Study of Different Lemma on Imaginary Quadratic Fields

IHARA'SLEMMA IN THE VERSION STATED IN [6] ASSERTS THAT

THE KERNEL OF THE MAP IS EISENSTEIN IF (N,P) = 1. HERE JO(N') DENOTES THE JACOBIAN OF THE COMPACTIFIED MODULAR CURVE AND A IS THE SUM OF THE TWO STANDARD P- DEGENERACY MAPS FROM TO THE ORIGINAL PROOF OF THE RESULT IS DUE TO IHARA [4] AND USES ALGEBRAIC GEOMETRY. IN [6] RIBET GAVE A DIFFERENT PROOF WITHOUT APPEALING TO ALGEBRO-GEOMETRIC METHODS. THE RESULT WAS LATER IMPROVED UPON BY KHARE [5] TO DISPOSE OF THE CONDITION THAT N BE COPRIME TO P. KHARE ALSO GIVES A REARRANGED PROOF IN THE CASE WHEN (N,P) = 1 USING THE METHOD OF MODULAR SYMBOLS (CF. [5], REMARK 4). WE WILL USE HIS APPROACH TO GENERALIZE THE RESULT TO IMAGINARY- QUADRATIC FIELDS, WHERE ALGEBRO-GEOMETRIC TECHNIQUES ARE NOT AVAILABLE.

Let F denote an imaginary quadratic extension ofandits ring of integers. The reason why over F the algebro-geometric machinery is not available is the fact that the symmetric space on which automorphic forms are defined is the hyperbolic- 3-space, the product ofandand the analogues Xn of the modular curves are not algebraic varieties (cf. section 2). However, [5] uses only group cohomology and his method may be adapted to the situation over an imaginary quadratic field. In this setting the Jacobians are replaced with certain sheaf cohomology groups and for a primep we have analogues of the two standard p-degeneracy maps whose sumwe will call(For precise definitions see section 2.) The main result of this note (Theorem 3.1) then asserts that the kernel of a is Eisenstein (for definition of "Eisenstein" see section 3). Originally Ihara's lemma had been used by Ribet [6] to prove the existence of congruences between modular forms of level N and those of level Np. His result, valid for forms of weight 2, was later generalized to arbitrary weight by- Diamond [2], who used the language of cohomology like we chose to. A crucial ingredient in Diamond's proof is the self-duality of, M). Over imaginary- quadratic fields, as overthere is a connection between the space of automorphic forms and the cohomology groupscalled the Eichler-Shimura-Harder isomorphism (cf. [10]). However, there seems to be no obvious way to adapt the approach of Ribet and Diamond to our situation asis not self-dual. Ihara's lemma was also used in the proof of modularity of Galois representations attached to elliptic curves over([11], [1])- Thanks to the work of Taylor [9] one can attach Galois representations to a certain class of automorphic forms on One could hope

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that Ihara's lemma in our formulation could be useful in proving the converse to Taylor's theorem, i.e., that ordinary Galois representations of(satisfying appropriate conditions) arise from automorphic forms, but at this moment this is a mere speculation as too many other important ingredients of a potential proof seem to be missing. The author would like to thank Trevor Arnold, Tobias Berger, Brian Conrad, Chandrashekhar Khare and Chris

Skinner for many helpful and inspiring discussions.

Let F be an imaginary quadratic extension ofand denote byits ring of integers. Letbe an ideal ofsuch that the Z-idealhas a generator greater than 3. Letbe a prime ideal such thatDenote by C1F the class group of F and choose representatives of distinct ideal classes to be prime idealsrelatively prime to bothandLetbe a uniformizer of the completion Fp (resp. Fpi) of F at the primeand put to be the idele whereoccurs at the p-th place (resp.place). We also put For eachwe define compact open subgroups of Heredenotes the finite adeles of F andthe ring of integers ofFor we also set For any compact open subgroup K ofwe put whereis the center of GL2(C) and (here 'bar' denotes complex conjugation and h stands for the 2 x 2-identity matrix). If K is sufficiently large (which will be the case for all compact open subgroups we will consider) this space is a disjoint union ofconnected components where and . To ease notation we put We have the following diagram:

