A generalization of Mazur's theorem (Ogg's conjecture) for number fields
aa r X i v : . [ m a t h . N T ] S e p A GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FORNUMBER FIELDS
DEBARGHA BANERJEE, NARASIMHA KUMAR, AND DIPRAMIT MAJUMDAR
Abstract.
In this article, we prove a generalization of a theorem (Ogg’s conjecture) due to Bary Mazurfor arbitrary N ∈ N and for number fields . The main new observation is a modification of a theoremdue to Glenn Stevens for the congruence subgroups of the form Γ ( N ) for any N ∈ N . This in turn helpus to determine the relevant part of the cuspidal subgroups without dependence on Shimura subgroups. Introduction
Let
Jac ( X ( N )) be the Jacobian abelian variety associated to the modular curve of the form X ( N )for any N ∈ N [2, Definition 6. 1.1]. In [6, Conjecture 2], Ogg conjectured that the rational torsionsubgroups of the Jacobian variety Jac ( X ( N )) co-incide with the cuspidal subgroups if N is a prime. TheOgg’s conjecture has been proved in a landmark paper by Barry Mazur [4]. This is in fact an intermediatestep in the proof of now famous “uniform boundedness theorem for torsion points of elliptic curves over Q ”. It shows that there are only finite many groups that can appear as torsion subgroups for ellipticcurves over Q [4, Theorem 8].There is a recent interest to generalize the work to arbitrary N ∈ N by series of papers by MasamiOhta [7], [8], Hwajong Yoo [15] for square free N ∈ N . Yuan Ren [9] computed the same by reducing itto integers of the form N = DC ∈ N such that D is square free and C | D but for Q -rational points ofJacobian variety of modular curves. It is natural to ask if similar conclusions can be made for arbitrary N ∈ N and for torsion subgroups inside the K -rationals points of Jacobian varieties for fields bigger than Q . Our theorem 2 is a partial generalization of Mazur’s theorem (Ogg’s conjecture) for general N andfor number fields bigger than Q .The key ingredient in Bary Mazur’s proof of the Ogg’s conjecture is the calculation of the indices ofEisenstein ideals inside the Hecke algebras. These indices are also of independent interest as it producescongruences between Eisenstein series and cusp forms (generalization of Ramanujan congruences). Inthis paper, we study the torsion points of the Jacobian of modular curves in the same spirit. We modifythe argument of [13, Theorem 1.3] and prove a variant of the theorem [cf. Theorem 10] that holds forthe congruence subgroup Γ ( N ) for any N ∈ N . This is a key observation in our endeavor.We now describe our results. For a Dirichlet character ϕ , let us denote by G ( ϕ ) the classical Gausssum associated to ϕ and B is the second Bernoulli polynomial. Let ξ be the primitive Dirichletcharacter associated to the Dirichlet character ϕ of conductor n . For a divisor d | N , let µ d be a Mathematics Subject Classification.
Primary: 11F67, Secondary: 11F11, 11F20, 11F30.
Key words and phrases.
Eisenstein series, Modular symbols, Special values of L -functions. primitive d -th root of 1 and let us consider the ring R = Z [ Nφ ( N ) ][ µ d ]. Consider the Eisenstein series E = E d ( ϕ ) ∈ E (Γ ( N )) [cf. Section 3.1, [13, page 541]]. This is a holomorphic Eisenstein series whoseFourier co-efficients are not rational. We now consider the subgroup C Γ ( N ) ( E ) inside the cuspidal group C (Γ ( N )) as defined in [13, p. 523]. The following theorem of this paper gives a description of the finitegroup C Γ ( N ) ( E ) for the Eisenstein series as above [cf. [13, page 541, Example 4.8 ] for Γ ( N )]. For anyfinite abelian group A and q ∈ N , let A [ q ] (respectively A ( q ) ) be the the q part (respectively primes to q ) part of A . For any N ∈ N , let φ ( N ) be the Euler’s totient function. Theorem 1.
Let = d be such that d | N and consider the Stevens’ Eisenstein series E = E d ( ϕ ) forany character ϕ : ( Z /d Z ) × → C × . The cuspidal subgroup associated to E is given by C ( Nφ ( N ))Γ ( N ) ( E ) = R/N um ( β ) , where β = N nd Q p | N (cid:16) − ξ ( p ) p (cid:17) G ( ϕ ) G ( ϕ ) G ( ξ ) B ( ¯ ξ ) and N um ( β ) = ideal ( β ) ∩ R . We can compute the torsion points of the Jacobian modular varieties and this can be seen as ageneralization of Ogg’s conjecture [9, Conjecture 1 .
1] for number fields bigger than Q . In fact, the nexttheorem determines the primes that appear in the support of torsion similar to [9, Theorem 1.2]. Theorem 2.
Let d = 1 be such that d | N . For a character ϕ : ( Z /d Z ) × → C × , let C d,ϕ = | C Γ ( N ) ,d ( E ) | ,where E = E d ( ϕ ) . For q ∤ C d,ϕ N φ ( N ) , the q ∞ -torsion subgroup of Q [ µ d ] rational points of Jac ( X ( N ) ,satisfies the following property: Jac ( X ( N )) tor ( Q [ µ d ])[ ϕ ][ q ∞ ] = 0 . Corollary 3.
Let d = 1 be such that d | N . Let q be a prime such that q ∤ N φ ( N ) Q ϕ C d,ϕ , where C d,ϕ is defined as in Theorem 2. Then, we have, Jac ( X ( N )) tor ( Q [ µ d ])[ q ∞ ] = 0 . By Mordell-Weil theorem, there is a finite set S d for which Corollary 3 is true. The importance ofthe theorem lies in the fact that we can make the set S d explicit and it depends on a carefully chosenEisenstein series invented by Glenn Stevens. However, there is a possibility that the actual support oftorsion subgroup in question is smaller than S d .In [13], Glenn Stevens computed the order of the subgroup C Γ ( E ) of the cuspidal subgroup associatedto any Eisenstein series E ∈ E (Γ ( N )) ⊂ E (Γ ( N )) for the congruence subgroup Γ = Γ ( N ) from theorder of the cuspidal subgroup for Γ = Γ ( N ). We can compute the order of the cuspidal subgroup upto the order of intersection of the said group with the Shimura subgroup.In the present paper, we modify the argument of [13, Theorem 1.3] and prove a variant of theabove theorem [cf. Theorem 10] that holds for the congruence subgroup Γ ( N ) for any N ∈ N . Asa consequence, we can compute the order of cuspidal subgroup associated to an Eisenstein series E ∈ E (Γ ( N )) without dependence on the order of the Shimura subgroups. Using the above mentionedtheorem of Stevens, we can only compute the order of certain component [13, Example 4. 9, page 542]of the cuspidal group associated to the Eisenstein series rather than the whole group because of the GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FOR NUMBER FIELDS 3 dependence on the Shimura subgroups. It is possible to compute the whole cuspidal subgroup using thevariant of the theorem proved in this paper for the congruence subgroup of the form Γ ( N ).We compute the index of Eisenstein ideal for a particular Eisenstein series for the congruence subgroupΓ ( p ) in Proposition 4. We expect that the variant of Stevens’ theorem proved in this paper should beuseful to calculate the index of Eisenstein ideal for arbitrary N ∈ N .Let T R be the Hecke algebra acting on the space of cusps forms over R in the sense of Serre. Considerthe annihilator of the Eisenstein series E as above inside the Hecke algebra T R := T Z ⊗ R and we denotethe same by I R . We now prove the main theorem of this paper that is a generalization of [4, Proposition9.7, p. 96]. Proposition 4.
