THE EISENSTEIN IDEAL WITH SQUAREFREE LEVEL. Contents 1. Introduction 1 2. Modular forms The pseudodeformation ring 15 4.

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1 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL PRESTO WAKE AD CARL WAG-ERICKSO Abstract. We use seudodeformation theory to study the anaogue of Mazur s Eisenstein idea with certain squarefree eves. Given a rime number > 3 and a squarefree number satisfying certain conditions, we study the Eisenstein art of the -adic Hecke agebra for Γ 0 (), and show that it is a oca comete intersection and isomorhic to a seudodeformation ring. We aso show that in certain cases, the Eisenstein idea is not rincia and that the cusida quotient of the Hecke agebra is not Gorenstein. As a coroary, we rove that mutiicity one fais for the moduar Jacobian in these cases. In a articuar case, this roves a conjecture of Ribet. Contents 1. Introduction 1 2. Moduar forms The seudodeformation ring Toward R = T The case ɛ = ( 1, 1, 1,..., 1) The case ɛ = ( 1, 1) Generators of the Eisenstein idea 32 Aendix A. Comarison with the Hecke agebra containing U 39 Aendix B. Comutation of some cu roducts 42 Aendix C. Agebra 43 References Introduction In his andmark study [Maz77] of the Eisenstein idea with rime eve, Mazur named five secia settings in which it woud be interesting to deveo the theory of the Eisenstein idea in a broader context [g. 39, oc. cit.], the first of which is the setting of squarefree eve. In this aer, we deveo such a theory in certain cases Mazur s resuts and their squarefree anaogues. Let 3 and be rimes, and et T be the -adic Eisenstein cometion of the Hecke agebra acting on moduar forms of weight 2 and eve, and et T T 0 be the cusida quotient. Let I 0 T0 be the Eisenstein idea, and et m0 = (, I0 ) be the maxima idea. Mazur roved the foowing resuts [Maz77]: (1) T 0 /I0 = Z /( 1 12 )Z, Date: Ari 27,

2 2 PRESTO WAKE AD CARL WAG-ERICKSO (2) I 0 is rincia, (3) T 0 is Gorenstein, (4) dim F (J 0 ()[m 0 ]) = 2, and (5) if q is a rime such that q 1 (mod ) and such that q is not a -th ower moduo, then T q (q + 1) generates I 0. Mazur cas a rime q as in (5) a good rime for (, ). We note that, of course, (5) imies (2) imies (3). We aso note that (2) imies that T is Gorenstein aso. The anaogue of (1) has been roven for squarefree eves by Ohta [Oht14]. However, as has been noted by many authors, notaby Ribet and Yoo [Rib15, Yoo15], the statements (2)-(5) are not true in the squarefree setting. Sti, in this aer, we rove, in certain cases, anaogues of (2)-(5). amey, we count the minima number of generators of the Eisenstein idea, count the dimension of the Eisenstein kerne of the Jacobian, and give sufficient (and sometimes aso necessary) conditions for a ist of eements T q (q + 1) to generate the Eisenstein idea Pseudomoduarity. Our main technica resut is an R = T theorem, where R is a deformation ring for Gaois seudoreresentations and T is the Eisenstein art of the Hecke agebra. The strategy is simiar to that of our revious works [WWE17c, WWE17a], where we gave new roofs and refinements of Mazur s resuts. However, there are severa oints of interest that are new in this setting. (a) In the case of rime eve, Caegari and Emerton [CE05] have aready aied deformation theory to study Mazur s Eisenstein idea. Their method is to rigidify the deformation theory of Gaois reresentations using auxiiary data coming from the rime eve. In the case of squarefree eve, a simiar strategy wi not work: the deformation robem at rime eve is aready rigid, and cannot be further rigidified to account for the additiona rimes. (b) In the case of squarefree eve, there are mutie Eisenstein series, and one has to account for the ossibiity of congruences among them. (c) At squarefree eve, unike rime eve, the Tate modue of the Jacobian may not be free over the Hecke agebra. Since this Tate modue is the natura way to construct Gaois reresentations, it is reay necessary to work with seudoreresentations. (d) We rove R = T even in some cases where the Gaois cohomoogy grous controing the tangent sace of R are a non-cycic (see Remark 1.4.8). In this case, the universa seudodeformation cannot arise from a reresentation. To address issue (a), we have to deveo a theory of Cayey-Hamiton reresentations and seudoreresentations with squarefree eve, which has the required fexibiity; for this, we drew insiration from our revious joint works [WWE15, WWE17c, WWE17a] and the work of Caegari-Secter [CS16]. The ideas are discussed ater in this introduction in 1.8. To address issue (b), we make extensive use of an idea of Ohta [Oht14]: we use the Atkin-Lehner invoutions at to define T, rather than the usua Hecke oerators U Setu. We introduce notation in order to state our main resuts recisey. Throughout the aer we fix a rime > 3 and et denote a squarefree integer with distinct rime factors 0, 1,..., r. The case is not excuded Eisenstein series and Hecke agebras. The Eisenstein series of weight two and eve have a basis {E ɛ 2, }, abeed by eements ɛ = (ɛ 0,..., ɛ r ) in the set E =

3 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL 3 {±1} r+1 \ {(1, 1,..., 1)}. The E2, ɛ are characterized in terms of Hecke eigenvaues by the roerties that (1) T n E2, ɛ = t E2, ɛ for a n with gcd(n, ) = 1, and 0<t n (2) w i E ɛ 2, = ɛ ie ɛ 2, for the Atkin-Lehner invoutions w 0,..., w r, together with the normaization a 1 (E2, ɛ ) = 1. The constant coefficients satisfy (1.3.1) a 0 (E2, ɛ ) = 1 r (ɛ i i + 1). 24 Based on the hiosohy that congruences between Eisenstein series and cus forms shoud haen when the constant term is divisibe by, we exect the most interesting congruences to occur when i ɛ i (mod ) for many i. Consider the Hecke agebra of weight 2 and eve generated by a T n with gcd(n, ) = 1 and by a Atkin-Lehner invoutions w 0,..., w r. Let T ɛ denote the cometion of this agebra at the maxima idea generated by together with the annihiator of E2, ɛ. Let I ɛ denote the annihiator of E2, ɛ in Tɛ, so Tɛ /Iɛ = Z, and et m ɛ = (I ɛ, ) be the maxima idea of T ɛ. For a Hecke modue M, et M m ɛ denote the tensor roduct of M with T ɛ over the Hecke agebra. In articuar, et M 2() m ɛ (res. S 2 () m ɛ) denote the resuting modue of moduar forms (res. cusida forms). Let T ɛ,0 denote the cusida quotient of Tɛ, and et Iɛ,0 be the image of I ɛ in T ɛ,0. i= Another Hecke agebra. In contrast with our aroach, one often studies a different Hecke agebra T ɛ,u, containing the oerators U instead of w, and with Eisenstein idea IU ɛ generated by T q (q + 1) for q and U i ɛ i +1 2 i for i = 0,..., r. We rove that T ɛ,u = Tɛ in some of the cases that we consider see Aendix A. Our main resuts together with Aendix A can be used to rove resuts about T ɛ,u that are cosey reated to the resuts of authors incuding Ribet [Rib10, Rib15], Yoo ([Yoo15, Yoo17b, Yoo17a] and others) and Hsu [Hsu18]. In genera, when T ɛ Tɛ U,, we beieve that Tɛ is more natura and better behaved, so we mosty consider T ɛ The number fieds K i. Let be a rime such that ±1 (mod ). Then there is a unique degree Gaois extension K /Q(ζ ) such that (1) Ga(Q(ζ )/Q) acts on Ga(K /Q(ζ )) via the character ω 1, (2) the rime (1 ζ ) of Q(ζ ) sits cometey in K, and (3) ony the rimes above ramify in K /Q(ζ ). For each i such that i ±1 (mod ), et K i = K i (see aso Definition ) Structure of the Hecke agebra. Our main resuts concern the structure of the Hecke agebra T ɛ. Theorem Assume that ɛ = ( 1, 1,..., 1). Let S = {i {1,..., r} i 1 and et s = #S. Then (1) T ɛ is a comete intersection ring. (2) T 0,ɛ is Gorenstein if and ony if Iɛ is rincia. (mod )}

