aa r X i v : . [ m a t h . N T ] S e p ON SHARIFI’S CONJECTURE: EXCEPTIONAL CASE
SHENG-CHI SHIH AND JUN WANG
Abstract.
In the present article, we study the conjecture of Sharifi on the surjectivityof the map ̟ θ . Here θ is a primitive even Dirichlet character of conductor Np , whichis exceptional in the sense of Ohta. After localizing at the prime ideal p of the Iwasawaalgebra related to the trivial zero of the Kupota–Leopoldt p -adic L -function L p ( s, θ − ω ) ,we compute the image of ̟ θ, p in local Galois cohomology groups and prove that it is anisomorphism. Also, we prove that the residual Galois representations associated to thecohomology of modular curves are decomposable after taking the same localization. Contents
1. Introduction 12. Sharifi’s Conjecture 43. Galois cohomology and cohomology of modular curves 74. Suejectivity of ̟ θ, p Introduction
Let N be a positive integer, and let p ≥ N φ ( N ) , where φ ( N ) is the order of the group ( Z / N Z ) × . Set H ∶= lim ←Ð H ( X ( N p r ) / Q , Z p ) ord , where ord denotes the ordinary part for the Hecke operator U ∗ p , and denote by H − the subgroup of H on which the complex conjugation acts via −
1. Let h ∗ be the cuspidal Hecke algebra actingon H , and let I ∗ be the Eisenstein ideal in h ∗ generated by T ∗ l − l ⟨ l ⟩ − − l ∤ N p and by U ∗ q − q ∣ N p . For a Dirichlet character θ of conductor N p , we denoteby H θ the θ -eigenspace of H and set Λ θ ∶= O ⟦ T ⟧ for some extension O of Z p containing allof the values of θ . Let ̟ ∶ H − ( ) → lim ←Ð H ( Z [ ζ Np r , p ] , Z p ( )) + ∶= S be the homomorphism constructed by Sharifi in [13, Section 5.3] (or see Section 2.1 for thedefinition). It was further conjectured by Sharifi (Conjecture 5.8 in loc. cit. ) and proved Date : September 17, 2020. by Fukaya–Kato ([6, Theorem 5.2.3]) that the kernel of ̟ contains the Eisenstein ideal I ∗ . Combining this with a conjecture of McCallum–Sharifi [9], it is conjectured that thehomomorphism ̟ induces a surjective homomorphism of Λ θ -modules ̟ θ ∶ H − θ ( )/ I ∗ θ H − θ ( ) ↠ S θ for all primitive even Dirichlet characters θ of conductor N p . When θ is not exceptional(in the sense of Ohta [12]), it was conjectured by Sharifi in loc. cit. that ̟ θ is indeed anisomorphism together with the inverse homomorphism Υ θ . Moreover, he also proved thatthere is a canonical isomorphism of Λ θ -modules S θ ≅ X θ ( ) (Lemma 4.11 in loc. cit. ), where X is the Galois group of the maximal abelian unramified pro- p extension of Q ( ζ Np ∞ ) . Thus,the conjecture of ̟ θ being an isomorphism is a refinement of the Iwasawa main conjectureand describes the Λ θ -module structure of X θ . Some partial results on this conjecture wereproved by Fukaya–Kato loc. cit. , Fukaya–Kato–Sharifi [7], and Wake–Wang-Erickson [15].Their ideas are to prove that the homomorphism Υ θ is an isomorphism and is the inverse of ̟ θ . The requirement of θ being not exceptional is essential to the construction of Υ θ as inthis case, one can apply a work of Ohta [12] to show that one has a short exact sequence ofΛ θ [ G Q ] -modules 0 → H − θ / I ∗ θ H − θ → H θ / I ∗ θ H θ → H + θ / I ∗ θ H + θ → . When θ is exceptional, there is no literature discussing the homomorphism ̟ θ . One ofdifficulties in this case is that it is not clear whether one of H − θ / I ∗ θ H − θ and H + θ / I ∗ θ H + θ is G Q -sable, and hence, one can not construct Υ θ following Sharifi’s construction.One of the goals in this paper is to study the homomorphism ̟ θ when θ is exceptional.We will prove that it is an isomorphism after taking localization at a certain height oneprime of Λ θ corresponding to the trivial zero of the Kubolta–Leopoldt p -adic L -function L p ( s, θ − ω ) without constructing Υ θ (Theorem 1.1). It is known that the leading coefficientof this p -adic L -function involves a certain L -invariant. Since the Sharifi’s conjecture isa refinement of the Iwasawa main conjecture, it is nature to ask how such an L -invariantis related to Sharifi’s conjecture. This question will be addressed in Theorem 1.2 below.Another goal is to study the residual Galois representation attached to H θ = H − θ ⊕ H + θ aftertaking the same localization (Theorem 1.3). An advantage of taking such localization is thatthe image of ̟ θ belongs to a certain local Galois cohomology so that we are able to computeit explicitly. Another advantage is that the cuspidal Hecke algebra is Gorenstein by a workof Betina–Dimitrov–Pozzi [1], which makes the study of the cohomology of modular curvesmodulo Eisenstein ideals easier (for example, see Proposition 3.5). These two advantages areessential in our study. For example, the former and the later respectively help us to showthe surjectivity and the injectivicity of ̟ θ, p in Theorem 1.1.To state our results, we let ω be the Teichm¨uller character, κ ∶ Gal ( Q ( ζ Np ∞ )/ Q ( ζ N )) → Z × p be the p -adic cyclotomic character, γ be a topological generator of Gal ( Q ( ζ Np ∞ )/ Q ( ζ Np )) and N SHARIFI’S CONJECTURE: EXCEPTIONAL CASE 3 θ be a primitive even Dirichlet character of conductor N p such that the character χ ∶ = θω − is trivial on ( Z / p Z ) × and χ ∣ ( Z / N Z ) × ( p ) = θ being exceptional in thesense of Ohta). We denote by L ( χ ) the L -invariant attached to χ (for example, see [1, (15)]for the definition). Let p be the prime ideal of Λ θ generated by T + − κ ( γ ) , which is relatedto the trivial zero of the Kubolta–Leopoldt p -adic L -function L p ( s, χ − ω ) (see Section 3.2).We denote by Λ θ, p the localization of Λ θ at p with residue field k θ, p ∶ = Λ θ, p / p , and for aΛ θ -module M , we set M p ∶ = M ⊗ Λ θ Λ θ, p . The following is the first main result in this paper. Theorem 1.1 (Theorem 2.2) . Suppose p ≥ with p ∤ N φ ( N ) . If θ is exceptional, then ̟ θ induces an isomorphism of k θ, p -vector spaces ̟ θ, p ∶ H − θ, p ( )/ I ∗ θ, p H − θ, p ( ) ≅ S θ, p . When θ is not exceptional, assuming Greenberg’s conjecture, an analog result of the abovetheorem has been proved by Wake–Erickson-Wang [15, Corollary C]. Their idea is first toshow that for each height one prime ideal q of Λ θ , the map Υ θ, q is an isomorphism and then,by a work of Fukaya–Kato [6], it is the inverse of ̟ θ, q . When θ is exceptional, their methodcan be adapted if one consider the localization at height one prime ideals q other than p . Inthis case (or in general even without taking any localization), it is difficult to show that ̟ θ, q is an isomorphism without using Υ θ, q , since it is defined by the cup product of two cyclotomicunits and is difficult to be computed.A crucial point in the proof of Theorem 1.1 is that we are able to compute the imageof ̟ θ, p via local explicit reciprocity law by showing that the image of ̟ θ, p is non-trivial incertain local cohomology groups. There are three steps in the proof: (1) Show that both H − θ, p ( )/ I ∗ θ, p H − θ, p ( ) and S θ, p are 1-dimensitonal k θ, p -vector spaces (Propositions 3.2 and 3.5).(2) Show that ( U ∗ q − ){ , ∞} DM,θ is a basis of H − θ / I ∗ θ H − θ whose image under ̟ θ in S θ is thecup product ( q, − ζ Np r ) r ≥ ,θ for all primes q ∣ N p (Corollary 3.4). Here { , ∞} is the modularsymbol attached to the cusps 0 and ∞ . (3) Show that the cup product ( q, − ζ Np r ) r,θ, p isnon-zero for all big enough positive integers r , which is the second main result in this article. Theorem 1.2 (Corollary 4.3) . For each positive integer r , we have ( ℓ, − ζ Np r ) r,θ, p = ( p − ) log p ( ℓ ) pφ ( N ) ω ( N ) τ ( χ − ) L ( , χ ) ∈ ( O / p r O )( ) for all ℓ ∣ N and ( p, − ζ Np r ) r,θ, p = − ( p − ) pφ ( N ) ω ( N ) τ ( χ − ) L ( χ ) L ( , χ ) ∈ ( O / p r O )( ) . In particular, ( q, − ζ Np r ) r,θ, p is non-zero for all r big enough and for all q ∣ N p . By the second step of the proof of Theorem 1.1 mentioned above, one can show that H − θ, p / I ∗ θ, p H − θ, p is G Q -stable by using a Λ θ -adic perfect pairing on H θ constructed by Ohta (seeSection 5). We show that H + θ, p / I ∗ θ, p H + θ, p is also G Q -stable. SHENG-CHI SHIH AND JUN WANG
Theorem 1.3 (Theorem 5.1) . Let the notation be as above. Then the following short exactsequence of k θ, p [ G Q ] -modules splits → H − θ, p / I ∗ θ, p H − θ, p → H θ, p / I ∗ θ, p H θ, p → H + θ, p / I ∗ θ, p H + θ, p → . If one considers the localization other than p (or the case that θ is not exceptional), itwas shown by Ohta [11, Section 3.3] (see [10, Section 5.3] for not exceptional case) thatthe correpsonding short exact sequence of G Q -modules in Theorem 1.3 does not split andthe associated ordinary Galois representation is p -distinguished. Thus, one can construct ananalog of the homomorphism Υ θ following Sharifi’s construction. This construction can notbe adapted to the situation of Theorem 1.3, since it follows from the assumption χ ( p ) = p -distinquished.1.1. Outline.
In Section 2, we briefly review the construction of ̟ θ following [13] and sketchthe strategy of the proof of Theorem 1.1.In Section 3, we first show that S θ, p is a 1-dimensional k θ, p -vector space (Proposition 3.2).This phenomena is opposite to the case of θ being not exceptional. In this case, it is knownthat S θ ≅ X K, Σ ,χ ( ) [13, Lemma 4.11]. Second, we construct some elements in H − θ bycomputing the congruence modules attached to cohomology of modular curves (Theorem 3.3).This plays an important role in computing the image of ̟ θ, p . When θ is not exceptional,this was done by Ohta [12, Section 3.5] by showing that the desired congruence module isessentially isomorphic to the congruence module attached to (3.5). His argument can notbe adapted when the character is exceptional since in this case, it is not clear whether theshort exact sequences (3.4.6) in loc. cit. split as Hecke-modules. Our idea is to use theDrinfeld–Manin modification and a result of Lafferty in [8]. Third, we will show that thesource of ̟ θ, p is a 1-dimensional k θ, p -vector space by a result of Betina–Dimitrov–Pozzi [1]on the Gorenstainess of the cuspidal Hecke algebra after localizing at p .Section 4 is devoted to computing the image of ̟ θ, p and completing the proof of Theo-rem 1.1. The main goal of Section 5 is to prove Theorem 1.3. Acknowledgments.
The first author has been supported by the Labex CEMPI under GrantNo. ANR-11-LABX-0007-01, by I-SITE ULNE under Grant No. ANR-16-IDEX-0004, andby Austrian Science Fund (FWF) under Grant No. START-Prize Y966. The second authorwould like to thank Sujatha Ramdorai for supporting his postdoctoral studies.2.