Let N = p and E = E p ( ψ ) , the indices of Eisenstein ideals are given by: | T R /I R | ≃ | C ( Nφ ( N )) E | . It is expected that the above theorem is true for general N also, however we use the fact that C E is a cyclic Z [ µ p ]-module. In general, this is not true for arbitrary N ∈ N and the structure of C E iscomplicated for general N ; hence we give the proof of the Proposition 4 only for N = p . We stronglybelieve that one should be able to prove the similar equality of sets using the variant of Steven’s theoremproved in this paper.There is a possibility to shrink the set S d for general N even further. This stems from the fact thatwe can only compute the bound of the order of | T R /I R | rather than actual order for general N . However,we can compute the same for N = p . 2. Acknowledgements
The first named author was partially supported by the SERB grants YSS/2015/001491 and MTR/2017/000357.The third named author was partially supported by NFIG grant MAT/16-17/839/NFIG/DIPR.3. preliminaries
Eisenstein series of Glenn Stevens for Γ ( N ) . Let 1 = d be such that d | N . For any character ϕ : ( Z /d Z ) × → C × , we construct the Eisenstein series E ϕ from [13, page 541] by observing that in thecase of Γ ( N ), the characters satisfy the condition that ǫ = ǫ − = ϕ . Hence the conductors of ǫ and ǫ are equal. Since d | N , we can take N = d and N = d . Let T ( N ) = { (cid:0) l m (cid:1) } ⊆ GL ( Z /N Z ) be thestandard torus of GL ( Z /N Z ), one can associate a character ψ : T ( N ) → C × for any homomorphism ϕ defined by ψ ( (cid:0) r s (cid:1) ) = ϕ ( rs ). We define the Eisenstein series [13, Definition 4.6(a)]: E ϕ = 12 d − X r =1 d − X s =1 ϕ ( rs ) φ ( rd , sd ) with φ ( rd , sd ) basis for Eisenstein series E (Γ( d )). We now define the Stevens’ Eisenstein series for Γ ( N )[cf. [13, Definition 4.6(b)]]: E := E d ( ϕ ) = E ϕ | Y p | Nd (1 − ϕ ( p ) p (cid:0) p
00 1 (cid:1) ) Y p | d (1 − ϕ ( p ) p (cid:0) p (cid:1) ) · (cid:0) d
00 1 (cid:1) . DEBARGHA BANERJEE, NARASIMHA KUMAR, AND DIPRAMIT MAJUMDAR
Consider the Z [ ψ ] = Z [ ϕ ] = Z [ µ d ] submodule of D Γ ,d ⊗ Z [ ψ ] on which T ( N ) acts via ψ and call it D Γ ,d ( ψ ). Consider now the direct sum D Γ ( ψ ) = P e | N D Γ ,e ( ψ ). The following Proposition [13, page540] is useful for our purpose: Proposition 5. [13, Proposition 4.5] We have a following description: • D Γ ,d ( ψ ) = 0 . • The ψ -part of the cuspidal group is given by D Γ ,d ( ψ ) = P ϕ ( ab ) " adb with the sum running overall cusps " adb of divisor d . The following proposition is a simple modification of [13, Proposition 4.7, page 541] for the modularcurves of the form X ( N ) for any N ∈ N . Proposition 6.
For the congruence subgroup Γ ( N ) , the Eisenstein series E = E d ( ϕ ) satisfy the fol-lowing properties: (1) For all l ∤ N , we have T l ( E ) = ( ϕ ( l ) + lϕ ( l )) E. (2) The correspodning L -function associated to the Eisenstein series is given by: L ( s, E ) = − G ( ϕ ) Y p | Nd (cid:18) − ϕ ( p ) p s (cid:19) Y p | d (1 − ϕ ( p ) p s − ) L ( s − , ϕ ) L ( s, ϕ ) . (3) For all χ ∈ X − ǫ ( − S , Λ( E, χ,
1) = − ϕ ( m χ ) χ ( d ) Y p | Nd (cid:18) − ϕχ ( p ) p (cid:19) Y p | d (cid:18) − ϕ ¯ χ ( p ) p (cid:19) B ( ϕ ¯ χ ) B ( ϕχ ) . (4) Consider the following quantity β = ϕ ( − N nd Q p | N (1 − ξ ( p ) p ) G ( ϕ ) G ( ξ ) B ( ξ ) . The boundary of thecorresponding Eisentein series is given by: δ Γ ( N ) ( E ) = β D Γ ( N ) ,d ( ψ ) . Proof. (1)-(3) follows from [13, Proposition 4.7, page 541]. For part (4), we change the proof a bit sothat it works for Γ ( N ) rather than on Γ ( N ) as in the loc. cit. As in [13, Equation (5.2)], we have δ Γ ( N ) ( E ) = X n c ( n ) · D Γ ( N ) ,n ( ψ ) , where the sum is over n > d | n, n | Nd . We start with the case N = N N = d . In this case, δ Γ ( N ) ( E ) = c ( N ) · D Γ ( N ) ,N ( ψ ) , with c ( N ) = r Γ ( N ) ,E ( { N } Γ ( N ) ) = e Γ ( N ) ( { N } Γ ( N ) ) a ( E | " N ) . Note that e Γ ( N ) ( { ab } Γ ( N ) ) = Ndt ,where d = ( b, N ) and t = ( d, N/d ). GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FOR NUMBER FIELDS 5
Following Stevens [13, Equation (5.5)], we define E ′ = P N − a =0 ϕ ( a ) E | " aN and note that a ( E | " N ) = | ( Z /N Z ) ∗ | a ( E ′ ) . Finally, following Stevens [13, c.f page 545-547] we compute a ( E ′ ) = − L (0 , E ′ ) = ϕ ( − N N n G ( ϕ ) G ( ξ ) B ( ξ ) Y p | N (1 − ξ ( p ) p ) Y p | N (1 − p − ) . Putting everything together we obtain (when N = N N and N = N = d ) δ Γ ( N ) ( E ) = N ϕ ( − nN G ( ϕ ) G ( ξ ) B ( ξ ) Y p | N (1 − ξ ( p ) p ) · D Γ ( N ) ,N ( ψ ) . The proof in the general case follows from induction. We write A = N N C · ...C k with C , C .. · C k primes integers. The proof follows from induction on k . If k = 0, we already provedthe result. Assume the result is true for a natural number k . We prove it for k + 1. We denote C k +1 by l following Stevens. We denote by π : X ( Al ) → X ( A ) the map induced from injection of Γ ( Al ) → Γ ( A )and π ∗ l : X ( Al ) → X ( A ) the map induced from γ " l
