American Mathematical Society

On the modularity of elliptic curves over bold upper 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 upper E Subscript slash bold upper Q is an elliptic curve, then upper E is modular.

Theorem B

If rho overbar colon upper G a l left-parenthesis bold upper Q overbar slash bold upper Q right-parenthesis right-arrow upper G upper L 2 left-parenthesis bold upper F 5 right-parenthesis is an irreducible continuous representation with cyclotomic determinant, then rho overbar is modular.

We will first remind the reader of the content of these results and then briefly outline the method of proof.

If upper N is a positive integer, then we let normal upper Gamma 1 left-parenthesis upper N right-parenthesis denote the subgroup of upper S upper L 2 left-parenthesis bold upper Z right-parenthesis consisting of matrices that modulo upper N are of the form

Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column asterisk 2nd Row 1st Column 0 2nd Column 1 EndMatrix period

The quotient of the upper half plane by normal upper Gamma 1 left-parenthesis upper N right-parenthesis , acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve upper Y 1 left-parenthesis upper N right-parenthesis Subscript slash bold upper C . This curve has a natural model upper Y 1 left-parenthesis upper N right-parenthesis Subscript slash bold upper Q , which for upper N greater-than 3 is a fine moduli scheme for elliptic curves with a point of exact order upper N . We will let upper X 1 left-parenthesis upper N right-parenthesis denote the smooth projective curve which contains upper Y 1 left-parenthesis upper N right-parenthesis as a dense Zariski open subset.

Recall that a cusp form of weight k greater-than-or-equal-to 1 and level upper N greater-than-or-equal-to 1 is a holomorphic function f on the upper half complex plane German upper H such that

for all matrices Start 2 By 2 Matrix 1st Row 1st Column a 2nd Column b 2nd Row 1st Column c 2nd Column d EndMatrix element-of normal upper Gamma 1 left-parenthesis upper N right-parenthesis

and all z element-of German upper H , we have f left-parenthesis left-parenthesis a z plus b right-parenthesis slash left-parenthesis c z plus d right-parenthesis right-parenthesis equals left-parenthesis c z plus d right-parenthesis Superscript k Baseline f left-parenthesis z right-parenthesis ;

and StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue squared left-parenthesis upper I m z right-parenthesis Superscript k is bounded on German upper H .

The space upper S Subscript k Baseline left-parenthesis upper N right-parenthesis of cusp forms of weight k and level upper N is a finite-dimensional complex vector space. If f element-of upper S Subscript k Baseline left-parenthesis upper N right-parenthesis , then it has an expansion

f left-parenthesis z right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts c Subscript n Baseline left-parenthesis f right-parenthesis e Superscript 2 pi i n z

and we define the upper L -series of f to be

upper L left-parenthesis f comma s right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts c Subscript n Baseline left-parenthesis f right-parenthesis slash n Superscript s Baseline period

For each prime p does-not-divide upper N there is a linear operator upper T Subscript p on upper S Subscript k Baseline left-parenthesis upper N right-parenthesis defined by

left-parenthesis f vertical-bar upper T Subscript p Baseline right-parenthesis left-parenthesis z right-parenthesis equals p Superscript negative 1 Baseline sigma-summation Underscript i equals 0 Overscript p minus 1 Endscripts f left-parenthesis left-parenthesis z plus i right-parenthesis slash p right-parenthesis plus p Superscript k minus 1 Baseline left-parenthesis c p z plus d right-parenthesis Superscript negative k Baseline f left-parenthesis left-parenthesis a p z plus b right-parenthesis slash left-parenthesis c p z plus d right-parenthesis right-parenthesis

for any

Start 2 By 2 Matrix 1st Row 1st Column a 2nd Column b 2nd Row 1st Column c 2nd Column d EndMatrix element-of upper S upper L 2 left-parenthesis bold upper Z right-parenthesis

with c identical-to 0 mod upper N and d identical-to p mod upper N . The operators upper T Subscript p for p does-not-divide upper N can be simultaneously diagonalised on the space upper S Subscript k Baseline left-parenthesis upper N right-parenthesis and a simultaneous eigenvector is called an eigenform. If f is an eigenform, then the corresponding eigenvalues, a Subscript p Baseline left-parenthesis f right-parenthesis , are algebraic integers and we have c Subscript p Baseline left-parenthesis f right-parenthesis equals a Subscript p Baseline left-parenthesis f right-parenthesis c 1 left-parenthesis f right-parenthesis .

Let lamda be a place of the algebraic closure of bold upper Q in bold upper C above a rational prime script l and let bold upper Q overbar Subscript lamda denote the algebraic closure of bold upper Q Subscript script l thought of as a bold upper Q overbar algebra via lamda . If f element-of upper S Subscript k Baseline left-parenthesis upper N right-parenthesis is an eigenform, then there is a unique continuous irreducible representation

rho Subscript f comma lamda Baseline colon upper G a l left-parenthesis bold upper Q overbar slash bold upper Q right-parenthesis long right-arrow upper G upper L 2 left-parenthesis bold upper Q overbar Subscript lamda Baseline right-parenthesis

such that for any prime p does-not-divide upper N l , rho Subscript f comma lamda is unramified at p and trace rho Subscript f comma lamda Baseline left-parenthesis upper F r o b Subscript p Baseline right-parenthesis equals a Subscript p Baseline left-parenthesis f right-parenthesis . The existence of rho Subscript f comma lamda is due to Shimura if k equals 2 ReferenceSh2, to Deligne if k greater-than 2 ReferenceDe and to Deligne and Serre if k equals 1 ReferenceDS. Its irreducibility is due to Ribet if k greater-than 1 ReferenceRi and to Deligne and Serre if k equals 1 ReferenceDS. Moreover rho is odd in the sense that det rho of complex conjugation is negative 1 . Also, rho Subscript f comma lamda is potentially semi-stable at script l in the sense of Fontaine. We can choose a conjugate of rho Subscript f comma lamda which is valued in upper G upper L 2 left-parenthesis script upper O Subscript bold upper Q overbar Sub Subscript lamda Subscript Baseline right-parenthesis , and reducing modulo the maximal ideal and semi-simplifying yields a continuous representation

rho overbar Subscript f comma lamda Baseline colon upper G a l left-parenthesis bold upper Q overbar slash bold upper Q right-parenthesis long right-arrow upper G upper L 2 left-parenthesis bold upper F overbar Subscript script l Baseline right-parenthesis comma

which, up to isomorphism, does not depend on the choice of conjugate of rho Subscript f comma lamda .

