Access the full text.
Sign up today, get DeepDyve free for 14 days.
H. Carayol (1986)
Sur la mauvaise réduction des courbes de ShimuraCompositio Mathematica, 59
F. Jarvis (1999)
Mazur's Principle for Totally Real Fields of Odd DegreeCompositio Mathematica, 116
Chuangxun Cheng (2013)
IHARA’S LEMMA FOR SHIMURA CURVES OVER TOTALLY REAL FIELDS AND MULTIPLICITY TWO
(1989)
Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday
Toby Gee (2008)
Automorphic lifts of prescribed typesMathematische Annalen, 350
R. Gow (1981)
On the Schur Indices of Characters of Finite Classical GroupsJournal of The London Mathematical Society-second Series
M. Kisin (2007)
Potentially semi-stable deformation ringsJournal of the American Mathematical Society, 21
Haining Wang (2015)
Anticyclotomic Iwasawa theory for Hilbert modular forms
(1973)
On modular curves over finite fields, Discrete subgroups of Lie groups and applications to moduli
Frank Calegari, D. Geraghty (2012)
Modularity lifting beyond the Taylor–Wiles methodInventiones mathematicae, 211
(2005)
an-ticyclotomic Z p -extensions
M. Emerton, David Helm (2011)
The local Langlands correspondence for GL_n in familiesarXiv: Number Theory
M. Emerton, Toby Gee, David Savitt (2013)
Lattices in the cohomology of Shimura curvesInventiones mathematicae, 200
Toby Gee, Tongyin Liu, David Savitt (2013)
THE WEIGHT PART OF SERRE’S CONJECTURE FOR $\text{GL}(2)$Forum of Mathematics, Pi, 3
K. Ribet (1990)
On modular representations of $$(\bar Q/Q)$$ arising from modular formsInventiones mathematicae, 100
Commentarii Helvetici (2012)
Anticyclotomic Iwasawa ’ s Main Conjecture for Hilbert modular forms
J. Shotton (2015)
The Breuil-Mézard conjecture when l is not equal to p
M. Kisin (2009)
Moduli of finite flat group schemes, and modularityAnnals of Mathematics, 170
A. Wiles (1995)
Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?), 34
Kevin Buzzard, Fred Diamond, F. Jarvis (2009)
ON SERRE’S CONJECTURE FOR MOD ℓ GALOIS REPRESENTATIONS OVER TOTALLY REAL FIELDS
Richard Taylor (2006)
On the Meromorphic Continuation of Degree Two L-FunctionsDocumenta Mathematica
C. Breuil, Fred Diamond (2012)
FORMES MODULAIRES DE HILBERT MODULO p ET VALEURS D'EXTENSIONS GALOISIENNESAnnales Scientifiques De L Ecole Normale Superieure, 47
(2017)
Local deformation rings for 2-adic representations of GQl , l = 2, (2017) Appendix to Yongquan Hu, Vytautas Paškūnas, On crystabelline deformation rings of Gal(Qp/Qp
K. Ribet (2003)
MULTIPLICITIES OF GALOIS REPRESENTATIONS IN JACOBIANS OF SHIMURA CURVES
Fred Diamond (1997)
The Taylor-Wiles construction and multiplicity oneInventiones mathematicae, 128
Fred Diamond, Richard Taylor (1994)
Non-optimal levels of modl modular representationsInventiones mathematicae, 115
M. Bertolini, H. Darmon (2005)
Iwasawa's Main Conjecture for elliptic curves over anticyclotomic Zp-extensionsAnnals of Mathematics, 162
Fred Diamond, Richard Taylor (1994)
Lifting modular $\mod \ell$ representationsDuke Mathematical Journal, 74
Y. Morita (1981)
Reduction modulo ~ss~ of Shimura curvesHokkaido Mathematical Journal, 10
Przemyslaw Chojecki, C. Sorensen (2013)
Weak local-global compatibility in the p-adic Langlands program for U(2)arXiv: Number Theory
Fred Diamond, M. Flach, Li Guo (2004)
THE TAMAGAWA NUMBER CONJECTURE OF ADJOINT MOTIVES OF MODULAR FORMSAnnales Scientifiques De L Ecole Normale Superieure, 37
Thomas Barnet-Lamb, Toby Gee, D. Geraghty, Richard Taylor (2010)
Potential automorphy and change of weightAnnals of Mathematics, 179
Richard Taylor (1989)
On galois representations associated to Hilbert modular formsInventiones mathematicae, 98
Jeffrey Manning (2019)
Patching and multiplicity 2k for Shimura curvesarXiv: Number Theory
(2017)
Stacks Project
Toby Gee, James Newton (2016)
PATCHING AND THE COMPLETED HOMOLOGY OF LOCALLY SYMMETRIC SPACESJournal of the Institute of Mathematics of Jussieu, 21
J. Shotton (2016)
The Breuil--M\'{e}zard conjecture when $l \neq p$arXiv: Number Theory
Yong Hu, Vytautas Paškūnas (2017)
On crystabelline deformation rings of $$\mathrm {Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)$$Gal(Q¯p/Qp) (with an appendix by Jack Shotton)Mathematische Annalen, 373
Masataka Chida, Ming-Lun Hsieh (2013)
On the anticyclotomic Iwasawa main conjecture for modular formsCompositio Mathematica, 151
L. Clozel, M. Harris, Richard Taylor (2008)
Automorphy for some l-adic lifts of automorphic mod l Galois representationsPublications mathématiques, 108
B. Edixhoven (1992)
The weight in Serre's conjectures on modular formsInventiones mathematicae, 109
(1984)
Congruence relations between modular forms, Proceedings of the International Congress of Mathematicians, Vol
birthday, Part II (Ramat Aviv, 1989), Israel Math. Conf. Proc., vol. 3, Weizmann, Jerusalem, 1990, pp. 221–236
M. Longo (2012)
Anticyclotomic Iwasawa's Main Conjecture for Hilbert modular formsCommentarii Mathematici Helvetici, 87
J. Shotton (2013)
Local deformation rings for GL2 and a Breuil–Mézard conjecture when l≠p.Algebra & Number Theory, 10
P. Scholze (2015)
On the p-adic cohomology of the Lubin-Tate towerarXiv: Algebraic Geometry
Richard Taylor (2008)
Automorphy for some l-adic lifts of automorphic mod l Galois representations. IIPublications mathématiques, 108
(1989)
Commutative ring theory, second ed
(2011)
Local-global compatibility in the p-adic Langlands programme for GL2/Q, 2011, draft available at http://www.math.uchicago.edu/~emerton/preprints.html
Przemyslaw Chojecki, C. Sorensen (2014)
Strong local-global compatibility in the p-adic Langlands program for U(2)
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
We prove Ihara’s lemma for the mod l cohomology of Shimura curves, localized at a maximal ideal of the Hecke algebra, under a large image hypothesis on the associated Galois representation. This was proved by Diamond and Taylor, for Shimura curves over Q, under various assumptions on l. Our method is totally different and can avoid these assumptions, at the cost of imposing the large image hypothesis. It uses the Taylor–Wiles method, as improved by Diamond and Kisin, and the geometry of integral models of Shimura curves at an auxiliary prime. 1 Introduction Let = (N ) be the usual congruence subgroup of SL (Z),for some N ≥ 1, 0 2 and let p be a prime not dividing N . Write = ∩ (p).If X and X are the compactiﬁed modular curves of levels and , then there are two degeneracy maps π ,π : X → X 1 2 −1 p 0 p 0 induced by the inclusions → and → .If l is another 01 01 prime, then we have a map ∗ ∗ ∗ 1 2 1 π = π + π : H (X , F ) → H (X , F ). l l 1 2 Communicated by Wei Zhang. B Jack Shotton jack.g.shotton@durham.ac.uk Jeffrey Manning jmanning@math.ucla.edu Mathematics Department, UCLA, Math Sciences Building 6164, Los Angeles, CA 90095, USA Department of Mathematical Sciences, Durham University, Lower Mountjoy, Stockton Road, Durham DH1 3LE, UK 123 188 J. Manning, J. Shotton As a consequence of a result of Ihara—[22] Lemma 3.2, and see also the proof of [30] Theorem 4.1—the kernel of π may be determined. In particular: Theorem (Ihara’s Lemma) If m is a non-Eisenstein maximal ideal of the Hecke algebra acting on these cohomology groups (that is, m corresponds to an irreducible Galois ∗ 1 representation), then the map π is injective after localizing at m. This was used by Ribet in [30] to prove a level-raising result for modular forms: if f ∈ S () is a cuspidal eigenform such that ρ is irreducible and the Fourier coefﬁcient a satisﬁes a ≡±(1 + p)(mod l), p−new then there is a cuspidal eigenform g ∈ S ( ) such that ρ ρ . 2 f g Now suppose that F is a totally real number ﬁeld and that D is a quaternion division algebra over F ramiﬁed at all but one inﬁnite place. For K ⊂ (D ⊗ A ) a compact F , f open subgroup, p a ﬁnite place of F at which K and D are unramiﬁed, and l aprime, there is an obvious (conjectural) generalisation of Theorem 1 with X replaced by the Shimura curve X . We refer to this as “Ihara’s Lemma at p for X , localized at m”; it K K depends on K and on a maximal ideal m of the Hecke algebra acting on H (X , F ), K l to which is associated a Galois representation ρ : G → GL (F ). The purpose of F 2 l this paper is to prove: Theorem 1.1 Suppose that l > 2 and that the image of ρ contains a subgroup of GL (F ) conjugate to SL (F ) (and satisﬁes an additional Taylor–Wiles hypothesis 2 l 2 l if l = 5 and 5 ∈ F). Then Ihara’s Lemma at p for X , localized at m,istrue. Ihara’s method of proof does not generalise, since it relies on the “congruence sub- group property of SL (Z[ ])”, the analogue of which is a longstanding conjecture of Serre in the quaternionic case. In [13], Diamond and Taylor overcame this difﬁculty for Shimura curves over Q using the good reduction of Shimura curves at l and com- parison of mod l de Rham and étale cohomology. This necessitates various conditions on l: • p does not divide l; • D and K must be unramiﬁed at l; k−2 • if the result is formulated with coefﬁcients Sym F , then the weight k satisﬁes k ≤ l − 1. It seems likely that the approach of [13] can be adapted to the totally real case with sim- ilar conditions on l, as in Cheng’s draft [8] (which the author tells us is not complete), but this has not yet been carried out in full detail. 1 ∗ In fact, if we instead take = (N ) then π is already injective. For us, however, localizing at a maximal ideal of the Hecke algebra will be crucial. See the end of [12]for k = l − 1. 123 Ihara’s Lemma for Shimura curves... 189 Our method of proof is entirely different, and requires no such conditions on l. On the other hand, we have to impose a more stringent condition on ρ —rather than merely being irreducible, its image must contain the subgroup SL (F ). 2 l Our starting point is that Ihara’s Lemma is known (and easy) for the “Shimura sets” associated to deﬁnite quaternion algebras. Following a strategy introduced by Ribet in [31] we introduce an auxiliary prime q, at which both K and D are unramiﬁed. Then there is a totally deﬁnite quaternion algebra D ramiﬁed at the same ﬁnite places q × as D, together with q, and a compact open subgroup K ⊂ (D ⊗ A ) agreeing F , f with K at all places besides q and maximal at q. Our goal will then be to reduce the statement of Ihara’s Lemma for X at m to the corresponding (known) statement for the Shimura set Y corresponding to K . The link between X and Y is given by the geometry of integral models of the K K Shimura curve X , with (q)-level structure. Speciﬁcally, the special ﬁbre of K (q) 0 X at q consists of two components, both of which are isomorphic to the special K (q) ﬁbre of X , and has singularities at a ﬁnite set of points which are in bijection with Y . This results in a ﬁltration of H (X , F ) whose graded pieces are two copies K K (q) l 0 1 ⊕2 of H (Y , F ) and one copy of H (X , F ) . This idea has been extensively studied K l K l by Mazur, Ribet [31], Jarvis [23] and others. Unfortunately, the existence of this ﬁltration does not directly imply any relation 1 0 between the Hecke module structures of H (X , F ) and H (Y q , F ). For example, K l K l the ﬁltration could be split (in the sense that 1 1 ⊕2 0 ⊕2 H (X , F ) H (X , F ) ⊕ H (Y q , F ) K (q) l K l K l as Hecke modules) which would not impose any relations between H (X , F ) and K l 0 1 0 H (Y q , F ). So in order to deduce anything about H (X , F ) and H (Y q , F ), K l K l K l we need to have additional information about the Hecke module structure of H (X , F ) and its interaction with the ﬁltration. K (q) l The novelty of this paper, then, is to obtain this extra information. It takes the form of a certain “ﬂatness” statement, which we formulate and prove by using the Taylor– Wiles–Kisin patching method. To our knowledge, this is the ﬁrst time that patching has been combined with the geometry of integral models in this way. Brieﬂy, the Taylor–Wiles–Kisin method considers a ring R , which is a power series ring over the completed tensor product of various local Galois deformation 1 0 1 rings, and relates the Hecke modules H (X , F ), H (Y , F ) and H (X , F ) K l K l K (q) l to certain maximal Cohen–Macaulay “patched” modules over R . Our method proves 1 3 that the “patched” module corresponding to H (X , F ) is ﬂat over some speciﬁc K (q) l local deformation ring at the prime q. Using this and some commutative algebra we are able to deduce Ihara’s Lemma for X from the corresponding result for Y . K K Our strategy for proving this ﬂatness is inspired by Taylor’s “Ihara avoidance” argument, used in the proof of the Sato–Tate conjecture [40]. We impose the condition that our auxiliary prime q satisﬁes Nm(q) ≡ 1 (mod l), and consider a certain tamely ps ps ramiﬁed principal series deformation ring, R = R , which is a quotient of the ρ | ,O This is a slight simpliﬁcation. 