(2.1)

where the horizontal and diagonal arrows are inclusions and the vertical arrows are conjugation by the maps. Diagram (2.1) is not commutative, but it is "vertically commutative", by which we mean that given two objects in the diagram, two directed paths between those two objects define the same map if and only if the two paths contain the same number of vertical arrows. Diagram (2.1) induces the following vertically commutative diagram of the corresponding symmetric spaces:

(2.2)

The horizontal and diagonal arrows in diagram (2.2) are the natural projections and the vertical arrows are maps given by-

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Let M be a torsion abelian group of exponent relatively prime toendowed with a GL2(F)-action. Denote bythe sheaf of continuous sections of the topological covering GL2(F) \ (GL2(Af)/K ■where GL2(F) acts diagonally on. Here M is equipped with the discrete topology. Since we will only be concerned with the case when M is a trivial GL2(F)- module, we assume itfrom now on. This means thatis a constant sheaf. As above, we putand Givena surjective mapwe get an isomorphism of sheaves which yields a map on cohomology Hence diagram (2.2) gives rise to a vertically commutative diagram of cohomology groups:

(2.3)

These sheaf cohomology groups can be related to the group cohomology of andwith coefficients in M. In fact, for each compact open subgroup K with corresponding decompositionwe have the following com mutative diagram in which the horizontal maps are inclusions:

(2.4)

Heredenotes the image of the cohomology with compact support insideanddenotes the parabolic cohomology, i.e., whereis the set of Borel subgroups of GL^iF) andThe vertical arrows in diagram (2.4) are isomorphisms provided that there exists a

torsion-free normal subgroup ofof finite index relatively prime to the exponent of M. If K = Kn

orthis condition is satisfied because of our assumption that has a generator greater than 3 and the exponent of M is relatively prime to(cf. [10], section 2.3). In what follows we may therefore identify the sheaf cohomology with the group cohomology. Note that all maps in diagram (2.3) preserve parabolic cohomology. The mapsare the natural restriction mapson group cohomology, so in particular they preserve the decomposition Using the identifications of diagram (2.4) we can prove the following result which will be useful later: Lemma 2.1. The mapis injective. Proof. Using the isomorphism between group and sheaf cohomology all we need to prove is that the restriction maps res, :are injective. Since M is a trivialthe cohomology groups are just Homs, so it is enough to show the following statement: if G denotes the smallest normal subgroup ofcontainingthen. For this we use the decomposition where the matrixis chosen so that C and D are relatively prime elements of Of withandsatisfy AD — BC = 1. Here denotes a set of representatives in Op of the distinct residue classes ofLet withThen for anywe have

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and the matrixhence G containsand thus We can augment diagram (2.1) on the right by introducing one more group: The group K-1 is not compact, but we can still define forAfter identifying the sheaf cohomology groupsand with the groupsandrespectively, using diagram (2.4), we can augment diagram (2.3) on the right in the following way-

(2.5)

Here we putand the mapsand are direct sums of the restriction maps. The sheaf and group cohomologies are in a natural way modules over the corresponding Hecke algebras. (For the definition of the Hecke action on cohomology, see [10] or [3]). Here we will only consider the subalgebra T of the full Hecke algebra which is generated over Z by the double cosetsK and p ofsuch thatThe algebra T acts on all the cohomology groups in p the group cohomology respects the decomposition, where or P.