4 4 PRESTO WAKE AD CARL WAG-ERICKSO (3) There is a short exact sequence r (1.4.2) 0 Z /( i + 1)Z I ɛ /I ɛ2 Z /( 0 1)Z 0. i=1 (4) The minima number of generators of I ɛ is s + δ where { 1 if 0 sits cometey in K δ = i for a i S, or 0 otherwise. Proof. Parts (1) and (3) are roved in 5 (see eseciay Theorem 5.2.6). It is known to exerts that Part (2) foows from (1) (see Lemma 2.4.2). Part (4) is Theorem Remark In fact, we show that, uness s = r, there are no newforms in M 2 () m ɛ, so we can easiy reduce to the case s = r (i.e. the case that i 1 (mod ) for a i > 0). When s = r, one coud use this theorem to rove that there are newforms in M 2 () m ɛ, but this is known (see [Rib15], [Yoo17b, Thm. 1.3(3)]). Remark The criterion of Part (4) determines whether or not the extension cass defined by the sequence (1.4.2) is -cotorsion. In fact, one can describe this extension cass exacty in terms of agebraic number theory, but we content ourseves with the simer statement (4). Theorem Assume r = 1 and ɛ = ( 1, 1) and that 0 1 (mod ) but 1 1 (mod ). If 1 is not a -th ower moduo 0, then there are no newforms in M 2 () m ɛ. In articuar, I ɛ is rincia, and generated by T q (q + 1) where q is a good rime (of Mazur) for ( 0, ). Proof. This is Theorem Remark In the case 1, this is a theorem of Ribet [Rib10] and Yoo [Yoo17b, Thm. 2.3]. Yoo has informed us that the method shoud work for the case 1 = as we. In any case, our method is cometey different. Theorem Assume r = 1 and ɛ = ( 1, 1) and that (mod ). Assume further that Then i is not a -th ower moduo j for (i, j) {(0, 1), (1, 0)}. (1) there are newforms in M 2 () m ɛ. (2) T ɛ is a comete intersection ring. (3) T ɛ,0 is not a Gorenstein ring. (4) I ɛ,0 /I ɛ,02 = Z /( 0 1)Z Z /( 1 1)Z. Proof. Parts (2) and (4) are roven in Theorem Part (1), the recise meaning of which is given in Definition 6.4.3, foows from Part (2) by Theorem Part (3) foows from (2) and (4) by Lemma Remark The roof of this theorem may be of articuar interest for exerts in the deformation theory of Gaois reresentations. The roof is the first (as far as we are aware) exame of an R = T theorem, where R is a universa seudodeformation ring, and where we do not rey on certain Gaois cohomoogy grous being cycic. (This cycicity ensures that the seudoreresentations come from true

5 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL 5 reresentations.) In fact, with the assumtions of the theorem, the reevant cohomoogy grous are not cycic. However, see [BK15, Thm. 8.2], where R = T is roved, where R is a certain quotient of a universa seudodeformation ring. Remark Outside of the cases considered in these theorems, we cannot exect that T ɛ is a comete intersection ring, as the exames in 1.9 beow iustrate. Our method, which aies Wies s numerica criterion [Wi95], roves that T ɛ is a comete intersection ring as a byroduct. A new idea is needed to roceed beyond these cases Aications to mutiicity one. For an aication of the main resut, we et J 0 () be the Jacobian of the moduar curve X 0 (). Coroary In the foowing cases, we can comute dim F J 0 ()(Q )[m ɛ ]: (1) With the assumtions of Theorem 1.4.1, we have dim F J 0 ()(Q )[m ɛ ] = 1 + s + δ, where s and δ are as in Theorem (2) With the assumtions of Theorem 1.4.5, we have dim F J 0 ()(Q )[m ɛ ] = 2. (3) With the assumtions of Theorem 1.4.7, we have dim F J 0 ()(Q )[m ɛ ] = 3. Proof. This foows from the named theorems together with Lemma (which is known to exerts). One says that mutiicity one hods if dim F J 0 ()(Q )[m ɛ ] = 2. This coroary imies that mutiicity one hods in case (1) if and ony if s + δ = 1, aways hods in case (2), and aways fais in case (3) Ribet s Conjecture. Previous works on mutiicity one have used a different Hecke agebra T ɛ,u, defined in (see, for exame, [Yoo15]). Let mɛ U = (IU ɛ, ) Tɛ,U be its maxima idea. The revious coroary together with Proosition A.2.3 give the foowing Coroary (Generaized Ribet s Conjecture). With the assumtions of Theorem 1.4.1, assume in addition that i 1 (mod ) for i > 0. Then where s and δ are as in Theorem dim F J 0 ()(Q )[m ɛ U ] = 1 + s + δ, The case s = r = 1 of this coroary was conjectured by Ribet [Rib15] (see aso [Yoo17a, g. 4]). Remark After we tod Yoo about the resuts of this aer, he found an aternate roof of this coroary in the case s = r = 1, under the assumtion that IU ɛ is rincia if and ony if T0,ɛ,U is Gorenstein (this assumtion foows from Theorem and Proosition A.2.3). Yoo s roof invoves a deicate study of the geometry of J 0 () and, unike our roof, does not make use of the fact that T ɛ is Gorenstein. The fact that our roof is simer demonstrates the ower of the Gorenstein roerty and is a reason for our interest in using T ɛ rather than Tɛ,U.