123 190 J. Manning, J. Shotton universal local deformation ring R = R . The standard map from R to the q q ρ | ,O m G ps mod l Hecke algebra acting on H (X , F ) then factors through the quotient R , K (q) l q even though the map from R to the integral Hecke algebra acting on H (X , Z ) q K (q) l does not. In our situation, the assumption on the image of ρ allows us to choose the auxiliary prime q so that ρ (Frob ) = . ps In this case, the ring R is a regular local ring (a calculation carried out in [34]). This is what enables us to gain a foothold—it is a standard principle going back to Diamond [11] that regular local deformation rings give rise to important structural results about Hecke modules. We apply a version of the miracle ﬂatness criterion to ps prove that a particular patched module is ﬂat over R , which is the key fact needed to make our argument work. The advantage of this argument, as opposed to that of [13], is that we do not need to make any assumptions about the structure of the local deformation rings at primes dividing l, or indeed at any primes besides q, beyond knowing that they have the correct dimension (a fact which certainly holds in the generality we need). This is the reason we do not need to impose any of the restrictions on the prime l appearing in earlier results. 1.1 Applications We brieﬂy survey some of the applications of Ihara’s Lemma (for modular or Shimura curves, or Shimura sets) that are in the literature. 1.1.1 Representation theoretic reformulation p × Suppose that K ⊂ (D ⊗ A ) is a compact open subgroup, and let F , f V = lim H (X p , F ) K K l − → where the limit runs over compact open subgroups K ⊂ GL (F ). Then V is a p 2 p smooth admissible representation of GL (F ). Suppose that m is a maximal ideal of 2 p the Hecke algebra acting on V . Then we have: Proposition 1.2 Suppose that, for K = K K with SL (O ) ⊂ K ⊂ GL (O ) p 2 F ,p p 2 F ,p a compact open subgroup, Ihara’s Lemma is true for X at m. Then the representation V of G L (F ) has no one-dimensional subrepresentations. m 2 p Suppressing minor issues due to framing and ﬁxed determinants. Provided that one carefully controls the ramiﬁcation in the coefﬁcient ring O. 123 Ihara’s Lemma for Shimura curves... 191 Remark 1.3 If l = p then there is a notion of genericity for smooth representations of GL (F ) (see, for instance, [16]); when n = 2, the non-generic smooth irreducible n p representations are precisely the one-dimensional ones. It is this “no non-generic subrepresentations” property that conjecturally generalises to higher rank (see [9]). 1.1.2 Freeness results If T is the algebra of Hecke operators acting on H = H (X , O) , including those 1 K m at primes at which K ramiﬁes, then we can ask whether H is free as a T-module. For modular curves results along these lines were proved by Mazur, Ribet and others (see, for instance, [14] Theorem 9.2 and [6] Theorem 4.8). In the case of Shimura curves, there are results starting with [32]. Note that it is not always the case that H is free; in many cases this can be explained by the geometry of local deformation rings, as in work of the ﬁrst author [27]. In [11] section 3.2, it is explained how the Taylor–Wiles method and a ‘numerical criterion’ may be used to prove freeness results at minimal and non-minimal level for modular curves (some limited freeness results for Shimura curves are also given in [11] section 3.3). At non-minimal level, this relies crucially on Ihara’s Lemma, and so using our result we can extend these freeness results. For instance, we have the following, in which denotes the set of places where we are allowing non-minimal level. Theorem 1.4 Let F be a totally real number ﬁeld, D be a quaternion algebra over F ramiﬁed at exactly one inﬁnite place, a ﬁnite set of ﬁnite places of F, and l > 2 be aprime. Let K = K ⊂ (D ⊗ A ) be a compact open subgroup and let k ≥ 2 be v F , f k−2 2 an integer. Let H = H (X , Sym (O ⊗ Z )), and let T be the Hecke algebra 1 K l K acting on H generated by the T and S for v lat which K is maximal compact and v v v D is split, and the U for each v ∈ . Let m be a maximal ideal of T containing l. Suppose that the Galois representation ρ attached to m has non-exceptional image, and that the following conditions hold. (1) For all ﬁnite places v | lofF,F /Q is unramiﬁed and D is split at v. v l (2) For all ﬁnite places v ∈ not dividing l, D is split and ρ is unramiﬁed at v. (3) For all ﬁnite places v lofF, ρ| has minimal Artin conductor n among all G v its twists by characters of G . (4) For all ﬁnite places v l of F at which D splits, either: • v/ ∈ and K = U (v );or v 1 • v ∈ and K = U (v ). v 1 (See (2.4) below for the deﬁnition of U (v )). (5) For all ﬁnite places v of F at which D ramiﬁes, K is the group of units in a maximal order of D ⊗ F , and if ρ is unramiﬁed at v then either: • Nm(v) ≡±1 (mod l); • Nm(v) ≡ 1 (mod l) and ρ(Frob ) is not scalar; or 123 192 J. Manning, J. Shotton • Nm(v) ≡−1 (mod l), and tr(ρ(Frob )) = 0. (6) If v l is a place of F at which D splits and Nm(v) ≡−1 (mod l), then either ρ| is reducible or ρ(I ) has order divisible by l. G F F v (7) One of the following holds. • (the Fontaine–Laffaille case) 2 ≤ k ≤ l − 1 and K is a maximal compact subgroup for each v | l; or • (the ordinary case)k = 2 and, for each v | l, either: v/ ∈ and K is maximal ∼ ∼ compact; or v ∈ ,F = Q ,K = U (v), and ρ| = . v l v 0 I Then H is free of rank 2 over T . m K ,m min × Proof (sketch) For v ∈ ,let K ⊂ (D ⊗ F ) be a maximal compact subgroup; min min min otherwise, let K = K .Let K = K . The numbered conditions were v v v chosen to ensure that all the relevant local deformation rings corresponding to forms min of level K are formally smooth. Thus the Taylor–Wiles method gives a result min analogous to [11] Theorem 3.1 at level K . The result at level K now follows exactly as in the proof of [11] Theorem 3.4, using Ihara’s Lemma at each prime in . See also [39] Theorem 3.2 for a similar result in the deﬁnite case. Remark 1.5 (1) In the ‘Fontaine–Laffaille case’, at least if (k − 1)[F : Q]≤ l − 2, the version of Ihara’s lemma required would presumably follow from the method of [13], as in [8], and so the condition on the image of ρ could be relaxed to a Taylor–Wiles hypothesis. In the ‘ordinary case’ we require Ihara’s lemma at places of dividing l, which is apparently not accessible by the method of [13]. (2) Without a condition such as (5) where D ramiﬁes, the module may genuinely not be free, see [27]. (3) Conditions (3) and (6) could probably be omitted, and the set of non-minimal places could probably be allowed to contain places where ρ ramiﬁes. (4) The requirement that the weights are parallel is for convenience. The restriction to the Fontaine–Laffaille range is not required for our version of Ihara’s lemma, but is required to prove minimal freeness results using the method of [11]. Nevertheless, in other situations where the multiplicity at minimal level can be determined (even if this multiplicity is not one), it seems plausible that Ihara’s Lemma could be used to deduce information about the multiplicity at non-minimal levels. 1.1.3 Local-global compatibility In theworkofEmerton [17] on local-global compatibility in the p-adic Langlands progam, Ihara’s lemma is essential to obtain results with integral coefﬁcients. Gen- eralisations of Emerton’s result to compact forms of U (2) over totally real ﬁelds in which l splits have been proved in [10]—the compactness assumption ensuring that Ihara’s Lemma is known. We expect that our results (and those of [13]) could be used to prove analogues of Emerton’s Theorem 1.2.6 for the completed cohomology of Shimura curves, at least in settings where multiplicity one still holds. 123 Ihara’s Lemma for Shimura curves... 193 1.1.4 Iwasawa theory In [1], Bertolini and Darmon proved one divisibility in the anticyclotomic Iwasawa Main Conjecture for (certain) elliptic curves over imaginary quadratic ﬁelds. The result of [13] on Ihara’s Lemma for Shimura curves was an important technical tool in the proof. Contingent on Ihara’s Lemma for Shimura curves over totally real ﬁelds, Longo [26] generalises Bertolini and Darmon’s work to the setting of Hilbert modular forms of parallel weight two; our results therefore make his results unconditional in many cases. Further generalisations are made by Chida and Hsieh [7] and Wang [41], and our work may be able to weaken some of their hypotheses. 1.1.5 Level raising The works [30] and [13] apply Ihara’s Lemma to the problem of level-raising for modu- lar forms—that is, of determining at which non-minimal levels there is a newform with a given residual Galois representation. Nowadays, there is an argument of Gee [18] using the Taylor–Wiles–Kisin method and a lifting technique of Khare–Wintenberger. Combined with the results of [19] and of [4], this gives (under a Taylor–Wiles hypoth- esis) level raising theorems for Hilbert modular forms in arbitrary weight. We thank Toby Gee for explaining this point to us. Since we also require the Taylor–Wiles hypothesis, it is unlikely that our theorem gives substantial new level raising results. 1.2 Outline of the paper In Sect. 2 we recall the deﬁnitions of Shimura curves and Hecke operators. We also deﬁne the Shimura sets we will need, and recall the necessary results on integral models. Most of Sect. 3 is taken up with the calculation of local deformation rings at the aux- iliary prime q. We also precisely deﬁne lattices in certain inertial types (representations of GL (O )). 2 F ,q Section 4 carries out the Taylor–Wiles–Kisin patching method. We use the formal- ism of patching functors, introduced in [15]. This is mostly standard, and we include it because we don’t know a reference for the fact that the ﬁltrations of homology coming from integral models may be patched. Section 5 contains calculations in commutative algebra over the local deformation rings at q that are at the technical heart of the proof. Section 6 contains the precise statement and proof of our theorem. A sensible order to read this article in would be to skim Sect. 2, to ﬁx notation, and then turn to Sect. 6, referring back to the other sections as needed. 1.3 Notation If k is a local or global ﬁeld, then G will denote its absolute Galois group. If l is a prime distinct from the characteristic of k, then we write : G → Z for the l-adic k l cyclotomic character and for its reduction modulo l. 123 194 J. Manning, J. Shotton If l is a prime and M is a Z -module, then we write M for its Pontrjagin dual. If M is a ﬁnite free Z -module (resp. an F -vector space, resp. a Q -vector space), then l l l we write M = Hom (M , Z ) (resp. Hom (M , F ), resp. Hom (M , Q )). Z l F l Q l l l l 2 Shimura curves 2.1 Let F be a totally real number ﬁeld of degree d and let O be the ring of integers of F. We write A for the ﬁnite adeles of F.If v is a place of F then we write k F , f v for its residue ﬁeld, for a ﬁxed choice of uniformizer in F , and A for the ﬁnite v v F , f adeles of F with the factor F dropped. If l is a rational prime then we write for v l the set of places of F above l; we write for the set of inﬁnite places of F. 2.2 Let D be a quaternion division algebra over F split at either no inﬁnite places (the deﬁnite case) or exactly one inﬁnite place, τ (the indeﬁnite case), and let O be a maximal order in D. We write for the set of ﬁnite places of F at which D ramiﬁes. We assume that if F = Q and we are in the indeﬁnite case then is nonempty. We write G for the algebraic group over O associated to O , and Z for its centre. For every place v at which D splits we ﬁx an isomorphism κ : O ⊗ O − → v D O F ,v M (O ). We also denote by κ the various isomorphisms, such as (D ⊗ F ) − → 2 F ,v v F v GL (F ), obtained from it. 2 v 2.3 We ﬁx a rational prime l and a ﬁnite place p of F such that p ∈ / ; we do allow the possibility that p | l. 2.4 Let K be a compact open subgroup of G(A ).If v is a ﬁnite place of F F , f v v v then when it is possible to do so we will write K = K K for K ⊂ G(A ) and F , f K ⊂ G(F ). A compact open subgroup K of G(A ) is unramiﬁed at v if v/ ∈ v v F , f v v and K = K G(O ) for some K , and that it is ramiﬁed otherwise. We let F ,v (K ) = ∪{v : K is ramiﬁed at v}. If v/ ∈ is a ﬁnite place of F, and n ≥ 1, then we deﬁne U (v ) to be the subgroup ab n −1 n U (v ) = κ ∈ G(O ) : c ≡ 0 (mod ) 0 F ,v v v cd of G(O ), and F ,v ab n −1 n n U (v ) = κ ∈ U (v ) : d ≡ 1 (mod ) . 1 0 v v cd If K is unramiﬁed at v then we write v v K (v) = K U (v) ⊂ K = K G(O ). 0 0 F ,v 2.5 Suppose that we are in the indeﬁnite case. Letting H = C \ R be acted on by GL (R) in the usual way, via κ we get an action of G(F ) GL (R) on H.We 2 τ τ 2 123 Ihara’s Lemma for Shimura curves... 