Now suppose that rho colon upper G Subscript bold upper Q Baseline right-arrow upper G upper L 2 left-parenthesis bold upper Q overbar Subscript script l Baseline right-parenthesis is a continuous representation which is unramified outside finitely many primes and for which the restriction of rho to a decomposition group at script l is potentially semi-stable in the sense of Fontaine. To rho vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript script l Subscript slash bold upper Q Sub Subscript script l Subscript right-parenthesis Baseline we can associate both a pair of Hodge-Tate numbers and a Weil-Deligne representation of the Weil group of bold upper Q Subscript script l . We define the conductor upper N left-parenthesis rho right-parenthesis of rho to be the product over p not-equals script l of the conductor of rho vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript p Subscript slash bold upper Q Sub Subscript p Subscript right-parenthesis Baseline and of the conductor of the Weil-Deligne representation associated to rho vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript script l Subscript slash bold upper Q Sub Subscript script l Subscript right-parenthesis Baseline . We define the weight k left-parenthesis rho right-parenthesis of rho to be 1 plus the absolute difference of the two Hodge-Tate numbers of rho vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript script l Subscript slash bold upper Q Sub Subscript script l Subscript right-parenthesis Baseline . It is known by work of Carayol and others that the following two conditions are equivalent:

rho tilde rho Subscript f comma lamda for some eigenform f and some place lamda vertical-bar script l ;

rho tilde rho Subscript f comma lamda for some eigenform f of level upper N left-parenthesis rho right-parenthesis and weight k left-parenthesis rho right-parenthesis and some place lamda vertical-bar script l .

When these equivalent conditions are met we call rho modular. It is conjectured by Fontaine and Mazur that if rho colon upper G Subscript bold upper Q Baseline right-arrow upper G upper L 2 left-parenthesis bold upper Q overbar Subscript script l Baseline right-parenthesis is a continuous irreducible representation which satisfies

rho is unramified outside finitely many primes,

rho vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript script l Subscript slash bold upper Q Sub Subscript script l Subscript right-parenthesis Baseline 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,

then rho is modular ReferenceFM.

Next consider a continuous irreducible representation rho overbar colon upper G a l left-parenthesis bold upper Q overbar slash bold upper Q right-parenthesis right-arrow upper G upper L 2 left-parenthesis bold upper F overbar Subscript script l Baseline right-parenthesis . Serre ReferenceSe2 defines the conductor upper N left-parenthesis rho overbar right-parenthesis and weight k left-parenthesis rho overbar right-parenthesis of rho overbar . We call rho overbar modular if rho overbar tilde rho overbar Subscript f comma lamda for some eigenform f and some place lamda vertical-bar script l . We call rho overbar strongly modular if moreover we may take f to have weight k left-parenthesis rho overbar right-parenthesis and level upper N left-parenthesis rho overbar right-parenthesis . It is known from work of Mazur, Ribet, Carayol, Gross, Coleman, Voloch and others that for script l greater-than-or-equal-to 3 , rho overbar is strongly modular if and only if it is modular (see ReferenceDi1). Serre has conjectured that all odd, irreducible rho overbar are strongly modular ReferenceSe2.

Now consider an elliptic curve upper E Subscript slash bold upper Q . Let rho Subscript upper E comma script l (resp. rho overbar Subscript upper E comma script l ) denote the representation of upper G a l left-parenthesis bold upper Q overbar slash bold upper Q right-parenthesis on the script l -adic Tate module (resp. the script l -torsion) of upper E left-parenthesis bold upper Q overbar right-parenthesis . Let upper N left-parenthesis upper E right-parenthesis denote the conductor of upper E . It is known that the following conditions are equivalent:

(1)

The upper L -function upper L left-parenthesis upper E comma s right-parenthesis of upper E equals the upper L -function upper L left-parenthesis f comma s right-parenthesis for some eigenform f .

(2)

The upper L -function upper L left-parenthesis upper E comma s right-parenthesis of upper E equals the upper L -function upper L left-parenthesis f comma s right-parenthesis for some eigenform f of weight 2 and level upper N left-parenthesis upper E right-parenthesis .

(3)

For some prime script l , the representation rho Subscript upper E comma script l is modular.

(4)

For all primes script l , the representation rho Subscript upper E comma script l is modular.

(5)

There is a non-constant holomorphic map upper X 1 left-parenthesis upper N right-parenthesis left-parenthesis bold upper C right-parenthesis right-arrow upper E left-parenthesis bold upper C right-parenthesis for some positive integer upper N .

(6)

There is a non-constant morphism upper X 1 left-parenthesis upper N left-parenthesis upper E right-parenthesis right-parenthesis right-arrow upper E which is defined over bold upper Q .

The implications (2) right double arrow (1), (4) right double arrow (3) and (6) right double arrow (5) are tautological. The implication (1) right double arrow (4) follows from the characterisation of upper L left-parenthesis upper E comma s right-parenthesis in terms of rho Subscript upper E comma script l . The implication (3) right double arrow (2) follows from a theorem of Carayol ReferenceCa1. The implication (2) right double arrow (6) follows from a construction of Shimura ReferenceSh2 and a theorem of Faltings ReferenceFa. The implication (5) right double arrow (3) seems to have been first noticed by Mazur ReferenceMaz. When these equivalent conditions are satisfied we call upper E modular.

It has become a standard conjecture that all elliptic curves over bold upper 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 rho overbar Subscript upper E comma 5 Baseline vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar slash bold upper Q left-parenthesis StartRoot 5 EndRoot right-parenthesis right-parenthesis Baseline is irreducible, we show that rho Subscript upper E comma 5 is modular.

(2)

If rho overbar Subscript upper E comma 5 Baseline vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar slash bold upper Q left-parenthesis StartRoot 5 EndRoot right-parenthesis right-parenthesis Baseline is reducible, but rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar slash bold upper Q left-parenthesis StartRoot negative 3 EndRoot right-parenthesis right-parenthesis Baseline is absolutely irreducible, we show that rho Subscript upper E comma 3 is modular.

(3)

If rho overbar Subscript upper E comma 5 Baseline vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar slash bold upper Q left-parenthesis StartRoot 5 EndRoot right-parenthesis right-parenthesis Baseline is reducible and rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar slash bold upper Q left-parenthesis StartRoot negative 3 EndRoot right-parenthesis right-parenthesis Baseline is absolutely reducible, then we show that upper E is isogenous to an elliptic curve with j -invariant 0 , left-parenthesis 11 slash 2 right-parenthesis cubed , or minus 5 left-parenthesis 29 right-parenthesis cubed slash 2 Superscript 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 rho overbar Subscript upper E comma script l is modular and then that rho Subscript upper E comma script l 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 upper E obtains semi-stable reduction over a tame extension of bold upper Q Subscript script l and the deduction of the modularity of rho Subscript upper E comma script l from that of rho overbar Subscript upper E comma script l 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 rho overbar be as in that theorem. Twisting by a quadratic character, we may assume that rho overbar vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript 3 Subscript slash bold upper Q 3 right-parenthesis Baseline falls into one of the following cases (see §2.2):

(1)

rho overbar is unramified at 3 .

(2)

ModifyingAbove rho With bar left-parenthesis upper I 3 right-parenthesis has order 5 .

(3)

ModifyingAbove rho With bar left-parenthesis upper I 3 right-parenthesis has order 4 .

(4)

ModifyingAbove rho With bar left-parenthesis upper I 3 right-parenthesis has order 12 and rho overbar vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript 3 Subscript slash bold upper Q 3 right-parenthesis Baseline has conductor 27 .