Sharifi’s Conjecture
Throughout this paper, we will denote by N a positive integer, denote by p ≥ N φ ( N ) . The goal of this section is to review Sharifi’s conjecture on themap ̟ following [13] and to state Theorem 1.1. We refer the reader to loc. cit. for moredetails on Sharifi’s conjecture. N SHARIFI’S CONJECTURE: EXCEPTIONAL CASE 5
The map ̟ . Set H r ∶ = H ( X ( N p r ) / Q , Z p ) ord and set ̃ H r = H ( Y ( N p r ) / Q , Z p ) . Onecan identify ̃ H r with a certain relative homology group by Poincar´e duality and the com-parison between Betti (co)homology groups and ´etale (co)homology groups. Namely, onehas(2.1) ̃ H r ( ) ≅ H ( Y ( N p r )( C ) , C ( N p r ) , Z p ) , where C ( N p r ) is the set of cusps for Γ ( N p r ) (see Sections 3.4 and 3.5 of loc. cit. for moredetails). We denote by ̃ H + r (resp. ̃ H − r ) the subgroup of ̃ H r on which the complex conjugationacts trivially (resp. via −
1) and denote by H + r and H − r in the same manner.Form the discussion in Section 3.1 of loc. cit. , the group H ( Y ( N p r )( C ) , C ( N p r ) , Z p ) + is generated by the adjusted Manin symbols [ u, v ] + r for all u, v ∈ Z / N p r Z with ( u, v ) = loc. cit. . We will identify these symbols as elements of ̃ H r ( ) + via (2.1) without further notice. Let ̃ H r, ( ) + be the subgroup of ̃ H r ( ) + generatedby [ u, v ] + r for all u, v ∈ Z / N p r Z − { } with ( u, v ) =
1. Let H ( Z [ ζ Np r , Np ] , Z p ( )) ○ be definedas at the end of p. 31 of loc. cit. It was proved in Proposition 5.7 of loc. cit. that there existsa homomorphism ̃ ̟ r ∶ ̃ H r, ( ) + → H ( Z [ ζ Np r , Np ] , Z p ( )) ○ , sending [ u, v ] + r to the cup product ( − ζ uNp r , − ζ vNp r ) ○ r and satisfying ̃ ̟ r ○ ⟨ j ⟩ r = σ − j ○ ̃ ̟ r for all j ∈ ( Z / N p r Z ) × . Here ⟨ j ⟩ is the diamond operator and σ j ∈ Gal ( Q ( ζ Nr )/ Q ) satisfying σ j ( ζ Np r ) = ζ jNp r . Since ̃ H r ( ) + ≅ ̃ H − r ( ) and since H ( X ( N p r ) / Q , Z p ) − ( ) is contained in ̃ H − r, ( ) , the homomorphism ̃ ̟ r induces via restriction a homomorphism ̟ r ∶ H ( X ( N p r ) / Q , Z p ) − ( ) → H ( Z [ ζ Np r , Np ] , Z p ( )) ○ whose image is contained in H ( Z [ ζ Np r , p ] , Z p ( )) ○ by [6, Theorem 5.3.5].Let I ∗ r be the ideal of the Hecke algebra H ∗ r (acting on of ̃ H r, ) generated by T ∗ l − − l ⟨ l ⟩ − for all l ∤ N p r and U ∗ q − q ∣ N p , and denote by I ∗ r the image of I ∗ r in the cuspidal Heckealgebra h ∗ r acting on H ( X ( N p r ) / Q , Z p ) under the natural map H ∗ r ↠ h ∗ r . It was conjecturedby Sharifi (Conjecture 5.8 in loc. cit. ) and proved by Fukaya–Kato [6, Theorem 5.2.3] thatthe map ̃ ̟ r satisfies(2.2) ̃ ̟ r ( ηx ) = η ∈ I ∗ r and x ∈ ̃ H − r, ( ) . Moreover, they also proved that the following diagramcommutes: H ( X ( N p r + ) / Q , Z p ) − ( ) H ( Z [ ζ Np r + , p ] , Z p ( )) ○ H ( X ( N p r ) / Q , Z p ) − ( ) H ( Z [ ζ Np r , p ] , Z p ( )) ○ , ̟ r + ̟ r SHENG-CHI SHIH AND JUN WANG where the left and right vertical maps are the trace map and the norm map, respectively.This diagram induces a map by taking projective limit ̟ ∶ lim ←Ð H ( X ( N p r ) / Q , Z p ) − ( ) → lim ←Ð H ( Z [ ζ Np r , p ] , Z p ( )) ○ = ∶ S. It follows from (2.2) that this map factors through the quotient by I ∗ ∶ = lim ←Ð I ∗ r . Since H ( X ( N p r ) / Q , Z p )/ I ∗ r H ( X ( N p r ) / Q , Z p ) is isomorphic to H r / I ∗ r H r for all r ∈ Z ≥ , weobtain a homomorphism ̟ ∶ H − ( )/ I ∗ H − ( ) → S. Set Λ ∶ = Z p [[ lim ←Ð ( Z / N p r Z ) × ]] ≅ Z p [( Z / N p Z ) × ]⟦( + p Z p )⟧ . Both H − ( ) and S are Λ-moduleson which lim ←Ð ( Z / N p r Z ) × acts respectively via diamond operators and Galois actions. More-over, the map ̟ satisfies ̟ ○ ⟨ j ⟩ = σ − j ○ ̟ for all j ∈ ( Z / N p Z ) × × ( + p Z p ) . The followingconjecture is the conjecture of McCallum-Sharifi in [9]. Conjecture 2.1.
The Λ -module homomorphism ̟ ∶ H − ( )/ I ∗ H − ( ) ↠ S is surjective. Main result.
For a Dirichlet character θ of modulus N p , we denote by H − θ the subgroupof H − such that the action of ( Z / N p ) × is via θ and denote by S θ in the same manner. Let O be a finite extension of Z p containing values of θ and set Λ θ ∶ = Λ ⊗ Z p [( Z / N Z ) × ] ,θ O . Let K = Q ( ζ Np ∞ ) . We fix a topological generator γ of Gal ( K / Q ( ζ N )) and identify Λ θ with O ⟦ T ⟧ via the map γ ↦ + T . Denote by κ ∶ Gal ( K / Q ( ζ N )) → Z × p the universal cyclotomiccharacter. For a height 1 prime p of Λ θ , we will denote by Λ θ, p the localization at p anddenote by k θ, p the residue field of Λ θ, p .It follows from the assumption p ∤ N φ ( N ) that one has a decompositions H − ( ) = ⊕ θ H − θ ( ) and S = ⊕ θ S θ . Here, the decompositions run through all even Dirichlet char-acters of modulus N p . Thus, Conjecture 2.1 is equivalent to the surjectivity of(2.3) ̟ θ ∶ H − θ ( )/ I θ H − θ ( ) ↠ S θ for all even Dirichlet characters θ of modulus N p . The following theorem gives a partialresult of the conjecture when θ is exceptional. Theorem 2.2.
Suppose p ∤ N φ ( N ) . Let p be the height prime of Λ θ generated by + T − κ ( γ ) , and let other notation be as above. If θ is exceptional, then after taking localization at p , the Λ θ -module homomorphism ̟ θ induces an isomorphism of Λ θ, p -modules (2.4) ̟ θ, p ∶ H − θ, p ( )/ I ∗ θ, p H − θ, p ( ) ≅ S θ, p . Proof.