195 −1 −1 say that K is sufﬁciently small if the action of G(F ) ∩ gKg /Z (F ) ∩ gKg on H is free for every g ∈ G(A ). We will assume throughout that all our compact open F , f subgroups are sufﬁciently small. We let X (C) = G(Q)\ G(A )/K × H , K F , f a compact Riemann surface. By the theory of Shimura varieties, there is a smooth projective curve X over F such that, when F is considered as a subﬁeld of C via τ , the C-points of X are given by the above formula. For F a sheaf of abelian groups i i on X (C) we write H (X , F) = H (X (C), F). K K K 2.6 Write [γ, x ] for the point in X (C) corresponding to γ ∈ G(A ) and x ∈ H. K F , f If K ⊂ K ⊂ G(A ) are compact open subgroups then there is a map X → X F , f K K given on complex points by [γ, x ] →[γ, x ].For g ∈ G(A ) there is a map F , f ρ : X → X −1 given on C-points by g K g Kg ρ ([γ, x ]) =[γ g, x ], The maps ρ deﬁne a right action of G(A ) on the inverse system (X ) ;if g F , f K K −1 g Kg ⊂ K then we will also write ρ for the composite map X − → X → X . −1 K K g Kg 2.7 Let M be an abelian group. Suppose that K , K ⊂ G(A ) are sufﬁciently 1 2 F , f small and that g ∈ G(A ). Then, as in [3] section 4, there are double coset operators F , f i i [K gK ]: H (X , M ) → H (X , M ) 1 2 K K 2 1 for i = 0, 1, 2. If v/ ∈ (K ) ∪ then we deﬁne the Hecke operators T and S to ∞ v v be the double coset operators T = K K and S = K K . If A is a ring and S is a ﬁnite set of places containing ∪ then we write T = A[T , S : v/ ∈ S], v v a polynomial ring in inﬁnitely many variables which acts on H (X , M ) for any K for which (K ) ⊂ S and any A-module M. 123 196 J. Manning, J. Shotton If v/ ∈ , then we deﬁne the Hecke operator U to be the double coset operator K K acting on any H (K , M ) for M an abelian group (note that U = T if K is unramiﬁed v v at v). 2.8 Now suppose that we are in the deﬁnite case. A compact open subgroup K ⊂ G(A ) is sufﬁciently small if, for every g ∈ G(A ),wehave F , f F , f −1 G(F ) ∩ g Kg ⊂ Z (F ). Again, we will always assume that our compact open subgroups are sufﬁciently small. We deﬁne Y = G(F )\G(A )/K K F , f which is a ﬁnite set. Exactly as in the indeﬁnite case, we deﬁne an action of G(A ) F , f on the inverse system (Y ) , and actions of double coset operators [K gK ] and K K 1 2 Hecke operators T , S and U on the groups H (Y , M ), for any abelian group M. v v v K S 0 In particular, we obtain an action of T on H (Y , M ) for any ﬁnite set of places S containing (K ),ring A, and A-module M. 2.9 Suppose that we are in the deﬁnite or indeﬁnite case, and that A is a ﬁnite Z - algebra, so that the residue ﬁeld of any maximal ideal of A is a ﬁnite extension of F . Deﬁnition 2.1 A maximal ideal m of T is G-automorphic of level K if it is in the i i support of H (X , A) (in the indeﬁnite case) or H (Y , A) (in the deﬁnite case) for K K some i.Itis G-automorphic if it is G-automorphic of level K for some K . If m is a G-automorphic maximal ideal of T then there is an associated semisimple representation ρ : G → GL T /m m F 2 characterised by char (X ) = X − T X + Nm(v)S for all v/ ∈ S ∪ . ρ (Frob ) v v l m v Deﬁnition 2.2 An G-automorphic maximal ideal of T is non-Eisenstein if ρ is absolutely irreducible, and Eisenstein otherwise. A T -module is Eisenstein if every maximal ideal in its support is Eisenstein. It is non-exceptional if ρ (G ) contains a subgroup of GL (F ) conjugate to m F 2 l SL (F ); equivalently if it is non-Eisenstein and the image of ρ contains an element 2 l m of order l.Otherwise,itis exceptional. Proposition 2.3 Suppose that we are in the indeﬁnite case. The T -modules H (X , A) and H (X , A) are Eisenstein. 123 Ihara’s Lemma for Shimura curves... 197 Proof Let ν : G → G be the reduced norm. There is (see [5] section 1.2) a m,F bijection × ×,+ π (X (C)) → A F ν(K ) 0 K F , f ×,+ × where F is the set of totally positive elements of F . Write C for the group on the right. If g ∈ G(A ) then C = C −1 and the diagram F , f K g Kg π (X (C)) − −−− → C 0 K K ⏐ ⏐ ⏐ ⏐ g ·ν(g) π (X −1 (C)) − −−− → C 0 K g Kg S 0 commutes. This implies that T acts on H (X (C), A) = A[C ] via the homomor- K K phism T → A[C ] given by T → (Nm(v) + 1)[ ] v v S →[ ], wherewewrite [g] for the basis element of A[C ] corresponding to g.If n is a maximal ideal of A[C ] with residue ﬁeld F, corresponding to a character χ : C → F , K K then T and S act on A[C ]/n as (Nm(v) + 1)χ ( ) and χ( ) respectively. If v v K v ψ : G → F is the character of G associated to χ by class ﬁeld theory, and F F ρ = ψ ⊕ ψ, then T and Nm(v)S act on A[C ]/n by the scalars tr(ρ(Frob )) v v K v and det(ρ(Frob )), so that the action of T on A[C ]/n factors through an Eisenstein v K S 0 maximal ideal as required. It follows that the action of T on H (X , A) is Eisenstein. The statement for H follows from Poincaré duality 2 0 ∨ ∨ H (X , A) H (X , A ) K K ∗ −1 ∗ −1 and the formulae S = S and T = S T for the adjoints of T and S . v v v v v v v 2.10 Let A be a ﬁnite Z -algebra. There is an exact functor M → L from the l M category of A[K ]-modules on which K ∩ Z (F ) acts trivially, to the category of local systems of A-modules on X (C) or Y .If S is a ﬁnite set of places of F containing K K (K )∪ ∪∪ , and such that the action of K on M factors through K , then l ∞ v v∈S S i we obtain an action of the Hecke algebra T on each cohomology group H (X , L ) K M or H (Y , L ). K M Proposition 2.4 Suppose that we are in the indeﬁnite case. For any A, M and S as above, the T -module H (X , L ) is Eisenstein for i = 0, 2. K M Proof This is proved just as in Proposition 2.3. 123 198 J. Manning, J. Shotton 2.11 Suppose that K is unramiﬁed at p.Let ω = . Then since −1 ωK (p)ω ⊂ K , we have two degeneracy maps π ,π deﬁned (in the notation 0 1 2 of 2.6) by π = ρ : X → X 1 e K (p) K π = ρ : X → X −1 2 K (p) K ω 0 (with similar formulae in the deﬁnite case). If A is an abelian group then we obtain maps ∗ ∗ i i π ,π : H (X , A) → H (X , A) K K (p) 1 2 0 with, again, similar formulae in the deﬁnite case. We write ∗ ∗ ∗ i 2 i π = π + π : H (X , A) → H (X , A). K K (p) 1 2 0 If M, L , and S are as in 2.10 and if p ∈ / (K ) ∪ is such that the action of K on M ∞ M factors through K , then we can similarly deﬁne ∗ ∗ ∗ i 2 i π = π + π : H (X , L ) → H (X , L ) K M K (p) M 1 2 0 (and analogous maps in the deﬁnite case). 2.12 Deﬁne the ﬁnite abelian (class) group by = Z (A )/Z (F )(K ∩ Z (A )). K F , f F , f It acts freely on X and Y by our assumption that K is sufﬁciently small. K K × × Suppose that A is a ﬁnite Z -algebra and that ψ is a character A /F → A F , f that vanishes on K ∩ Z (A ) (regarded as a subgroup of Z (A ) = A ), so that F , f F , f F , f we may consider ψ as a character of .For M any A[ ]-module, we write M [ψ ] K K for the largest submodule of M on which acts as ψ and M for the largest quotient K ψ module of M on which acts as ψ. Lemma 2.5 Let A be as above, and let m be a non-Eisenstein maximal ideal of T . 0 ∨ 1 ∨ Then H (Y , A ) and H (X , A ) are injective A[ ]-modules. K m K m K Proof In the indeﬁnite case, we use the Hochschild–Serre sequence and fact that m is non-Eisenstein. Let V be an A[ ]-module and let L be the local system on X / associated to V . The action of the Hecke operators away from ramiﬁed K K i ∨ primes descends to an action on H (X / , L ). Then K K 0 ∨ ∨ H (X / , L ) = Hom (H (X , A), V ) K K 0 K V K is Eisenstein by Proposition 2.4, and the same is true for H (X / , L ) by Poincaré K K duality. As H (X , L ) vanishes by Proposition 2.4, K m 123 Ihara’s Lemma for Shimura curves... 199 1 ∨ ∨ Hom (V , H (X , A ) ) = Hom (H (X , A) , V ) K m 1 K m K K 1 ∨ = H (X / , L ) K K m and the latter is an exact functor of V as m is non-Eisenstein. In the deﬁnite case the proof is similar but easier (and the assumption on m is not actually necessary). 2.13 For the rest of this section we suppose that we are in the indeﬁnite case, and ﬁx a ﬁnite place q ∈ / ∪ of F,let O be the localization of O at q,let k be the residue l (q) F ﬁeld of q, and let k be an algebraic closure of k.Bya model of X we will mean a proper ﬂat O -scheme X equipped with an isomorphism X ⊗ F − → X . (q) K K O K (q) We will consider K that are (sufﬁciently small and) of the form K GL (O ) or 2 F ,q K U (q). For such K , there are models X of X constructed by Morita [29] (in the 0 K K ﬁrst case) and by Jarvis [23], following Carayol [5] (in the second). They have the following properties: Theorem 2.6 Suppose that K is unramiﬁed at q. (1) The curve X is smooth over O . K (q) (2) The curve X is regular and X ⊗ k is the union of two curves, each K (q) K (q) O 0 0 (q) isomorphic to X ⊗ k, that intersect transversely at a ﬁnite set of points. K O (q) Remark 2.7 We will use implicitly the functoriality of these models. For instance, if K ⊂ K are as above then the morphism X → X extends uniquely to a ﬁnite K K ﬂat morphism between the models. If K is ﬁxed, then the action of G(A ) on the F , f q q inverse system (X ) extends uniquely to the inverse system of models. This K K K action is compatible with varying K , and with the maps X ⊗ k → X ⊗ k q K O K (q) O (q) (q) implicit in part 2 of the theorem. 2.14 Suppose that K is unramiﬁed at q. Deﬁnition 2.8 The set of points where the two components of X ⊗ k intersect K (q) maps injectively to X ⊗ k under the natural map X → X . The image is a ﬁnite K K (q) K ss set of points called the supersingular points and is denoted X . There is an adelic description of this set that we now explain. Let D be a quaternion algebra over F ramiﬁed at ∪{q,τ } and let G be the algebraic group over F associated to D . We ﬁx a continuous isomorphism q q ι : D ⊗ A − → D ⊗ A . F F F , f F , f Let O be the unique maximal order of D ⊗ F . Then we write F q D,q Y q = Y . −1 q ι (K )O D,q Remark 2.9 It follows from the Jacquet–Langlands correspondence that, if K is unram- iﬁed at q and m is in the support of H (Y q , A), then m is in the support of H (X , A). K (q) 123 200 J. Manning, J. Shotton Theorem 2.10 ([5] (11.2)) There is a G(A )-equivariant isomorphism of inverse F , f systems ss q q q (X ) − → (Y ) . K K K 2.15 Suppose that K is unramiﬁed at q and that F is a ﬁnite extension of F . The geom- etry of X and the theory of vanishing cycles allow us to relate H (X , F), K (q) K (q) 0 0 1 0 H (X , F) and H (Y , F). In the case at hand, this is worked out in [23], sec- K K tions 14-18. We recall the result in our notation: Theorem 2.11 Suppose that K is unramiﬁed at q. Let S be a ﬁnite set of places con- taining (K ) ∪{q}∪ ∪ and let m be a non-Eisenstein maximal ideal of T . Then there is a ﬁltration 0 ⊂ V ⊂ V ⊂ V = H (X , F) 1 2 K (q) m together with isomorphisms V − → H (Y , F) , 1 K m 1 ⊕2 V /V − → H (X , F) 2 1 K and V /V − → H (Y , F) . 2 K m The ﬁltration, and isomorphisms, are compatible with the transition morphisms for varying K and with the action of the Hecke operators T and S forv/ ∈ (K )∪{q}∪ v v and U for v/∈{q}∪ . Proof As mentioned, this is proved in [23]: we give references to that paper. The key diagram is that at the end of section 14, which relates Hecke-modules X (H ), Y (H ), ˜ ˜ X (H ), Y (H ), M (H ), and R(H ). In particular, there is a ﬁltration of M (H ) with graded ˜ ˜ pieces X (H ), R(H ), and Y (H ). Choosing the group H in that paper appropriately, taking the sheaf there called F to be the constant sheaf F, and after localizing at m,we 1 1 ⊕2 have that M (H ) is our H (X , F) , while R(H ) is our H (X , F) (see [23] m K (q) m m K 0 m Corollary 16.3). A choice of ordering of the irreducible components of each connected component of the special ﬁbre of X gives, by Theorem 2.10, an isomorphism K (q) between Y (H ) and H (Y q , F) . By Proposition 2.3,or[23] Lemma 18.1, we m K m ∼ ˜ have Y (H ) Y (H ) .By[23] Proposition 17.4 and Lemma 18.2, we have (Hecke- m m ˜ ∼ ∼ equivariant) isomorphisms X (H ) X (H ) Y (H ) . The result follows. = = m m m It follows from Lemma 2.5 that we can take ψ-parts in the ﬁltration of Theorem 2.11 1 0 to obtain a ﬁltration of H (X , F) [ψ ] with graded pieces H (Y q , F) [ψ ], K (q) m K m 1 ⊕2 0 H (X , F) [ψ ], H (Y q , F) [ψ ] for any non-Eisenstein maximal ideal m of T . K K m 123 Ihara’s Lemma for Shimura curves... 