6 6 PRESTO WAKE AD CARL WAG-ERICKSO Gorensteinness, and mutiicity one for the generaized Jacobian. The foowing observations are not used (nor roven) in this aer (athough they are famiiar to exerts), but we incude them to iustrate the the arithmetic significance of the Gorenstein roerty for T ɛ roved in Theorems 1.4.1, and We earned this oint of view from aers of Ohta, eseciay [Oht05]. As is we-known, and as we exain in 2.4, mutiicity one hods if and ony if T 0,ɛ is Gorenstein. The nomencature mutiicity one comes from reresentation theory. It is reated to the question of whether Hét 1 (X 0() Q, Z (1)) m ɛ is a free T 0,ɛ attice in the free T 0,ɛ [ 1 ]-modue H1 ét (X 0() Q, Q (1)) m ɛ. There is another natura attice to consider, namey Hét 1 (Y 0() Q, Z (1)) m ɛ,dm, the image of Hét 1 (Y 0() Q, Z (1)) m ɛ under the Drinfed-Manin sitting H 1 ét(y 0 () Q, Q (1)) m ɛ H 1 ét(x 0 () Q, Q (1)) m ɛ. In a simiar manner to the roof of Lemma 2.4.1, one can show that T ɛ is Gorenstein if and ony if Hét 1 (Y 0() Q, Z (1)) m ɛ,dm is a free T 0,ɛ -modue, if and ony if dim F GJ 0 ()(Q )[m ɛ ] = 2, where GJ 0 () is the generaized Jacobian of J 0 () reative to the cuss (see e.g. [Oht99, 3] for a discussion of generaized Jacobians). Hence our resut that T ɛ is Gorenstein can be thought of as a mutiicity one resut for GJ 0 (). Finay, we note that these ideas iustrate why the faiure of mutiicity one in Coroary is reated to the faiure of I ɛ to be rincia: if T ɛ is Gorenstein, Hét(X 1 0 () Q, Z (1)) m ɛ Hét(Y 1 0 () Q, Z (1)) mɛ,dm has the form, as T 0,ɛ -modues, of T 0,ɛ I0,ɛ T 0,ɛ T0,ɛ. Hence Hét 1 (X 0() Q, Z (1)) m ɛ is free if and ony if I 0,ɛ is rincia Good rimes. We aso rove anaogues of Mazur s good rime criterion (statement (5) of 1.1). In the situation of Theorem 1.4.1, the ist of conditions is cumbersome to write down, so we are not recise here. We refer the reader to for some secific exames and 7.2 for the comete criterion. Theorem With the assumtions of Theorem 1.4.1, we can secify sufficient conditions on a set of rimes q 1,..., q s+δ not dividing such that the eements T q1 (q 1 + 1),..., T qs+δ (q s+δ + 1) together generate I ɛ. Remark We can aso write down a necessary and sufficient condition, but cannot comute with it, so we doubt its ractica use. In the situation of Theorem 1.4.7, the sufficient condition is very sime to state, and aso necessary. To state it, we et og : (Z/Z) F denote an arbitrary surjective homomorhism, for any rime that is congruent to 1 moduo (the statement beow wi not deend on the choice). -

7 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL 7 Theorem With the assumtions of Theorem 1.4.7, fix rimes q 0, q 1 not dividing (but ossiby dividing ). Then the eements T q0 (q 0 + 1) and T q1 (q 1 + 1) together generate I ɛ if and ony if ( ) og0 (q (q 0 1)(q 1 1) det 0 ) og 0 (q 1 ) F og 1 (q 0 ) og 1 (q 1 ). Remark For a singe rime, Mazur s criterion for q to be a good rime can be written as (q 1) og (q) F, so this is a natura generaization Reation to Hida Hecke agebras. The reader wi note that we have aowed for the ossibiity that. When, in Aendix A, we aso consider a reated Hecke agebra T ɛ,h that contains U instead of w (but sti has a other w for ) and show that, in many cases we consider, Tɛ,H = Tɛ. This is reated to Hida theory because (as is we-known for the Hecke agebra T ɛ,u ) there is a Hida-theoretic Hecke agebra Tɛ Λ that is a free modue of finite rank over Λ Z [T ] that satisfies a contro theorem with resect to T ɛ,h : there is an eement ω 2 Λ such that T ɛ,h = Tɛ Λ /ω 2T ɛ Λ. (A roof of this contro theorem wi aear in forthcoming work of the first-named author with Rob Poack.) Then our resuts about T ɛ (incuding its Gorensteinness and the number of generators of its Eisenstein idea) transate directy to T ɛ Λ. Subsequenty, these resuts can be seciaized into higher weights, as is usua in Hida theory Method of seudodeformation theory. Like our revious work [WWE17c], the method of roof of the theorems in 1.4 is to construct a seudodeformation ring R and rove that R = T using the numerica criterion. The ring R is the deformation ring of the residua seudoreresentation D = ψ(ω 1) associated to E2, ɛ that is universa subject to certain conditions (here ψ is the functor associating a seudoreresentation to a reresentation, and ω is the mod cycotomic character). These conditions incude incude the conditions considered in our revious works [WWE15, WWE17c] (having cycotomic determinant, being fat at, being ordinary at ), but they aso incude new conditions at dividing that are of a different favor, as we now exain The Steinberg at condition. Fix = i, assume, and et G G Q be a decomosition grou at. Let f be a normaized cusida eigenform of weight 2 and eve Γ 0 () and ρ f : G Q GL 2 (O f ) is the associated Gaois reresentation, where O f is a finite extension of Z. If f is od at, then ρ f G is unramified. If f is new at, we have ( ) λ(a (f))κ (1.8.1) ρ f G cyc 0 λ(a (f)) where λ(x) is the unramified character of G sending a Frobenius eement σ to x, and a (f) is the coefficient of q in the q-exansion of f (see Lemma 2.3.1). ote that since det(ρ f ) = κ cyc, we have λ(a (f)) 2 = 1. In fact, a (f) is the negative of the w -eigenvaue of f. We ca such reresentations (1.8.1) ±1-Steinberg at, where ±1 = a (f) is the w -eigenvaue of f. ow assume in addition that f S 2 () m ɛ, so that the semi-simification of the residua reresentation of ρ f is ω 1 and w f = ɛf, where ɛ = ɛ i. We want to imose a condition on seudoreresentations that encasuates the condition that

8 8 PRESTO WAKE AD CARL WAG-ERICKSO ρ f G is either unramified or ɛ-steinberg. The main observation is the foowing, and is insired by the work of Caegari-Secter [CS16]. Observation Suose that ρ : G GL 2 (O) is either unramified or ɛ- Steinberg. Then (1.8.3) (ρ(σ) λ( ɛ)κ cyc (σ))(ρ(τ) λ( ɛ)(τ)) = 0 for a σ, τ G with at east one of σ or τ in the inertia grou I. This is cear if ρ is unramified: the factor invoving the one of σ or τ that is in I wi be zero. If ρ is ɛ-steinberg, then the given roduct (1.8.3) wi have the form ( ) ( ) and any such roduct is zero (note that the order is imortant!). To imose the unramified-or-ɛ-steinberg condition on the seudodeformation ring R, we imose the condition (1.8.3) on the universa Cayey-Hamiton agebra, using the theory of [WWE17a] (see 3) The ordinary at condition. When and f S 2 () m ɛ is a newform, then ɛ = 1 and the reresentation ρ f G is ordinary. In this aer, we define ordinary seudoreresentation exacty as we define the unramified-or-ɛ-steinberg, foowing ideas of Caegari-Secter. In our revious aer [WWE15], we gave a different definition of ordinary, and we rove in this aer that the two definitions coincide (see Lemma 3.7.4). This gives an answer to a question of Caegari-Secter [CS16, g. 2] Exames. We concude this introduction with exames that iustrate the theorems and show that the hyotheses are necessary. For exames where we show that T ɛ is not Gorenstein, it is hefu to note that Tɛ is Gorenstein if and ony if Soc(T ɛ /Tɛ ) is 1-dimensiona, where Soc(Tɛ /Tɛ ) is the annihiator of the maxima idea (see C.1). A comutations are using agorithms we have written for the Sage comuter agebra software [S + 18] Exames iustrating Theorem Exame Let = 5, 0 = 41, 1 = 19, so = 19 41, and et ɛ = ( 1, 1). In this case, we comute that K 19 is the fied cut out by x 20 x 19 7x x x x x x x x 11 91x x 9 439x x x 6 58x x x x x + 16 and that 41 sits cometey in K 19. The theorem says that I ɛ has 2 generators. Moreover, Theorem says, in this case, that I ɛ is generated by T q0 (q 0 + 1) and T q1 (q 1 + 1) where q 0 is a good rime for (41, 5) and where q 1 satisfies (a) q 1 is a rime such that q 1 1 (mod 5), (b) 41 is not a 5-th ower moduo q 1, and (c) q 1 does not sit cometey in K 19.

