'/K$ in $\overline{\mathbf{Q}}_\ell$ is weakly of type $\tau$(resp. weakly of extended $\ell$-type$\tau '$) if and only if $M$ is of type $\tau$(resp. of extended type $\tau '$).
If $\tau$ is a tamely ramified $\ell$-type, then we expect that it is frequently the case that $\tau$ is acceptable for residual representations ${\overline{\rho }}$, as in Conjectures 1.2.2 and 1.2.3 of ReferenceCDT. On the other hand if $\tau$ (resp. $\tau '$) is a wildly ramified $\ell$-type (resp. wildly ramified extended $\ell$-type), then we expect that it is rather rare that $\tau$ (resp. $\tau '$) is acceptable for a residual representation ${\overline{\rho }}$. In this paper we will be concerned with a few wild cases for the prime $\ell =3$ which do turn out to be acceptable.
1.2. Types for admissible representations
From now on we assume that $\ell$ is odd. If $F$ is a finite extension of $\mathbf{Q}_\ell$ we will identify $F^\times$ with $W_F^{\operatorname {ab}}$ via the Artin map. Let $U_0(\ell ^r)$ denote the subgroup of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ consisting of elements with upper triangular $\bmod \, \ell ^r$ reduction. Also let $\widetilde{U}_0(\ell )$ denote the normaliser of $U_0(\ell )$ in ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$. Thus $\widetilde{U}_0(\ell )$ is generated by $U_0(\ell )$ and by
If $\tau$ is an $\ell$-type, set $U_\tau = {\operatorname {GL}}_2(\mathbf{Z}_\ell )$ if $\tau$ is reducible and $U_\tau =U_0(\ell )$ if $\tau$ is irreducible. If $\tau '$ is an extended $\ell$-type with $\tau '|_{I_\ell }$ irreducible, set $U_{\tau '} = \widetilde{U}_0(\ell )$. In this subsection we will associate to an $\ell$-type$\tau$ an irreducible representation $\sigma _\tau$ of $U_\tau$ over $\overline{\mathbf{Q}}_\ell$ with open kernel, and to an extended $\ell$-type$\tau '$ with $\tau '|_{I_\ell }$ irreducible an irreducible representation $\sigma _{\tau '}$ of $U_{\tau '}$ over $\overline{\mathbf{Q}}_\ell$ with open kernel. We need to consider several cases, which we treat one at a time.
First suppose that $\tau = \chi _1|_{I_\ell } \oplus \chi _2|_{I_\ell }$ where each $\chi _i$ is a character of $W_{\mathbf{Q}_\ell }$. Let $a$ denote the conductor of $\chi _1/\chi _2$. If $a=0$, then set
where $\operatorname {St}$ denotes the Steinberg representation of $\operatorname {PGL}_2(\mathbf{F}_\ell )$. Now suppose that $a>0$. Let $\sigma _\tau$ denote the induction from $U_0(\ell ^a)$ to ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ of the character of $U_0(\ell ^a)$ which sends
This is irreducible and does not depend on the ordering of $\chi _1$ and $\chi _2$.
For the next case, let $F$ denote the unramified quadratic extension of $\mathbf{Q}_\ell$ and $s$ the non-trivial automorphism of $F$ over $\mathbf{Q}_\ell$. Suppose that $\tau$ is the restriction to $I_\ell$ of the induction from $W_F$ to $W_{\mathbf{Q}_\ell }$ of a character $\chi$ of $W_F$ with $\chi \neq \chi ^s$. Let $a$ denote the conductor of $\chi /\chi ^s$, so that $a>0$. Choose a character $\chi '$ of $W_{\mathbf{Q}_\ell }$ such that $\chi '|_{W_F}^{-1}\chi$ has conductor $a$. If $a=1$ we set
where $\Theta (\cdot )$ is the irreducible representation of ${\operatorname {GL}}_2(\mathbf{F}_\ell )$ defined on page 532 of ReferenceCDT.
To define $\sigma _\tau$ for $a>1$ we will identify ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ with the automorphisms of the $\mathbf{Z}_\ell$-module${\mathcal {O}}_F$. If $a$ is even, then we let $\sigma _\tau$ denote the induction from ${\mathcal {O}}_F^\times (1+\ell ^{a/2}{\mathcal {O}}_F s)$ to ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ of the character $\varphi$ of ${\mathcal {O}}_F^\times (1+\ell ^{a/2}{\mathcal {O}}_F s)$, where, for $\alpha \in {\mathcal {O}}_F^\times$ and $\beta \in (1+\ell ^{a/2}{\mathcal {O}}_F s)$,
If $a>1$ is odd, then we let $\sigma _\tau$ denote the induction from ${\mathcal {O}}_F^\times (1+\ell ^{(a-1)/2}{\mathcal {O}}_F s)$ to ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ of $\eta$, where $\eta$ is the $\ell$-dimensional irreducible representation of ${\mathcal {O}}_F^\times (1+\ell ^{(a-1)/2}{\mathcal {O}}_F s)$ such that $\eta |_{{\mathcal {O}}_F^\times (1+\ell ^{(a+1)/2}{\mathcal {O}}_F s)}$ is the direct sum of the characters
for $\alpha \in {\mathcal {O}}_F^\times$ and $\beta \in (1+\ell ^{(a+1)/2}{\mathcal {O}}_F s)$, where $\chi ''$ runs over the $\ell$ non-trivial characters of ${\mathcal {O}}_F^\times / \mathbf{Z}_\ell ^\times (1+\ell {\mathcal {O}}_F)$.