201 3 Types and local deformation rings For this section, let L be a local ﬁeld of characteristic 0, with residue ﬁeld k of order q. Let be the absolute Galois group of L, I its inertia subgroup, and P its wild inertia subgroup. Let σ ∈ I be a lift of a topological generator of I /P, and let φ ∈ be a −1 q lift of arithmetic Frobenius. Then we have the well-known relation φσ φ = σ in /P. For this section and Sect. 4 we assume that l > 2. By a coefﬁcient system we will mean a triple (E , O, F) where: E /Q is a ﬁnite extension, with ring of integers O, uniformizer , and residue ﬁeld F = O/ . For now, we will take an arbitrary coefﬁcient system; later we will impose further conditions on E /Q . Let C (resp. C ) be the category of Artinian (resp. complete Noetherian) local O- algebras with residue ﬁeld F. We say that a functor F : C → Set is pro-represented by some R ∈ C if F is naturally isomorphic to Hom (R, −). Now ﬁx a continuous representation ρ : → GL (F). The primary goal of this section is to introduce various deformation rings of ρ. Many treatments of this material assume that the coefﬁcient ring O is sufﬁciently ramiﬁed. For our purposes, it will be necessary to precisely control the ramiﬁcation of O, and so a little more care will be needed in certain parts. Consider the (framed) deformation functor C → Set deﬁned on objects A by A →{continuous lifts ρ : → GL (A) of ρ} It is well-known that this functor is pro-representable by some R ∈ C . Fur- ρ,O O thermore, ρ admits a universal lift ρ : → GL (R ). ρ,O For any continuous homomorphism, x : R → E, we obtain a Galois repre- ρ,O ρ x sentation ρ : → GL (E ) lifting ρ, from the composition −→ GL (R ) − → x 2 2 ρ,O GL (E ). ,ψ × −1 For any character ψ : → O with det ρ = ψ (mod ) deﬁne R to ρ,O ,ψ −1 be the quotient of R on which det ρ = ψ . Equivalently, R is the ring ρ,O ρ,O −1 pro-representing the functor of lifts of ρ with determinant ψ . As l > 2, we have an isomorphism ,ψ R ⊗R = R (1) det(ρ),O ρ,O ρ,O where R is the universal deformation ring of the character det(ρ). det(ρ),O 3.1 Deformation rings when l q ,ψ For this subsection, we assume that l q. In this case, the O-algebras R and R ρ,O ρ,O are ﬂat of relative dimensions 3 and 4, respectively. The second statement follows from [36] Theorem 2.5. The ﬁrst statement follows from the second, the isomor- phism (1), and the ﬂatness of the deformation ring of a character (see for example 123 202 J. Manning, J. Shotton [35] Lemma 2.5). Shortly, we will analyse these rings in more detail in a particular case. 3.2 Deformation rings when l|q Now assume that l|q, so that l is the residue characteristic of L.If L /L is any ﬁnite ,L -st extension, then by [24] there is a quotient R of R such that a continuous O- ρ,O ρ,O ,L -st algebra homomorphism x : R → E factors through R if and only if ρ | x G ρ,O ρ,O L is semistable with parallel Hodge–Tate weights {0, 1}.For ψ a ﬁnite order character of ,ψ,L -st ,L -st that factors through Gal(L /L), there is a quotient R of R on which ρ,O ρ,O ,ψ,L -st −1 we additionally impose the condition det(ρ) = ψ . We have that Spec R ρ,O is equidimensional of dimension 3 +[L : Q ]. 3.3 Deformation rings at the auxiliary prime q In this subsection, we study the speciﬁc local deformation ring R = R that ρ | ,O m F will occur at the auxiliary prime q in our argument, and deﬁne and compute certain quotients of it. From now on assume that q ≡ 1 (mod l) (so that in particular l q), and let ρ : → GL (F) be the unramiﬁed representation with ρ(φ) = . Note that both and det(ρ) are the trivial character. We will now impose a hypothesis on our coefﬁcient system: −1 Hypothesis 3.1 The coefﬁcient system (E , O, F) is such that O = W (F)[ζ + ζ ] for a primitive lth root of unity ζ ∈ O. Under this hypothesis, we write W = W (F) be the ring of Witt vectors and let E = W [1/l], so that E is an unramiﬁed extension of Q with residue ﬁeld F.Weﬁx 0 0 l ζ ∈ E a primitive lth root of unity. We also let −1 2 −1 2 π = (ζ − ζ ) = (ζ + ζ ) − 4 ∈ O, and note that this is a uniformizer of O. We deﬁne the following quotients of R in terms of the subfunctors that they ρ,O represent: nr • R parametrises lifts ρ of ρ that are unramiﬁed. ρ,O • R parametrises lifts ρ of ρ such that ρ,O char (T ) = (T − 1) ρ(σ ) and 2 2 (tr ρ(φ)) q = (q + 1) det ρ(φ). 123 Ihara’s Lemma for Shimura curves... 203 unip • R parametrises lifts ρ of ρ such that ρ,O char (T ) = (T − 1) ρ(σ ) and 2 2 (tr ρ(φ)) q − (q + 1) det ρ(φ) · (ρ(σ ) − 1) = 0. ps • R parametrises lifts ρ of ρ such that ρ,O 2 −1 char (T ) = T − (ζ + ζ )T + 1 ρ(σ ) −1 = (T − ζ)(T − ζ ). 2 2 Remark 3.2 The relation “q tr(φ) = (q + 1) det(φ)” should be thought of as saying that the eigenvalues of ρ(φ) are in the ratio q : 1, which is the case for all characteristic zero lifts of ρ for which the image of inertia is non-trivial and unipotent. ps Remark 3.3 It is important for us that R be deﬁned over O and not just O[ζ ]. ρ,O Fix an unramiﬁed character ψ : → O lifting the trivial character det(ρ).Note that, on each of these quotients, we have that det(ρ ) is unramiﬁed, and so agrees −1 with ψ on I.For ? ∈{N , nr, unip, ps}, we make the following deﬁnitions: ?,ψ ? −1 • R is the quotient of R on which det(ρ ) = ψ ; ρ,O ρ,O • R = R ⊗ F; ρ,O ?,ψ ?,ψ • R = R ⊗ F. ρ,O We will need somewhat explicit descriptions of these rings, which were obtained in Proposition 5.8 of [34] and its proof. Let AB ρ (σ ) = 1 + CD and 1 + P 1 + R ρ (φ) = . S 1 + Q We will choose more convenient coordinates. We may replace B by X = , Q 1+R by T = tr(ρ (φ)) − 2, and S by δ = det(ρ (φ)) − 1. By this we mean that the natural map O[[A, X , C , D, P, T , R,δ]] → R ρ,O 123 204 J. Manning, J. Shotton is surjective, which follows from the formulae B = (1 + R)X, Q = T − P, and −1 S = (1 + R) (T + P(T − P) − δ). Then we may replace T by either Y = (tr ρ (φ)) − 4 det ρ (φ) or 2 2 Y = (tr ρ (φ)) q − (q + 1) det ρ (φ), by which we mean that the natural maps O[[A, X , C , D, P, R,δ, Y ]] → R ρ,O are surjections. This follows from the equation T = 4 + Y + 4δ − 2 in the ﬁrst case—where the square root is deﬁned by a convergent Taylor series, as l > 2—and a similar expression in the second. We have maps α : O[[X , Y , P, R,δ]] → R . i i ρ,O −1 Remark 3.4 Write γ = (φ) ψ(φ) − 1 ∈ O. Then the maps α descend to maps, also denoted α , ,ψ α : O[[X , Y , P, R]] = O[[X , Y , P, R,δ]]/(δ − γ) → R . i i i ρ,O In the proofs of all of the following propositions we work without ﬁxing determinants. For each ? ∈{N , nr, unip, ps} we already have that det(ρ ) is unramiﬁed on the quotient R . This means that to get the ﬁxed determinant versions in the statements, O,ρ we simply quotient by δ − γ . ps,ψ Proposition 3.5 The ring R is isomorphic (via α )to ρ,O O[[X , Y , P, R]]/(X Y − π). 1 1 In particular, it is regular. Proof This follows from the proof of [34] Proposition 5.8 part 2. The quantity denoted y in the proof of that proposition is here equal to 1. The variables X ,... X in that 1 5 proof are our variables X , Y , P, R, 2P − T , but by the above remarks we can replace 2P − T with δ and obtain that α deﬁnes an isomorphism ps O[[X , Y , P, R,δ]]/(X Y − π) R . 1 1 ρ,O The result with ﬁxed determinant follows. 123 Ihara’s Lemma for Shimura curves... 205 unip,ψ Proposition 3.6 The ring R is isomorphic (via α )to ρ,O O[[X , Y , P, R]]/(XY ) 2 2 nr,ψ N ,ψ and its quotients R and R are, respectively, ρ,O ρ,O O[[X , Y , P, R]]/(X ) and O[[X , Y , P, R]]/(Y ). 2 2 2 In particular, these last two deformation rings are formally smooth. unip Proof This is not quite in [34] Proposition 5.8, as the quotient R is not considered ρ,O there, but the method of proof extends easily—we will be brief. The proof shows unip (q−1)(2+T ) that, if we write U = P − Q and α(T ) = , then R is cut out of q+1 ρ,O O[[A, X , C , D, U , T , R, S]] by the following equations: A + D = 0 A + (1 + R)XC = 0 2 2 (4(1 + R)S + (U − α(T ) )) = 0 A = X (U − α(T )) 2AS − C (U + α(T )) = 0 C = Aα(T ) + XS (q − 1)(AU + (1 + R)XS + (1 + R)C ) = 0. Here denotes each of A, X , C , D, so that the third line is really four equations. Note that the third line can be rewritten as Y = 0. The ﬁrst, fourth and sixth lines show that A, C and D may be written in terms of X , T , S and U. Making these substitutions we see that this set of equations is equivalent to the single equation 2 2 X (4(1 + R)S + (U − α(T ) )) = 0. But if we now replace T , S and U by Y , δ and unip P as discussed above, we obtain that R is the quotient of O[[X , Y , P, R,δ]] by O,ρ XY = 0 as required. nr N The expressions for the quotients R and R follow immediately, and ﬁnally ρ,O ρ,O we eliminate δ by imposing the ﬁxed determinant condition. ,ψ Proposition 3.7 The images of Y and Y are equal in R . Denoting this common 1 2 image by Y , the diagram ∼ ps,ψ F[[X , Y , P, R]]/(X Y ) − −−− → R ⏐ ⏐ ⏐ ⏐ ∼ unip,ψ F[[X , Y , P, R]]/(XY ) − −−− → R commutes. 123 206 J. Manning, J. Shotton Proof That the images of Y and Y are equal is immediate from q ≡ 1 (mod l).The 1 2 diagram commutes since α and α are equal as maps F[[X , Y , P, R]] → R . 1 2 Remark 3.8 In [34] it is assumed that ζ ∈ O, which is not the case for us—however, −1 this assumption is not used (only the assumption that ζ + ζ ∈ O, which is required ps to even deﬁne R ). O,ρ Remark 3.9 The proofs above show that each of our deformation rings R turns out ρ,O to be reduced and l-torsion free, and therefore is one of the ﬁxed-type deformation rings deﬁned by a Zariski closure operation in [34]. 3.4 Types Next we deﬁne various representations of GL (O ) over W (or extensions of W ). 2 L Let G = GL (k) and B be its subgroup of upper triangular matrices. We will always regard representations of G as representations of GL (O ) by inﬂation. If A is a ring, 2 L then we will write 1 for A with the trivial action of any group under consideration. Since q + 1 =[G : B] is invertible in W , the natural map 1 → Ind 1 W W splits, and so we deﬁne St by the formula Ind 1 = 1 ⊕ St . W W W If A is a W -algebra, then deﬁne St = St ⊗ A; then we have Ind 1 = 1 ⊕St . A W W A A A × −1 Now let E = E [ζ ] and χ : k → E be a non-trivial character. Let χ ⊗ χ : 1 1 B → E be the character xz −1 −1 (χ ⊗ χ ) = χ(x )χ (y). 0 y Let ps G −1 σ = Ind (χ ⊗ χ ). E B ps −1 If E = E [ζ + ζ ] as before then σ is isomorphic to its conjugate under the −1 nontrivial element of Gal(E /E ), which switches χ and χ . It therefore has a model ps σ over E, by the calculation of the Schur index of a character of a ﬁnite general linear group in [21] Theorem 2a—see also Lemma 3.1.1 of [15]. By [15] Lemma 4.1.1, there ps ps is a unique O-lattice σ in σ such that there is a nonsplit short exact sequence O E ps 0 → F → σ ⊗ F → St → 0. (2) ps ps For A an O-algebra, we let σ = σ ⊗ A. 123 Ihara’s Lemma for Shimura curves... 207 3.5 The local Langlands correspondence Suppose ﬁrst that we are in the setting of Sect. 3.3.For ρ : G → GL (E ) a con- L 2 0 tinuous representation, let π(ρ) be the smooth admissible representation of GL (L) associated to ρ by the local Langlands correspondence, and let x : R → E be the ρ,O associated homomorphism. Then we have: nr Proposition 3.10 (1) If π(ρ)| contains 1 , then x factors through R . GL (O ) 2 L E ρ,O unip (2) If π(ρ)| contains St , then x factors through R . GL (O ) 2 L E ρ,O (3) If π(ρ) is discrete series and π(ρ)| contains St , then x factors through GL (O ) 2 L R . ρ,O ps ps (4) If π(ρ)| contains σ , then x factors through R . GL (O ) 2 L ρ,O Now suppose that we are in the setting of Sect. 3.2. Suppose that D is a quaternion algebra over L and K is a compact open subgroup of D .If π is an irreducible admis- sible representation of D over E, then by the local Langlands and Jacquet–Langlands correspondences there is an associated Weil–Deligne representation (r , N ).Wemay π π and do choose a ﬁnite extension L /L such that, for all π having a K -ﬁxed vector, the restriction r | is unramiﬁed. It follows that, if π has a K -ﬁxed vector and π G ρ : G → GL (E ) is a de Rham representation of parallel Hodge–Tate weights L 2 ss {0, 1} such that WD(ρ) = (r , N ), then ρ| is semistable and so corresponds π π G ,L -st to a point of R . We write ρ,O ,ψ,K -st ρ,O for ,ψ,L -st R . ρ,O We will say that a lift ρ : → GL (A) of ρ is K -semistable if the associated map ,K -st R → A factors through R . ρ,O ρ,O 4 Patching The goal of this section is to summarize the Taylor–Wiles–Kisin patching construction, and to prove the results about it that will be needed for the proof of Theorem 6.5.We choose a coefﬁcient system (E , O, F), which we will eventually require to satisfy Hypothesis 3.1. 4.1 Ultrapatching In this section we summarize the commutative algebra behind the patching method. For convenience we will use the “ultrapatching” construction introduced by Scholze in [33]; we follow closely the exposition of [27] section 4. 123 208 J. Manning, J. Shotton From now on, ﬁx a nonprincipal ultraﬁlter F on the natural numbers N (it is well known that such an F must exist, provided we assume the axiom of choice). For convenience, we will say that a property P(n) holds for F-many i if there is some I ∈ F such that P(i ) holds for all i ∈ I . For any sequence of sets A ={A } , we deﬁne their ultraproduct to be the n n≥1 quotient U(A ) = A / ∼ n=1 where we deﬁne the equivalence relation ∼ by (a ) ∼ (a ) if a = a for F-many i. n n n i n i If the A ’s are sets with an algebraic structure (eg. groups, rings, R-modules, R- algebras, etc.) then U(A ) naturally inherits the same structure. If each A is a ﬁnite set, and the cardinalities of the A ’s are bounded (this is n n the only situation we will consider in this paper), then U(A ) is also a ﬁnite set and there are bijections U(A ) − → A for F-many i. Moreover if the A ’s are sets with i n an algebraic structure, such that there are only ﬁnitely many distinct isomorphism classes appearing in {A } (which happens automatically if the structure is deﬁned n n≥1 by ﬁnitely many operations, eg. groups, rings or R-modules or R-algebras over a ﬁnite ring R) then these bijections may be taken to be isomorphisms. This is merely because our conditions imply that there is some A such that A A for F-many i and hence U(A ) is isomorphic to the “constant” ultraproduct U {A} which is easily seen to n≥1 be isomorphic to A if A is a ﬁnite set. Lastly, in the case when each A is a module over a ﬁnite local ring R, there is a simple algebraic description of U(A ). Speciﬁcally, the ring R = R contains a n=1 ∼ ∼ unique maximal ideal Z ∈ Spec R for which R R and A U A = = ( ) F Z n n=1 as R-modules. This shows that U(−) is a particularly well-behaved functor in our situation. In particular, it is exact. For the rest of this section, ﬁx a power series ring S = O[[z ,..., z ]] and ∞ 1 d consider the ideal n = (z ,..., z ). Fix a sequence of ideals I ⊆ S such that 1 d n ∞ for any open ideal a ⊆ S we have I ⊆ a for all but ﬁnitely many n. Also deﬁne ∞ n S = S /( ) = F[[z ,..., z ]] and I = (I + ( ))/( ) ⊆ S . ∞ ∞ 1 d n n ∞ For any ﬁnitely generated S -module M, we will say that the S -rank of M, ∞ ∞ denoted by rank M, is the cardinality of a minimal generating set for M as an S -module. We can now make our main deﬁnitions: Deﬁnition 4.1 Let M ={M } be a sequence of ﬁnitely generated S -modules n n≥1 ∞ with I ⊆ Ann M for all but ﬁnitely many n. n S n • We say that M is a weak patching system if the S -ranks of the M ’s are uniformly ∞ n bounded. If we further have M = 0 for all but ﬁnitely many n, we say that M is a residual weak patching system 123 Ihara’s Lemma for Shimura curves... 