We will prove in Proposition 3.2 and Proposition 3.5 that H − θ, p ( )/ I θ, p H − θ, p ( ) and S θ, p are 1-dimensional k θ, p -vector spaces. Moreover, we will prove in Corollary 4.3 that ̟ θ, p isnon-trivial, and thus, it is an isomorphism. ∎ N SHARIFI’S CONJECTURE: EXCEPTIONAL CASE 7 Galois cohomology and cohomology of modular curves
From now on, we assume that the Dirichlet character θ is exceptional. The aim of Sec-tion 3.1 is to show that k θ, p -vector space S θ, p is 1-dimensional. In Section 3.2, we constructelements in H − θ ( ) by computing congruence modules attached to (3.8). In addition, we showthat the k θ, p -vector space H − θ, p ( )/ I ∗ θ, p H − θ, p ( ) is a 1-dimensional. Recall that χ ∶ = θω − andthat K = Q ( ζ Np ∞ ) .3.1. Localization of S θ at p . Let Σ be the set of the finite places of Q ( ζ Np ∞ ) above p , andlet X K (resp. X K, Σ ) be the Galois group of the maximal abelian unramified pro- p extensionof K (resp. in which all primes above those in Σ split completely).By [13, Lemma 2.1], We have the following exact sequence(3.1) 0 → X K, Σ → lim ←Ð H ( Z [ ζ Np r , p ] , Z p ( )) → ⊕ v ∈ Σ H ( Q ( ζ Np ∞ ) v , Z p ( )) inv Ð→ Z p → . Note that one haslim ←Ð H ( Z [ ζ Np r , p ] , Z p ( )) θ = lim ←Ð H ( Z [ ζ Np r , p ] , Z p ( )) χ ( ) and(3.2) H ( Q ( ζ Np ∞ ) v , Z p ( )) ≅ Z p for all v ∈ Σ. Denote by ( ⊕ v ∈ Σ Z p ) the kernel of inv in (3.1) under the identification (3.2).After taking χ components and Tate twist on the exact sequence (3.1), we have the followingshort exact sequence(3.3) 0 → X K, Σ ,χ ( ) → S θ → ( ⊕ v ∈ Σ Z p ) χ ( ) → . The following lemma shows that the first term in (3.3) is trivial after localizing at p . Lemma 3.1.
Let the notation be as above. Then, the characteristic ideal of X K,χ / X K, Σ ,χ is ( T ) . Moreover, one has X K, Σ ,χ ( ) p = .Proof. From the discussion in the last paragraph of p. 99 in [3, Section 4], one can see that X K,χ / X K, Σ ,χ is a direct summand of the Pontryagin dual of the direct limit of D − n (see thedefinition in loc. cit. ) whose characteristic ideal is a power of ( T ) . Since θ is exceptional,it was also shown in loc. cit. that ( T ) divides the characteristic ideal of X K,χ exactly once.Hence, the characteristic ideal of X K,χ / X K, Σ ,χ is ( T ) . Thus, X K, Σ ,χ ⊗ Λ θ Λ θ, ( T ) = ∎ Proposition 3.2.
We have isomorphisms of Λ θ, p -modules S θ, p ≅ (( ⊕ v ∈ Σ Z p ) χ ( )) p ≅ Λ θ, p / p . SHENG-CHI SHIH AND JUN WANG
Proof.
The first isomorphism follows from Lemma 3.1 and (3.3). We now prove the sec-ond isomorpihsm. Note that as Z p [ χ ] -module, ( ⊕ v ∈ Σ Z p ) χ is isomorphic to Z p [ χ ] . For σ ∈ Gal ( Q ( ζ p )/ Q ) , it acts on ( ⊕ v ∈ Σ Z p ) χ ( ) via ω ( σ ) . Moreover, the topological generator γ of Gal (( Q ( ζ Np ∞ )/ Q ( ζ Np ))) acts on ( ⊕ v ∈ S Z p ) χ ( ) via κ ( γ ) . Therefore, one has ( ⊕ v ∈ Σ Z p ) χ ( ) ≅ Λ θ / p , and hence, the assertion follows after taking localization at p . ∎ Cohomology of modular curves.
The Λ θ -adic Eisenstein series E ( θ, ) is defined by2 − G θ ( T ) + ∞ ∑ n = ⎛⎝ ∑ d ∣ n, ( d,p ) = θ ( d )( + T ) s ( d ) d ⎞⎠ e πinz , where s ( d ) ∶ = log p ( d ) log p ( γ ) and G θ ( T ) ∈ Λ θ is the power series expression of the Kubolta–Leopoldt p -adic L -function L p ( s, θω ) satisfying G θ ( γ s − ) = L p (− s − , θω ) . Recall that the Kubolta –Leopoldt p -adic L -function L p ( s, ψ ) satisfies the interpolation prop-erty L p ( − k, ψ ) = ( − ψω − k ( p ) p k − ) L ( − k, ψω − k ) for all k ∈ Z > and for all Dirichlet character ψ . Set ξ θ ( T ) ∶ = G θ − (( + T ) − − ) . Then, itfollows from the assumption θω − ( p ) = T + − κ ( γ ) is a zero of ξ θ ( T ) , called the trivialzero.Let M ord ( N, θ ; Λ θ ) be the space of ordinary Λ θ -adic modular forms of type θ , and let S ord ( N, θ ; Λ θ ) be the subspace of M ord ( N, θ ; Λ θ ) consisting of Λ θ -adic cusp forms. Recallthat we denote by C ( N p r ) the set of cusps for Γ ( N p r ) . It was proved by Ohta [12, (2.4.4)]that one has the following short exact sequence of free Λ θ -modules(3.4) 0 → S ord ( N, θ ; Λ θ ) → M ord ( N, θ ; Λ θ ) Res
ÐÐ→ e ⋅ Λ θ ⟦ C ∞ ⟧ → , where e = lim n → ∞ U n ! p is the ordinary projector and Λ θ ⟦ C ∞ ⟧ ∶ = lim ←Ð Λ θ [ C ( N p r )] for whichthe projective limit is with respect to the natural projection C ( N p r ) ↠ C ( N p s ) for r ≥ s .Let e ( θ, ) ∈ e ⋅ Λ θ ⟦ C ∞ ⟧ be defined in [8, Proposition 3.2.1] such that Λ θ ⋅ e ( θ, ) is a directsummand of e ⋅ Λ θ ⟦ C ∞ ⟧ and Res ( E ( θ, )) = G θ ( T ) ⋅ e ( θ, ) . Let M Λ θ be the preimage of Λ θ ⋅ e ( θ, ) under the map Res, and set S Λ θ ∶ = S ord ( N, θ ; Λ θ ) . Then we obtain a short exact sequence offree Λ θ -modules(3.5) 0 → S Λ θ → M Λ θ Res