(5)

ModifyingAbove rho With bar left-parenthesis upper I 3 right-parenthesis has order 3 .

(6)

rho overbar vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript 3 Subscript slash bold upper Q 3 right-parenthesis Baseline is induced from a character chi colon upper G a l left-parenthesis bold upper Q overbar Subscript 3 Baseline slash bold upper Q 3 left-parenthesis StartRoot negative 3 EndRoot right-parenthesis right-parenthesis right-arrow bold upper F 25 Superscript times such that chi left-parenthesis negative 1 right-parenthesis equals negative 1 andchi left-parenthesis StartRoot negative 3 EndRoot right-parenthesis equals chi left-parenthesis 1 plus 3 StartRoot negative 3 EndRoot right-parenthesis minus chi left-parenthesis 1 minus 3 StartRoot negative 3 EndRoot right-parenthesis comma

where we use the Artin map (normalised to take uniformisers to arithmetic Frobenius) to identify chi with a character of bold upper Q 3 left-parenthesis StartRoot negative 3 EndRoot right-parenthesis Superscript times .

We will refer to these as the f equals 1 comma 3 comma 9 comma 27 comma 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 upper E Subscript slash bold upper Q with the following properties (see §2.2):

rho overbar Subscript upper E comma 5 Baseline tilde rho overbar ,

rho overbar Subscript upper E comma 3 is surjective onto upper G upper L 2 left-parenthesis bold upper F 3 right-parenthesis ,

and

(1)

in the f equals 1 case, either rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper I 3 Baseline circled-times bold upper F 9 tilde omega 2 circled-plus omega 2 cubed orrho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper I 3 Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega 2nd Column asterisk 2nd Row 1st Column 0 2nd Column 1 EndMatrix

and is peu ramifié;

(2)

in the f equals 3 case,rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper I 3 Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega 2nd Column asterisk 2nd Row 1st Column 0 2nd Column 1 EndMatrix semicolon

(3)

in the f equals 9 case, rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper I 3 Baseline circled-times bold upper F 9 tilde omega 2 circled-plus omega 2 cubed ;

(4)

in the f equals 27 case,rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper I 3 Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega 2nd Column asterisk 2nd Row 1st Column 0 2nd Column 1 EndMatrix

and is très ramifié;

(5)

in the f equals 81 case,rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper I 3 Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column asterisk 2nd Row 1st Column 0 2nd Column omega EndMatrix

and is très ramifié;

(6)

in the f equals 243 case,rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript 3 Subscript slash bold upper Q 3 right-parenthesis Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega 2nd Column asterisk 2nd Row 1st Column 0 2nd Column 1 EndMatrix

is non-split over bold upper Q overbar Subscript 3 Superscript kernel rho overbar 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 mod 3 cyclotomic character and omega 2 the second fundamental character upper I 3 right-arrow bold upper F 9 Superscript times , i.e.

omega 2 left-parenthesis sigma right-parenthesis identical-to sigma left-parenthesis RootIndex 8 StartRoot 3 EndRoot right-parenthesis slash RootIndex 8 StartRoot 3 EndRoot mod RootIndex 8 StartRoot 3 EndRoot period

We will often use the equality omega equals omega Superscript negative 1 without further remark.) We emphasise that for a general elliptic curve over bold upper Q with rho overbar Subscript upper E comma 5 Baseline approximately-equals rho overbar , the representation rho overbar Subscript upper E comma 3 does not have the above form, rather we are placing a significant restriction on upper E .

In each case our strategy is to prove that rho Subscript upper E comma 3 is modular and so deduce that rho overbar tilde rho overbar Subscript upper E comma 5 is modular. Again we use the Langlands-Tunnell theorem to see that rho overbar Subscript upper E comma 3 is modular and then an analogue of the arguments of ReferenceWi and ReferenceTW to conclude that rho Subscript upper E comma 3 is modular. This was carried out in ReferenceDi2 in the cases f equals 1 and f equals 3 , and in ReferenceCDT in the case f equals 9 . (In these cases the particular form of rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper I 3 Baseline is not important.) This leaves the cases f equals 27 , 81 and 243 , which are complicated by the fact that upper E now only obtains good reduction over a wild extension of bold upper Q 3 . In these cases our argument relies essentially on the particular form we have obtained for rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript 3 Subscript slash bold upper Q 3 right-parenthesis Baseline (depending on rho overbar Subscript upper E comma 5 Baseline vertical-bar Subscript upper I 3 Baseline ). We do not believe that our methods for deducing the modularity of rho Subscript upper E comma 3 from that of rho overbar Subscript upper E comma 3 would work without this key simplification. It seems to be a piece of undeserved good fortune that for each possibility for rho overbar vertical-bar Subscript upper I 3 Baseline we can find a choice for rho overbar Subscript upper E comma 3 Baseline vertical-bar Subscript upper G a l left-parenthesis bold upper Q overbar Sub Subscript 3 Subscript slash bold upper Q 3 right-parenthesis Baseline for which our methods work.

Following Wiles, to deduce the modularity of rho Subscript upper E comma 3 from that of rho overbar Subscript upper E comma 3 , we consider certain universal deformations of rho overbar Subscript upper E comma 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 bold upper 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 left-parenthesis rho overbar Subscript upper E comma 5 Baseline comma rho overbar Subscript upper E comma 3 Baseline right-parenthesis .

Our arguments follow closely the arguments of ReferenceCDT. There are two main new features. Firstly, in the f equals 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 equals 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 upper G upper L 2 left-parenthesis bold upper Q 3 right-parenthesis in §3. We see no direct connection, but cannot help thinking that some such connection should exist.

Notation

In this paper script l 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 script l equals 3 .

If upper F is a field we let upper F overbar denote a separable closure, upper F Superscript a b the maximal subextension of upper F overbar which is abelian over upper F and upper G Subscript upper F the Galois group upper G a l left-parenthesis upper F overbar slash upper F right-parenthesis . If upper F 0 is a p -adic field (i.e. a finite extension of bold upper Q Subscript p ) and upper F prime slash upper F 0 a (possibly infinite) Galois extension, then we let upper I Subscript upper F prime slash upper F 0 denote the inertia subgroup of upper G a l left-parenthesis upper F prime slash upper F 0 right-parenthesis . We also let upper I Subscript upper F 0 denote upper I Subscript upper F overbar Sub Subscript 0 Subscript slash upper F 0 , upper F r o b Subscript upper F 0 Baseline element-of upper G Subscript upper F 0 slash upper I Subscript upper F 0 denote the arithmetic Frobenius element and upper W Subscript upper F 0 denote the Weil group of upper F 0 , i.e. the dense subgroup of upper G Subscript upper F 0 consisting of elements which map to an integer power of upper F r o b Subscript upper 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 script upper O Subscript upper F 0 denote the ring of integers of upper F 0 , normal script upper P Subscript upper F 0 the maximal ideal of script upper O Subscript upper F 0 and k Subscript upper F 0 the residue field script upper O Subscript upper F 0 Baseline slash normal script upper P Subscript upper F 0 . We write simply upper G Subscript p for upper G Subscript bold upper Q Sub Subscript p , upper I Subscript p for upper I Subscript bold upper Q Sub Subscript p and upper F r o b Subscript p for upper F r o b Subscript bold upper Q Sub Subscript p . We also let bold upper Q Subscript p Sub Superscript n denote the unique unramified degree n extension of bold upper Q Subscript p in bold upper Q overbar Subscript p . If k is any perfect field of characteristic p we also use upper F r o b Subscript p to denote the p Superscript t h -power automorphism of k and its canonical lift to the Witt vectors upper W left-parenthesis k right-parenthesis .