9 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL 9 A quick search yieds that q 0 = 2 and q 1 = 11 satisfy these criteria. And indeed, we comute that there is an isomorhism F 5 [x, y] (y 2 2x 2 T ɛ, xy) /5T ɛ, (x, y) (T 2 3, T 11 12). Exame Let = 5, 0 = 11, 1 = 19, 2 = 29, so = , and et ɛ = ( 1, 1, 1). In this case, 11 does not sit cometey in either of the fieds K 19, K 29, and the theorem says that I ɛ has 2 generators. Moreover, Theorem says, in this case, that I ɛ is generated by T q0 (q 0 + 1) and T q1 (q 1 + 1) where q 0 is a good rime for (11, 5) (for exame q 0 = 2) and where the rime q 1 satisfies: (a) q 1 1 (mod 5), (b) 11 is not a 5-th ower moduo q 1, (c) q 1 does not sit cometey in K 19, and (d) q 1 does sit cometey in K 29. In this case, K 19 is the fied comuted in the revious exame and K 29 is the fied cut out by x 20 x 19 11x x x x x x x x x x x x x x x x x 2 832x A quick search finds that q 1 = 181 satisfies the conditions (a)-(d). And indeed, we comute that there is an isomorhism F 5 [x, y] (x 3 + 2x 2, y 3, xy + y 2 T ɛ ) /5T ɛ, (x, y) (T 2 3, T ). ote that these conditions are far from necessary. For exame T 2 3 and T 7 8 aso generate the Eisenstein idea Exames reated to Theorem We give exames iustrating that the assumtion is necessary. In fact, it seems that the assumtion is necessary even for the Gorensteinness of T ɛ. Exame Let = 5, 0 = 11, 1 = 23, so = 11 23, and et ɛ = ( 1, 1). Then 1 1 (mod 11) is a 5-th ower so the theorem does not ay. We can comute that F 5 [x, y] (x 2, xy, y 2 T ɛ ) /5T ɛ, (x, y) (T 2 3, T 3 4) has dimension 3. Since T 0 11 = Z 5, we see that the sace of odforms has dimension 2, so there must be a newform at eve. Moreover, Soc(T ɛ /5Tɛ ) = xf 5 yf 5, so T ɛ is not Gorenstein. Exame Let = 5, 0 = 31, 1 = 5, so = 5 31, and et ɛ = ( 1, 1). Then note that 1 = (mod 31), so the theorem does not ay. We can comute that F 5 [x, y] (x 3, xy, y 2 T ɛ ) /5T ɛ, (x, y) (T 2 3, 2T 2 + T 3 ) has dimension 4. Since rank Z5 (T 0 31) = 2, we see that the sace of odforms has dimension 3, and there must be a newform at eve. Moreover, Soc(T ɛ /5Tɛ ) = x 2 F 5 yf 5, so T ɛ is not Gorenstein.

10 10 PRESTO WAKE AD CARL WAG-ERICKSO In this ast exame, the reader may think that 0 = 31 is secia because the rank of T 0 31 is 2. However, we can take = 1 = 5 and 0 = 191 (note that T = Z ). oting that (mod 191), we again see that the theorem does not ay, and we can comute that T ɛ is aso not Gorenstein in this case Exames reated to Theorem First, we give exames iustrating that the assumtion is necessary. Again, it seems that the assumtion is necessary even for the Gorenstein roerty of T ɛ. Exame Let = 5, 0 = 11, 1 = 61, so = 11 61, and et ɛ = ( 1, 1). Then note that (mod 61) so the theorem does not ay (but note that 61 is not a 5-th ower moduo 11). We can comute that F 5 [x, y] (x 2, xy, y 3 T ɛ ) /5T ɛ, (x, y) (T 3 T 2 1, T 2 3). We see that Soc(T ɛ /5Tɛ ) = xf 5 y 2 F 5, so T ɛ is not Gorenstein. Exame Let = 5, 0 = 31, 1 = 191, so = , and et ɛ = ( 1, 1). We have (mod 31) and (mod 191), so the assumtion of the theorem fais most sectacuary. We can comute that F 5 [x, y] ((x, y) 4, 2x 3 + xy 2 + 3y 3, x 3 x 2 y + 2y 3 T ɛ /5T ɛ, ) (x, y) (T 2 3, T 7 8). Letting m ɛ denote the maxima idea of T ɛ /5Tɛ, we see that ( mɛ ) 4 = 0 but that ( m ɛ ) 3 is 2-dimensiona, so dim F5 Soc(T ɛ /5Tɛ ) > 1 and Tɛ is not Gorenstein. Finay, we give an exame iustrating Theorem Exame Let = 5, 0 = 11, 1 = 41, so = 11 41, and et ɛ = ( 1, 1). We see that neither of 11 or 41 is a 5-th ower moduo the other, so Theorem aies. We consider the rimes 2, 3, 7 and 13, none of which are congruent to 1 moduo 5. q Is 5-th ower moduo 11? Is 5-th ower moduo 41? 2 o o 3 o Yes 7 o o 13 o o From this we see that ( ) og11 (3) og det 11 (q) = og og 41 (3) og 41 (q) 11 (3) og 41 (q) 0. for any q {2, 7, 13}. By Theorem 1.6.3, {T 3 4, T q (q + 1)} generates I ɛ for any q {2, 7, 13}, and we can see by direct comutation that this is true. More subty, we can comute that det ( og11 (2) og 11 (7) og 41 (2) og 41 (7) ) 0, ( og11 (2) og det 11 (13) og 41 (2) og 41 (13) ) = 0. By Theorem 1.6.3, this imies that {T 2 3, T 7 8} generates I ɛ, but that {T 2 3, T 13 14} does not, and we again verify this by direct comutation.

11 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL Acknowedgements. We thank Akshay Venkatesh for interesting questions that stimuated this work, and Ken Ribet for his insiring tak [Rib15]. We thank Hwajong Yoo for bringing our attention to his work on the subject, and Frank Caegari for carifying the rovenance of Ribet s Conjecture. We thank Shekhar Khare for hefu discussions about the Steinberg condition, Matt Emerton for encouragement and for asking us about imications for newforms, and Rob Poack for asking us about the case. We thank Joë Beaïche, Tobias Berger, Frank Caegari, Kȩstutis Česnavičius, Emmanue Lecouturier, Barry Mazur, Rob Poack, Ken Ribet, and Hwajong Yoo and for comments on and corrections to an eary draft. P.W. was suorted by the ationa Science Foundation under the Mathematica Sciences Postdoctora Research Feowshi o C.W.E. was suorted by Engineering and Physica Sciences Research Counci grant EP/L025485/ otation and Conventions. We et ij denote the Kronecker symbo, which is 1 if i = j and 0 otherwise. For each rime, we fix G G Q, a decomosition grou at, and et I G denote the inertia subgrou. We fix eements σ G whose image in G /I = Ga(F /F ) is the Frobenius. For, we fix eements γ I such that the image in the maxima ro--quotient I ro (which is we-known to be rocycic) is a tooogica generator. Let γ I be an eement such that the image of γ in Ga(Q nr ( )/Q nr ) is non-trivia and ω(γ ) = 1. When = i for i {0,..., r} (i.e. ), we aso write σ i := σ i and γ i := γ i for these eements. We write G Q,S for the Gaois grou of the maxima extension of Q unramified outside of the set aces S of Q suorting, and use the induced mas G G Q,S. For rimes q, we write Fr q G Q,S for a Frobenius eement at q. As in the theory of reresentations, Cayey-Hamiton reresentations, actions on modues, seudoreresentations, and cochains/cocyces/cohomoogy of rofinite grous G discussed in [WWE17a], these objects and categories are imicity meant to be continuous without further comment. Here a of the targets are finitey generated A-modues for some oetherian oca (continuous) Z -agebra A with idea of definition I, and the I-adic tooogy is used on the target. Profinite grous used in the seque satisfy the Φ -finiteness condition (i.e. the maxima ro- quotient of every finite-index subgrou is tooogicay finitey generated), which aows the theory of [WWE17a] to be aied. We write H i (Z[1/], M) = H i (C (Z[1/], M)) = Zi (Z[1/], M) B i (Z[1/], M) for (continuous) cohomoogy of a G Q,S -modue M, together with this notation for cochains, cocyces, and coboundaries. We write x 1 x 2 C (Z[1/], M 1 M 2 ) for the cu roduct of x i C (Z[1/], M i ), and a 1 a 2 H (Z[1/], M 1 M 2 ) for cu roduct of cohomoogy casses a i H (Z[1/], M i ). 2. Moduar forms In this section, we reca some resuts about moduar curves and moduar forms. Our reference is the aer of Ohta [Oht14] Moduar curves, moduar forms, and Hecke agebras. The statements given here are a we-known. We review them here to fix notation.