Now suppose $\tau '$ is an extended type such that $\tau '|_{I_\ell }$ is irreducible. There is a ramified quadratic extension $F/\mathbf{Q}_\ell$ and a character $\chi$ of $W_F$ such that the induction from $W_F$ to $W_{\mathbf{Q}_\ell }$ of $\chi$ is $\tau '$ (see §2.6 of ReferenceG). Let $s$ denote the non-trivial field automorphism of $F$ over $\mathbf{Q}_\ell$ and also let $\wp _F$ denote the maximal ideal of the ring of integers ${\mathcal {O}}_F$ of $F$. Let $a$ denote the conductor of $\chi /\chi ^s$, so $a$ is even and $a \geq 2$. We may choose a character $\chi '$ of $W_{\mathbf{Q}_\ell }$ such that $\chi '|_{W_F}^{-1} \chi$ has conductor $a$. We will identify ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$ with the automorphisms of the $\mathbf{Q}_\ell$ vector space $F$. We will also identify $U_0(\ell )$ with the stabiliser of the pair of lattices $\wp _F^{-1} \supset {\mathcal {O}}_F$. We define $\sigma _{\tau '}$ to be the induction from $F^\times (1+\wp _F^{a/2}s)$ to $\widetilde{U}_0(\ell )$ of the character $\varphi$ of $F^\times (1+\wp _F^{a/2}s)$, where
with $\alpha \in F^\times$ and $\beta \in (1+\wp _F^{a/2}s)$, where $\chi ''$ is a character of $F^\times /({\mathcal {O}}_F^\times )^2$ defined as follows. Let $\psi$ be a character of $\mathbf{Q}_\ell$ with kernel $\mathbf{Z}_\ell$. Choose $\theta \in F^\times$ such that for $x \in \wp _F^{a-1}$ we have
where $\varpi$ is a uniformiser in ${\mathcal {O}}_F$.
Finally if $\tau$ is an irreducible $\ell$-type, choose an extended $\ell$-type$\tau '$ which restricts to $\tau$ on $I_\ell$ and set $\sigma _\tau = \sigma _{\tau '}|_{U_0(\ell )}$.
We remark that these definitions are independent of any choices (see ReferenceG).
Recall that by the local Langlands conjecture we can associate to an irreducible admissible representation $\pi$ of ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$ a two-dimensional representation $WD(\pi )$ of $W_{\mathbf{Q}_\ell }$. (See §4.1 of ReferenceCDT for the normalisation we use.)
Lemma 1.2.1
Suppose that $\tau$ is an $\ell$-type and that $\tau '$ is an extended $\ell$-type with $\tau '|_{I_\ell }$ irreducible. Suppose also that $\pi$ is an infinite-dimensional irreducible admissible representation of ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$ over $\overline{\mathbf{Q}}_\ell$. Then:
(1)
$\sigma _\tau$ and $\sigma _{\tau '}$ are irreducible.
The case that $\tau$ extends to a reducible representation of $W_{\mathbf{Q}_\ell }$ follows from the standard theory of principal series representations for ${\operatorname {GL}}_2(\mathbf{Q}_\ell )$. The case that $\tau$ is reducible but does not extend to a reducible representation of $W_{\mathbf{Q}_\ell }$ follows from Theorem 3.7 of ReferenceG. The case of $\tau '$ follows from Theorem 4.6 of ReferenceG.
Thus, suppose that $\tau$ is an irreducible $\ell$-type and that $\tau '$ is an extension of $\tau$ to an extended $\ell$-type. If $\delta$ denotes the unramified quadratic character of $W_{\mathbf{Q}_\ell }$, then $\tau ' \not \sim \tau ' \otimes \delta$ and so we deduce that
Thus $\sigma _{\tau '}|_{\mathbf{Q}_{\ell }^\times U_0(\ell )}$ is irreducible. It follows that $\sigma _\tau$ is irreducible. The second and third part of the lemma for $\tau$ follow similarly.