We write epsilon for the script l -adic cyclotomic character and sometimes omega for the reduction of epsilon modulo script l . We write omega 2 for the second fundamental character upper I Subscript script l Baseline right-arrow bold upper F Subscript script l squared Superscript times , i.e.

omega 2 left-parenthesis sigma right-parenthesis identical-to sigma left-parenthesis script l Superscript 1 slash left-parenthesis script l squared minus 1 right-parenthesis Baseline right-parenthesis slash script l Superscript 1 slash left-parenthesis script l squared minus 1 right-parenthesis Baseline mod script l Superscript 1 slash left-parenthesis script l squared minus 1 right-parenthesis Baseline period

We also use omega and omega 2 to denote the Teichmuller lifts of omega and omega 2 .

We let bold 1 denote the trivial character of a group. We will denote by upper V Superscript logical-or the dual of a vector space upper V .

If g colon upper A right-arrow upper B is a homomorphism of rings and if upper X Subscript slash upper S p e c upper A is an upper A -scheme, then we sometimes write Superscript g Baseline upper X for the pullback of upper X by upper S p e c g . We adopt this notation so that Superscript g Baseline left-parenthesis Superscript h Baseline upper X right-parenthesis equals Superscript g h Baseline upper X . Similarly if theta colon upper X right-arrow upper Y is a morphism of schemes over upper A we will sometimes write Superscript g Baseline theta for the pullback of theta by upper S p e c g .

By finite flat group scheme we always mean commutative finite flat group scheme. If upper F 0 is a field of characteristic 0 with fixed algebraic closure upper F overbar Subscript 0 we use without comment the canonical identification of finite flat upper F 0 -group schemes with finite discrete upper G a l left-parenthesis upper F overbar Subscript 0 Baseline slash upper F 0 right-parenthesis -modules, and we will say that such objects correspond. If upper R is a Dedekind domain with field of fractions upper F of characteristic 0 , then by a model of a finite flat upper F -group scheme upper G we mean a finite locally free upper R -group scheme script upper G and an isomorphism i colon upper G ModifyingAbove right-arrow With tilde script upper G times upper F prime . As in Proposition 2.2.2 of ReferenceRa the isomorphism classes of models for upper G form a lattice ( left-parenthesis script upper G comma i right-parenthesis greater-than-or-equal-to left-parenthesis script upper G prime comma i prime right-parenthesis if there exists a map of finite flat group schemes script upper G right-arrow script upper G prime compatible with i and i prime ) and we can talk about the inf and sup of two such models. If upper R is also local we call the model left-parenthesis script upper G comma i right-parenthesis local-local if its special fibre is local-local. When the ring upper R is understood we sometimes simply refer to left-parenthesis script upper G comma i right-parenthesis , or even just script upper G , as an integral model of upper G .

We use Serre’s terminology peu ramifié and très ramifié; see ReferenceSe2.

1. Types

1.1. Types of local deformations

By an script l -type we mean an equivalence class of two-dimensional representations

tau colon upper I Subscript script l Baseline right-arrow upper G upper L left-parenthesis upper D right-parenthesis

over bold upper Q overbar Subscript script l which have open kernel and which can be extended to a representation of upper W Subscript bold upper Q Sub Subscript script l . By an extended script l -type we shall simply mean an equivalence class of two-dimensional representations

tau prime colon upper W Subscript bold upper Q Sub Subscript script l Subscript Baseline right-arrow upper G upper L left-parenthesis upper D prime right-parenthesis

over bold upper Q overbar Subscript script l with open kernel.

Suppose that tau is an script l -type and that upper K is a finite extension of bold upper Q Subscript script l in bold upper Q overbar Subscript script l . Recall from ReferenceCDT that a continuous representation rho of upper G Subscript script l on a two-dimensional upper K -vector space upper M is said to be of type tau if

(1)

rho is Barsotti-Tate over upper F for any finite extension upper F of bold upper Q Subscript script l such that tau vertical-bar Subscript upper I Sub Subscript upper F Subscript Baseline is trivial;

(2)

the restriction of upper W upper D left-parenthesis rho right-parenthesis to upper I Subscript script l is in tau ;

(3)

the character epsilon Superscript negative 1 Baseline det rho has finite order prime to script l .

(For the definition of “Barsotti-Tate” and of the representation upper W upper D left-parenthesis rho right-parenthesis associated to a potentially Barsotti-Tate representation, see §1.1 and Appendix B of ReferenceCDT.) Similarly if tau prime is an extended script l -type, then we say that rho is of extended type tau prime if

(1)

rho is Barsotti-Tate over upper F for any finite extension upper F of bold upper Q Subscript script l such that tau prime vertical-bar Subscript upper I Sub Subscript upper F Subscript Baseline is trivial;

(2)

upper W upper D left-parenthesis rho right-parenthesis is equivalent to tau prime ;

(3)

the character epsilon Superscript negative 1 Baseline det rho has finite order prime to script l .

Note that no representation can have extended type tau prime unless det tau prime is of the form chi 1 chi 2 where chi 1 has finite order prime to script l and where chi 2 is unramified and takes an arithmetic Frobenius element to script l ; see Appendix B of ReferenceCDT. (Using Theorem 1.4 of ReferenceBr2, one can show that for script l odd one obtains equivalent definitions of “type tau and “extended type tau prime if one weakens the first assumption to simply require that rho is potentially Barsotti-Tate.)

Now fix a finite extension upper K of bold upper Q Subscript script l in bold upper Q overbar Subscript script l over which tau (resp. tau prime ) is rational. Let script upper O denote the integers of upper K and let k denote the residue field of script upper O . Let

rho overbar colon upper G Subscript script l Baseline long right-arrow upper G upper L left-parenthesis upper V right-parenthesis

be a continuous representation of upper G Subscript script l on a two-dimensional k -vector space upper V and suppose that upper E n d Subscript k left-bracket upper G Sub Subscript script l Subscript right-bracket Baseline upper V equals k . One then has a universal deformation ring upper R Subscript upper V comma script upper O for rho overbar (see, for instance, Appendix A of ReferenceCDT).

We say that a prime ideal German p of upper R Subscript upper V comma script upper O is of type tau (resp. of extended type tau prime ) if there exist a finite extension upper K prime of upper K in bold upper Q overbar Subscript script l and an script upper O -algebra homomorphism upper R Subscript upper V comma script upper O Baseline right-arrow upper K prime with kernel German p such that the pushforward of the universal deformation of rho over upper R Subscript upper V comma script upper O to upper K prime is of type tau (resp. of extended type tau prime ).