12 12 PRESTO WAKE AD CARL WAG-ERICKSO Moduar curves. Let Y 0 () /Z be the Z -scheme reresenting the functor taking a Z -scheme S to the set of airs (E, C), where E is an eitic curve over S and C E[] is a finite-fat subgrou scheme of rank and cycic (in the sense of Katz-Mazur [KM85]). Let X 0 () /Z be the usua comactification of Y 0 () /Z, and et {cuss} denote the comement of Y 0 () /Z in X 0 () /Z, considered as an effective Cartier divisor on X 0 () /Z. Finay, et X 0 () = X 0 () /Z Q Moduar forms and Hecke agebras. The ma X 0 () /Z Sec(Z ) is known to be LCI, and we et Ω be the sheaf of reguar differentias. Let S 2 (; Z ) = H 0 (X 0 () /Z, Ω), M 2 (; Z ) = H 0 (X 0 () /Z, Ω({cuss})) Let T and T 0 be the subagebras of End Z (M 2 (; Z )), End Z (S 2 (; Z )), resectivey, generated by the standard Hecke oerators T n with (, n) = 1, and a Atkin-Lehner oerators w for (we do not incude any U for ). These are semi-sime commutative Z -agebras (see, e.g. [AL70]) Eisenstein series and Eisenstein arts. For each ɛ {±1} r+1 \{(1, 1,..., 1)}, there is a eement E2, ɛ M 2(; Z ) that is an eigenform for a T n with (, n) = 1, and has q-exansion (2.1.1) E ɛ 2, = 1 24 r (ɛ i i + 1) + a n q n i=0 where a n = 0<d n t when gcd(n, ) = 1 (in articuar, a 1 = 1), and w i E2, ɛ = ɛ i E2, ɛ (see [Oht14, Lem ]). Let I ɛ = Ann T (E2, ɛ ), and et Tɛ be the cometion of T at the maxima idea (I ɛ, ), and et T 0,ɛ = T 0 T Tɛ. Let Iɛ = I ɛ T ɛ and et I0,ɛ be the image of I ɛ in T 0,ɛ. For a T -modue M, et M Eis ɛ = M T Tɛ. The ma Tɛ Z induced by E2, ɛ is a surjective ring homomorhism with kerne Iɛ. We refer to this as the augmentation ma for T ɛ. ote that we have w i = ɛ i as eements of T ɛ. Indeed, this foows from w2 i = 1, w i ɛ i I ɛ, and 2: consider (w i ɛ i )(w i + ɛ i ) = 0 and observe that w i + ɛ i (T ɛ ). Consequenty, T ɛ is generated as a Z -agebra by T q for q. If, et U T ɛ denote the unit root of the oynomia X 2 T X + = 0, which exists and is unique by Hense s emma. Since T ( + 1) I ɛ, we see that U 1 I ɛ. Moreover, we see that T = U + U Duaity. As in [Oht14, Thm ], there are erfect airings of free Z - modues n=1 (2.1.2) M 2 (; Z ) ɛ Eis T ɛ Z, S 2 (; Z ) ɛ Eis T 0,ɛ Z given by (f, t) a 1 (t f), where a 1 ( ) refers to the coefficient of q in the q- exansion. In articuar, M 2 (; Z ) ɛ Eis (res. S 2(; Z ) ɛ Eis ) is a duaizing Tɛ - modue (res. T 0,ɛ -modue).

13 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL Odforms and stabiizations. If is a rime and f S 2 (/; Z ) is an eigenform for a T n with (n, /) = 1, then the subsace {g S 2 (; Z ) : a n (g) = a n (f) for a (n, /) = 1} has rank two, with basis f(z), f(z). If we et f ± (z) = f(z)±f(z), then w f ± (z) = ±f ± (z). ote that, since 2, we have f + f (mod ). In articuar, if ɛ {±1} r is the tue obtained from ɛ by deeting the entry corresonding to, then there are injective homomorhisms given by f f ɛ, M 2 (/; Z ) ɛ Eis M 2 (; Z ) ɛ Eis and S 2 (/; Z ) ɛ Eis S 2 (; Z ) ɛ Eis Congruence number. We reca this theorem of Ohta, and reated resuts. Theorem (Ohta). There is an isomorhism T ɛ,0 /Iɛ,0 = Z /a 0 (E ɛ 2, )Z. This is [Oht14, Thm ]. His method of roof actuay can be used to give the foowing stronger resut, exacty as in [WWE17c, Lem ]. See Lemma C.2.1 for a discussion of fiber roducts of rings. Lemma The comosition of the augmentation ma T ɛ Z with the quotient ma Z Z /a 0 (E2, ɛ )Z factors through T 0,ɛ and induces an isomorhism T ɛ T 0,ɛ Z /a 0(E2, ɛ Z. )Z In articuar, ker(t ɛ T0,ɛ ) = Ann T ɛ (Iɛ ) Eigenforms and associated Gaois reresentations. Let ν : T 0,ɛ denote the normaization of T 0,ɛ. T 0,ɛ Lemma We record facts about (1) Letting q vary over rimes q, there is an isomorhism h : T 0,ɛ T 0,ɛ and associated Gaois reresentations. f Σ O f, ν(t q ) (a q (f)) f Σ, where Σ S 2 (; Q ) ɛ Eis is the set of normaized eigenforms, and O f is the vauation ring of the finite extension Q (a q (f) q )/Q. (2) For each f Σ, there is an absoutey irreducibe reresentation ρ f : G Q,S GL 2 (O f [1/]) such that the characteristic oynomia of ρ f (Fr q ) is X 2 a q (f)x+ q for any q. (3) Assume i. The reresentation ρ f Gi is unramified if f is od at i. Otherwise, f is new at i and there is an isomorhism ( ) λ(ai (f))κ (2.3.2) ρ f Gi cyc, 0 λ(a i (f)) where a i (f) = ɛ i. (4) There is an isomorhism ( λ(a (f) (2.3.3) ρ f G 1 )κ cyc 0 λ(a (f)) ). Moreover, (a) ρ f G is finite-fat if and ony if either (i), in which case h : ν(u ) (a (f)) f Σ, or (ii) and f is od at.