00 1 γ " l −
00 1 .Since the map π l : X ( Al ) → X ( A ) is given by z → lz , hence the pull-back map on Picard group[2, p. 228] satisfy π ∗ l ( δ Γ ( N ) ( E )) = δ Γ ( N ) ( E ( lz )) = δ Γ ( N ) ( E ( " l
00 1 z )). Notice that, we also have π ∗ ( δ Γ ( A ) ( E )) = δ Γ ( Al ) ( E ).We denote by Γ = Γ ( A ) and Γ ′ = Γ ( Al ). Observe that π ∗ ( D Γ ,d ( ψ )) = P k c ( k ) D Γ ′ ,k ( ψ ) , with thesum is over k such that d | k and k | Ald . By the definition of the pull-back maps on Picard groups [2, p.228], π ∗ D Γ ,d ( ψ ) = P ϕ ( ab ) π ∗ { adb } = P ϕ ( ab )( P x ∈ π − adb e ( π ; x ) x ).It is clear that only non-zero k ’s are k = d or k = d and dl depending on l . Note that ( db, A ) = d implies ( b, Ad ) = 1.Observe that, e Γ ′ ( { d } Γ ′ ) = Ald , e Γ ( { d } Γ ) = Ad hence e ( π, { d } Γ ′ ) = l. Similarly we see that if l ∤ Ad , e Γ ′ ( { dl } Γ ′ ) = Ald l , e Γ ( { dl } Γ ) = Ad hence e ( π, { dl } Γ ′ ) = 1 . Case 1:- If l | Ad , we compute ( db, Al ) = ( db, Ad ld ) = d ( b, Ad l ) = d . The last equality follows from thefact that l | Ad and hence ( b, Ad l ) = 1. In this case, π ∗ D Γ ,d ( ψ ) = c ( d ) D Γ ′ ,d ( ψ ). Comparing the co-efficientsfor D Γ ,d ( ψ ) and D Γ ′ ,d ( ψ ), we get ϕ ( d ) e ( π, { d } Γ ′ ) = c ( d ) ϕ ( d ) and hence c ( d ) = e ( π, { d } Γ ′ ) = l .Case 2:- For l ∤ Ad , we compute that " dbAl = " db Ad ld = d " b Ad l . If l | b then ( b, Ad l ) = l and if l ∤ b then( b, Ad l ) = 1 since ( b, Ad ) = 1. In this case, π ∗ D Γ ,d ( ψ ) = c ( d ) D Γ ′ ,d ( ψ ) + c ( dl ) D Γ ′ ,dl ( ψ ). We now proceedto compute c ( d ) and c ( dl ) determined by ramification indices and value of ϕ .Now if l ∤ A , then " ldb Al → " ldb A . Comparing the co-efficients for D Γ ,d ( ψ ) (cusps of level d ) and D Γ ′ ,dl ( ψ ) (cusps of level ld ), we get ϕ ( l ) ϕ ( b ) e ( π, { dl } Γ ′ ) = c ( dl ) ϕ ( b ) and hence c ( dl ) = ϕ ( l ). Combining DEBARGHA BANERJEE, NARASIMHA KUMAR, AND DIPRAMIT MAJUMDAR these facts, we obtain: π ∗ ( D Γ ,d ( ψ )) = l D Γ ′ ,d ( ψ ) if l | Ad ,l D Γ ′ ,d ( ψ ) + ϕ ( l ) D Γ ′ ,ld ( ψ ) if l ∤ Ad . Similarly, we have π ∗ l ( D Γ ,d ( ψ )) = P k c ( k ) D Γ ′ ,k ( ψ ) with co-efficients c ( k ) determined by the corre-sponding coefficient of { k } Γ ′ in π ∗ l ( D Γ ,d ( ψ )) and D Γ ′ ,k ( ψ ). As in the case of π , we get an equality usingthe pull-back maps on Picard groups [2, p. 228]: X ϕ ( ab )( X x ∈ π − l adb e ( π l ; x ) x ) = X k c ( k ) X ϕ ( ab ) " akb Γ ′ . We now compute the ramification indices for the map π l . Recall the description of local charts at thecusps of modular curves [2, p. 62]. We will show that if the cusp of level ld is mapped to a cusp of level d by π l , then the ramification index is l . Consider the subsets V, V ′ ⊂ H ∪ P ( Q ) around the cusps { ld } and { d } respectively and consider the corresponding neighborhood U, e U on the modular curves X ( Al )and X ( A ) respectively and U ′ , f U ′ are the neighborhood of the point 0 of the unit disc. We have adiagram: V π (cid:15) (cid:15) π l / / V ′ e π (cid:15) (cid:15) U φ (cid:15) (cid:15) e π l / / e U e φ (cid:15) (cid:15) U ′ π l / / f U ′ . We will show that the map π l : U ′ → f U ′ is z → z l . Observe that π l is achieved by the matrix τ l = (cid:16) √ l √ l (cid:17) ∈ SL ( R ). Note that δ ld = (cid:0) − ld (cid:1) ∈ SL ( Z ) (respectively δ d = (cid:0) − d (cid:1) ∈ SL ( Z )) is suchthat δ ld ( ld ) = ∞ and δ d ( d ) = ∞ . A small computation using matrices shows that δ d ◦ τ l ◦ δ − ld = τ l .For a cusp s , let δ s ∈ SL ( Z ) be such that δ s ( s ) = ∞ and h s be the width of the cusp s . Considerthe wrapping map ρ s ( z ) = e πizhs and hence ρ − s ( z ) = h s πi log( z ). In particular, let h ld (respectively h d )be the width of the cusp ld (respectively d ) for the modular curves X ( Al ) (respectively X ( A )). By[11, Section 1.4.1], we conclude that h d = C C ...C k = h dl . We now obtain the ramification index by GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FOR NUMBER FIELDS 7 the following computation: π l ( z ) = e φ ◦ e π ◦ π l ◦ ( φ ◦ π ) − = ρ d ◦ δ d ◦ π l ◦ ( ρ ld ◦ δ ld ) − = ρ d ◦ δ d ◦ τ l ◦ ( ρ ld ◦ δ ld ) − = ρ d ◦ τ l ◦ ( ρ ld ) − = e l hldhd log( z ) = e l log( z ) = z l . On the other hand, we claim that if the cusp d goes to the cusp ld and l ∤ d , then the ramification indexis 1. Observe that, in this case h ld = C C ..C k l and h d = C C ..C k l . As above using δ d ◦ τ l ◦ δ − ld = τ l ,we get by a computation similar to above π l ( z ) = z and hence the ramification index is 1 in this case.If l | d , then the cusp of