Let tau be an script l -type and tau prime an irreducible extended script l -type. If there do not exist any prime ideals German p of type tau (resp. of extended type tau prime ), we define upper R Subscript upper V comma script upper O Superscript upper D Baseline equals 0 (resp. upper R Subscript upper V comma script upper O Superscript upper D prime Baseline equals 0 ). Otherwise, define upper R Subscript upper V comma script upper O Superscript upper D (resp. upper R Subscript upper V comma script upper O Superscript upper D prime ) to be the quotient of upper R Subscript upper V comma script upper O by the intersection of all German p of type tau (resp. of extended type tau prime ). We will sometimes write upper R Subscript upper V comma script upper O Superscript tau (resp. upper R Subscript upper V comma script upper O Superscript tau prime ) for upper R Subscript upper V comma script upper O Superscript upper D (resp. upper R Subscript upper V comma script upper O Superscript upper D prime ). We say that a deformation of rho overbar is weakly of type tau (resp. weakly of extended type tau prime ) if the associated local script upper O -algebra map upper R Subscript upper V comma script upper O Baseline right-arrow upper R factors through the quotient upper R Subscript upper V comma script upper O Superscript upper D (resp. upper R Subscript upper V comma script upper O Superscript upper D prime ). We say that tau (resp. tau prime ) is weakly acceptable for rho overbar if either upper R Subscript upper V comma script upper O Superscript upper D Baseline equals 0 (resp. upper R Subscript upper V comma script upper O Superscript upper D prime Baseline equals 0 ) or there is a surjective local script upper O -algebra map script upper O mathematical left white bracket upper X mathematical right white bracket two headed right-arrow upper R Subscript upper V comma script upper O Superscript upper D (resp. script upper O mathematical left white bracket upper X mathematical right white bracket two headed right-arrow upper R Subscript upper V comma script upper O Superscript upper D prime ). We say that tau (resp. tau prime ) is acceptable for rho overbar if upper R Subscript upper V comma script upper O Superscript upper D Baseline not-equals 0 (resp. upper R Subscript upper V comma script upper O Superscript upper D prime Baseline not-equals 0 ) and if there is a surjective local script upper O -algebra map script upper O mathematical left white bracket upper X mathematical right white bracket two headed right-arrow upper R Subscript upper V comma script upper O Superscript upper D (resp. script upper O mathematical left white bracket upper X mathematical right white bracket two headed right-arrow upper R Subscript upper V comma script upper O Superscript upper D prime ).

If upper K prime is a finite extension of upper K in bold upper Q overbar Subscript script l with valuation ring script upper O prime and residue field k prime , then script upper O prime circled-times Subscript script upper O Baseline upper R Subscript upper V comma script upper O Superscript upper D (resp. script upper O prime circled-times Subscript script upper O Baseline upper R Subscript upper V comma script upper O Superscript upper D prime ) is naturally isomorphic to upper R Subscript upper V circled-times Sub Subscript k Subscript k Sub Superscript prime Subscript comma script upper O Sub Superscript prime Subscript Superscript upper D (resp. upper R Subscript upper V circled-times Sub Subscript k Subscript k Sub Superscript prime Subscript comma script upper O Sub Superscript prime Subscript Superscript upper D prime ). Thus (weak) acceptability depends only on tau (resp. tau prime ) and rho overbar , and not on the choice of upper K . Moreover tau (resp. tau prime ) is acceptable for rho overbar if and only if tau (resp. tau prime ) is acceptable for rho overbar circled-times Subscript k Baseline k prime .

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 script l -type and tau prime an absolutely irreducible extended script l -type. A deformation rho colon upper G Subscript script l Baseline right-arrow upper G upper L left-parenthesis upper M right-parenthesis of rho overbar to the ring of integers script upper O prime of a finite extension upper K prime slash upper K in bold upper Q overbar Subscript script l is weakly of type tau (resp. weakly of extended script l -type tau prime ) if and only if upper M is of type tau (resp. of extended type tau prime ).

If tau is a tamely ramified script l -type, then we expect that it is frequently the case that tau is acceptable for residual representations rho overbar , as in Conjectures 1.2.2 and 1.2.3 of ReferenceCDT. On the other hand if tau (resp. tau prime ) is a wildly ramified script l -type (resp. wildly ramified extended script l -type), then we expect that it is rather rare that tau (resp. tau prime ) is acceptable for a residual representation rho overbar . In this paper we will be concerned with a few wild cases for the prime script l equals 3 which do turn out to be acceptable.

1.2. Types for admissible representations

From now on we assume that script l is odd. If upper F is a finite extension of bold upper Q Subscript script l we will identify upper F Superscript times with upper W Subscript upper F Superscript a b via the Artin map. Let upper U 0 left-parenthesis script l Superscript r Baseline right-parenthesis denote the subgroup of upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis consisting of elements with upper triangular mod script l Superscript r reduction. Also let ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis denote the normaliser of upper U 0 left-parenthesis script l right-parenthesis in upper G upper L 2 left-parenthesis bold upper Q Subscript script l Baseline right-parenthesis . Thus ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis is generated by upper U 0 left-parenthesis script l right-parenthesis and by

StartLayout 1st Row with Label left-parenthesis 1.2 .1 right-parenthesis EndLabel w Subscript script l Baseline equals Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column negative 1 2nd Row 1st Column script l 2nd Column 0 EndMatrix period EndLayout

If tau is an script l -type, set upper U Subscript tau Baseline equals upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis if tau is reducible and upper U Subscript tau Baseline equals upper U 0 left-parenthesis script l right-parenthesis if tau is irreducible. If tau prime is an extended script l -type with tau prime vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline irreducible, set upper U Subscript tau prime Baseline equals ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis . In this subsection we will associate to an script l -type tau an irreducible representation sigma Subscript tau of upper U Subscript tau over bold upper Q overbar Subscript script l with open kernel, and to an extended script l -type tau prime with tau prime vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline irreducible an irreducible representation sigma Subscript tau prime of upper U Subscript tau prime over bold upper Q overbar Subscript script l with open kernel. We need to consider several cases, which we treat one at a time.

First suppose that tau equals chi 1 StartAbsoluteValue circled-plus chi 2 EndAbsoluteValue Subscript upper I Sub Subscript script l Baseline Subscript upper I Sub Subscript script l where each chi Subscript i is a character of upper W Subscript bold upper Q Sub Subscript script l . Let a denote the conductor of chi 1 slash chi 2 . If a equals 0 , then set

sigma Subscript tau Baseline equals upper S t circled-times left-parenthesis chi 1 ring det right-parenthesis equals upper S t circled-times left-parenthesis chi 2 ring det right-parenthesis comma

where upper S t denotes the Steinberg representation of upper P upper G upper L 2 left-parenthesis bold upper F Subscript script l Baseline right-parenthesis . Now suppose that a greater-than 0 . Let sigma Subscript tau denote the induction from upper U 0 left-parenthesis script l Superscript a Baseline right-parenthesis to upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis of the character of upper U 0 left-parenthesis script l Superscript a Baseline right-parenthesis which sends

Start 2 By 2 Matrix 1st Row 1st Column alpha 2nd Column beta 2nd Row 1st Column script l Superscript a Baseline gamma 2nd Column delta EndMatrix long right-arrow from bar left-parenthesis chi 1 slash chi 2 right-parenthesis left-parenthesis alpha right-parenthesis chi 2 left-parenthesis alpha delta minus script l Superscript a Baseline beta gamma right-parenthesis period

This is irreducible and does not depend on the ordering of chi 1 and chi 2 .