14 14 PRESTO WAKE AD CARL WAG-ERICKSO (b) If and f is new at, then a (f) = ɛ = +1, i.e. ɛ = 1. Proof. For (1)-(3) and (4a) see, for exame, [DDT94, Thm. 3.1]. In (4b), the fact that a (f) = ɛ is [AL70, Thm. 3]. To see that ɛ = 1, note that the semi-sime residua reresentation ρ ss f is ω 1, but (2.3.3) imies ρss f G = λ( ɛ )ω λ( ɛ ). Since ω G is ramified, this imies that λ( ɛ ) = 1, so ɛ = 1. Combining Lemmas and 2.3.1, we obtain an injective homomorhism (2.3.4) T ɛ Z T 0,ɛ Z f Σ O f determined by sending T q to (q + 1, a q (f) f Σ ) for q and, if, sending U to (1, a (f) f Σ ) The kerne of m ɛ on the moduar Jacobian and the Gorenstein condition. In this section, we use some resuts of Ohta (foowing ideas of Mazur) to reate the structure of the rings T ɛ and T0,ɛ to the geometry of the éron mode J 0 () /Z of the Jacobian of X 0 (). Let J 0 () = J 0 () /Z Q. For a Z -modue M, et Ta (M) = Hom(Q /Z, M) be the Tate modue of M, et M = Hom Z (M, Q /Z ) be the Pontrjagin dua, and et M = Hom Z (M, Z ) be the Z -dua. If M is a free Z -modue, there is an identification M = Ta (M). Let T = Hét 1 (X 0() Q, Z (1)) = Ta (J 0 ()(Q )). Lemma There is an exact sequence of T 0,ɛ [I ]-modues 0 T 0,ɛ (1) T m ɛ (T0,ɛ ) 0. The sequence sits as T 0,ɛ -modues. In articuar, we have dim F J 0 ()[m ɛ ](Q ) = dim F (T /m ɛ T ) = 2 + δ(t 0,ɛ ) where δ(t 0,ɛ ) is the Gorenstein defect of T0,ɛ. (See C.1 for a discussion of Gorenstein defect.) Proof. Ohta has shown in [Oht14, Pro ] that dim F J 0 () /Z (F )[m ɛ ] 1. This imies the resut, foowing [Maz77, II.7-II.8] (see aso [Maz97]). Lemma Suose that T ɛ is Gorenstein. Then there is an isomorhism of T ɛ -modues I ɛ (T 0,ɛ ). In articuar, the minima number of generators of I ɛ is δ(t 0,ɛ ) + 1, and Iɛ is rincia if and ony if T 0,ɛ is Gorenstein. Proof. Like the roof of [Oht14, Lem ], there is an exact sequence of T ɛ - modues 0 S 2 (; Z ) ɛ Eis M 2 (; Z ) ɛ Res Eis Z 0 where T ɛ acts on Z via the augmentation ma T ɛ Tɛ /Iɛ = Z. Since we assume that T ɛ is Gorenstein, we see by the duaity (2.1.2) that M 2(; Z ) ɛ Eis is a free T ɛ -modue of rank 1. We may choose a generator f of M 2(; Z ) ɛ Eis such that Res(f) = 1. Then we obtain a surjective T ɛ -modue homomorhism T ɛ Z, T Res(T f)

15 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL 15 whose kerne is isomorhic to S 2 (; Z ) ɛ Eis. Because this ma sends 1 to 1, it is a ring homomorhism, and it must be the augmentation ma T ɛ Z. Thus I ɛ = S 2 (; Z ) ɛ Eis, so duaity (2.1.2) yieds the isomorhism of the emma. The remaining arts foow from C.1. Combining the receding two emmas, we obtain the foowing Lemma Assume that T ɛ is Gorenstein. Then dim F J 0 ()[m ɛ ](Q ) = 1 + dim F (I ɛ /m ɛ I ɛ ). 3. The seudodeformation ring In this section, we set u the deformation theory of Gaois seudoreresentations modeing those that arise from Hecke eigenforms of weight 2 and eve that are congruent to the Eisenstein series E2, ɛ. These are the Gaois reresentations of Lemma See 1.8 for further introduction Theory of Cayey-Hamiton reresentations. This section is a summary of [WWE17a]. Ony for this section, we work with a genera rofinite grou G satisfying condition Φ (of 1.11). A seudoreresentations are assumed to have dimension 2, for simicity Pseudoreresentations. A seudoreresentation D : E A is the data of an associative A-agebra E aong with a homogeneous mutiicative oynomia aw D from E to A. This definition is due to Chenevier [Che14]; see [WWE17a] and the references therein. Desite the notation, the seudoreresentation D incudes the data of a mutiicative function D : E A, but is not characterized by this function aone. It is characterized by the air of functions Tr D, D : E A, where Tr D is defined by the characteristic oynomia: (3.1.1) D(x t) = t 2 Tr D (x)t + D(x) A[t]. A seudoreresentation D : E A is said to be Cayey-Hamiton if, for every commutative A-agebra B, every eement x E A B satisfies its characteristic oynomia. We aso denote by D : G A a seudoreresentation D : A[G] A Cayey-Hamiton reresentations. In the category of Cayey-Hamiton reresentations of a rofinite grou G, an object is a trie (ρ : G E, E, D : E A), and sometimes referred to more briefy as ρ. Here ρ is a homomorhism (continuous, as aways), E is an associative A-agebra that is finitey generated as an A-modue, (A, m A ) is a oetherian oca Z -agebra, and D is a Cayey-Hamiton seudoreresentation. We ca A the scaar ring of E. The induced seudoreresentation of ρ is D ρ : G A, aso denoted ψ(ρ). The functor ψ is essentiay surjective. The Cayey-Hamiton reresentation ρ is said to be over ψ(ρ) A A/m A, and ψ(ρ) is said to be a seudodeformation of ψ(ρ) A A/m A. Given a seudoreresentation D : G F for a fied F, there is a universa object in the category of Cayey-Hamiton reresentations over D. This is denoted by (ρ ū D : G E D, E ū D, D E ū D : Eū D Rū D ), and the induced seudoreresentation D ū D := ψ(ρū D ) is the universa seudodeformation of D.

16 16 PRESTO WAKE AD CARL WAG-ERICKSO Generaized matrix agebras (GMA). An imortant exame of a Cayey- Hamiton agebra is a generaized matrix agebra (GMA). An A-GMA E is given by the data (B, C, m) where B and C are finitey-generated R-modues, m : B R C R is an R-modue homomorhism satisfying certain conditions, and E = ( ) R B C R (see [WWE17a, Exame 3.1.7]). There is a Cayey-Hamiton seudoreresentation D : E A given by the usua formua for characteristic oynomia. We write a homomorhism ρ : G E as ρ = ( ρ 1,1 ρ 1,2 ) ρ 2,1 ρ 2,2. If D is mutiicity-free (see [WWE17a, Defn ]), then E ū has a GMA D structure whose associated seudoreresentation is D E ū [WWE17a, Thm ]. D Reducibiity. We wi refer to the condition that a Cayey-Hamiton reresentation or a seudoreresentation is reducibe. We aso refer to the reducibiity idea in rings receiving a seudoreresentations. For these definitions, see [WWE17a, 4.2] or [WWE15, 5.7]. The imortant case for this aer is that, if (ρ, E, D : E A) is a Cayey-Hamiton reresentation where E is the GMA associated to (B, C, m), then the reducibiity idea of D is the image of m. There are aso universa objects, denoted ρ red, etc Conditions on Cayey-Hamiton reresentations. We consider two favors of conditions P imosed on Cayey-Hamiton reresentations of G: (1) P is a condition that certain eements vanish, e.g. Definition (2) P is a roerty aying to finite-ength Z [G]-modues and satisfying a basic stabiity condition, e.g In case (1), one roduces a universa Cayey-Hamiton ρ P D reresentation of G satisfying P by taking the quotient by the two-sided idea of E D generated by the reevant eements, and then taking a further quotient so that a seudoreresentation exists. This fina quotient is known as the Cayey-Hamiton quotient of ρ ū for P. D See [WWE17a, Defn ] for detais; cf. aso [WWE15, Defn ]. In case (2), we consider E ū as a G-modue using its eft action on itsef by D mutiication, and find in [WWE17a, 2.4] that the maxima eft quotient modue satisfying P can be defined and is an agebra quotient. The subsequent Cayey- Hamiton quotient is then shown to satisfy the desired roerties of ρ P D Conditions on seudoreresentations. As discussed in [WWE17a, 2.5], one says that a seudoreresentation D of G satisfies P if there exists a Cayey-Hamiton reresentation ρ of G such that ψ(ρ) = D and ρ satisfies P. Then the universa seudodeformation of D with roerty P turns out to be ψ(ρ P D) Universa Cayey-Hamiton reresentations of Gaois grous. Let be a rime. Reca from 1.11 the decomosition grou G G Q,S. Let D : G Q,S F denote the seudoreresentation ψ(f (1) F ). We denote by (ρ D : G Q,S E D, E D, D E D : E D R D) the universa Cayey-Hamiton reresentation of G Q,S over D. The scaar ring R D is the universa seudodeformation ring of D, with universa seudoreresentation D D := ψ(ρ D). Simiary, we et the trie (ρ : G E, E, D E : E R )