ÐÐ→ Λ θ ⋅ e ( θ, ) → . Let H θ and h θ be the Hecke algebras acting on M Λ θ and S Λ θ , respectively. Also, let I θ ∶ = Ann H θ ( E ( θ, )) be the Eisenstein ideal, and let I θ be the image of I θ under the naturalhomomorphism H θ ↠ h θ . Denote by s ′ ∶ M Λ θ ⊗ Λ θ Q ( Λ θ ) → S Λ θ ⊗ Λ θ Q ( Λ θ ) the unique Hecke N SHARIFI’S CONJECTURE: EXCEPTIONAL CASE 9 equivariant splitting map and set M Λ θ ,DM ′ ∶ = s ′ ( M Λ θ ) . Here Q ( Λ θ ) denotes the quotient fieldof Λ θ . It was proved in Section 3 of loc. cit. that one has isomomorphisms of Λ θ -modules(3.6) M Λ θ ,DM ′ / S Λ θ ≅ Λ θ /( G θ ( T )) ≅ h θ / I θ . Recall that in Section 2, we set H θ ∶ = lim ←Ð H ( X ( N p r ) / Q , Z p ) ord θ . By [10, (4.3.12)], onehas the short exact sequence of free Λ θ [ G Q ] -modules(3.7) 0 → H θ → lim ←Ð H ( Y ( N p r ) / Q , Z p ) ord θ → e ⋅ Λ θ ⟦ C ∞ ⟧( − ) → . Let ̃ H θ ⊂ lim ←Ð H ( Y ( N p r ) / Q , Z p ) ord θ be the preimage of Λ ⋅ e ( θ, ) . Then, one obtains from(3.7) a short exact sequence of free Λ θ [ G Q ] -modules(3.8) 0 → H θ → ̃ H θ → Λ θ ⋅ e ( θ, ) ( − ) → . We denote by H ∗ θ (resp. h ∗ θ ) the Hecke algebra acting on ̃ H θ (resp. on H θ ). Also, we denoteby I ∗ θ and I ∗ θ the corresponding Eisnenstein ideals in H ∗ θ and h ∗ θ , respectively. Note that thelast map in (3.8) commutes the action of H ∗ θ on ̃ H θ and the action of H θ on Λ θ ⋅ e ( θ, ) ( − ) .Moreover, by (3.6), we obtain an isomorphism of Λ θ -modules(3.9) h ∗ θ / I ∗ θ ≅ Λ θ /( ξ θ ) induced by the canonical isomorphism h θ ≅ h ∗ θ sending U q → U ∗ q for all q ∣ N p , T l to T ∗ l and ⟨ l ⟩ to ⟨ l ⟩ − for all l ∤ N p .The Drinfield–Manin modification ̃ H θ,DM of ̃ H θ is defined as ̃ H θ ⊗ H ∗ θ h ∗ θ . For each r ∈ Z > ,we denote by s r ∶ H ( Y ( N p r ) , Q p ) → H ( X ( N p r ) , Q p ) the Drinfield–Manin splitting whichinduces a splitting map s ∶ lim ←Ð H ( Y ( N p r ) , Q p ) → lim ←Ð H ( X ( N p r ) , Q p ) . Set ̃ H θ,DM ′ = s ( ̃ H θ ) . Proposition 3.3.
Let the notation be as above. We have Λ θ -modules (3.10) ̃ H θ,DM / H θ ≅ Λ θ /( ξ θ ) and (3.11) ̃ H θ,DM ′ / H θ ≅ Λ θ /( ξ θ ) In particular, we have an isomorphism of Λ θ -modules ̃ H θ,DM ′ ≅ ̃ H θ,DM , and the Λ θ /( ξ θ ) -module ̃ H θ,DM / H θ is generated by { , ∞ } θ,DM , where { , ∞ } is the modular symbol attachedto the cusps and ∞ .Proof. We first prove the isomorphism (3.10). Note that one has a natural isomorphism ofΛ θ -modules H ∗ θ / I ∗ θ ≅ H θ / I θ H ≅ Λ θ ⋅ e θ, . Thus, the sequence (3.8) yields the following shortexact sequence of free Λ θ -modules0 Ð→ H θ Ð→ ̃ H θ Ð→ H ∗ θ / I ∗ θ Ð→ . By tensoring h ∗ θ over H ∗ θ on the above sequence, one obtains0 Ð→ H θ Ð→ ̃ H θ,DM Ð→ h ∗ θ / I ∗ θ Ð→ , which yields (3.10) by (3.9).Next, we prove the isomorphism (3.11). It follows from [10, Lemma 1.1.4] that the con-gruence module ̃ H θ,DM ′ / H θ is isomorphic to Λ θ /( f ) for some f ∈ Λ θ . The Drinfield–Maninsplitting induces a surjective homomorphism of Λ θ -modules ̃ H θ,DM ↠ ̃ H θ,DM ′ (see the proofof [13, Lemma 4.1]), which yields another surjective homomorphism of Λ θ -modules ̃ H θ,DM / H θ ↠ ̃ H θ,DM ′ / H θ . It follows from (3.10) that f ∣ ξ θ . We claim that one also has ξ θ ∣ f , which implies (3.11)by the above discussion. It was proved by Ohta that one has a surjectiv homomorphisms ̃ H θ ↠ M Λ and H θ ↠ S Λ on which the action of T ∗ l and U ∗ q on the left commute with T l and U q on the right for all l ∤ N p and for all q ∣ N p . They induce surjective homomorphisms ̃ H θ,DM ′ ↠ M Λ ,DM ′ and ̃ H θ,DM ′ / H θ ↠ M Λ ,DM ′ / S Λ . Thus, the claim follows from (3.6). Theisomorphism ̃ H θ,DM ′ ≅ ̃ H θ,DM follows from (3.10), (3.11), and the Snake lemma.Finially, we note that { , ∞ } θ is part of a basis of lim ←Ð H ( Y ( N p r ) / Q , Z p ) ord θ whose imageunder the boundary map is in Λ θ ⋅ e ( θ, ) . Therefore, { , ∞ } θ,DM is a part of basis of ̃ H θ,DM ,and hence, ̃ H θ,DM / H θ is generated by { , ∞ } θ,DM . ∎ The following corollary will be used in Section 4.
Corollary 3.4.
The elements ξ θ { , ∞ } DM,θ and ( U ∗ q − ) ⋅ { , ∞ } DM,θ for all q ∣ N p are in H − θ . Moreover, we have (3.12) ̟ θ (( U ∗ q − ) ⋅ { , ∞ } DM,θ ) = ( q, − ζ Np r ) r ≥ ,θ ∈ S θ for all q ∣ N p .Proof.