209 • We say that M is a patching system if it is a weak patching system, and we have Ann (M ) = I for all but ﬁnitely many n. S n n • We say that M is a residual patching system if it is a residual weak patching system, and we have Ann (M ) = I for all but ﬁnitely many n. n n • We say that M is MCM (resp. MCM residual) if M is a patching system (resp. residual patching system) and M is free over S /I (resp. S /I ) for all but n ∞ n ∞ n ﬁnitely many n. Furthermore, assume that R ={R } is a sequence of ﬁnite local S -algebras. n n≥1 ∞ • We say that R ={R } is a (weak, residual) patching algebra, if it is a (weak, n n≥1 residual) patching system. • If M is an R -module (viewed as an S -module via the S -algebra structure on n n ∞ ∞ R ) for all n we say that M ={M } is a (weak, residual) patching R-module n n n≥1 if it is a (weak, residual) patching system. Let wP be the category of weak patching systems, with the obvious notion of morphism. Note that this is naturally an abelian category. Now for any weak-patching system M , we deﬁne its patched module to be the S -module P(M ) = lim U(M /a) , ← − where the inverse limit is taken over all open ideals of S . We may treat P is as functor from wP to the category of S -modules. If R is a weak patching algebra and M is a weak patching R-module, then P(R) inherits a natural S -algebra structure, and P(M ) inherits a natural P(R)-module structure. In the above deﬁnition, the ultraproduct essentially plays the role of the pigeonhole principal in the classical Taylor–Wiles–Kisin construction, with the simpliﬁcation that it is not necessary to explicitly deﬁne a “patching datum” before making the construc- tion. Indeed, if one were to deﬁne patching data for the M /a’s (essentially, imposing extra structure on each of the modules M /a) then the machinery of ultraproducts would ensure that the patching data for U(M /a) would agree with that of M /a for inﬁnitely many n. It is thus easy to see that our deﬁnition agrees with the classical construction (cf. [33]). Thus the standard patching Lemmas (cf. [25], Proposition 3.3.1) can be rephrased as follows: Proposition 4.2 Let R be a weak patching algebra, and let M be an MCM patching R-module. Then: (1) P(R) is a ﬁnite type S -algebra, and P(M ) is a ﬁnitely generated free S - ∞ ∞ module. (2) The structure map S → P(R) (deﬁning the S -algebra structure) is injective, ∞ ∞ and thus dim P(R) = dim S . (3) The module P(M ) is maximal Cohen–Macaulay over P(R), and ( , z ,..., z ) 1 d is a regular sequence for P(M ). 123 210 J. Manning, J. Shotton Proposition 4.3 Let R be a weak patching algebra, and let M be an MCM residual patching R-module. Then: (1) P(R)/( ) is a ﬁnite type S -algebra, and P(M ) is a ﬁnitely generated free S -module. (2) The structure map S → P(R)/( ) is injective, and thus dim P(R)/( ) = dim S . (3) The module P(M ) is maximal Cohen–Macaulay over P(R)/( ), and (z , ..., z ) is a regular sequence for P(M ). Proposition 4.4 Let n = (z ,..., z ) ⊆ S , as above. Let R be a ﬁnite type O- 1 d ∞ 0 algebra, and let M be a ﬁnitely generated R -module. If, for each n ≥ 1, there are 0 0 ∼ ∼ ∼ isomorphisms R /n R of O-algebras and M /n M of R /n R -modules, = = = n 0 n 0 n 0 ∼ ∼ ∼ then we have P(R)/n R as O-algebras and P(M )/n M as P(R)/n R - = = = 0 0 0 modules. From the set up of Proposition 4.2 there is very little we can directly conclude about the ring P(R). However in practice one generally takes the rings R to be quotients of aﬁxedring R of the same dimension as S (and thus as P(R)). Thus we ∞ ∞ deﬁne a cover of a weak patching algebra R ={R } to be a pair (R , {ϕ } ), n n≥1 ∞ n n≥1 where R is a complete, topologically ﬁnitely generated O-algebra of Krull dimension dim S and ϕ : R → R is a surjective O-algebra homomorphism for each n.It ∞ n ∞ n is straightforward to show the following (cf. [27]) Proposition 4.5 If (R , {ϕ }) is a cover of a weak patching algebra R, then the ϕ ’s ∞ n n induce a natural continuous surjection ϕ : R P(R). ∞ ∞ Combining this with Propositions 4.2 and 4.3 we get the following (using the fact [37, Lemma 0AAD] that if f : A B is a surjection of noetherian local rings, then a B-module M is Cohen–Macaulay as an A-module if and only if it is Cohen–Macaulay as a B-module): Corollary 4.6 Let R be a weak patching algebra and let (R , {ϕ }) be a cover of R. ∞ n If M is an MCM patching R-module, then P(M ) is a maximal Cohen–Macaulay R -module. If M is an MCM residual patching R-module, then P(M ) is a maximal Cohen–Macaulay R /( )-module. In our arguments, it will be necessary to patch the ﬁltration from Theorem 2.11. This would certainly be possible if P were an exact functor. However, this is not true in general, but we can prove a weaker statement which sufﬁces for our purposes: Lemma 4.7 The functor P(−) is right-exact. Moreover, if 0 → A → B → C → 0 For an easy counterexample, assume that S /I is -torsion free for all n (a condition which will be ∞ n satisﬁed for our choice of I below) and let M ={S /I } .Deﬁne ϕ ={ϕ } : M → M by n ∞ n n≥1 n n≥1 ϕ (x ) = x.Then ϕ : M → M is injective, P(M ) = S ,and P(ϕ) : S → S is the zero map. n ∞ ∞ ∞ 123 Ihara’s Lemma for Shimura curves... 211 is an exact sequence of weak patching systems then 0 → P(A ) → P(B) → P(C ) → 0 is exact, provided that either: • C is MCM, or • A , B and C are all residual weak patching systems, and C is MCM residual. Proof Let Ab be the category of abelian groups. For any countable directed set I,let ﬁnAb be the category of inverse systems of ﬁnite abelian groups indexed by I . Now note that any (A , f : A → A ) ∈ ﬁnAb clearly satisﬁes the Mittag- i ji j i Lefﬂer condition: For any i ∈ I there is a j ≥ i for which im( f ) = im( f ) for all ki ji k ≥ j (since A is ﬁnite, and {im( f )} is a decreasing sequence of subgroups). i ji j ≥i Thus by [37, Lemma 0598] it follows that lim : ﬁnAb → Ab is exact. ← − Now assume that A , B and C are weak patching systems, and that we have an exact sequence 0 → A → B → C → 0 Then for any a ⊆ S , A /a → B/a → C /a → 0 is exact, so by the exactness of U(−) we get the exact sequence U(A /a) → U(B/a) → U(C /a) → 0. Thus we have an exact sequence of inverse systems U A /a → U B/a → U C /a → 0 ( ) ( ) ( ) a a a But now as U(A /a), U(B/a) and U(C /a) are all ﬁnite, and there are only countably many open ideals of S , the above argument shows that taking inverse limits preserves exactness, and so indeed P(A ) → P(B) → P(C ) → 0 is exact. Now assume that one of the further conditions of the lemma holds. Write A = {A } , B ={B } and C ={C } . Then letting I = Ann C (so that n n≥1 n n≥1 n n≥1 n S n either I = I or I for all n 0), we get that for all n 0, n n n 0 → A → B → C → 0 n n n is an exact sequence of S /I -modules, and C is a free S /I -module (this is true ∞ n n ∞ n regardless of which case we are in). It follows that S /I ∞ n Tor (C , S /a) = 0 n ∞ 123 212 J. Manning, J. Shotton for all a ⊆ S , and so 0 → A /a → B /a → C /a → 0 n n n is exact for all n 0. The same argument as above now shows that 0 → P(A ) → P(B) → P(C ) → 0 is exact. This now implies that P preserves ﬁltrations in the cases that will be relevant to us: Corollary 4.8 Let V be a residual weak patching system with a ﬁltration 0 1 r 0 = V ⊆ V ⊆ ··· ⊆ V = V k k k k−1 by residual weak patching systems V .For k = 1,...,rlet M = V /V . Assume that the M ’s are all MCM residual. Then P(V ) has a ﬁltration 0 1 r 0 = P(V ) ⊆ P(V ) ⊆ ··· ⊆ P(V ) = P(V ) k k−1 k with P(V )/P(V ) = P(M ) for all k = 1,..., r. One can also make an analogous statement about ﬁltrations of weak patching sys- tems, instead of residual weak patching systems, but we will not need that result. Proof For any k ≥ 1 we have an exact sequence k−1 k k 0 → V → V → M → 0. k k−1 k As M is MCM residual, Lemma 4.7 implies that the map P(V ) → P(V ) is k k−1 k an inclusion, and that P(V )/P(V ) P(M ). The result follows. 4.2 Global deformation rings We ﬁx the following data: • a quaternion division algebra D over F split at exactly one inﬁnite place, as in Sect. 2; • a coefﬁcient system (E , O, F) satisfying Hypothesis 3.1; • a non-Eisenstein maximal ideal m ⊆ T (for some set S, which we will not ﬁx yet) which is G-automorphic; • a ﬁnite order character ψ : G → O for which ψ ≡ det ρ (mod ).Wealso × ab write ψ for the character ψ ◦ Art, where Art : A /F → G is the global F , f F Artin map. 123 Ihara’s Lemma for Shimura curves... 213 Enlarging F if necessary, we assume that the residue ﬁeld of m is F. By deﬁnition, m is G-automorphic of some level K ⊂ G(A ), which we ﬁx temporarily. Now we m F , f ﬁx, for the rest of this section: • a ﬁnite place q ∈ / ∪ (K ) of F at which ρ is unramiﬁed; l m • a ﬁnite set of ﬁnite places of F that contains ∪{q}∪ (K ) (which means l m that we can, and will, regard m as a maximal ideal of T rather than T ); O O • for each v ∈ , a compact open subgroup K ⊂ K ∩ G(F ). l m v We will use S to denote a ﬁnite set of places of F. In the following, S and K will sometimes vary but we will always impose the following hypotheses on the pair (S, K ): Hypotheses 4.9 • m is G-automorphic of level K ; • S contains ∪ (K ) ∪ ; • F (K ∩ Z (A )) ⊂ ker(ψ ) (this implies that ψ is unramiﬁed outside of S); F , f • for all v ∈ , K ∩ G(F ) ⊃ K ; l v q q • K has the form K K for some K ⊂ G(A ) and K ⊂ G(F ). q q q F , f Let ρ = ρ : G → GL (F), and note that ρ is absolutely irreducible and F 2 unramiﬁed outside of S. For any place v of F,let ρ = ρ| . By taking a quadratic v F extension of F if necessary, we will assume that for each g ∈ G , all of the eigenvalues of ρ(g) lie in F . As in [25, section 3.2], deﬁne R (ρ) ∈ C to be the O-algebra pro-representing F ,S O the functor D (ρ) : C → Set which sends A to the set of equivalence classes of F ,S tuples (ρ, (β ) ) (3) v v∈ where: • ρ : G → GL (A) is a continuous lift of ρ; F ,S 2 • for each v ∈ , β ∈ 1 + M (m ) (we think of this as basis for A lifting the v 2 A standard basis of F ); • for each v | l the restriction ρ | is K -semistable, in the notation of Sect. 3.5; F v • two such collections (ρ , (β ) ) and (ρ ,(β ) ) are equivalent if there is v v∈ v∈ −1 γ ∈ 1 + M (m ) such that ρ = γργ and β = γβ for all v ∈ . 2 A v ,ψ Now let D (ρ) : C → Set be the subfunctor of D (ρ) consisting of the F ,S F ,S ,ψ −1 ∧ tuples ρ, (β ) with det ρ = ψ , and let R (ρ) ∈ C be the O-algebra ( ) v v∈ F ,S O ,ψ pro-representing D (ρ). F ,S Also deﬁne the unframed deformation ring R (ρ) to be the O-algebra pro- F ,S representing the functor C → Set which sends A to the set of equivalence classes of lifts ρ : G → GL (A) such that ρ| is K -semistable for all v | l, two such F ,S 2 G F v lifts being equivalent if they are conjugate by an element of 1 + M (m ). Finally, 2 A deﬁne R (ρ) to be the quotient of R (ρ) on which det ρ(g) = ψ(g) for all F ,S F ,S g ∈ G . The unframed deformation rings R (ρ) and R (ρ) exist because ρ is F ,S F ,S F ,S 123 214 J. Manning, J. Shotton univ absolutely irreducible. We will let ρ : G → GL (R (ρ)) be a representa- F ,S 2 F ,S tive for the universal equivalence class of lifts of ρ, which induces a homomorphism univ ρ : G → GL (R (ρ)). F ,S 2 S,ψ F ,S ψ ,ψ There is a ‘forgetful’ map R (ρ) → R (ρ), which by [25, (3.4.11)] is F ,S F ,S ,ψ formally smooth of dimension j = 4||− 1, and so we may identify R = F ,S R [[w ,...,w ]]. 1 j F ,S ,ψ | ,ψ,K -st v v Lastly, for any v ∈ ,let R = R if v l and R = R if v|l.If v v ρ| ,O ρ,O Fv −1 (ρ , (β ) ) is as in equation (3) then, for each v ∈ , β ρβ is a lift of ρ that only v v∈ v −1 depends on the equivalence class of (ρ , (β ) ). Restricting each β ρβ to G v v∈ v F v v induces a map ,ψ ⊗ R → R (ρ). v∈ v F ,S We write R for ⊗ R . loc v∈ v The Taylor–Wiles–Kisin patching construction relies on carefully picking sets of auxiliary primes to add to the level, using the following lemma (see [25] Proposi- tion 3.2.5). Lemma 4.10 Assume that ρ satisﬁes the following conditions: (1) ρ| is absolutely irreducible. F (ζ ) (2) If l = 5 and the image of the projective representation proj ρ : G → GL (F ) PGL (F ) is isomorphic to PG L (F ), then ker proj ρ G . 