For the next case, let upper F denote the unramified quadratic extension of bold upper Q Subscript script l and s the non-trivial automorphism of upper F over bold upper Q Subscript script l . Suppose that tau is the restriction to upper I Subscript script l of the induction from upper W Subscript upper F to upper W Subscript bold upper Q Sub Subscript script l of a character chi of upper W Subscript upper F with chi not-equals chi Superscript s . Let a denote the conductor of chi slash chi Superscript s , so that a greater-than 0 . Choose a character chi prime of upper W Subscript bold upper Q Sub Subscript script l such that chi prime vertical-bar Subscript upper W Sub Subscript upper F Subscript Baseline Superscript negative 1 Baseline chi has conductor a . If a equals 1 we set

sigma Subscript tau Baseline equals normal upper Theta left-parenthesis chi prime vertical-bar Subscript upper W Sub Subscript upper F Subscript Baseline Superscript negative 1 Baseline chi right-parenthesis circled-times left-parenthesis chi prime ring det right-parenthesis comma

where normal upper Theta left-parenthesis dot right-parenthesis is the irreducible representation of upper G upper L 2 left-parenthesis bold upper F Subscript script l Baseline right-parenthesis defined on page 532 of ReferenceCDT.

To define sigma Subscript tau for a greater-than 1 we will identify upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis with the automorphisms of the bold upper Z Subscript script l -module script upper O Subscript upper F . If a is even, then we let sigma Subscript tau denote the induction from script upper O Subscript upper F Superscript times Baseline left-parenthesis 1 plus script l Superscript a slash 2 Baseline script upper O Subscript upper F Baseline s right-parenthesis to upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis of the character phi of script upper O Subscript upper F Superscript times Baseline left-parenthesis 1 plus script l Superscript a slash 2 Baseline script upper O Subscript upper F Baseline s right-parenthesis , where, for alpha element-of script upper O Subscript upper F Superscript times and beta element-of left-parenthesis 1 plus script l Superscript a slash 2 Baseline script upper O Subscript upper F Baseline s right-parenthesis ,

phi left-parenthesis alpha beta right-parenthesis equals left-parenthesis chi prime vertical-bar Subscript upper W Sub Subscript upper F Subscript Baseline Superscript negative 1 Baseline chi right-parenthesis left-parenthesis alpha right-parenthesis chi prime left-parenthesis det alpha beta right-parenthesis period

If a greater-than 1 is odd, then we let sigma Subscript tau denote the induction from script upper O Subscript upper F Superscript times Baseline left-parenthesis 1 plus script l Superscript left-parenthesis a minus 1 right-parenthesis slash 2 Baseline script upper O Subscript upper F Baseline s right-parenthesis to upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis of eta , where eta is the script l -dimensional irreducible representation of script upper O Subscript upper F Superscript times Baseline left-parenthesis 1 plus script l Superscript left-parenthesis a minus 1 right-parenthesis slash 2 Baseline script upper O Subscript upper F Baseline s right-parenthesis such that eta vertical-bar Subscript script upper O Sub Subscript upper F Sub Superscript times Subscript left-parenthesis 1 plus script l Sub Superscript left-parenthesis a plus 1 right-parenthesis slash 2 Subscript script upper O Sub Subscript upper F Subscript s right-parenthesis Baseline is the direct sum of the characters

alpha beta long right-arrow from bar left-parenthesis chi prime vertical-bar Subscript upper W Sub Subscript upper F Subscript Baseline Superscript negative 1 Baseline chi chi Superscript double-prime Baseline right-parenthesis left-parenthesis alpha right-parenthesis chi prime left-parenthesis det alpha beta right-parenthesis

for alpha element-of script upper O Subscript upper F Superscript times and beta element-of left-parenthesis 1 plus script l Superscript left-parenthesis a plus 1 right-parenthesis slash 2 Baseline script upper O Subscript upper F Baseline s right-parenthesis , where chi double-prime runs over the script l non-trivial characters of script upper O Subscript upper F Superscript times Baseline slash bold upper Z Subscript script l Superscript times Baseline left-parenthesis 1 plus script l script upper O Subscript upper F Baseline right-parenthesis .

Now suppose tau prime is an extended type such that tau prime vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline is irreducible. There is a ramified quadratic extension upper F slash bold upper Q Subscript script l and a character chi of upper W Subscript upper F such that the induction from upper W Subscript upper F to upper W Subscript bold upper Q Sub Subscript script l of chi is tau prime (see §2.6 of ReferenceG). Let s denote the non-trivial field automorphism of upper F over bold upper Q Subscript script l and also let normal script upper P Subscript upper F denote the maximal ideal of the ring of integers script upper O Subscript upper F of upper F . Let a denote the conductor of chi slash chi Superscript s , so a is even and a greater-than-or-equal-to 2 . We may choose a character chi prime of upper W Subscript bold upper Q Sub Subscript script l such that chi prime vertical-bar Subscript upper W Sub Subscript upper F Subscript Baseline Superscript negative 1 Baseline chi has conductor a . We will identify upper G upper L 2 left-parenthesis bold upper Q Subscript script l Baseline right-parenthesis with the automorphisms of the bold upper Q Subscript script l vector space upper F . We will also identify upper U 0 left-parenthesis script l right-parenthesis with the stabiliser of the pair of lattices normal script upper P Subscript upper F Superscript negative 1 Baseline superset-of script upper O Subscript upper F . We define sigma Subscript tau prime to be the induction from upper F Superscript times Baseline left-parenthesis 1 plus normal script upper P Subscript upper F Superscript a slash 2 Baseline s right-parenthesis to ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis of the character phi of upper F Superscript times Baseline left-parenthesis 1 plus normal script upper P Subscript upper F Superscript a slash 2 Baseline s right-parenthesis , where

phi left-parenthesis alpha beta right-parenthesis equals left-parenthesis chi prime vertical-bar Subscript upper W Sub Subscript upper F Subscript Baseline Superscript negative 1 Baseline chi chi Superscript double-prime Baseline right-parenthesis left-parenthesis alpha right-parenthesis chi prime left-parenthesis det alpha beta right-parenthesis comma

with alpha element-of upper F Superscript times and beta element-of left-parenthesis 1 plus normal script upper P Subscript upper F Superscript a slash 2 Baseline s right-parenthesis , where chi double-prime is a character of upper F Superscript times Baseline slash left-parenthesis script upper O Subscript upper F Superscript times Baseline right-parenthesis squared defined as follows. Let psi be a character of bold upper Q Subscript script l with kernel bold upper Z Subscript script l . Choose theta element-of upper F Superscript times such that for x element-of normal script upper P Subscript upper F Superscript a minus 1 we have

left-parenthesis chi prime vertical-bar Subscript upper W Sub Subscript upper F Subscript Baseline Superscript negative 1 Baseline chi right-parenthesis left-parenthesis 1 plus x right-parenthesis equals psi left-parenthesis trace Subscript upper F slash bold upper Q Sub Subscript script l Subscript Baseline left-parenthesis theta x right-parenthesis right-parenthesis period