We will say that a maximal ideal n of the Hecke algebra T is Eisenstein if (mod n) for all ideals l ofwhich are trivial as elements of the ray- class group of conductor n. Such ideals l are principal and have a generator I with n denotes the ideal norms From now on we fix a non-Eisenstein maximal ideal m of the Hecke algebra T. Our main result is the following theorem. Theorem 3.1. Consider the mapdefined as The localizationofis injective. We prove Theorem 3.1 in two steps. Define a map and note thatby the ver tical commutativity of diagram (2.5), i.e., ker. We first prove Proposition 3.2. ker Then we show

Propositions 3.2 and 3.3 imply Theorem 3.1. The idea of the proof is due to Khare [5] and uses modular symbols, which we now define. Let D denote the free abelian group on the setof all Borel subgroups of GL2(F). The action of GL2(F) onby conjugation gives rise to a Z-linear action of GL2(F) on D. We sometimes identifywith on which GL2(F) acts by the linear fractional transformations. Letbe the subset of elements of degree zero. Ifthen for eachthe exact sequence gives rise to an exact sequence

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The groupis called the group of modular symbols. Lemma 3.4. LetT be a group acting on the setof Borel subgroups o/GL2(F) and let C denote a set representatives for the T-orbits of. Then for any trivial F-module W, whereis the stabilizer of c in Proof. Thestructure on Homz(D,W) is defined via and onvia Note that we have aisomorphism given byThus since the action ofstabilizesW for every. The last group is in turn isomorphic toby Shapiro's Lemma. By taking the direct sum of the exact sequences (3.1) and using Lemma 3.4, we obtain the exact sequence where the last group is isomorphic to Remark 3.5. The space of modular symbolsis also a Hecke module in a natural way. In fact it can be shown (at least if N is square-free) that the localized mapis an isomorphism, but we will not need this fact.

For K C K' two compact open subgroups offor which and withwe have a commutative diagram

(4.1)

where the mapsanddenote the appropriate connecting homomorphisms from exact sequence (3.2). So far we have shown that

(4.2)

We identify g with a tupleand defineandsimilarly. Equality (4.2) translates to(4.3) Fixand regard it as an element of Homz(-Do) M) invariant under To,,. Using diagram (4.1) withand equality (4.3) we

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conclude that there existssuch that regarded as elements of Homz(D0,M). Henceis invariant under both and Lemma 4.1. Forthe groupsandgenerate Using Lemma 4.1 we conclude that Again, by the commutativity of diagram (4.1) with andwe haveHence HF , AF_i) satisfies Thus By the vertical comutativity of diagram (2.5) we have Hence Completing the proof of Proposition 3.2

In this section we prove that for a principal idealsuch that (mod N) we haveon elements For such an ideal [, the operators T[ preserve each direct sum- mandThe restriction of T{ tois given by the usual action of the double coseton group cohomology (see, e.g., [3]). For we putTo describe the action ofexplicitly we use the following lemma. Lemma 5.1. Let l = (I) be a principal ideal ofand,Then wheredenotes a set of representatives ofin Proof. This is easy. Lemma 5.2. Letbe an odd integer. Every ideal class c of F contains infinitely many prime ideals q such that (Nq — l,n) = 1. Proof. We assumethe other case being easier. Let andwhere H denotes the Hilbert class field of F. We have the following diagram of fields

(5.1)

corresponds to an elementwithmodulo any of the divisors of n, andcorresponds to the ideal class c. By the Chebotarev density theorem thereexist infinitely many primesof the ring q oflying under suchsatisfy the condition of the lemma, i.e.,and (Wq-l,n) = l. By Lemma 5.2 we may assume that the ideals p were chosen so thatis relatively prime to the exponent of M for all Proof of Proposition 3.3. Letand letbe a principal ideal ofwith. We will prove thatBy the definition of parabolic cohomology, we havefor all

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Moreover, as the exponent of M is relatively prime toit is enough to prove thatwherePut We first show that on the z-th principal congruence subgroup by the definition of parabolic cohomology. So on the smallest normal subgroup H ofcontaining matrices of the form withBy a theorem of Serre [7], Thus onPut Since PNPi is a normal subgroup ofof index have by Lemma 5.2 and our choice of p. On one hand ft is zero on the elements of the form(again by the definition of parabolic cohomology) and on the other hand elements of this form together withgenerateso as asserted. Thus descends to the quotientHowever, on this quotient all and act as the identity, sinceand we can always

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