2 5 2 5 2 5 F (ζ ) m 5 (This condition holds automatically whenever 5 ∈ / F.) Suppose that S = ∪ . Then there exist integers r , g ≥ 0 such that for each n ≥ 1, there is a ﬁnite set Q of primes of F for which: • #Q = r. • Q ∩ S =∅. • For any v ∈ Q , Nm(v) ≡ 1 (mod l ). • For any v ∈ Q , ρ(Frob ) has two distinct eigenvalues in F . n v ,ψ • There is a surjection R [[x ,..., x ]] R (ρ) extending the map R → loc 1 g loc F ,S∪Q ,ψ R (ρ). F ,S∪Q Moreover, we have dim R = r + j − g + 1. loc From now on, ﬁx integers r , g and a sequence Q ={Q } of sets of primes satis- n n≥1 ψ ,ψ fying the conclusions of Lemma 4.10. Deﬁne R = R (ρ), R = R (ρ) F ,S∪Q F ,S∪Q n n for n ≥ 1 and R = R [[x ,..., x ]] = ⊗ R [[x ,..., x ]], ∞ loc 1 g v∈ v 1 g so that we have surjections R R for all n.Alsolet R = R (ρ) and R = ∞ 0 n F ,S 0 ,ψ R (ρ). Note that R R [[w ,...,w ]] for all n ≥ 0 and dim R = r + j + 1. n 1 j ∞ F ,S n 123 Ihara’s Lemma for Shimura curves... 215 4.3 Patched modules over Shimura curves and sets As before, we use S to denote a ﬁnite set of places of F containing ∪ , and K to denote a compact open subgroup of G(A ), such that S and K satisfy Hypothe- F , f ses 4.9. In particular, there is a maximal ideal m of T that is G-automorphic of level S 1 K.Let T(K , S) denote the image of T in End (H (X , O) [ψ ]). Then T(K , S) O K m O,m is a ﬁnite rank free O-algebra which is local with maximal ideal m. Note that T(K , S) depends on the choices of m and ψ but we suppress these from the notation. As in section 6 of [15] we have the following: Lemma 4.11 For any compact open K and set S as above, there exists a natural univ surjection R (ρ) T(K , S) with the property that ρ (tr(Frob )) → T and v v F ,S S,ψ univ ρ (det(Frob )) → Nm(v)S for any v/ ∈ S. These maps are compatible with the v v S,ψ restriction maps T(K , S ) → T(K , S) for K ⊆ K and S ⊂ S . If S ⊂ S are sets as above, then by Lemma 4.11 and the deﬁnitions we have a commutative diagram R − −−− → T(K , S ) F ,S ⏐ ⏐ ⏐ ⏐ R − −−− → T(K , S) F ,S where the left hand vertical map and the horizontal maps are surjections. It follows that the right hand vertical map, injective by deﬁnition, is an isomorphism. We therefore drop S from the notation and write T = T(K , S) for any K and S satisfying Hypotheses 4.9. These Hecke algebras also act on the spaces H (Y , O) [ψ ], by the following K m lemma. Lemma 4.12 For any compact open K and set S as above such that K is unramiﬁed S 0 at q,the map T → End(H (Y , O) [ψ ]) factors through the quotient T . K m K (q) O,m 0 Proof As H (Y , O) [ψ ] is torsion-free, we may check this after inverting l.Itis K m then a consequence of the Jacquet–Langlands correspondence and the semisimplicity 0 S of H (Y , O) [ψ ] as a module over T . K m O,m We now ﬁx S to be the union of ∪ , and let Q ={Q } be the sequence of ∞ n n≥1 sets of places provided by Lemma 4.10. For any n ≥ 1, let be the maximal l-power quotient of k . Consider the ring = O[ ], and note that: n n v∈Q O[[y ,..., y ]] 1 r e(n,1) e(n,r ) l l (1 + y ) − 1,...,(1 + y ) − 1 1 r 123 216 J. Manning, J. Shotton e(n,i ) × where l is the l-part of Nm(v) − 1 = #k , so that e(n, i ) ≥ n by assumption. Let a = (y ,..., y ) ⊆ be the augmentation ideal. Also deﬁne n 1 r n F[[y ,..., y ]] F[[y ,..., y ]] 1 r 1 r = ⊗ F = = n n e(n,1) e(n,1) e(n,r ) e(n,r ) l l l (1 + y ) − 1,...,(1 + y ) − 1 y ,..., y 1 r 1 r ⎛ ⎞ ⎝ ⎠ Now let H = ker k . For any ﬁnite place v of F, there is a n n v∈Q ab × −1 group homomorphism U (v) → k given by → ad (mod v).Now let cd U (Q ) ⊆ U (v) be the preimage of H ⊆ k under the map H n 0 n v∈Q v∈Q n n U (v) k v∈Q v∈Q n n Finally, for any K (satisfying 4.9 for the set S), let K be the preimage of U (Q ) n H n under K → G(A ) G(F ). F , f v v∈Q We also let K = K , and remark that for n ≥ 1, K and S ∪ Q satisfy 4.9;in 0 n n particular, K = K K . For any n ≥ 0, let T = T . n n q n,K K Now for any n ≥ 1 consider the O-algebra T [U ] ⊆ End (H (X , O) [ψ ]). n,K v v∈Q O K m n n Now for each v ∈ Q ﬁx a choice α ∈ F of eigenvalue for ρ(Frob ) (recall that n v v by assumption, for each v ∈ Q ρ(Frob ) has two distinct eigenvalues in F , and so n v |Q | there are 2 ways to pick the system (α ) ). Now deﬁne the ideal v v∈Q m = (m, U − α ) ⊆ T [U ] . n v v n,K v v∈Q Now for each n ≥ 1, deﬁne T = T U . Also deﬁne T = T [ ] n,K n,K v v∈Q 0,K 0,K n m and m = m. As in [39, section 2] we have: Lemma 4.13 The ring T is a ﬁnite T -algebra and m is a maximal ideal of it n,K n,K n lying over m. The composite map R → T → T n n,K n,K is surjective. Moreover, there exist O-algebra maps → R and → T n n n n,K making the above map a surjection of -algebras. 123 Ihara’s Lemma for Shimura curves... 217 1 1 By deﬁnition, T [U ] acts on H (X , O) [ψ ] and H (X , F) [ψ ] (the n,K v v∈Q K m K m n n n latter through its quotient T [U ] ⊗ F). Also, by Theorem 2.11,if K is n,K v v∈Q O unramiﬁed at q then T [U ] ⊗ F acts on H (Y , F) [ψ ]. n,K (q) v v∈Q O m 0 n K So now for any n ≥ 0 we can deﬁne 1 ∗ M = H (X , O) −1 = H (X , O) [ψ ] , n,K 1 K K m m ,ψ n n n 1 ∗ M = M ⊗ F = H (X , F) −1 = H (X , F) [ψ ] , n,K n,K 1 K m ,ψ K m n n n and 0 ∗ q q q N = H (Y , O) −1 = H (Y , O) [ψ ] , n,K 0 m m ,ψ n K n K n n 0 ∗ q q N q = H (Y , F) = H (Y , F) [ψ ] . −1 n,K 0 m K m ,ψ K n n n The reason for dualizing is that the patching argument works more naturally with homology rather than cohomology. Note that M and M are naturally T -modules and, if K is unramiﬁed at q, n,K n,K n,K q q then N and N are naturally T -modules by Lemma 4.12. In particular n,K n,K n,K (q) we may regard them all as R -modules. We now have the following result, a standard ingredient in the patching argument (see for instance [2,25], and [15]): Proposition 4.14 For any n ≥ 1 and any K , the map → R from Lemma 4.13 n n makes M and N into ﬁnite rank free -modules. In particular, the maps n,K n,K n → R and → T are injective. Moreover, the natural maps deﬁne an n n n n,K ∼ ∼ ∼ isomorphism R /a = R and isomorphisms M /a = M and N /a = n n 0 n,K n 0,K n,K n N of R -modules. 0,K 0 Similarly M and N are ﬁnite rank free -modules and we have n,K n,K n ∼ ∼ q q M /a = M and N /a = N . n,K n 0,K n,K n 0,K In particular, rank R = rank R , n O 0 rank M = rank M = rank M , n,K n,K O 0,K and q q q rank N = rank N = rank N n,K n,K F 0,K for all n ≥ 1, and so these ranks are independent of n. We can now deﬁne framed versions of all of these objects. First let O[[y ,..., y ,w ,...,w ]] 1 r 1 j = [[w ,...,w ]] n 1 j e(n,1) e(n,r ) l l (1 + y ) − 1,...,(1 + y ) − 1 1 r F[[y ,..., y ,w ,...,w ]] 1 r 1 j = [[w ,...,w ]] n 1 j e(n,1) e(n,r ) y ,..., y 123 218 J. Manning, J. Shotton Now deﬁne M = M ⊗ R = M ⊗ = M [[w ,...,w ]] n,K R n,K n,K 1 j n,K n n n n ∼ ∼ and deﬁne M , N , and N similarly. Also note that R R ⊗ q q = = n,K n,K n n n n,K R [[w ,...,w ]]. n 1 r Now let S = O[[y ,..., y ,w ,...,w ]] and consider the ideals ∞ 1 r 1 j e(n,1) e(n,r ) l l I = (1 + y ) − 1,...,(1 + y ) − 1 ⊆ S . n 1 r ∞ Note that: Lemma 4.15 For any open ideal a ⊆ S , we have I ⊆ a for all but ﬁnitely many n. ∞ n Proof As S /a is ﬁnite, and the group 1 + m is pro-l, the group (1 + m )/a = ∞ S S ∞ ∞ im(1 + m → S S /a) is a ﬁnite l-group. Since 1 + y ∈ 1 + m for all i, S ∞ ∞ i S ∞ ∞ there is an integer k ≥ 0 such that (1 + y ) ≡ 1 (mod a) for all i = 1,..., r. Then for any n ≥ k, e(n, i ) ≥ n ≥ k for all i, and so indeed I ⊆ a by deﬁnition. Thus we may apply the results of Sect. 4.1 with this ring S and these ideals I . ∞ n Note that dim S = 1 + r + j = dim R . ∞ ∞ Let n = (y ,..., y ,w ,...,w ) ⊆ S , and identify with S /I via the 1 r 1 j ∞ ∞ n above isomorphism. Tensoring everything in Proposition 4.14 with , we get that M is free of rank n n,K ∼ ∼ rank M over for all n with M /n = M /a = M . Similar statements O 0,K n,K n 0,K n,K hold for M , N , and N q. n,K n,K n,K Summarizing the results of this section in the language of Sect. 4.1,wehave: Proposition 4.16 The sequence R ={R } is a patching algebra and R is a n≥1 ∞ cover of R . The sequences M ={M } and N ={N } q q n≥1 n≥1 K n,K K n,K are MCM patching R -modules, and the sequences M ={M } and N q ={N q } n≥1 n≥1 K n,K K n,K are MCM residual patching R -modules. ∼ ∼ ∼ For all n ≥ 1 we have R /n R and M /n M , M /n M , = = = 0 0,K 0,K n n,K n,K ∼ ∼ N /n N q and N /n N q as R -modules. q = q = 0,K 0,K 0 n,K n,K 123 Ihara’s Lemma for Shimura curves... 219 So now deﬁne the patched modules: M = P(M ), ∞,K M = P(M ), ∞,K N = P(N ), and ∞,K N = P(N q ). ∞,K All of these modules are technically framed objects but, following standard convention, we are suppressing the in our notation. By Corollary 4.6 it follows that M and N are maximal Cohen–Macaulay ∞,K ∞,K R -modules, and M and N are maximal Cohen–Macaulay R = ∞ ∞,K ∞,K ∞ R /( )-modules. ∼ ∼ Moreover, Proposition 4.4 gives that M /n = M , M /n = M , ∞,K 0,K ∞,K 0,K ∼ ∼ q q q N /n = N , and N /n = N ,as R -modules. q 0,K ∞,K 0,K 0 n,K Now consider the ﬁltration from Theorem 2.11. By dualizing this, completing at m, and applying −⊗ we get a ﬁltration n n 0 = V ⊆ V ⊆ V ⊆ V = M 0 1 2 3 n,K (q) of R -modules, with isomorphisms V − → N , n,K ⊕2 V /V − → (M ) , 2 1 n,K and V /V − → N q 3 2 n,K q q for all n ≥ 1, where we are writing K = K G(O ) and K (q) = K U (q) as in F ,q 0 0 Sect. 2. Thus Corollary 4.8 and the above work give the following: Theorem 4.17 There is a ﬁltration 0 = V ⊆ V ⊆ V ⊆ V = M 0 1 2 3 ∞,K (q) of R -modules, with isomorphisms V − → N q , 1 ∞,K ⊕2 V /V − → (M ) 2 1 ∞,K 123 220 J. Manning, J. Shotton and V /V − → N . 3 2 ∞,K 4.4 Patching functors Theorem 4.17 provides a link between the modules M and N . However, in ∞,K ∞,K order to use this to deduce properties of M from those of N we will need ∞,K ∞,K additional information about the structure of M , namely a ﬂatness statement ∞,K (q) for a particular submodule of M (St ) ⊂ M . ∞,K F ∞,K (q) To prove this, we will ﬁrst need to introduce the notion of a patching functor, σ → M (σ ). We will largely follow the presentation in [15]. ∞,K We consider pairs (S, K ) satisfying 4.9, and we take K to be of the form K K q q q for a ﬁxed K ⊆ G(A ). For any n ≥ 0let K ⊆ G(A ) be as in Sect. 4.3. F , f F , f We note that ∨ ∨ 1 M = H (X q , O) = H (X q , E /O) [ψ ] 1 −1 m n,K K K m,ψ K K n q n q for any n ≥ 0. Deﬁne ⎡ ⎤ ⎡ ⎤ ∨ ∨ ∨ 1 ⎣ ⎦ ⎣ ⎦ q q = lim M q = lim H (X , E /O) [ψ ] n,K m n,K K K K q n q − → − → K K q q where the direct limit is taken over all compact open subgroups K ⊆ G(O ).Note q F ,q that this carries a continuous action of G(O ) = GL (O ). F ,q 2 F ,q S 1 q q As the action of T on H (X , O) [ψ ] factors through T , the action of K K m K K m q q T on factors through q q T = lim T K K K ← − Note that by Lemma 4.11 we have natural surjections R (ρ) T for all K , K K q F ,S and so we have a surjection R (ρ) T . F ,S Now following [15], let C be the category of ﬁnitely generated O-modules with a continuous action of G(O ).Let ψ = (det ρ| ) ◦ Art : O → F be the F ,q I q F ,q character corresponding to det ρ| : I → F via local class ﬁeld theory. Write I q Z = Z (G(O )) O and let C be the subcategory of C consisting of those F ,q Z F ,q σ ∈ C possessing a central character which lifts ψ and agrees with ψ on I (in other ﬁn words, is unramiﬁed). Also let C be the subcategory of ﬁnite length objects of C . Remark 4.18 In [15], the condition that the central character of σ agrees with ψ is not imposed; this necessitates a ‘twisting’ argument. We only need to patch σ with unramiﬁed central character, so we avoid this technicality. 123 Ihara’s Lemma for Shimura curves... 