17 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL 17 denote the universa Cayey-Hamiton reresentation of G over D G, so that D := ψ(ρ ) : G R is the universa seudodeformation of D G. Definition ote that D is mutiicity-free, and that, if 1 (mod ), then D G is mutiicity-free. In this case, E and E D have the structure of a GMA. In this aer, whenever we fix such a structure, we assume that (ρ ) 1,1 R F = ω G (res. (ρ D) 1,1 R D F = ω) Case : unramified. For, we want Gaois reresentations to be unramified at. We imose this by considering reresentations of G Q,S, as oosed to Ga(Q/Q) Case and : the unramified-or-steinberg condition. In this subsection, we write for one of the factors of referred to esewhere in this manuscrit as i. Likewise, we write ɛ for ɛ i. Definition Let (ρ : G E, E, D E : E A) be a Cayey-Hamiton reresentation of G over D G. We ca ρ unramified-or-ɛ -Steinberg (or US ɛ ) if (3.4.2) V ɛ ρ (σ, τ) := (ρ(σ) λ( ɛ )(σ)κ cyc (σ))(ρ(τ) λ( ɛ )(τ)) E is equa to 0 for a (σ, τ) ranging over the set I G G I G G. Write V ɛ ρ for the set of a eements V ɛ ρ (σ, τ) over this range. A seudodeformation D : G A of D G is caed US ɛ if there exists a US ɛ Cayey-Hamiton reresentation ρ of G such that ψ(ρ) = D. Definition Let (E ɛ, D E ɛ of (E, D ) by V ɛ ρ. Let (ρ ɛ : E ɛ R ɛ ) be the Cayey-Hamiton quotient : G (E ɛ ), E ɛ, D E ɛ : E ɛ R ɛ ), be the corresonding Cayey-Hamiton reresentation, with induced seudoreresentation of G denoted D ɛ := ψ(ρ ɛ ) : G R ɛ. By the theory of 3.1.5, ρ ɛ over D G, and D ɛ is the universa USɛ Cayey-Hamiton reresentation is the universa US ɛ seudodeformation of D G. Lemma If, then, for any ɛ, we have D ɛ (τ) = 1 and Tr D ɛ (τ) = 2 for a τ I. That is, (D ɛ ) I = ψ(1 1). Proof. Let τ I. We see in (3.4.2) that V ɛ ρ ɛ (τ, τ) = (ρ ɛ (τ) 1)2 = 0. Thus by [Che14, Lem. 2.7(iv)], we see Tr ɛ D (τ 1) = D ɛ (τ 1) = 0. As traces are additive, we have Tr ɛ D (τ) = Tr ɛ D (1) = 2. Aying (3.1.1) with x = τ and t = 1, we find that D ɛ (τ) = 1. Lemma Suose that ɛ = +1 and 1, 0 (mod ). Then ρ ɛ is unramified (i.e. ρ ɛ I = 1). Proof. Let σ G be the eement σ defined in By definition of E ɛ, V ɛ ρ ɛ (τ, σ) = (ρ ɛ (τ) 1)(ρɛ (σ) + 1) = 0, for any τ I. To rove the emma, it suffices to show that (ρ ɛ (σ) + 1) (Eɛ ).

18 18 PRESTO WAKE AD CARL WAG-ERICKSO By the Cayey-Hamiton roerty, we know that any eement x E ɛ satisfies x 2 Tr ɛ D (x)x + D ɛ (x) = 0. In articuar, we see that x (Eɛ ) if and ony if D ɛ (x) (Rɛ ). Hence it wi suffice to show that D ɛ (σ + 1) (Rɛ ). Writing m R ɛ for the maxima idea, we know that D ɛ D (mod m), so it wi suffice to show that D(σ + 1) F. Because and D = ψ(ω 1), we ay (3.1.1) with x = σ and t = 1, cacuating that D(σ + 1) = 2( + 1) F. This is a unit because is odd and 1 (mod ) The finite-fat case: = and. A finite-ength Z [G ]-modue V is said to be finite-fat when it arises as G(Q ), where G is a finite fat grou scheme over Z. In [WWE17a, 5.2] we check that the theory of can be aied to the finite-fat condition. This theory gives us (ρ fat : G (E fat ), E fat, D E fat : E fat R fat ), the universa finite-fat Cayey-Hamiton reresentation of G over D G. The seudoreresentation D fat := ψ(ρ fat ) : G R fat is the universa finite-fat seudodeformation of D G. Consider a GMA structure on E fat as in Definition 3.2.1, which we write as ( ) ( ) ρ ρ fat fat,1,1 ρ = fat,1,2 R fat B : G fat. ρ fat,2,1 ρ fat,2,2 C fat R fat Lemma For any such GMA structure on E, C fat = 0. Proof. The roof is imicit in [WWE17c] but not stated in this form there. One simy combines the foowing facts. See [WWE17c, B.4] for the notation. As the maxima idea of R fat contains the reducibiity idea, we have Hom R fat(c fat, F ) = Ext 1 ffgs/z (µ, Z/Z), where ffgs/z is the category of finite fat grous schemes over Z, by [WWE17a, Thm ]. We see in [WWE17c, Lem (1)] that Ext 1 ffgs/z (µ, Z/Z) = 0. As C fat is a finitey-generated R fat -modue, this imies that C fat = 0. ow that we know that C fat = 0, ρ fat,i,i are Rfat -vaued characters of G, for i = 1, 2. Simiary to [WWE17c, 5.1], using the fact that ω G 1, we see the foowing Lemma A seudodeformation D of D G is finite-fat if and ony if D = ψ(κ cyc χ 1 χ 2 ) where χ 1, χ 2 are unramified deformations of the trivia character The finite-fat case: =,, and ɛ = +1. By Lemma 2.3.1(4), we see that, if ɛ = +1, then the residuay Eisenstein cus forms are od at with associated G Q,S -reresentation being finite-fat at. We imose this condition exacty as in 3.5. amey, we say that a Cayey-Hamiton reresentation of G is unramified-or-(+1)-steinberg (or US +1 ) if it is finite-fat The ordinary case: =,, and ɛ = 1. Based on the form of Gaois reresentations arising from -ordinary eigenforms given in Lemma 2.3.1(4), we roceed exacty as in the case given in 3.4. Definition We say that a Cayey-Hamiton reresentation or a seudodeformation over D G is ordinary (or US 1 ) when it satisfies Definition 3.4.1, simy etting =.