Note that U ∗ q − I ∗ θ for all q ∣ N p . By Proposition 3.3, the proof of the firstassertion is essentially the same as the proof of [13, Lemma 4.8]. For q = p , (3.12) wasproved in [6, Section 10.3] so it reminds to deal with the case q ∣ N . Given any r ∈ Z > , bythe definition of the Hecke actions on the set of cusps (see [11, Section 2.1] for example), wehave ( − U ∗ q ) ⋅ { , ∞ } r = ∑ q − i = { iq , ∞ } r . From the discussion in [13, Section 3.1] and using(3.1)-(3.3) in loc. cit. , one can write the above modular symbols as Manin symbols. Namely,one obtains ∑ q − i = { iq , ∞ } r = ∑ q − i = [ N ′ p r , ] r , where N ′ = N / q . Furthermore, by the definitionof ̟ (see Section 2.1), one has ̟ ( q − ∑ i = [ N ′ p r , ] r ≥ ) = q − ∑ i = { − ζ iq , − ζ Np r } r ≥ = { q − ∏ i = ( − ζ iq ) , ζ Np r } r ≥ = { q, − ζ Np r } r ≥ . Then, by taking θ -components and taking Drinfield–Manin modification, (3.12) for q ∣ N fol-lows. ∎ N SHARIFI’S CONJECTURE: EXCEPTIONAL CASE 11
To close this section, we next show that H − θ, p / I ∗ θ, p H − θ, p is a 1-dimensional vector space over k θ, p . It was proved by Ferrero–Greenberg [3] that the trivial zero of Kubota–Leopoldt p -adic L -functions is a simple zero if it exists. Therefore, I ∗ θ, p is the maximal ideal of h ∗ θ, p by (3.9) andhence, it coincides with p , since it was proved by Betina–Dimitrov–Pozzi [1] that h ∗ θ, p = Λ θ, p .One can deduce from this that H θ, p is a free h ∗ θ, p -module of rank 2, since it is known that H θ is a torsion free h ∗ θ -module and H θ ⊗ h ∗ θ Q ( h ∗ θ ) is a 2-dimensional vector space over Q ( h ∗ θ ) ,where Q ( h ∗ θ ) is the quotient field of h ∗ θ . Therefore, both H + θ, p and H − θ, p are free h ∗ θ, p -modulesof rank 1. The following proposition follows from the above discussion immediately. Proposition 3.5.
One has dim k θ, p H − θ, p / I ∗ θ, p H − θ, p = = dim k θ, p H + θ, p / I ∗ θ, p H + θ, p . Suejectivity of ̟ θ, p The goal of this section is to show that the image of ̟ θ, p is non-trivial, which completesthe proof of Theorem 2.2.We first recall some useful formulas for later use. For a positive integer a < N , the partialzeta function associated to a is defined by ζ a ( Np r ) ( s ) ∶ = ∑ n ≡ a mod Np r n − s . It absolutely converges when Re ( s ) > C − { } which has a simple pole at s = / N [5, Section 1.3.1]. Lemma 4.1.
Let the notation be as above. Then the following assertions hold.(1) If a = a ′ p r for some positive integer a ′ < N , then ζ a ( Np r ) ( s ) = p − rs ζ a ′ ( N ) ( s ) .(2) For any g N ∈ ( Z / N Z ) × and g p r ∈ ( Z / p r Z ) × , we have ζ g N N ζ g pr p r ζ g N N ζ g pr p r − = − ∑ a ∈ Z / Np r Z ζ a ( Np r ) ( )( ζ g N N ζ g pr p r ) a . (3) For r ∈ Z ≥ , one has ∑ i ∈ ( Z / p r Z ) × ζ ip r = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ − if r = , otherwise . Proof.
Note that for each positive integer n with n ≡ a mod N p r , one can write n as a + iN p r for some i ∈ Z ≥ . Suppose a = a ′ p r for some positive integer a ′ < N . Then, one has ζ a ( Np r ) ( s ) = ∑ n ≡ a ′ p r mod Np r n − s = ∞ ∑ i = ( a ′ p r + iN p r ) − s = p − rs ∞ ∑ i = ( a ′ + iN ) − s = ζ a ′ ( N ) ( s ) . This proves the assertion (1). The assertion (2) follows from [5, Lemma 1.3.15(1)] by taking r = t = ζ g N N ζ g pr p r . To prove the assertion (3), it is known that ∑ p r − i = ζ ip r =
0. When r = this implies that ∑ p − i = ζ ip = −
1. If r ∈ Z > , then one has ∑ p ∣ i, ≤ i ≤ p r − ζ ip r = p r − − ∑ i = ζ p r − = − . which implies ∑ i ∈ ( Z / p r Z ) × ζ ip r = ∎ For each r ∈ Z ≥ and each q ∣ N , the cup product ( q, − ζ p − r N ζ p r ) r,θ, p can be identified with thelocal Hilber symbol via the first isomorphism in Proposition 3.2. In the following theorem,those Hilbert symbols will be computed by local explicit reciprocity law. Theorem 4.2.
For each positive integer r and each prime ℓ ∣ N , we have ( ℓ, − ζ p − r N ζ p r ) r,θ, p = ( p − ) log p ( ℓ ) pφ ( N ) τ ( χ − ) L ( , χ ) ∈ ( O / p r O )( ) . Proof.
Note that the action of ( Z / p Z ) × on the group of p r th root of unity is given by a ⋅ ζ p r = ζ ω ( a ) p r for all a ∈ ( Z / p Z ) × . Then, one has ( ℓ, − ζ p − r N ζ p r ) r,θ, p = φ ( N p ) − ∑ g N ,g p χ ( g − N ) ω ( g − p )( ℓ, − ζ p − r g N N ζ ω ( g p ) p r ) r, p . Here g N and g p run through all elements in ( Z / N Z ) × and ( Z / p Z ) × , respectively. Set G N = p − r g N . Using the assumption that χ ( p ) =
1, one can rewrite the above summation as(4.1) φ ( N p ) − ∑ G N ,g p χ ( G − N ) ω ( g − p )( ℓ, − ζ G N N ζ ω ( g p ) p r ) r, p . Here G N runs through all elements in ( Z / N Z ) × , since it follows from the assumption p ∤ φ ( N ) that p r ∈ ( Z / N Z ) × .Set β Np r = ζ G N N ζ ω ( g p ) p r −
1. The the Coleman power series g β Npr ( T ) associated to β Np r isdefined by g β Npr ( T ) = ζ G N N ( + T ) ω ( g p ) − . Then, one has g β Npr ( ζ p r − ) = β Np r . Set δg β Npr ∶ = ( + T ) dg βNpr ( T ) dT × g β Npr ( T ) − . Then, onehas δg β Npr ( ζ p r − ) = ω ( g p ) ζ GNN ζ ω ( gp ) pr ζ GNN ζ ω ( gp ) pr − . By local explicit reciprocity law [4, Theorem 8.18] or [2, Theorem I.4.2] (taking λ ( X ) = + X )one can write (4.1) as(4.2) 1 p r φ ( N p ) ∑ G N ,g p χ ( G − N ) ω ( g − p ) Tr Q p ( µ pr )/ Q p ⎛⎜⎝ log p ( ℓ ) ⋅ ω ( g p ) ζ G N N ζ ω ( g p ) p r ζ G N N ζ ω ( g p ) p r − ⎞⎟⎠ . One can simplify (4.2) by computing the trace as(4.3) log p ( ℓ ) p r φ ( N p ) ∑ G N ,g p χ ( G − N ) ∑ g pr ∈ ( Z / p r Z ) × ζ G N N ζ ω ( g p ) g pr p r ζ G N N ζ ω ( g p ) g pr p r − . N SHARIFI’S CONJECTURE: EXCEPTIONAL CASE 13