level ld are in bijection with the cusps of level d by τ l . Hence, we have anisomorphsim D Γ ′ ,dl ( ψ ) τ l −→ D Γ ,d ( ψ ). Comparing the co-efficients, we get c ( dl ) = e ( π l ; dlb ) = l . Hence,we get π ∗ l ( D Γ ,d ( ψ )) = l D Γ ′ ,dl ( ψ ).On the other hand if l ∤ d , then the some cusp of level ld and some cusps of level d are mappedthe cusps of level d by τ l . In other words, D Γ ′ ,dl ( ψ ) ⊕ D Γ ′ ,d ( ψ ) τ l −→ D Γ ,d ( ψ ). If the cusp of level d ismapped to cusps of level d , then the map is: " adb → " ladb . To compute the co-efficient c ( d ), observethat ϕ ( lab ) e ( π l ; d ) = c ( d ) ϕ ( ab ) and c ( d ) = ϕ ( l ) if l ∤ d . Hence, we get: π ∗ l ( D Γ ,d ( ψ )) = l D Γ ′ ,dl ( ψ ) if l | d,l D Γ ′ ,ld ( ψ ) + ϕ ( l ) D Γ ′ ,d ( ψ ) if l ∤ d. Following [13, Lemma 5.3], we now define E = E if l | Ad ,E | (1 − ϕ ( l ) l l
00 1 ) if l ∤ Ad , and E = E | l
00 1 if l | d,E | ( l
00 1 − ϕ ( l ) l ) if l ∤ d. By a computation as in [13, Lemma 5.3], we get δ Γ ′ ( E ) = lD Γ ′ ,d ( ψ ) if l | A,l (1 − ξ ( l ) l ) D Γ ′ ,d ( ψ ) if l ∤ A, DEBARGHA BANERJEE, NARASIMHA KUMAR, AND DIPRAMIT MAJUMDAR and δ Γ ′ ( E ) = lD Γ ′ ,ld ( ψ ) if l | A,l (1 − ξ ( l ) l ) D Γ ′ ,ld ( ψ ) if l ∤ A. From this, it follows that the statement holds for Al = d C . . . C k +1 . (cid:3) The group A Γ ( N ) ( E ) of Stevens. In the section, we recall the basic properties of the periods ofthe Eisenstein series for the congruence subgroup Γ ( N ). Definition 7.
Fix a point z ∈ H and let c ( γ ) be the geodesic in Y ( N ) joining z and γ ( z ). Theintegral π E ( γ ) = Z c ( γ ) E ( z ) dz is the period of the Eisenstein series E .The group P Γ ( N ) is the free abelian group generated by π E ( γ ) with γ ∈ Γ ( N ). The details aboutthe group P Γ ( N ) ( E ) can be found in [9, Page 3]. The following assertion follows from [12, page 52] forthe congruence subgroup Γ ( N ). Proposition 8.
Let K be a number field with O K being it’s ring of integers. For any γ = (cid:0) a bc d (cid:1) ∈ Γ ( N ) , (1) we have a homomorphism π E : H ( Y ( N ) , K ) → O K . (2) The homomorphism is given by π E ( γ ) = a + dc a ( E ) − πi L ( E [ (cid:0) − d c (cid:1) ] , if c = 0 and π E ( γ ) = 0 if c = 0 . The above proposition shows that the group P Γ ( N ) ( E ) is the free abelian group generated by the L -values L ( E [ (cid:0) − d c (cid:1) ] , . Computation of R Γ ( N ) ( E ) . Let C ( N ) be the cuspidal subgroup corresponding to the congruencesubgroup Γ ( N ). By [12, page 36], there is an isomorphism δ : E (Γ ( N )) −→ C ( N ). For all cusps x ,let e Γ ( N ) ( x ) denote the ramification index of x over X (1). The Eisenstein series E corresponds to thedivisor δ ( E ) = X x ∈ Cusps { Γ ( N ) } e Γ ( N ) ( x ) a ( E [ x ]) { x } . By [13, Page 538, equation (4.5)], we can compute the ramification index e Γ ( N ) ( x ) explicitely. Wedenote by R Γ ( N ) ( E ) the Z -submodule of C generated by the co-efficients of δ ( E ). Corollary 9.
For E as in Proposition 6, R Γ ( N ) ( E ) is a finitely generated Z module and it is generatedby Rϕ ( k ) for ϕ : ( Z /d Z ) × → C × and R as in Proposition 6.Proof. Recall that R ( E ) is the free Z -modules generated by the coefficients of δ ( E ). By Proposition 6,the module R ( E ) is generated by M ϕ ( k ) for ϕ : ( Z /d Z ) × → C × . (cid:3) By [13, Theorem 1.2], there is a pairing C Γ ( N ) × A Γ ( N ) ( E ) → Q / Z . Since both are finite groups,there is a non-canonical isomorphism: C Γ ( N ) ≃ A Γ ( N ) ( E ). GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FOR NUMBER FIELDS 9 A variant of Stevens’ theorem for Γ ( N )Let S be the set of primes satisfying(1) q ≡ , ( q, N ) = 1 for all q ∈ S ,(2) S intersects every arithmetic progression of the form {− N kr | r ∈ Z } .Consider the set X S of primitive Dirichlet character χ such that conductor of χ is a power of q and q ∈ S and χ is not quadratic. We denote by X + S (resp. X − S ) then set of even (resp. odd) characters of X S .We denote by χ q ( p ) = ( pq ) the quadratic residue modulo q . For a primitive nonquadratic Dirichletcharacter χ whose conductor is power of m , defineΛ ± ( E, χ,
1) = 12 (Λ(
E, χ, ± Λ( E, χχ q , . Observe that χ q is odd character as q ≡ χ ∈ X − ǫ ( − S , the character ¯ ǫ ¯ χ ¯ χ q is a evencharacter. As noted in proof of Theorem 4.2(b), for a even character θ , B ( θ ) = 0, thus, B ( ¯ ǫ ¯ χ ¯ χ q ) = 0.For any M ∈ N , consider the multiplicative closed set S M = { , M, M ... } of Z . Note that S − M Z = Z [ M ] and the prime ideals of Z [ M ] are in bijection with prime ideals q ∈ Z − S M [5, Proposition 11.1,p. 65]. Note that this ring is also a Dedekind domain [5, Proposition 11.4, p. 68]. We now give a proofof Theorem 10 that in turn is the generalization of [[13], Th 2.1, p. 525] for the congruence subgroupΓ ( N ). Theorem 10.