19 THE EISESTEI IDEAL WITH SQUAREFREE LEVEL 19, D E ord) be the Cayey-Hamiton quotient, D E ord : E ord R ord ) be the corresonding Cayey-Hamiton reresentation. As er 3.1.5, ρ ord is the universa ordinary Cayey-Hamiton reresentation over D G, and D ord := ψ(ρ ord ) : G R ord is the universa ordinary seudodeformation of D G. Simiary to Definition 3.4.3, et (E ord, and et (ρ ord, E ord of (E, D E ) by V 1 ρ Remark If one aies V ρ +1 = 0 in the case ɛ = +1, one does not get the the desired finite-fat condition of 3.6 that agrees with Lemma 2.3.1(4b). Instead, one finds that E +1 = 0 (i.e. no deformations of D satisfy this condition). We set u the foowing notation, which incudes a cases: ɛ = ±1 or. Definition For any and ɛ, we estabish notation { (ρ ord (ρ ɛ, E ɛ, D E ɛ, R ɛ, D ɛ, E ord, D E ord, R ord, D ord ) if, ɛ = 1, ) := (ρ fat, E fat, D E fat, R fat, D fat ) otherwise. In [WWE15, 5], we deveoed an aternative definition of ordinary Cayey- Hamiton agebra. (This definition aies to genera weight, which we seciaize to weight 2 here.) Choose a GMA structure on E, as in Definition Let J ord E be the two-sided idea generated by the subset As in [WWE15, Lem ], J ord ρ,2,1 (G ) (ρ,1,1 κ cyc )(I ) (ρ,2,2 1)(I ). is indeendent of the choice of GMA-structure. Lemma The Cayey-Hamiton quotient of E by J ord is equa to E ord. Proof. Let (Vρ ord ) denote the kerne of E E ord, which contains (but may not be generated by) V ord (see 3.1.5). It wi suffice to show that (Vρ ord ) = J ord. The incusion (Vρ ord ) J ord is straightforward: see the cacuations in [WWE15, 5.9], from which it is evident that the Cayey-Hamiton quotient of ρ by J ord is a Cayey-Hamiton reresentation that is ordinary (in the sense of Definition 3.4.1). It remains to show that J ord (Vρ ord ). First we wi show that D ord I = ψ(κ cyc 1) I Z R ord. For any τ I, ρ ord (τ) satisfies both oynomias T 2 Tr D ord (τ)t Dord (τ) and (T κ cyc (τ))(t 1), the first by the Cayey-Hamiton condition and the second by Definition If ω(τ) 1, Hense s emma imies that these two oynomias are identica. For such τ, we have D ord (τ) = κ cyc (τ) and Tr (τ) = κ D ord cyc(τ) + 1. ow choose an arbitrary eement of I and write it as στ with ω(σ), ω(τ) 1. We immediatey see that D ord (στ) = κ cyc (στ), since both sides are mutiicative. Let r σ = ρ ord (σ) and r τ = ρ ord (τ). Since E ord is Cayey-Hamiton, we have (t σ r σ + t τ r τ ) 2 Tr (t D ord σr σ + t τ r τ )(t σ r σ + t τ r τ ) + D ord (t σ r σ + t τ r τ ) = 0 in the oynomia ring E ord [t σ, t τ ]. We can exand D ord (t σ r σ + t τ r τ ) using [Che14, Exame 1.8]. Taking the coefficient of t σ t τ and writing Tr = Tr D ord for brevity, r σ r τ + r τ r σ Tr(σ)r τ Tr(τ)r σ Tr(στ) + Tr(σ)Tr(τ) = 0.

20 20 PRESTO WAKE AD CARL WAG-ERICKSO Substituting for r σ r τ using Vρ ord (σ, τ) = 0 and for r τ r σ using Vρ ord (τ, σ) = 0, one obtains the desired concusion Tr(στ) = κ cyc (στ) + 1. Let σ I, and et τ I be such that ω(τ) 1. Using the fact that ρ ord I is reducibe, we see that the (1, 1)-coordinate of V ord (σ, τ) is ρ ord (ρ ord,1,1(σ) κ cyc (σ))(ρ ord,1,1(τ) 1) = 0 Since ρ ord,1,1 is a deformation of ω, we have ρ ord,1,1(τ) 1 (R ord ), so this imies ρ ord,1,1(σ) κ cyc (σ) = 0. This shows that (ρ,1,1 κ cyc )(I ) (Vρ ord ), and a simiar argument gives (ρ ord,2,2 1)(I ) (Vρ ord ). It remains to show that ρ ord,2,1(g ) = 0. Let m R ord be the maxima idea. In fact, we wi show that C ord /mc ord = 0, which is equivaent because ρ ord,2,1(g ) generates the finitey generated R ord -modue C ord. We work with ρ ord := ρ ord (mod m). Since ρ ord is reducibe, we can consider ρ ord 2,1 Z 1 (G, C ord /mc ord F F ( 1)), and [BC09, Thm ] imies that there is an injection Hom F (C ord /mc ord, F ) H 1 (G, F ( 1)) /mc ord sending φ to the cass of the cocyce φ ρ ord 2,1. So to show that C ord it is enough to show that ρ ord 2,1 is a coboundary, or, equivaenty, that ρ ord for a σ ker(ω) G. However, we comute that the (2, 1)-entry of V ord ρ ρ ord,2,1(σ)(ρ ord,1,1(τ) 1) + (ρ ord,2,2(σ) κ cyc (σ))ρ ord,2,1(τ). is zero, 2,1 (σ) = 0 (σ, τ) is Taking σ ker(ω) and τ I such that ω(τ) 1, we see that ρ ord,1,1(τ) 1 ω(τ) 1 0 (mod m) and ρ ord,2,2(σ) κ cyc (σ) m, so this imies ρ ord 2,1 (σ) = 0. We have the foowing consequence, foowing [WWE15, 5.9]. Proosition A Cayey-Hamiton reresentation (ρ : G E, E, D : E A) over D G is ordinary if and ony if it admits a GMA structure such that (1) it is uer trianguar, i.e. ρ 2,1 = 0, and (2) the diagona character ρ 1,1 (res. ρ 2,2 ) is the roduct of κ cyc Z A (res. the constant character A) and an unramified A-vaued character. Coroary Any finite-fat Cayey-Hamiton reresentation of G over D G is ordinary. The resuting morhism of universa Cayey-Hamiton reresentations of G, (ρ ord, E ord, D E ord) (ρ fat, E fat, D E fat), induces an isomorhism on universa seudodeformation rings R ord R fat. The universa seudodeformations D ord = D fat of D G have the form ψ(κ cyc χ 1 χ 2 ), where χ 1, χ 2 are unramified deformations of the trivia character 1 : G F. Proof. The Cayey-Hamiton reresentation ρ fat satisfies conditions (1) and (2) of Proosition by Lemmas and 3.5.2, resectivey. The isomorhism of universa seudoreresentations becomes evident by comaring Lemma and Proosition 3.7.5(2) Goba formuation. We now combine the oca constructions to define what it means for a goba Cayey-Hamiton reresentation or seudoreresentation to be unramified-or-steinberg of eve and tye ɛ.