We impose the following conditions which determine chi double-prime :

chi double-prime is a character of upper F Superscript times Baseline slash left-parenthesis script upper O Subscript upper F Superscript times Baseline right-parenthesis squared ;

chi double-prime vertical-bar Subscript script upper O Sub Subscript upper F Sub Superscript times Baseline is non-trivial;

andchi double-prime left-parenthesis minus theta left-parenthesis upper N Subscript upper F slash bold upper Q Sub Subscript script l Subscript Baseline pi right-parenthesis Superscript a slash 2 Baseline right-parenthesis equals sigma-summation Underscript x element-of bold upper Z slash script l bold upper Z Endscripts psi left-parenthesis x squared slash upper N Subscript upper F slash bold upper Q Sub Subscript script l Subscript Baseline pi right-parenthesis comma

where pi is a uniformiser in script upper O Subscript upper F .

Finally if tau is an irreducible script l -type, choose an extended script l -type tau prime which restricts to tau on upper I Subscript script l and set sigma Subscript tau Baseline equals sigma Subscript tau Sub Superscript prime Subscript Baseline vertical-bar Subscript upper U 0 left-parenthesis script l right-parenthesis Baseline .

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 upper G upper L 2 left-parenthesis bold upper Q Subscript script l Baseline right-parenthesis a two-dimensional representation upper W upper D left-parenthesis pi right-parenthesis of upper W Subscript bold upper Q Sub Subscript script l . (See §4.1 of ReferenceCDT for the normalisation we use.)

Lemma 1.2.1

Suppose that tau is an script l -type and that tau prime is an extended script l -type with tau prime vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline irreducible. Suppose also that pi is an infinite-dimensional irreducible admissible representation of upper G upper L 2 left-parenthesis bold upper Q Subscript script l Baseline right-parenthesis over bold upper Q overbar Subscript script l . Then:

(1)

sigma Subscript tau and sigma Subscript tau prime are irreducible.

(2)

If upper W upper D left-parenthesis pi right-parenthesis vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde tau (resp. upper W upper D left-parenthesis pi right-parenthesis tilde tau prime ), then upper H o m Subscript upper U Sub Subscript tau Baseline left-parenthesis sigma Subscript tau Baseline comma pi right-parenthesis approximately-equals bold upper Q overbar Subscript script l

(resp. upper H o m Subscript upper U Sub Subscript tau Sub Sub Superscript prime Sub Subscript Subscript Baseline left-parenthesis sigma Subscript tau Sub Superscript prime Subscript Baseline comma pi right-parenthesis approximately-equals bold upper Q overbar Subscript script l Baseline right-parenthesis period

(3)

If upper W upper D left-parenthesis pi right-parenthesis vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tau (resp. upper W upper D left-parenthesis pi right-parenthesis neither-approximately-nor-actually-equals tau prime ), then upper H o m Subscript upper U Sub Subscript tau Baseline left-parenthesis sigma Subscript tau Baseline comma pi right-parenthesis equals left-parenthesis 0 right-parenthesis

(resp. upper H o m Subscript upper U Sub Subscript tau Sub Sub Superscript prime Sub Subscript Subscript Baseline left-parenthesis sigma Subscript tau Sub Superscript prime Subscript Baseline comma pi right-parenthesis equals left-parenthesis 0 right-parenthesis right-parenthesis period

Proof.

The case that tau extends to a reducible representation of upper W Subscript bold upper Q Sub Subscript script l follows from the standard theory of principal series representations for upper G upper L 2 left-parenthesis bold upper Q Subscript script l Baseline right-parenthesis . The case that tau is reducible but does not extend to a reducible representation of upper W Subscript bold upper Q Sub Subscript script l follows from Theorem 3.7 of ReferenceG. The case of tau prime follows from Theorem 4.6 of ReferenceG.

Thus, suppose that tau is an irreducible script l -type and that tau prime is an extension of tau to an extended script l -type. If delta denotes the unramified quadratic character of upper W Subscript bold upper Q Sub Subscript script l , then tau prime not-tilde tau prime circled-times delta and so we deduce that

sigma Subscript tau Sub Superscript prime Subscript Baseline not-tilde sigma Subscript tau prime circled-times delta Baseline tilde sigma Subscript tau Sub Superscript prime Subscript Baseline circled-times left-parenthesis delta ring det right-parenthesis period

Thus sigma Subscript tau Sub Superscript prime Subscript Baseline vertical-bar Subscript bold upper Q Sub Subscript script l Sub Superscript times Subscript upper U 0 left-parenthesis script l right-parenthesis Baseline is irreducible. It follows that sigma Subscript tau is irreducible. The second and third part of the lemma for tau follow similarly.

1.3. Reduction of types for admissible representations

We begin by reviewing some irreducible representations of upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis , upper U 0 left-parenthesis script l right-parenthesis and ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis . Let sigma Subscript 1 comma 0 denote the standard representation of upper G upper L 2 left-parenthesis bold upper F Subscript script l Baseline right-parenthesis over bold upper F overbar Subscript script l . If n equals 0 comma 1 comma period period period comma script l minus 1 and if m element-of bold upper Z slash left-parenthesis script l minus 1 right-parenthesis bold upper Z , then we let sigma Subscript n comma m Baseline equals upper S y m m Superscript n Baseline left-parenthesis sigma Subscript 1 comma 0 Baseline right-parenthesis circled-times det Overscript m Endscripts . We may think of sigma Subscript n comma m as a continuous representation of upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis over bold upper F overbar Subscript script l . These representations are irreducible, mutually non-isomorphic and exhaust the irreducible continuous representations of upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis over bold upper F overbar Subscript script l .

If m 1 comma m 2 element-of bold upper Z slash left-parenthesis script l minus 1 right-parenthesis bold upper Z we let sigma prime Subscript m 1 comma m 2 denote the character of upper U 0 left-parenthesis script l right-parenthesis over bold upper F overbar Subscript script l determined by

Start 2 By 2 Matrix 1st Row 1st Column a 2nd Column b 2nd Row 1st Column script l c 2nd Column d EndMatrix long right-arrow from bar a Superscript m 1 Baseline d Superscript m 2 Baseline period

These representations are irreducible, mutually non-isomorphic, and exhaust the irreducible continuous representations of upper U 0 left-parenthesis script l right-parenthesis over bold upper F overbar Subscript script l .