For the congruence subgroup Γ ( N ) , let S be a set of prime considered as above. Let M ′ = N ϕ ( N ) = ϕ ( N ) and let A be a Z [ M ′ ][ X S ] module. Given a cohomology class φ : H ( X ( N ) , Z ) → A such that Λ ± ( φ, χ ) = 0 for all χ ∈ X + S , then we have φ = 0 .Proof. Choose a prime ideal P ⊆ Z [ M ′ ][ X S ] and consider the localization A P of A at P . Consider nowthe cohomology class ϕ P obtained by composing ϕ with the inclusion A → A P .We define Φ P : Γ ( N ) → A P by defining Φ P ( γ ) = ϕ P ( { x, γx } ). We show Φ P = 0 in the following 4steps.Step 1: q ∈ S , q = − N ) and P ∤ q +12 , then ϕ P ( { , Naq } Γ ( N ) ) = 0 whenever ( q, N a ) = 1.We remark that this step works for all P ∈ Z [ X S ], essentially for the same reason as in Step 1 of [13,Theorem 2.1]. However, there is a mistake that we point out in this article and write down the correctversion of it. For the sake of completeness, we recall the proof again. Consider(4.1) X χ =1 χ ( N a )Λ ± ( ϕ P , χ ) = X χ =1 χ ( N a ) q − X k =0 ,χ q ( Nk )= ± ¯ χ ( N k ) ϕ P ( { , N kq } )= q − X k =0 ,χ q ( Nk )= ± X χ =1 χ ( N a ) ¯ χ ( N k ) ϕ P ( { , N kq } )Now, split the case k ≡ a (mod q ) and k a (mod q ). When k ≡ a (mod q ), the sum becomes q − ϕ P ( { , Naq } ). In the other case, q − X k =1 ,k a (mod q ) ,χ q ( Nk )= ± ϕ P ( { , N kq } ) X χ =1 , even χ ( N a ) ¯ χ ( N k ) , Observe that χ ( N a ) ¯ χ ( N k ) = χ ( ak ) . Since the last sum is −
1, then (4.1) becomes X χ =1 χ ( N a )Λ ± ( ϕ P , χ ) = q − ϕ P ( { , N aq } ) − q − X k =1 ,k = a,χ q ( Nk )= ± ϕ P ( { , N kq } )We can rewrite it as X χ =1 χ ( N a )Λ ± ( ϕ P , χ ) = q − ϕ P ( { , N aq } ) + ϕ P ( { , N aq } ) − q − X k =1 ,χ q ( Nk )= ± ϕ P ( { , N kq } )this implies that ϕ P ( { , N aq } ) = 2 q + 1 X χ =1 χ ( N a )Λ ± ( ϕ P , χ ) + 2 q + 1 q − X k =1 ,χ q ( Nk )= ± ϕ P ( { , N kq } ) . By hypothesis Λ ± ( ϕ P , χ ) = 0 and hence: ϕ P ( { , N aq } ) = 2 q + 1 q − X k =1 ,χ q ( Nk )= ± ϕ P ( { , N kq } ) . Now the righthand side of the equation only depends on square classes of a modulo q .If χ q ( a ) = 1, then ϕ P ( { , N aq } ) = 2 q + 1 q − X k =1 ,χ q ( Nk )= ± ϕ P ( { , N kq } ) = ϕ P ( { , Nq } ) . If χ q ( a ) = −
1, then ϕ P ( { , N aq } ) = 2 q + 1 q − X k =1 ,χ q ( Nk )= ± ϕ P ( { , N kq } ) = ϕ P ( { , − Nq } ) , as χ q ( −
1) = −
1, since q is a prime congruent to 3 modulo 4. Hence without loss of generality, a = ± k ∈ Z such that q = − N k ∈ S and β = ± N k ! ∈ Γ ( N ). By the abovechoice of β ∈ Γ ( N ) we have: β · (0) = (0) and β · ( ∓ N ) = ( ± Nq ) . Hence ϕ P ( { , N aq } ) = ϕ P ( { , ± Nq } ) = ϕ P ( { β · (0) , β · ( ∓ N ) } )= ϕ P ( { , ∓ N } ) = Φ P ∓ N !! = 0 , since ∓ N ! ∈ Γ ( N ) is parabolic. GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FOR NUMBER FIELDS 11
Step 2:Φ P vanishes on the subgroup:Γ ′ ( N ) = n a bNcN d ! ∈ SL ( Z ) | a, d ≡ ± N ) o . We mostly follow the proof of Stevens, we make slight modifications while choosing q .Let γ = a bNcN d ! ∈ Γ ′ ( N ). Without loss of generality we can assume d = − N ), asotherwise, we can replace γ by − γ .Note that since P is a prime in Z [ Nφ ( N ) ], we have P ∤ N . Now we show that there is a prime q satisfying q = − N ) and q = − P ). Suppose there is no such prime satisfying theconditions, then if q = − N ), implies q = − P ) and hence q = − N P ). Thiswould imply that density of primes of the form − N and density of primes of the form − N P are same, which is false. In fact, since the density of such primes are different, it showsthere are infinitely many primes q satisfying q = − N ) and q = − P ).Let us choose q = d + N bk such that q = − N ) and q = − P ). Hence P ∤ q +12 and q ∈ S satisfies condition of Step 1.Let β = N k ! ∈ Γ ( N ). By the above choice of β ∈ Γ ( N ) we have: β · (0) = (0) and β · ( N bd ) = (
N bq ) . Thus Φ P ( γ ) = ϕ P ( { , N bd } ) = ϕ P ( { β · (0) , β · ( N dd ) } ) = ϕ P ( { , N aq } ) = 0 . Step 3: Let Γ ′ ( N ) = n a bNcN d ! o ⊂ Γ ( N ) , then Φ P vanishes on Γ ′ ( N ).Since we have ( a, N ) = ( d, N ) = 1, which is equivalent to saying ( a, N ) = ( d, N ) = 1, we see that, a φ ( N = d φ ( N = ± N ) . Let A = a bNcN d ! ≡ a d ! (mod N ). Then A φ ( N ≡ a φ ( N d φ ( N (mod N ) . Since a φ ( N = d φ ( N = ± N ), we see that A φ ( N ∈ Γ ′ ( N ). Now let γ ∈ Γ ′ ( N ), then γ φ ( N ∈ Γ ′ ( N ). Hence Φ P ( γ φ ( N ) = 0 by Step 2. Since Φ P is a homomorphism, we have,Φ P ( γ φ ( N ) = φ ( N )2 Φ P ( γ ) = 0 . Since P ∤ φ ( N )2 , φ ( N )2 is a unit in A P , hence Φ P ( γ ) = 0. Hence for P ∤ φ ( N )2 , Φ P = 0 on Γ ( N ) byStep 1.Step 4: Φ P = 0 . Let A ∈ Γ ( N ) be the matrix a bN c d ! , now consider the parabolic matrix x ! . Since( a, N ) = 1, we can choose x such that ax ≡ b (mod N ) has a solution. Clearly, we then have AB − ∈ Γ ′ ( N ). By Step 3, Φ P ( AB − ) = 0 and by by a well-know theorem due to Manin [3], we have Φ P ( B ) = 0.Hence, we conclude that Φ P ( A ) = Φ P ( AB − ) + Φ P ( B ) = 0.Since the homomorphism Γ ( N ) → H ( X ( N ) , Z ) is surjective, we see ϕ P = 0. Since ϕ P = 0 for allprimes P in Z [ M ′ ][ X S ], we get ϕ = 0. (cid:3) As a consequence, we prove the following Theorem that is a direct generalization of [[13], Th 1.3, p.524]:
Theorem 11.