Note that for a fixed g p ∈ ( Z / p Z ) × , ω ( g p ) ⋅ ( Z / p r Z ) × = ( Z / p r Z ) × . Hence, by setting G p r = ω ( g p ) g p r , one can simplify (4.3) as(4.4) ( p − ) log p ( ℓ ) p r φ ( N p ) ∑ G N χ ( G − N ) ∑ G pr ∈ ( Z / p r Z ) × ζ G N N ζ G pr p r ζ G N N ζ G pr p r − . By Lemma 4.1(3), one can write (4.4) as(4.5) ( p − ) log p ( ℓ ) p r φ ( N p ) ∑ G N ,G pr χ ( G − N ) ∑ a ∈ Z / Np r Z ζ a ( np r ) ( )( ζ G N N ζ G pr p r ) a . We next rewrite (4.5) in terms of L ( , χ ) by the following observation. ● If a is divisible by N , then ∑ G N ,G pr χ ( G − N ) ζ a ( np r ) ( )( ζ G N N ζ G pr p r ) a = ∑ G N χ ( G − N ) ∑ G pr ζ a ( np r ) ( ) ζ aG pr p r = , since ∑ G N χ ( G − N ) = a is not divisible by N . By setting G ′ N = G N ⋅ a , we have ∑ G N ,G pr χ ( G − N ) ζ a ( np r ) ( )( ζ G N N ζ G pr p r ) a = χ ( a ) ζ a ( Np r ) ( ) ⎛⎜⎝ ∑ G ′ N χ ( G ′ N ) − ζ G ′ N N ⎞⎟⎠ ⎛⎝ ∑ G pr ζ aG pr p r ⎞⎠ = χ ( a ) ζ a ( Np r ) ( ) τ ( χ − ) ∑ G pr ζ aG pr p r . ● If a is divisible by p i for some 0 ≤ i ≤ r −
1, by Lemma 4.1(5), one has ∑ G pr ζ aG pr p r = ● If a is divisible by p r , namely, a = ip r for some 1 ≤ i ≤ N −
1, then by Lemma 4.1(1)and using the assumption that χ ( p ) =
1, one has χ ( a ) ζ a ( Np r ) ( ) τ ( χ − ) ∑ G pr ζ aG pr p r = χ ( p r ) τ ( χ − ) χ ( i ) ζ i ( N ) ( ) . It is known that the Dirichlet L -function L ( s, χ ) satisfies L ( s, χ ) = N − ∑ a = χ ( a ) ζ a ( N ) ( s ) and has analytic continuation to whole complex plane as χ is not a trivial character (forexample, see [16, Ch. 4]). From the above discussion, one can simplify (4.5) as φ ( p r )( p − ) log p ( ℓ ) p r φ ( N p ) τ ( χ − ) ∑ a ∈ Z / N Z χ ( a ) ζ a ( n ) ( ) = ( p − ) log ( ℓ ) pφ ( N ) τ ( χ − ) L ( , χ ) . ∎ Corollary 4.3.
We have (4.6) ( ℓ, − ζ Np r ) r,θ, p = ( p − ) log p ( ℓ ) pφ ( N ) ω ( N ) τ ( χ − ) L ( , χ ) ∈ ( O / p r O )( ) for all ℓ ∣ N and (4.7) ( p, − ζ Np r ) r,θ, p = − ( p − ) pφ ( N ) ω ( N ) τ ( χ − ) L ( χ ) L ( , χ ) ∈ ( O / p r O )( ) . In particular, for all prime q ∣ N p , ( q, − ζ Np r ) r,θ, p is non-zero for all r big enough, and hence,the map ̟ θ, p is surjective.Proof. Let p r,N be the inverse of p r in ( Z / N Z ) × . For each r ∈ Z ≥ , we write ζ Np r = exp ( πi / N p r ) and ζ p − r N ζ p r = exp ( πip r,N / N + πi / p r ) . Then, for each ℓ ∣ N , one has ( ℓ, − ζ Np r ) r,θ, p = ( ℓ, − ( ζ p − r N ζ p r ) /( p r,N p r + N ) ) r,θ, p = θ ( p r,N p r + N ) − ( ℓ, − ( ζ p − r N ζ p r )) r,θ, p . Note that one can simplify θ ( p r,N p r + N ) − as ω ( N ) . Thus, (4.6) follows from Theorem 4.2,and hence, it is non-zero for all r big enough. (4.7) is obtained by (3.12) and by a consequenceof [1, Proposition 5.7], which asserts that for all prime ℓ divides N , one has ( U ∗ p − )/( U ∗ ℓ − ) = − L ( χ ) log p ( ℓ ) − ∈ h ∗ θ, p / I ∗ θ, p , where L ( χ ) ≠ L -invariant attached to χ . ∎ Galois representations attached to cohomology of modular curves
Throughout this section, we set h ∗ ∶ = h ∗ θ, p and I ∗ ∶ = I ∗ θ, p for simplicity. Let Λ θ be Λ θ onwhich G Q acts as follows. For σ ∈ G Q , σ acts as the multiplication by θ − ( σ )⟨ σ ⟩ − , where ⟨ σ ⟩ ∶ = ⟨ a ⟩ ∈ Λ θ for some a ∈ + p Z p satisfying σ ( ζ p r ) = ζ ap r for all r ≥
1. Recall that wehave seen in Section 3.2 that h ∗ = Λ θ, p , I ∗ = p , and both H + θ, p and H − θ, p are free h ∗ -modulesof rank 1. It was shown in the proof of [1, Proposition 2.1] that the Galois representationattached to H + θ, p ⊕ H − θ, p = H θ, p is reducible modulo ( I ∗ ) . Therefore, by fixing a basis of H + θ, p ⊕ H − θ, p over h ∗ , this Galois representation modulo I ∗ can be realized as either an uppertriangular matrix, a lower triangular matrix, or a diagonal matrix. The goal of this sectionis to determine which one is the case.Since the Galois representation attached to H + θ, p / I ∗ H + θ, p ⊕ H − θ, p / I ∗ H − θ, p is reducible, at leastone of H + θ, p / I ∗ H + θ, p and H − θ, p / I ∗ H − θ, p is stable under the action of G Q . We start by showingthat the minus part is Galois stable. Let ( ⋅ , ⋅ ) Λ θ ∶ H θ × H θ → Λ θ be the perfect pairing constructed by Ohta satisfying(5.1) ( σx, σy ) = κ ( σ ) − θ − ( σ )⟨ σ ⟩ − ( x, y ) for all σ ∈ G Q (see [6, Section 1.6.3] for the definition). Recall that we have shown inCorollary 3.4 that ξ θ { , ∞ } θ,DM is in H − θ . By the same argument as in Section 6.3.8 of loc. cit. , the above pairing induces a Λ θ [ G Q ] -module homomorphism ( ⋅ , ξ θ { , ∞ } θ,DM ) Λ θ ∶ H θ / I ∗ θ H θ → Λ θ /( ξ θ ) . N SHARIFI’S CONJECTURE: EXCEPTIONAL CASE 15
Let P be the kernel of the above homomogphism after localizing at p , and let Q ∶ = ( H θ, p / I ∗ H θ, p )/ P ≅ Λ θ, p /( ξ θ, p ) ≅ Λ θ, p / p be the quotient. Indeed, one has P ≅ ( H − θ, p / I ∗ H − θ, p )( − ) and Q ≅ H + θ, p / I ∗ H + θ, p as k θ, p [ G Q ] -modules, since ξ θ, p { , ∞ } DM,θ, p is a basis of H − θ, p / I ∗ H − θ, p over k θ, p . Thus, we obtain a shortexact sequence of k θ, p [ G Q ] -modules(5.2) 0 → H − θ, p / I ∗ H − θ, p → H θ, p / I ∗ H θ, p → H + θ, p / I ∗ H + θ, p → . As in [6, Section 9.6], we have an exact sequence of k θ, p [ G Q ] -modules(5.3) 0 → Q → ̃ H DM,θ, p ( ker ∶ H θ, p → Q ) → ̃ H DM,θ, p H θ, p → H ( Z [ Np , Λ θ, p /( ξ θ, p )( )) by (3.8) and (3.10). Since I ∗ = p isa principle ideal, one obtains from (5.3) by tensoring I ∗ /( I ∗ ) an exact sequence of k θ, p [ G Q ] -modules(5.4) 0 → I ∗ H + θ, p / I ∗ H + θ, p → ( I ∗ H + θ, p ⊕ H − θ, p )/ I ∗ ( I ∗ H + θ, p ⊕ H − θ, p ) → H − θ, p / I ∗ H − θ, p → . The following theorem describes the Galois representations attached to (5.2) and (5.4).