If m 1 comma m 2 element-of bold upper Z slash left-parenthesis script l minus 1 right-parenthesis bold upper Z , a element-of bold upper F overbar Subscript script l Superscript times and m 1 not-equals m 2 , then we let sigma prime Subscript StartSet m 1 comma m 2 EndSet comma a denote the representation of ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis over bold upper F overbar Subscript script l obtained by inducing the character of bold upper Q Subscript script l Superscript times Baseline upper U 0 left-parenthesis script l right-parenthesis which restricts to sigma prime Subscript m 1 comma m 2 on upper U 0 left-parenthesis script l right-parenthesis and which sends negative script l to a . If m element-of bold upper Z slash left-parenthesis script l minus 1 right-parenthesis bold upper Z and a element-of bold upper F overbar Subscript script l Superscript times , then we let sigma prime Subscript StartSet m EndSet comma a denote the character of ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis over bold upper F overbar Subscript script l which restricts to sigma prime Subscript m comma m on upper U 0 left-parenthesis script l right-parenthesis and which sends w Subscript script l to a . These representations are irreducible, mutually non-isomorphic and exhaust the irreducible, finite-dimensional, continuous representations of ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis over bold upper F overbar Subscript script l .

We will say that a reducible script l -type tau (resp. irreducible script l -type, resp. extended script l -type tau with irreducible restriction to upper I Subscript script l ) admits an irreducible representation sigma of upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis (resp. upper U 0 left-parenthesis script l right-parenthesis , resp. ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis ) over bold upper F overbar Subscript script l , if sigma Subscript tau (resp. sigma Subscript tau , resp. sigma Subscript tau prime ) contains an invariant script upper O Subscript bold upper Q overbar Sub Subscript script l -lattice normal upper Lamda and if sigma is a Jordan-Hölder constituent of normal upper Lamda circled-times bold upper F overbar Subscript script l . We will say that tau (resp. tau , resp. tau prime ) simply admits sigma if sigma is a Jordan-Hölder constituent of normal upper Lamda circled-times bold upper F overbar Subscript script l of multiplicity one.

For each of the bold upper F overbar Subscript script l -representations of upper G upper L 2 left-parenthesis bold upper Z Subscript script l Baseline right-parenthesis , upper U 0 left-parenthesis script l right-parenthesis and ModifyingAbove upper U With tilde Subscript 0 Baseline left-parenthesis script l right-parenthesis just defined, we wish to define notions of “admittance” and “simple admittance” with respect to a continuous representation rho overbar colon upper G Subscript script l Baseline right-arrow upper G upper L 2 left-parenthesis bold upper F overbar Subscript script l Baseline right-parenthesis . Let rho overbar be a fixed continuous representation upper G Subscript script l Baseline right-arrow upper G upper L 2 left-parenthesis bold upper F overbar Subscript script l Baseline right-parenthesis .

The representation sigma Subscript n comma m admits rho overbar if eitherrho overbar vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega 2 Superscript 1 minus script l n minus m left-parenthesis script l plus 1 right-parenthesis Baseline 2nd Column 0 2nd Row 1st Column 0 2nd Column omega 2 Superscript script l minus n minus m left-parenthesis script l plus 1 right-parenthesis EndMatrix

orrho overbar vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega Superscript 1 minus m Baseline 2nd Column asterisk 2nd Row 1st Column 0 2nd Column omega Superscript negative n minus m Baseline EndMatrix comma

which in addition we require to be peu-ramifié in the case n equals 0 . (Note that sigma Subscript n comma 0 admits rho overbar if and only if the Serre weight (see ReferenceSe2) of rho overbar Superscript logical-or Baseline circled-times omega is n plus 2 .)

The representation sigma Subscript n comma m simply admits rho overbar if sigma Subscript n comma m admits rho overbar .

The representation sigma prime Subscript m 1 comma m 2 admits rho overbar if eitherrho overbar vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega 2 Superscript 1 minus script l m Super Subscript i Superscript minus m Super Subscript j Superscript Baseline 2nd Column 0 2nd Row 1st Column 0 2nd Column omega 2 Superscript script l minus m Super Subscript i Superscript minus script l m Super Subscript j Superscript Baseline EndMatrix comma

where StartSet m Subscript i Baseline comma m Subscript j Baseline EndSet equals StartSet m 1 comma m 2 EndSet and m Subscript i Baseline greater-than-or-equal-to m Subscript j , orrho overbar vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega Superscript 1 minus m 1 Baseline 2nd Column asterisk 2nd Row 1st Column 0 2nd Column omega Superscript minus m 2 Baseline EndMatrix comma

orrho overbar vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega Superscript 1 minus m 2 Baseline 2nd Column asterisk 2nd Row 1st Column 0 2nd Column omega Superscript minus m 1 Baseline EndMatrix period

(Note that sigma prime Subscript m 1 comma m 2 admits rho overbar if and only if some irreducible constituent of upper I n d Subscript upper U 0 left-parenthesis script l right-parenthesis Superscript upper G upper L 2 left-parenthesis bold upper Z Super Subscript script l Superscript right-parenthesis Baseline sigma prime Subscript m 1 comma m 2 admits rho overbar .)

The representation sigma prime Subscript m 1 comma m 2 with m 1 not-equals m 2 simply admits rho overbar if eitherrho overbar vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega Superscript 1 minus m 1 Baseline 2nd Column asterisk 2nd Row 1st Column 0 2nd Column omega Superscript minus m 2 EndMatrix

orrho overbar vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega Superscript 1 minus m 2 Baseline 2nd Column asterisk 2nd Row 1st Column 0 2nd Column omega Superscript minus m 1 Baseline EndMatrix period

The representation sigma prime Subscript m comma m simply admits rho overbar ifrho overbar vertical-bar Subscript upper I Sub Subscript script l Subscript Baseline tilde Start 2 By 2 Matrix 1st Row 1st Column omega Superscript 1 minus m Baseline 2nd Column asterisk 2nd Row 1st Column 0 2nd Column omega Superscript negative m EndMatrix

is très ramifié.

The representation sigma prime Subscript StartSet m 1 comma m 2 EndSet comma a with m 1 not-equals m 2 admits rho overbar if either sigma prime Subscript m 1 comma m 2 or sigma prime Subscript m 2 comma m 1 admits rho overbar and if left-parenthesis omega Superscript negative 1 Baseline det rho overbar right-parenthesis vertical-bar Subscript upper W Sub Subscript bold upper Q Sub Sub Subscript script l Sub Subscript Subscript Baseline equals the central character of sigma prime Subscript StartSet m 1 comma m 2 EndSet comma a . (Note that in this case sigma prime Subscript StartSet m 1 comma m 2 EndSet comma a Baseline vertical-bar Subscript upper U 0 left-parenthesis script l right-parenthesis Baseline equals sigma prime Subscript m 1 comma m 2 Baseline circled-plus sigma prime Subscript m 2 comma m 1 .)

The representation sigma prime Subscript StartSet m 1 comma m 2 EndSet comma a with m 1 not-equals m 2