Consider the congruence subgroup Γ ( N ) for any N ∈ N . Let M ⊂ C be a finitelygenerated Z [ M ′ ] submodule. Then P Γ ( E ) ⊂ M if and only if (1) R Γ ( E ) ⊂ M . (2) Λ ± ( E, χ, ∈ M [ χ, q χ ] . Proof.
Suppose first that P Γ ( E ) ⊂ M . Since R Γ ( E ) ⊂ P Γ ( E ), we have R Γ ( E ) ⊂ M . That Λ ± ( E, χ, ∈ M [ χ, q χ ] . By Lemma 2. 2 (a) of [13] which works for Γ ( N ).We now prove that R Γ ( E ) ⊂ M and Λ ± ( E, χ, ∈ M [ χ, q χ ] then P Γ ( E ) ⊂ M . We consider the F asin the proof of [[13], Th 1.3, p. 524] which is the fraction field of M . For any field K linearly disjointfrom F we have a natural isomorphism M ⊗ K → M K . The rest of the proof goes exactly similar tothe proof of [[13], Th 1.3 (b)] by considering the prime P to be any prime ideal of Z [ M ′ ] and taking thelocalizations with respect to these prime ideals and then using Theorem 10 for the congruence subgroupΓ ( N ). (cid:3) We now prove the main theorem 1 of this paper.
Proof.
By Proposition 6, we have δ Γ ( N ) ( E ) = ϕ ( − N nd Y p | N (1 − ξ ( p ) p ) G ( ϕ ) G ( ξ ) B ( ξ ) D Γ ( N ) ,d ( ψ ) . Since R Γ ( N ) ( E ) is by definition the free Z -module generated by the co-efficients of δ Γ ( N ) ( E ), wededuce that R Γ ( N ) ( E ) = β Z [ ψ ]. In our case Z [ ψ ] = Z [ ϕ ]. Recall that for any finite abelian group A , A ( Nφ ( N )) = A ⊗ Z [ Nφ ( N ) ]. Hence R Γ ( N ) ( E ) ( Nφ ( N )) = β Z [ Nφ ( N ) ][ ψ ].By Proposition 6, for all χ ∈ X − ǫ ( − S , we have or all χ ∈ X − ǫ ( − S ,Λ( E, χ,
1) = − ϕ ( m χ ) χ ( d ) Y p | Nd (cid:18) − ϕχ ( p ) p (cid:19) Y p | d (cid:18) − ϕ ¯ χ ( p ) p (cid:19) B ( ¯ ϕ ¯ χ ) B ( ¯ ϕχ ) . GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FOR NUMBER FIELDS 13
We note that, Λ(
E, χχ m ,
1) = 0 as B ( ¯ ǫ ¯ χ ¯ χ m ) = 0. Hence,Λ ± ( E, χ,
1) = − ϕ ( m χ ) χ ( d ) Y p | Nd (cid:18) − ϕχ ( p ) p (cid:19) Y p | d (cid:18) − ϕ ¯ χ ( p ) p (cid:19) B ( ¯ ϕ ¯ χ )2 B ( ¯ ϕχ )2 . Now, by Theorem 4.2 (b), we see that Λ ± ( E, χ, ∈ Z [ Nφ ( N ) ][ ψ, χ, m χ ] for every χ ∈ X − ǫ ( − S .Here we remark that, Λ ± ( E, χ, ∈ Z [ M ][ ψ, χ, m χ ], where M = Q p | Nd ,p ∤ d p and hence Λ ± ( E, χ, ∈ Z [ Nφ ( N ) ][ ψ, χ, m χ ]. For m ∈ S and P ∤ m a prime in ¯ Q , Theorem 4.2 (a), (c) assures existence of χ with m power conductor such that Λ ± ( E, χ,
1) is a P − unit. Thus from Theorem 11, we obtain P Γ ( N ) ( E ) ( Nφ ( N )) ⊆ Z [ 1 N φ ( N ) ][ ψ ] + β Z [ 1 N φ ( N ) ][ ψ ] . Assume that 1 P Γ ( N ) ( E ). There exists a prime away from primes dividing M , say p , such that P Γ ( N ) ( E ) ⊆ ( p ). By Lemma 2.2 of [13], we see that Λ( E, χ, ≡ p ). Then, second last line ofpage of 537, says that(4.2) (1 − χ ( N )) B ( χ ) B ( ¯ χ ) ≡ p ) . By [13, Theorem 4.2(a)], we have that there are infinitely many χ such that p ∤ (1 − χ ( N )). By [13,Theorem 4.2(c)], we see that B ( χ ) is a p -adic unit for all but finitely many. That forces that 4.2 cannothold. Then it follows from that P Γ ( N ) ( E ) ( Nφ ( N )) = Z [ 1 N φ ( N ) ][ ψ ] + β Z [ 1 N φ ( N ) ][ ψ ] . Then, A ( Nφ ( N ))Γ ( N ) ( E ) = P Γ ( N ) ( E ) ⊗ Z [ 1 N φ ( N ) ] / R Γ ( N ) ( E ) ⊗ Z [ 1 N φ ( N ) ]= ( Z [ 1 N φ ( N ) ][ ψ ] + β Z [ 1 N φ ( N ) ][ ψ ]) /β Z [ ψ ] ∼ = Z [ 1 N φ ( N ) ][ ψ ] / ( Z [ 1 N φ ( N ) ][ ψ ] ∩ ( β )):= Z [ 1 N φ ( N ) ][ ψ ] / Num( β ) . By [[13], Th 1.2 (a), p. 522] valid for Γ ( N ), using the perfect duality pairing we conclude that: C ( Nφ ( N ))Γ ( N ) ( E ) ∼ = A ( Nφ ( N ))Γ ( N ) ( E ) ∼ = Z [ 1 N φ ( N ) ][ ψ ] /N um ( β ) , (cid:3) Index of the Eisenstein ideal for the give E Let R = Z [ Nφ ( N ) ][ µ d ] be a ring and S be any R algebra. We denote by S (Γ ( N )) the complex vectorspace of classical modular forms as in [2]. For f ∈ S (Γ ( N )), let f ( q ) denotes the Fourier expansion ofthe modular form at the cusp ∞ . For any Z algebra S , let us define the p -adic modular forms in thesense of Serre-Swinnerton-Dyer: M B (Γ ( N ) , S ) = { f | f ∈ M (Γ ( N ) and f ( q ) ∈ S [[ q ]] } and we denote by S B (Γ ( N ) , S ) the corresponding space of cusp forms. For l ∈ N , the Hecke operators T ( l ) act on the the space S (Γ ( N ) , Z ). Let T ⊂ End Z ( S B (Γ ( N ) , Z ) be Hecke algebra that is a Z -algebra generated by T ( l ) for all l ∈ N [8, p. 276] and let T ( N ) be corresponding algebra acting on thespace M B (Γ ( N ) , Z ). Consider the following ideal inside the Hecke algebra: e I R = ( T ( l ) − ( ϕ ( l ) + ϕ ( l ) l ) ⊂ T R := T ⊗ R. We now define the Eisenstein ideals:
Definition 12.