Theorem 5.1.
The short exact sequence (5.4) does not split as k θ, p [ G Q ] -modules, and theshort exact sequence (5.2) splits as k θ, p [ G Q ] -modules.Proof. To prove the first assertion, by the above discussion, it suffices to show that the exten-sion class (5.3) is non-trivial. It is known [6, Theorem 9.6.3] that the desired extension classcoincides with the image of the family ( − ζ Np r ) r ≥ ,θ, p under the canonical homomorphismlim ←Ð r ( Z [ Np , ζ Np r ]) θ, p → H ( Z [ Np ] , Λ θ, p /( ξ θ, p )( )) induced by the short exact sequence0 → Λ θ, p ( ) → Λ θ, p ( ) → Λ θ, p /( ξ θ, p )( ) → ←Ð r ( Z [ Np , ζ Np r ] × θ ⊗ Z p ) ≅ H ( Z [ Np ] , Λ θ ( )) . It is enough to prove that theimage of ( − ζ Np r ) r ≥ ,θ, p in H ( Q p , Λ θ, p /( ξ θ )( )) is not trivial. Let U θ be the θ eigenspaceof the project limit of local unit and C θ be the image of cyclotomic unit in U θ . By [14,Proposition 5.2(a)(ii)] , we know that U θ, p / p ≅ H ( Q p , Λ θ, p /( ξ θ )( )) . Note that both of thesevector spaces are dimension two. By Theorem 3.1(2) in loc. cit. , the image of C θ in U θ, p / p isnot trivial, which proves the first assertion.To prove the second assertion, let ( e − , e + ) be a basis of H − θ, p ⊕ H + θ, p . Recall that the maximalideal p = I ∗ is generated by f = T + − κ ( γ ) ∈ Λ θ, p . Then, it is clear that ( f e + , e − ) is a basisof I ∗ H + θ, p ⊕ H − θ, p . Thus, the second assertion follows from the first assertion as we have seenthat the Galois representation attached to I ∗ H + θ, p ⊕ H − θ, p is reducible modulo ( I ∗ ) . ∎ References [1]
A. Betina, M. Dimitrov, and A. Pozzi , On the failure of gorensteinness at weight eisenstein pointsof the eigencurve , arXiv:1804.00648, (2018).[2] E. de Shalit , Iwasawa theory of elliptic curves with complex multiplication , vol. 3 of Perspectives inMathematics, Academic Press, Inc., Boston, MA, 1987. p -adic L functions.[3] B. Ferrero and R. Greenberg , On the behavior of p -adic L -functions at s =
0, Invent. Math., 50(1978/79), pp. 91–102.[4]
K. Iwasawa , Local class field theory , Oxford Science Publications, The Clarendon Press, Oxford Univer-sity Press, New York, 1986. Oxford Mathematical Monographs.[5]
K. Kato , Lectures on the approach to Iwasawa theory for Hasse-Weil L -functions via B dR . I , in Arith-metic algebraic geometry (Trento, 1991), vol. 1553 of Lecture Notes in Math., Springer, Berlin, 1993,pp. 50–163.[6] K. Kato and T. Fukaya , On conjectures of Sharifi , Preprint, (2012).[7]
K. Kato, T. Fukaya, and R. Sharifi , Modular symbols and the integrality of zeta elements ,arXiv:1507.00495, (2015).[8]
M. Lafferty , Eichler-shimura cohomology groups and iwasawa main conjecture , Ph.D. thesis, (2015).[9]
W. G. McCallum and R. T. Sharifi , A cup product in the Galois cohomology of number fields , DukeMath. J., 120 (2003), pp. 269–310.[10]
M. Ohta , Ordinary p -adic ´etale cohomology groups attached to towers of elliptic modular curves , Compos.Math., 115 (1999), pp. 557–583.[11] , Ordinary p -adic ´etale cohomology groups attached to towers of elliptic modular curves. II , Math.Ann., 318 (2000), pp. 557–583.[12] , Congruence modules related to eisenstein series , Ann. Sci. ´Ecole Norm Sup. (4), 36 (2003), pp. 225–269.[13]
R. Sharifi , A reciprocity map and the two-variable p -adic L -function , Ann. of Math. (2), 173 (2011),pp. 251–300.[14] T. Tsuji , Semi-local units modulo cyclotomic units , J. Number Theory, 78 (1999), pp. 1–26.[15]
P. Wake and C. Wang-Erickson , Pseudo-modularity and iwasawa theory , Amer. J. Math., 140(4)(2018), pp. 977–1040.[16]
L. C. Washington , Introduction to cyclotomic fields , vol. 83 of Graduate Texts in Mathematics, Springer-Verlag, New York, second ed., 1997.
Fakultt fr Mathematik, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria.
E-mail address : [email protected] Morningside Center of Mathematics, No. 55, Zhongguancun East Road, Beijing, 100190,China
E-mail address ::