221 Now for any σ ∈ C and any n ≥ 0, deﬁne 1 ∨ M q (σ ) = H (X , L ∨ ) [ψ ] . n,K σ m K G(O ) n F ,q For any σ,thisisa T -module, and hence an R -module. Thus we may deﬁne the R -module: q q M (σ ) = M (σ ) ⊗ R = M (σ )[[w ,...,w ]]. n,K R n,K 1 j n,K n n ﬁn Now as in section 6 of [15], if σ ∈ C , M (σ ) ={M (σ )} is a weak patching q q n≥1 K n,K R -module and thus we may deﬁne M (σ ) = P(M (σ )). We can extend this ∞,K deﬁnition to all of C by setting q q M (σ ) = lim M (σ/ σ). ∞,K ∞,K ← − This deﬁnition agrees with the “patching functor” constructed in section 6.4 of [15], up to a technicality: the construction in [15] factors out the Galois representation in the indeﬁnite case, whereas we have not done so. In the notation of [15] the module M q (σ ) we have constructed is S(σ ) M (σ ) ⊗ ρ(σ ) . However, this ∞,K ∞ T(σ ) m m m ⊕2 is simply isomorphic to M (σ ) as a T(σ ) -module (again in the notation of [15]) ∞ m and so this does not present an issue. We therefore have: Theorem 4.19 ([15]) M q (σ ) satisﬁes the following properties: ∞,K (1) The functor σ → M q (σ ),from C to the category of ﬁnitely generated R - ∞,K Z ∞ modules, is exact. (2) For any σ ∈ C ,M q (σ )/n M q (σ ). Z ∞,K 0,K (3) If σ ∈ C is a ﬁnite free O-module, then M q (σ ) is maximal Cohen–Macaulay Z ∞,K over R . (4) If σ ∈ C is a ﬁnite dimensional F-vector space, then M q (σ) is maximal Z ∞,K Cohen–Macaulay over R . From now on assume that q satisﬁes the assumptions of Sect. 3.3. That is, Nm(q) ≡ 1 (mod l), ρ is unramiﬁed at q and ρ(Frob ) = . Thus the computations of Sect. 3.3 will apply to R . Under the map R → R → R , we may view any q q loc ∞ R -module as being a R -module. ∞ q In addition to the results listed in Theorem 4.19,[15] also describes the supports of M (σ ) as R -modules, for certain σ ’s corresponding to inertial types of F .In ∞,K q q order to avoid having to give a formal treatment of inertial types, we will simply state ps their results for the speciﬁc modules σ = 1 , St and σ ,for A = O, F, deﬁned in A A −1 section 3.5 (noting that we have assumed that O = W (F)[ζ + ζ ]): Proposition 4.20 ([15]) Viewing each M (σ ) as an R -module, ∞,K q nr nr (1) M q (1 ) (resp. M q (1 )) is supported on R (resp. R ), ∞,K O ∞,K F q q 123 222 J. Manning, J. Shotton unip unip (2) M q (St ) (resp. M q (St )) is supported on R (resp. R ), ∞,K O ∞,K F q ps ps ps ps q q (3) M (σ ) (resp. M (σ )) is supported on R (resp. R ). ∞,K ∞,K O F q Proof Follows from Proposition 3.10 and the fact that M (−) is a patching functor ∞,K in the sense of [15]. We also record the support of the modules N q and N q from Sect. 4.3 here. ∞,K ∞,K q q Proposition 4.21 As R -modules, N is supported on R and N is supported q ∞,K ∞,K on R . Proof As N q = N q ⊗ F and R = R ⊗ F, it sufﬁces to prove the ∞,K ∞,K O O q q statement for N . ∞,K By the deﬁnition of N it sufﬁces to prove that, for any n ≥ 1, the map ∞,K γ : R → R → End (N ) n q F n n,K factors through R R . We will prove this using Proposition 3.10. S∪Q N n Let T be the image of T in End(N ). Note that the map R → n,K n n,K O N , N N N End (N q ) factors through T . Deﬁne T = T ⊗ R T [[w ,..., O n,K R 1 n n n,K n,K n,K n,K N , q N w ]]; thus γ deﬁnes a map R → T . Since T is reduced and l-torsion free, it j n n,K n,K sufﬁces to show that, for every O-algebra homomorphism N , x : T → E , n,K the composition x ◦ γ factors through R . To x we have an associated homomorphism ρ : G → GL (E ) x F ,S∪Q 2 such that, for everyv/ ∈ S ∪ Q ,tr(ρ (Frob )) = x (T ). In particular, the isomorphism n x v v class of ρ is the Galois representation associated to x | . x T n,K The composition x ◦ γ is the homomorphism R → E corresponding to ρ | . n q x G By local-global compatibility and properties of the Jacquet–Langlands correspon- dence, ρ| is an inertially unipotent representation corresponding to a discrete series representation under the local Langlands correspondence. It follows that ρ| is a non- trivial unipotent representation, and therefore that x ◦ γ : R → E factors through n q R R by Proposition 3.10. The result follows. We ﬁnish this section by relating the patching functors of this section to the patched modules M considered in Sect. 4.3. ∞,K Proposition 4.22 For any compact open subgroup K ⊆ G(O ) we have q F ,q G(O ) F ,q q q M Ind 1 = M . ∞,K F ∞,K K 123 Ihara’s Lemma for Shimura curves... 223 q q In particular, letting K = K G(O ) and K (q) = K U (q), M F ,q 0 0 ∞,K M q (1 ) and ∞,K F ∼ ∼ M M q (1 ⊕ St ) M q (1 ) ⊕ M q (St ). = = ∞,K (q) ∞,K F F ∞,K F ∞,K F Proof By the fact that m is non-Eisenstein, we have G(O ) G(O ) F ,q F ,q M q Ind 1 = Hom H (X , F) [ψ ], Ind 1 n,K F G(O ) K G(O ) m F F ,q q F ,q K K q q = Hom H (X , F) [ψ ], 1 K K G(O ) m F q q F ,q = M . n,K K G(O ) F ,q It follows that M Ind 1 M and so q = q n,K K n,K K G(O ) G(O ) F ,q F ,q q q q M Ind 1 = P M q Ind 1 = P M = M . ∞,K F F K K ∞,K K ∞,K q q K K q q G(O ) G(O ) F ,q F ,q The last two statements follow from Ind 1 = 1 and Ind 1 = 1 ⊕ F F F F G(O ) U (q) F ,q 0 q q q St . The statement that M (1 ⊕ St ) = M (1 ) ⊕ M (St ) is just a F ∞,K F F ∞,K F ∞,K F consequence of the exactness of M (−). ∞,K Corollary 4.23 The R -module q q P = M (1 ) ⊕ M (St ) ∞,K F ∞,K F has a ﬁltration 0 = V ⊆ V ⊆ V ⊆ V = P 0 1 2 3 ⊕2 ∼ ∼ ∼ with V V /V N q and V /V M q (1 ) . = = = 1 3 2 ∞,K 2 1 ∞,K F Proof By Proposition 4.22 this is just a rephrasing of Theorem 4.17. 5 Commutative algebra lemmas The following is a mild generalisation of the “miracle ﬂatness criterion”, for which see [28] Theorem 23.1 or [37, Lemma 00R4]. A similar generalisation, in the setting of noncommutative completed group rings, also appears in [20]. Lemma 5.1 Let A → R be a local homomorphism of noetherian local rings, and let M be a ﬁnite R-module. Let m be the maximal ideal of A. Suppose that: (1) A is regular; (2) M is maximal Cohen–Macaulay; and (3) dim R = dim A + dim R/mR. 123 224 J. Manning, J. Shotton Then M is a ﬂat A-module. Proof The proof is essentially the same as that of [37, Lemma 00A4]. If M is zero, the result is clear; so suppose that M is nonzero. The proof is then by induction on d = dim A. The base case d = 0 is trivial, as then A is a ﬁeld. In general, suppose the lemma is true when dim A < d. Choose x ∈ m \ m . Then x is the ﬁrst element in a regular system of parameters (x , x ,..., x ) for A. 2 d The third condition implies that (x , x ,..., x ) extends to a system of parameters 2 d (x , x ,..., x , x ,..., x ) for R which is therefore also a system of parameters 2 d d+1 e for M (by the hypothesis that M is maximal Cohen–Macaulay). Since M is Cohen– Macaulay, this is a regular sequence on M. In particular, x is a non-zerodivisor on M. Now, A/xA is regular of dimension dim A − 1, dim(R/xR) = dim R − 1 (since x is part of a system of parameters for R), and M /xM is a maximal Cohen– Macaulay R/xR-module. So, by induction, M /xM is a ﬂat A/xA-module. Moreover, Tor (M , A/(x )) = 0as x is a non-zerodivisor on M. Therefore, by the local criterion for ﬂatness in the form of [37, Lemma 00ML], M is a ﬂat A-module. Lemma 5.2 Let A = F[[X , Y ]]/(X Y ) and let R be an A-algebra. Let 0 → L → M → N → 0 be a short exact sequence of R-modules such that (1) M is a ﬂat A-module; (2) (X ) ⊂ ann (L); (3) (XY ) ⊂ ann (N ). Then N = M ⊗ A/(XY ) and so N is a ﬂat A/(XY )-module. Moreover we have an isomorphism N /XN L of R-modules. Proof By the snake lemma, as multiplication by X is zero on L, there is an exact sequence of R modules 0 → L → M [X]→ N [X]→ L → M /XM → N /XN → 0. But we have an exact sequence 0 → (XY ) → A → (X ) → 0 (the second map being multiplication by X). As M is ﬂat this is still exact when tensored over A with M, and for any ideal I we can identify I ⊗ M with IM ⊂ M. Thus M [X]= XY M. But as N is killed by XY , this implies that the map M [X]→ N [X ] is zero. From the displayed exact sequence, we see that L = M [X]= XY M, and so N = M /L = M /XY M. This is ﬂat over A/(XY ). Now as L = XY M and M /XM is killed by X,the map L → M /XM in the above exact sequence is zero, which implies that the map N [X]→ L is an isomorphism of R-modules. ·Y ·X But now we have an exact sequence 0 → A/(X ) − → A/(XY ) − → XA/(XY ) → 0. Y X As N is ﬂat over A/(XY ), the sequence of R-modules 0 → N /XN − → N − → XN → ∼ ∼ 0 is exact, and so we get the desired isomorphism N /XN N [X ] L of R-modules. = = Lemma 5.3 Let B = F[[X , Y ]]/(XY ) and let R be a complete local noetherian B- algebra with residue ﬁeld F. Suppose that L, M, N and P are R-modules such that: 123 Ihara’s Lemma for Shimura curves... 225 (1) M is ﬂat over B; (2) (Y ) ⊂ ann (N ) and N is ﬂat over B/(Y ); (3) there is an isomorphism of R-modules L − → M /XM; (4) there is an isomorphism of R-modules α : P − → L ⊕ M; (5) there is a ﬁltration 0 ⊂ P ⊂ P ⊂ P by R-modules and isomorphisms of 1 2 ∼ ∼ ∼ R-modules P − → N, P /P − → L ⊕ L, and P/P − → N. 1 2 1 2 Then there is a short exact sequence of R-modules 0 → N → M /Y → N → 0. Proof. Since L is ﬂat over B/X by points (1) and (3), it has no Y -torsion, and so α induces an isomorphism P[Y ] − → M [Y ]. From the short exact sequence 0 → P = N → P → L ⊕ L → 0 1 2 of point (5), we have P [Y]= P [Y ] N . 2 1 From the other short exact sequence 0 → P → P → N → 0 of point (5), we get an exact sequence ∼ ∼ 0 → P [Y ] N → P[Y ] M [Y]→ N [Y]= N = = By the ﬂatness of M, we can identify M [Y ] with X · M, and so the image of M [Y ] in N is XN . Since N is ﬂat over B/(Y ), N = XN . Thus we have a short exact sequence ∼ ∼ ∼ 0 → P [Y ] = N → M [Y ] = XM → X · N = N → 0. Finally, since M is ﬂat over B there is an isomorphism M /YM − → XM. We get the desired short exact sequence: 0 → N → XM M /YM → N → 0. 6 Ihara’s lemma Let D be a quaternion division algebra over F ramiﬁed at exactly one inﬁnite place, so that we are in the indeﬁnite case of Sect. 2. Suppose that p is a ﬁnite place of F at which D is unramiﬁed. 123 226 J. Manning, J. Shotton 6.1 Statements Let K ⊆ G(A ) be unramiﬁed at p and sufﬁciently small, and let S be any ﬁnite F , f set of ﬁnite places of F containing (K ) ∪ ∪{p}∪ . There are two natural l ∞ degeneracy maps π ,π : X → X , deﬁned in section 2.11. 1 2 K (p) K Conjecture 6.1 Suppose that is the local system on X attached to a ﬁnite- dimensional continuous F -representation of K . Then for any non-Eisenstein maximal ideal m of T the map ∗ ∗ 1 1 1 π ⊕ π : H (X ,) ⊕ H (X ,) → H (X ,) K m K m K (p) m 1 2 0 is injective. For the constant sheaf F , this becomes: Conjecture 6.2 For any non-Eisenstein maximal ideal m of T ,the map ∗ ∗ 1 1 1 π ⊕ π : H (X , F ) ⊕ H (X , F ) → H (X , F ) K l m K l m K (p) l m 1 2 0 is injective. We also have an equivalent dualized version: Conjecture 6.3 For any non-Eisenstein maximal ideal m of T ,the map (π ,π ) : H (X , F ) → H (X , F ) ⊕ H (X , F ) 1,∗ 2,∗ 1 K (p) l m 1 K l m 1 K l m is surjective. Lemma 6.4 Conjecture 6.2 (or, equivalently, Conjecture 6.3) for all K implies Con- jecture 6.1 for all K . Proof Suppose that Conjecture 6.2 holds for all K . Suppose that and m are as in the statement of Conjecture 6.1, and that is associated to a representation V of K . p p p Let H ⊂ K be an open subgroup that acts trivially on V , and H = H K .Let f : X → X be the projection. The Hochschild–Serre spectral sequence provides H K a (Hecke-equivariant) exact sequence 1 0 ∗ 1 0 1 ∗ 0 → H (K /H , H (X , f )) → H (X ,) → H (K /H , H (X , f )). H K H After localizing at m, the ﬁrst term vanishes by Lemma 2.3. Noting that f is constant, we get an inclusion 1 1 dim V H (X ,) → H (X , F ) K m H m that commutes with the maps π . Since Conjecture 6.2 holds for the subgroup H by assumption, we deduce Conjecture 6.1 for the subgroup K . 123 Ihara’s Lemma for Shimura curves... 227 Our main result is the following: Theorem 6.5 If l > 2, then Conjectures 6.1, 6.2 and 6.3 are true for any non-Eisenstein maximal ideal m of T satisfying the conditions: (1) l|#ρ (G ). That is, m is not exceptional. (2) If l = 5 and the image of the projective representation proj ρ : G → GL (F ) PGL (F ) is isomorphic to PG L (F ), then ker proj ρ G . 2 5 2 5 2 5 F (ζ ) m 5 (This condition is automatically satisﬁed whenever 5 ∈ / F.) Remark 6.6 Condition (1) implies the Taylor–Wiles condition that ρ | is abso- F (ζ ) lutely irreducible. Condition (2) is simply the other Taylor–Wiles condition (see [25, 3.2.3]). The reason for including the stronger assumption that m is not exceptional, instead of just the usual Taylor–Wiles conditions, is that this assumption will be necessary for picking the auxiliary prime q. See Lemma 6.9 below. Remark 6.7 We have assumed that K is sufﬁciently small, for convenience. This assumption could be removed by the standard device of introducing auxiliary level structure at a place q at which there are no congruences, as in [27] section 4.2 or [15] section 6.2. 6.2 Definite quaternion algebras Let D be a totally deﬁnite quaternion algebra over F, unramiﬁed at p.Let G be the associated algebraic group. If H ⊂ G(A ) is a compact open subgroup unramiﬁed F , f at p then we have degeneracy maps π ,π : Y → Y .Let S be a ﬁnite set of 1 2 H (p) H places of F containing ∪ ∪{p} and all places at which H or D ramify. The l ∞ following version of Ihara’s Lemma is known: Theorem 6.8 If H ⊆ G(A ) is unramiﬁed at p, then for any non-Eisenstein maximal F , f ideal m of T ,the map ∗ ∗ ∗ 0 0 0 π = π + π : H (Y , F ) ⊕ H (Y , F ) → H (Y , F ) H l m H l m H (p) l m 1 2 0 is injective. Proof Versions of this have been proved by Ribet (over Q,[31] Theorem 3.15) and Taylor (over F,[38] Lemma 4). There it is proved that with Z coefﬁcients, with- out localizing at m, π has saturated image, from which the theorem may be easily deduced—but the method for doing this actually directly gives the result in the form we need. For Q this is carried out in [13] Lemma 2 and the general case is no harder. We include the proof for completeness. Suppose that ( f , g) is in the kernel of π .Regard f and g as H-invariant functions on G(F )\G(A ). Then f (x ) =−g(xω) for all x in this quotient, where ω = F , f (making use of the isomorphism G(F ) = GL (F )). Then f is invariant p 2 p 123 228 J. Manning, J. Shotton −1 p under H and ω H ω. These subgroups generate a subgroup containing H SL (F ), 2 p under which f is invariant. Let G be the subgroup of G of elements with reduced norm 1. Then by the strong approximation theorem in G , the function f factors through the reduced norm map: × × ν : G(F )\G(A )/G (A )H → F \A /ν(H ). F , f F , f F , f But the functions factoring through this map form a module over T that is supported on Eisenstein maximal ideals (the argument is similar to that of Proposition 2.3). The theorem follows. 