For any Eisenstein series E ∈ E (Γ ( N ); C ), let us define the Eisenstein ideal over R tobe the ideal I R := I R ( E ) = Ann T ( N ) ⊗ R ( E ) ⊂ T R .Clearly, we have an inclusion e I R ⊂ I R . Recall the following fundamental Proposition of Ribet [10,Theorem 2.2] and Diamond-Im [1, Proposition 12. 4. 13, p. 118]: Proposition 13.
For any Z algebra S , the pairing: S B (Γ ( N ) , S ) × T S → S given by ( f, T l ) → a ( T l f ) is a perfect pairing of A - modules. In particular, it shows that S B (Γ ( N ) , S ) ≃ Hom S ( T , S ) . By base changing over R , we get the correponding paring over R . Proposition 14. If ℓ ≥ then T ℓ can’t act as ϕ ( ℓ ) + lϕ − ( ℓ ) on the space of cuspforms.Proof. Suppose f is a Hecke eigenform with T ℓ ( f ) = [ ϕ ( ℓ ) + lϕ − ( ℓ )] f for all l ∤ N . By the well-known Deligne bound, we have | a ℓ ( f ) | ≤ √ ℓ . We deduce that | ϕ ( ℓ ) + ℓϕ − ( ℓ ) | ≤ ℓ and hence ( ϕ ( ℓ ) + lϕ − ( ℓ ))( ϕ − ( ℓ )+ lϕ ( ℓ )) ≤ ℓ . Let ϕ ( ℓ ) = x + iy with x + y = 1. We deduce that 1+ ℓ +2 ℓ (2 x − ≤ ℓ for some real number x with | x | ≤
1. We deduce that(1 − ℓ ) ≤ ℓ (1 − x ) ≤ ℓ. If ℓ ≥
7, then the inequality does not hold. This is a contradiction and T l can’t act as ϕ ( ℓ ) + lϕ − ( ℓ ) onthe space of cuspforms.Writing any cuspform f as linear combination of Hecke eigenforms, it is easy to see that correspondingeigen values of the Hecke eigenforms can’t be ϕ ( ℓ ) + lϕ − ( ℓ ). (cid:3) We remark that if ϕ is trivial or quadratic, the above proposition holds for all ℓ . If ϕ is cubic, thenthe above proposition holds for all ℓ ≥ Proposition 15.
We have an equality of ideals I R = e I R .Proof. We have an obvious inclusion e I R ⊂ I R . Note that δ Γ ( N ) ( T l ( E )) = T l · δ Γ ( N ) ( E ) [13, p. 539].This shows that if T ∈ I R then T ∈ Ann T ( δ Γ ( E )). Since T ∈ T R ( R algebra generated by the Heckeoperators T l ’s), hence we can write T = P nl =1 c l T l for Hecke operators T l ’s and c l ∈ R . The Heckeoperators T l ’s act by the multiplication by scalars b l = ϕ ( l ) + ϕ ( l ) l [14, p. 847]. Now T · δ Γ ( E ) = 0,implies that P nl =1 c l b l δ Γ ( E ) = 0. Hence, we deduce that P nl =1 c l b l β D Γ ( N ) ,d ( E ) = 0. We write down D Γ ( N ) ,d ( E ) explicitly [13, Proposition 4.5] and note that ϕ ( k ) (for all k ) and β are non-zero complexnumbers. Hence, we deduce that P nl =1 c l b l = 0. In other words, T = P nl =1 c l ( T l − b l ) and this impliesin turn I R ⊂ e I R . (cid:3) GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FOR NUMBER FIELDS 15
In the following proposition, we give a bound to indices of Eisenstein ideals.
Proposition 16.
The indices of Eisenstein ideals are bounded by the orders of the Eisenstein part ofthe cuspidal subgroup: | T R /I R | ≤ | C ( Nφ ( N )) E | . Proof.
We need to show that the natural map R → T R /I R is onto. Recall that modulo the ideal I R (since modulo e I R ), we have all T l ≡ ϕ ( l ) + lϕ ( l ) and hence it is image of an element from Z [ ϕ ] ⊂ R .Note that T R /I R can’t be isomorphic to R , since this will force that there is a cusp form over R ⊂ C with eigenvalue ϕ ( l ) + lϕ ( l ); a contradiction to Proposition 14. Hence, we show that T R /I R ≃ R/J forsome ideal J in R . By Proposition 13, we have a perfect pairing: S B (Γ ( N ) , R ) ≃ Hom R ( T R , R ) . We have following isomorphisms: S B (Γ ( N ) , R )[ I R ] ≃ Hom R ( T R , R )[ I R ] ≃ Hom R ( T R /I R , R )= Hom R ( R/J, R ) ≃ R/J.