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1.3. Reduction of types for admissible representations
We begin by reviewing some irreducible representations of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$,$U_0(\ell )$ and $\widetilde{U}_0(\ell )$. Let $\sigma _{1,0}$ denote the standard representation of ${\operatorname {GL}}_2(\mathbf{F}_\ell )$ over $\overline{\mathbf{F}}_\ell$. If $n=0,1,...,\ell -1$ and if $m \in \mathbf{Z}/(\ell -1)\mathbf{Z}$, then we let $\sigma _{n,m} =\operatorname {Symm}^n(\sigma _{1,0}) \otimes \det ^m$. We may think of $\sigma _{n,m}$ as a continuous representation of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ over $\overline{\mathbf{F}}_\ell$. These representations are irreducible, mutually non-isomorphic and exhaust the irreducible continuous representations of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ over $\overline{\mathbf{F}}_\ell$.
If $m_1,m_2 \in \mathbf{Z}/(\ell -1)\mathbf{Z}$ we let $\sigma '_{m_1,m_2}$ denote the character of $U_0(\ell )$ over $\overline{\mathbf{F}}_\ell$ determined by
$$\begin{equation*} \left( \begin{array}{cc} a & b \\\ell c & d \end{array} \right) \longmapsto a^{m_1} d^{m_2}. \end{equation*}$$
These representations are irreducible, mutually non-isomorphic, and exhaust the irreducible continuous representations of $U_0(\ell )$ over $\overline{\mathbf{F}}_\ell$.
If $m_1,m_2 \in \mathbf{Z}/(\ell -1)\mathbf{Z}$,$a \in \overline{\mathbf{F}}_\ell ^\times$ and $m_1 \neq m_2$, then we let $\sigma '_{\{ m_1,m_2\}, a}$ denote the representation of $\widetilde{U}_0(\ell )$ over $\overline{\mathbf{F}}_\ell$ obtained by inducing the character of $\mathbf{Q}_\ell ^\times U_0(\ell )$ which restricts to $\sigma '_{m_1,m_2}$ on $U_0(\ell )$ and which sends $-\ell$ to $a$. If $m \in \mathbf{Z}/(\ell -1)\mathbf{Z}$ and $a \in \overline{\mathbf{F}}_\ell ^\times$, then we let $\sigma '_{\{ m \},a}$ denote the character of $\widetilde{U}_0(\ell )$ over $\overline{\mathbf{F}}_\ell$ which restricts to $\sigma '_{m,m}$ on $U_0(\ell )$ and which sends $w_\ell$ to $a$. These representations are irreducible, mutually non-isomorphic and exhaust the irreducible, finite-dimensional, continuous representations of $\widetilde{U}_0(\ell )$ over $\overline{\mathbf{F}}_\ell$.
We will say that a reducible $\ell$-type$\tau$ (resp. irreducible $\ell$-type, resp. extended $\ell$-type$\tau$ with irreducible restriction to $I_\ell$)admits an irreducible representation $\sigma$ of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$ (resp. $U_0(\ell )$, resp. $\widetilde{U}_0(\ell )$) over $\overline{\mathbf{F}}_\ell$, if $\sigma _\tau$ (resp. $\sigma _\tau$, resp. $\sigma _{\tau '}$) contains an invariant ${\mathcal {O}}_{\overline{\mathbf{Q}}_\ell }$-lattice$\Lambda$ and if $\sigma$ is a Jordan-Hölder constituent of $\Lambda \otimes \overline{\mathbf{F}}_\ell$. We will say that $\tau$ (resp. $\tau$, resp. $\tau '$)simply admits$\sigma$ if $\sigma$ is a Jordan-Hölder constituent of $\Lambda \otimes \overline{\mathbf{F}}_\ell$ of multiplicity one.
For each of the $\overline{\mathbf{F}}_\ell$-representations of ${\operatorname {GL}}_2(\mathbf{Z}_\ell )$,$U_0(\ell )$ and $\widetilde{U}_0(\ell )$ just defined, we wish to define notions of “admittance” and “simple admittance” with respect to a continuous representation ${\overline{\rho }}:G_\ell \rightarrow {\operatorname {GL}}_2(\overline{\mathbf{F}}_\ell )$. Let ${\overline{\rho }}$ be a fixed continuous representation $G_\ell \rightarrow {\operatorname {GL}}_2(\overline{\mathbf{F}}_\ell )$.
which in addition we require to be peu-ramifié in the case $n=0$. (Note that $\sigma _{n,0}$ admits ${\overline{\rho }}$ if and only if the Serre weight (see ReferenceSe2) of ${\overline{\rho }}^\vee \otimes \omega$ is $n+2$.)
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The representation $\sigma _{n,m}$simply admits${\overline{\rho }}$ if $\sigma _{n,m}$ admits ${\overline{\rho }}$.
(Note that $\sigma _{m_1,m_2}'$ admits ${\overline{\rho }}$ if and only if some irreducible constituent of $\operatorname {Ind}_{U_0(\ell )}^{{\operatorname {GL}}_2(\mathbf{Z}_\ell )} \sigma _{m_1,m_2}'$ admits ${\overline{\rho }}$.)
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The representation $\sigma _{m_1,m_2}'$ with $m_1\neq m_2$simply admits${\overline{\rho }}$ if either