6.3 The auxiliary prime Recall our assumption that l|#ρ (G ). After conjugating ρ if necessary, we may m F m thus assume that ρ (G ) contains the matrix . We now get the following: Lemma 6.9 There are inﬁnitely many primes q for which: (1) q ∈ / ∪ (K ) ∪ ∪{p} (2) ρ is unramiﬁed at q (3) Nm(q) ≡ 1 (mod l) (4) ρ (Frob ) = Proof. All but ﬁnitely many primes satisfy (1) and (2), so it sufﬁces to ﬁnd inﬁnitely many primes satisfying (3) and (4). Pick a number ﬁeld L/F for which F (ζ ) ⊆ L and ρ : G → GL (F ) fac- l F 2 l tors through Gal(L/F ).Let : Gal(L/F ) Gal(F (ζ )/F)→ (Z/lZ) be the cyclotomic character. By the Chebotarev density theorem, it sufﬁces to ﬁnd some σ ∈ Gal(L/F ) for which ρ (σ ) = and (σ ) = 1 ∈ (Z/lZ) . Now by our assumption on the image of ρ , there is some σ ∈ Gal(L/F ) for 1−l which ρ (σ ) = .Let σ = σ ∈ Gal(L/F ). Then we indeed have m 0 1−l 11 11 1−l ρ (σ ) = ρ (σ ) = = m m 01 01 and l−1 × (σ ) = (σ ) = 1 ∈ (Z/lZ) . For the rest of the proof we ﬁx such a prime q. Note that it satisﬁes the requirements of sections 2.13 and 3.3.Welet D be a deﬁnite quaternion algebra ramiﬁed at ∪{q,τ }. 123 Ihara’s Lemma for Shimura curves... 229 6.4 The proof Choose F large enough that ρ is deﬁned over F, and let (E , O, F) be the coefﬁcient system satisfying Hypothesis 3.1.Let ψ : G → O be a ﬁnite order character lifting det(ρ ), and also write ψ for the character ψ ◦Art of A /F . We make sure that m F F , f F (K ∩ Z (A )) ⊂ ker(ψ ) and that the prime q is chosen so that ψ is unramiﬁed F , f at q. Let S be as in Sect. 6.1. Enlarging S if necessary (which is allowed, by Lemma 4.11), we assume that q ∈ S. We write for the set of ﬁnite places in S. The results of 1 S Sect. 2 imply that there is a ﬁltration of H (X , F) [ψ ] (by T -submodules) K (q) m whose graded pieces are 0 1 ⊕2 0 q q H (Y , F) [ψ ], H (X , F) [ψ ] , H (Y , F) [ψ ]. K m K m K m In Sect. 4 we explain how these cohomology groups and this ﬁltration (more pre- cisely, their duals) may be ‘patched’ using the Taylor–Wiles method. For each place v ∈ let R be ,ψ • if v l, the universal ﬁxed determinant framed deformation ring R of ρ | ,O ρ | ; m F • if v | l, the potentially semistable (over a ﬁxed extension depending only on K ∩ G(F ), and of parallel Hodge–Tate weights {0, 1}) deformation ring ,ψ,K ∩G(F )-st R deﬁned in Sect. 3.5. ρ | ,O For some integers g, d ≥ 0 (determined in Sect. 4, with d = r + j in the notation of that section) we let R = ⊗ R [[X ,..., X ]] ∞ v∈ v 1 g and S = O[[Y ,..., Y ]], ∞ 1 d and recall that d and g were chosen so that R and S have the same dimension. ∞ ∞ Then in Sect. 4.4 we constructed an injective homomorphism S → R , maximal ∞ ∞ q q Cohen–Macaulay R -modules M and N , and an exact functor M from ∞ ∞,K ∞,K ∞,K the category of ﬁnitely-generated O-modules with a continuous action of GL (O ) 2 F ,q (satisfying a condition on the central character) to the category of ﬁnitely-generated R -modules. Moreover, M has the property that if σ is a ﬁnite free O-module ∞ ∞,K (resp. a ﬁnite dimensional F-vector space) then M (σ ) is maximal Cohen– ∞,K Macaulay over R (resp. R = R ⊗ F). These are equipped with isomorphisms ∞ ∞ ∞ O M ⊗ F = H (X , F) −1 ∞,K S 1 K ∞ m,ψ and N q ⊗ F H (Y q , F) . = −1 ∞,K S 0 K ∞ m,ψ 123 230 J. Manning, J. Shotton Table 1 Supports of patched Patched module M Quotient R modules nr M q (1 ) R ∞,K O q unip M q (St ) R ∞,K O q N q R ∞,K q ps ps M q (σ ) R ∞,K In Table 1, for various patched modules, we write down a corresponding quotient R of R on which they are supported. Here ? is an element of {nr, N , unip, ps}, and we write ?,ψ R = R ρ| ,O and R = R ⊗ F, q q as shorthand for the rings deﬁned in Sect. 3.3. The claims of Table 1 follow from the properties of the Jacquet–Langlands correspondence and local-global compat- ibility, and are the content of Propositions 4.20 and 4.21. Furthermore, for ? ∈ {nr, N , unip, ps} we deﬁne the quotient ? ? R = R ⊗ ⊗ R [[X ,..., X ]] v∈\{q} v 1 g ∞ q of R . The ﬁltration provided by Theorem 2.11 may be patched as in Sect. 4. Thus (see Corollary 4.23) there is a ﬁltration of P = M = M q (1 ) ⊕ M q (St ) ∞,K (q) ∞,K F ∞,K F by R -modules 0 ⊂ P ⊂ P ⊂ P () 1 2 together with isomorphisms N q − → P , ∞,K 1 N − → P/P , ∞,K 2 and ⊕2 M q (1 ) − → P /P . ∞,K F 2 1 123 Ihara’s Lemma for Shimura curves... 231 To go further, we need the structure of the local deformation rings at q. The defor- ps nr N mation rings R , R and R are regular by Propositions 3.5 and 3.6. Therefore, by q q Lemma 5.1,wehave: nr Proposition 6.10 (1) M (1 ) is ﬂat over R . ∞,K O (2) N is ﬂat over R . ∞,K ps ps (3) M (σ ) is ﬂat over R . ∞,K By Proposition 3.7, there are isomorphisms unip ∼ R − → F[[X , Y , P, Q, R]]/(XY ) and ps ∼ R − → F[[X , Y , P, Q, R]]/(X Y ) ps unip compatible with the natural surjection R R and so that q q nr unip R = R /(X ) q q and N unip R = R /(Y ). q q By section 3.5, equation (2), we have an exact sequence ps 0 → M q (1 ) → M q (σ ) → M q (St ) → 0. ∞,K F ∞,K ∞,K F unip Proposition 6.11 The module M (St ) is ﬂat over R and there is an isomor- ∞,K F q phism nr ∼ q q q M (St ) ⊗ unip R − → M (1 ) = M . ∞,K F ∞,K F ∞,K Proof By Proposition 6.10 and the above exact sequence, the hypotheses of Lemma 5.2 unip apply with R = R ⊗ F (made into an F[[X , Y ]]/(X Y )-algebra in the evident ∞ O ps q q q way), L = M (1 ), M = M (σ ), and N = M (St ). The proposition ∞,K F ∞,K ∞,K F follows. Now we know that M (St ) is ﬂat, the ﬁltration () can be used to “transfer ∞,K F information” between N and M . More precisely, we have: ∞ ∞ Proposition 6.12 There is a short exact sequence of R -modules 0 → N q → M q (St ) ⊗ R → N q → 0. unip ∞,K ∞,K F ∞,K 123 232 J. Manning, J. Shotton Proof By Proposition 6.11 and the ﬁltration (), the hypotheses of Lemma 5.3 apply ps with R = R ⊗ F (made into an F[[X , Y ]]/(XY )-algebra in the evident way), ∞ O q q q L = M (1 ), M = M (St ), P = M , N = N , and P and P ∞,K F ∞,K F ∞,K (q) ∞,K 1 2 given by (). The proposition follows. Proof of Theorem 6.5 Now we are ready to prove our main result. We may carry out q q the constructions and arguments above equally well with K replaced by K (p) in a way compatible with the degeneracy maps π . We therefore obtain a commuting diagram 0 − −−−→ N q − −−−→ M q (St )/(Y ) − −−−→ N q − −−−→ 0 ∞,K (p) ∞,K (p) F ∞,K (p) 0 0 0 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ π π π ∗ ∗ ∗ ⊕2 ⊕2 ⊕2 q q q 0 − −−−→ N − −−−→ M (St ) /(Y ) − −−−→ N − −−−→ 0. ∞,K ∞,K F ∞,K By Theorem 6.8 the outer maps are surjective after applying ⊗ F, and so by Nakayama’s Lemma they are surjective. It follows that the middle map is surjective, and by Nakayama’s Lemma again that the map ⊕2 π : M q (St ) → M q (St ) ∗ ∞,K (p) F ∞,K F nr is surjective. Tensoring with R and applying Proposition 6.11 this gives that ⊕2 M → M is surjective. Applying ⊗ F, we see that ∞,K (p) ∞,K S 0 ∞ ⊕2 π : H (X , F) −1 → H (X , F) ∗ 1 K (p) 1 K −1 0 m,ψ m,ψ is surjective. By Nakayama’s Lemma, we obtain that ⊕2 π : H (X , F) → H (X , F) ∗ 1 K (p) m 1 K 0 m is surjective. This proves Conjecture 6.3 and hence Theorem 6.5 for this m. Acknowledgements Firstly, we thank Matthew Emerton for suggesting that we collaborate on this project and for many enlightening conversations. We also thank Chuangxun Cheng, Fred Diamond, Toby Gee, Yongquan Hu, David Loefﬂer, Matteo Longo, and Matteo Tamiozzo for useful comments or discussions. Part of this work was written up while the second author was at the Max Planck Institute for Mathematics, and he thanks them for their support. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 123 Ihara’s Lemma for Shimura curves... 233 References 1. Bertolini, M., Darmon, H.: Iwasawa’s main conjecture for elliptic curves over anticyclotomic Z - extensions. Ann. Math. (2) 162(1), 1–64 (2005) 2. Breuil, C., Diamond, F.: Formes modulaires de Hilbert modulo p et valeurs d’extensions entre caractères galoisiens. Ann. Sci. Éc. Norm. Supér. (4) 47(5), 905–974 (2014) 3. Buzzard, K., Diamond, F., Jarvis, F.: On Serre’s conjecture for mod Galois representations over totally real ﬁelds. Duke Math. J. 155(1), 105–161 (2010) 4. Barnet-Lamb, T., Gee, T., Geraghty, D., Taylor, R.: Potential automorphy and change of weight. Ann. Math. (2) 179(2), 501–609 (2014) 5. Carayol, H.: Sur la mauvaise réduction des courbes de Shimura. Composit. Math. 59(2), 151–230 (1986) 6. Calegari, F., Geraghty, D.: Modularity lifting beyond the Taylor–Wiles method. Invent. Math. 211(1), 297–433 (2018) 7. Chida, M., Hsieh, M.-L.: On the anticyclotomic Iwasawa main conjecture for modular forms. Compos. Math. 151(5), 863–897 (2015) 8. Cheng, C.: Ihara’s lemma for Shimura curves over totally real ﬁelds and multiplicity two, unpublished, available at https://www.math.uni-bielefeld.de/~ccheng/Research/Multi2.pdf 9. Clozel, L., Harris, M., Taylor, R.: Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes Études Sci. (2008), no. 108, 1–181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras 10. Chojecki, P., Sorensen, C.: Strong local-global compatibility in the p-adic Langlands program for U (2). Rend. Semin. Mat. Univ. Padova 137, 135–153 (2017) 11. Diamond, F.: The Taylor–Wiles construction and multiplicity one. Invent. Math. 128(2), 379–391 (1997) 12. Diamond, F., Taylor, R.: Lifting modular mod l representations. Duke Math. J. 74(2), 253–269 (1994) 13. Diamond, F., Taylor, R.: Nonoptimal levels of mod l modular representations. Invent. Math. 115(3), 435–462 (1994) 14. Edixhoven, B.: The weight in Serre’s conjectures on modular forms. Invent. Math. 109(3), 563–594 (1992) 15. Emerton, M., Gee, T., Savitt, D.: Lattices in the cohomology of Shimura curves. Invent. Math. 200(1), 1–96 (2015) 16. Emerton, M., Helm, D.: The local Langlands correspondence for GL in families. Ann. Sci. Éc. Norm. Supér. (4) 47(4), 655–722 (2014) 17. Emerton, M.: Local-global compatibility in the p-adic Langlands programme for GL /Q, 2011, draft available at http://www.math.uchicago.edu/~emerton/preprints.html 18. Gee, T.: Automorphic lifts of prescribed types. Math. Ann. 350(1), 107–144 (2011) 19. Gee, T., Liu, T., Savitt, D.: The weight part of Serre’s conjecture for GL(2). Forum Math. Pi 3, e2, 52 (2015) 20. Gee, T., Newton, J.: Patching and the completed homology of locally symmetric spaces, (2016) 21. Gow, R.: On the Schur indices of characters of ﬁnite classical groups. J. Lond. Math. Soc. (2) 24(1), 135–147 (1981) 22. Ihara, Y.: On modular curves over ﬁnite ﬁelds, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, Oxford Univ. Press, Bombay 1975, 161–202 (1973) 23. Jarvis, F.: Mazur’s principle for totally real ﬁelds of odd degree. Compos. Math. 116(1), 39–79 (1999) 24. Kisin, M.: Potentially semi-stable deformation rings. J. Am. Math. Soc. 21(2), 513–546 (2008) 25. Kisin, M.: Moduli of ﬁnite ﬂat group schemes, and modularity. Ann. Math. (2) 170(3), 1085–1180 (2009) 26. Longo, M.: Anticyclotomic Iwasawa’s main conjecture for Hilbert modular forms. Comment. Math. Helv. 87(2), 303–353 (2012) 27. Manning, J.: Patching and multiplicity 2 for Shimura curves, (2019), available at https://arxiv.org/ abs/1902.06878 28. Matsumura, H.: Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid 29. Morita, Y.: Reduction modulo P of Shimura curves. Hokkaido Math. J. 10(2), 209–238 (1981) 30. Ribet, K.A.: Congruence relations between modular forms, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, pp. 503–514 (1984) 123 234 J. Manning, J. Shotton 31. Ribet, K.A.: On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math. 100(2), 431–476 (1990) 32. Ribet, K.A.: Multiplicities of Galois representations in Jacobians of Shimura curves, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, : Israel Math. Conf. Proc., vol. 3. Weizmann, Jerusalem 1990, 221–236 (1989) 33. Scholze, P.: On the p-adic cohomology of the Lubin-Tate tower. Ann. Sci. Éc. Norm. Supér. (4) 51(4), 811–863 (2018) 34. Shotton, J.: Local deformation rings for GL and a Breuil-Mézard conjecture when = p. Algebra Number Theory 10(7), 1437–1475 (2016) 35. Shotton, J.: Local deformation rings for 2-adic representations of G , l = 2, (2017) Appendix to Yongquan Hu, Vytautas Paškunas, ¯ On crystabelline deformation rings of Gal(Q /Q ) 36. Shotton, J.: The Breuil-Mézard conjecture when l = p. Duke Math. J. 167(4), 603–678 (2018) 37. The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2017 38. Taylor, R.: On Galois representations associated to Hilbert modular forms. Invent. Math. 98(2), 265– 280 (1989) 39. Taylor, R.: On the meromorphic continuation of degree two L-functions, Doc. Math. (2006), no. Extra Vol., 729–779 40. Taylor, R.: Automorphy for some l-adic lifts of automorphic mod l Galois representations. II, Publ. Math. Inst. Hautes Études Sci. (2008), no. 108, 183–239 41. Wang, H.: Anticyclotomic Iwasawa theory for Hilbert modular forms, ProQuest LLC, Ann Arbor, MI, 2015, Thesis (Ph.D.)–The Pennsylvania State University Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.
Mathematische Annalen – Springer Journals
Published: Sep 25, 2020
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.