In other words, there is a cusp form f ∈ S (Γ ( N ) , C ) such that f = E d ( ψ ) + Jh for some h ∈ M (Γ ( N ) , C ).Note that the Fourier expansion of f at all cusps are zero. By Proposition 6, δ Γ ( N ) ( E ) = β D Γ ( N ) ,d ( ψ )and hence the coefficients of the Fourier expansion of E at all cusps are in the ring N um ( β ) ∩ R = J ′ .We conclude that the ideal J ′ of R is a subset of the ideal J . As a consequence, we get | C ( Nφ ( N )) E | ≥| T R /I R | . (cid:3) Corollary 17.
Index of the Eisenstein ideal divides the order of the cuspidal subgroup, | T R /I R | | | C ( Nφ ( N )) E | . This fact follows easily as J ′ ⊆ J as an ideal of R as seen in the proof of Proposition 16.We now prove the Proposition 4 of this paper that completely determines the indices of Eisensteinideals for N = p . Proof.
By Theorem 1, we have C ( Nφ ( N )) E = R/J ′ with R = Z [ ! p ( p − ][ ψ ] and ideal J ′ := N um ( β ) ⊂ R .Using [13, Proposition 4.5], we deduce that D Γ ( p ) ,p ( ψ ) is a cyclic Z [ ψ ] module generated by D Γ ( p ) ,p ( ψ ).Hence, arbitrary element of D Γ ( p ) ,p ( ψ ) is of the form a ( ψ ) D Γ ( p ) ,p ( ψ ). Since C E is the image of D Γ ( p ) ,p ( ψ ), hence this is also cyclic. We deduce that End ( C E ) = C E . Note that I R ( C E ) = 0 [13, p.539] and this implies the R -module R/J ′ is annihilated by J and hence J ⊂ J ′ ⊂ R . We conclude that | T R /I R | = | R/J | ≥ |
R/J ′ | = | C ( Nφ ( N )) E | . We deduce the theorem by above and Proposition 16. (cid:3) A generalization of Ogg’s conjecture
We now prove a generalization of Ogg’s conjecture for Γ ( N ). Proof.
For the field K = Q [ µ d ], consider the group scheme Jac ( X ( N ))( K )[ p ∞ ]. By the Eichler-Shimurarelation for the Γ ( N ) [7, Lemma 3. 1. 3], the Hecke operators T l ’s act by F rob l + lF rob − l for the group scheme J ( N ) / F l for all l ∤ N . The same is true for the group scheme Jac ( X ( N ))( K )[ p ∞ ]for primes p ∤ N by the same argument as in loc. cit. The Galois group Gal( K/ Q ) acts on V K = Jac ( X ( N ))( K ) and defines a representation Gal( K/ Q ) → GL( V K ). By Maschke’s theorem for fieldswith characteristic co-prime to | G | , every representation of a finite group is a direct sum of irreduciblerepresentations. Note that the Galois group Gal( K/ Q ) is abelian and by Schur’s Lemma, every irre-ducible representations of abelian groups are 1 dimensional. By definition, 1-dimensional subspaces areof the form: V K [ ϕ ] = Jac ( X ( N ))( K )[ ϕ ] = { P | P ∈ V K ; g · P = ϕ ( g ) P } . Hence, we have a decomposition V K = L ϕ :( Z /d Z ) × → K × V K [ ϕ ]. On the subspace V K [ ϕ ], we have T l ( P ) =( F rob l + lF rob − l ) · P = ( ϕ ( l ) + lϕ ( l ) − ) P . Hence, we conclude that I R P = 0.We claim that if P ∈ V K [ ϕ ][ p ∞ ] and P = 0, then p | X := | T R /I R | . If possible, p ∤ | T R /I R | then( p, | T R /I R | ) = 1. Hence, there exist integers l and m such that lp + mX = 1. Note that 1 ∈ T R and hence X ∈ I R (by the definition of index) and hence I R P = 0. Since P ∈ V K [ ϕ ][ p ∞ ], we deduce that P = 0; acontradiction. Hence, p | X := | T R /I R | . Now from Corollary 17, it follows that p | | C ( Nφ ( N )) E | . (cid:3) References [1] F. Diamond and J. Im,
Modular forms and modular curves , in Seminar on Fermat’s Last Theorem(Toronto, ON, 1993–1994), Vol. 17 of
CMS Conf. Proc. , 39–133, Amer. Math. Soc., Providence, RI(1995).[2] F. Diamond and J. Shurman, A first course in modular forms, Vol. 228 of
Graduate Texts inMathematics , Springer-Verlag, New York (2005).[3] J. I. Manin,
Parabolic points and zeta functions of modular curves , Izv. Akad. Nauk SSSR Ser. Mat. (1972) 19–66.[4] B. Mazur, Modular curves and the Eisenstein ideal , Inst. Hautes ´Etudes Sci. Publ. Math. (1977),no. 47, 33–186 (1978).[5] J. Neukirch, Algebraic number theory, Vol. 322 of
Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences] , Springer-Verlag, Berlin (1999), ISBN 3-540-65399-6. Translated from the 1992 German original and with a note by Norbert Schappacher, Witha foreword by G. Harder.[6] A. P. Ogg,
Diophantine equations and modular forms , Bull. Amer. Math. Soc. (1975) 14–27.[7] M. Ohta, Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties , J. Math.Soc. Japan (2013), no. 3, 733–772. GENERALIZATION OF MAZUR’S THEOREM (OGG’S CONJECTURE) FOR NUMBER FIELDS 17 [8] ———,
Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties II , TokyoJ. Math. (2014), no. 2, 273–318.[9] Y. Ren, Rational torsion subgroups of modular Jacobian varieties , J. Number Theory (2018)169–186.[10] K. A. Ribet,
Mod p Hecke operators and congruences between modular forms , Invent. Math. (1983), no. 1, 193–205.[11] W. Stein, Modular forms, a computational approach, Vol. 79 of Graduate Studies in Mathematics ,American Mathematical Society, Providence, RI (2007), ISBN 978-0-8218-3960-7; 0-8218-3960-8.With an appendix by Paul E. Gunnells.[12] G. Stevens, Arithmetic on modular curves, Vol. 20 of
Progress in Mathematics , Birkh¨auser Boston,Inc., Boston, MA (1982), ISBN 3-7643-3088-0.[13] ———,
The cuspidal group and special values of L -functions , Trans. Amer. Math. Soc. (1985),no. 2, 519–550.[14] S.-L. Tang, Congruences between modular forms, cyclic isogenies of modular elliptic curves andintegrality of p -adic L -functions , Trans. Amer. Math. Soc. (1997), no. 2, 837–856.[15] H. Yoo, On Eisenstein ideals and the cuspidal group of J ( N ), Israel J. Math. (2016), no. 1,359–377.(2016), no. 1,359–377.