Level compatibility in Sharifi’s conjecture

Abstract Romyar Sharifi has constructed a map 
$\varpi _M$
 from the first homology of the modular curve 
$X_1(M)$
 to the K-group 
$K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M+\zeta _M^{-1}, \frac {1}{M}]) \otimes \operatorname {\mathrm {\mathbf {Z}}}[1/2]$
 , where 
$\zeta _M$
 is a primitive Mth root of unity. Sharifi conjectured that 
$\varpi _M$
 is annihilated by a certain Eisenstein ideal. Fukaya and Kato proved this conjecture after tensoring with 
$\operatorname {\mathrm {\mathbf {Z}}}_p$
 for a prime 
$p\geq 5$
 dividing M. More recently, Sharifi and Venkatesh proved the conjecture for Hecke operators away from M. In this note, we prove two main results. First, we give a relation between 
$\varpi _M$
 and 
$\varpi _{M'}$
 when 
$M' \mid M$
 . Our method relies on the techniques developed by Sharifi and Venkatesh. We then use this result in combination with results of Fukaya and Kato in order to get the Eisenstein property of 
$\varpi _M$
 for Hecke operators of index dividing M.


Introduction and notation
Sharifi [8] has constructed a beautiful and explicit map (1.1) between modular symbols and a cyclotomic K-group.This map is conjecturally annihilated by a certain Eisenstein ideal.This conjecture, despite its apparent simplicity, turns out to be highly nontrivial and has led to much work in recent years, in particular by Fukaya and Kato [4] and more recently by Sharifi and Venkatesh [9].
This paper is devoted to the study of certain norm relations satisfied by Sharifi's map.This aspect has been studied before by Fukaya-Kato and Scott [11].Their results are, however, quite restrictive (cf.Remark 1.5 for a detailed comparison between their results and ours).We use the techniques developed by Sharifi and Venkatesh to remove most of these restrictions.
Our main motivation is to apply the results of the present note to obtain results toward the Birch and Swinnerton-Dyer conjecture in the "Eisenstein" case [6].We now set up some notation and describe our results in details.
Level compatibility in Sharifi's conjecture 1195

Homology of modular curves and Hecke operators
Let M ≥ 4 be an integer.Let and denote by X 1 (M) the compact modular curve (over C) of level 1 (M).Let be the set of cusps of X 1 (M), and let C 0 M be those cusps in C M of the form 1 (M) ⋅ a b with gcd(a, b) = 1 and a / ≡ 0 (modulo M) (in the case b = 0, we have the cusp Let H 1 (X 1 (M), C M , Z) be the singular homology of X 1 (M) relative to C M .If α and β are in P 1 (Q), let {α, β} ∈ H 1 (X 1 (M), C M , Z) be the class of the hyperbolic geodesic from α to β in X 1 (M).The group H 1 (X 1 (M), C M , Z) carries an action of various Hecke operators, which we now recall.
If is a prime number, the th Hecke operator T is the double coset operator This gives by functoriality an action of (Z /M Z) × / ± 1 on H 1 (X 1 (M), C M , Z).Note that diamond operators act on the set of cusps C M and that this action preserves C 0 M .We say that two cusps c and c ′ are in the same diamond orbit if there exists x ∈ (Z /M Z) × such that ⟨x⟩ ⋅ c = c ′ .
There are also dual Hecke operators: if T is one of the operators defined above, we let As is well known (cf.[1, Theorem 5.5.3]),we have ⟨x⟩ * = ⟨x⟩ −1 (for all x ∈ (Z /M Z) × ) and T * = ⟨ ⟩ −1 T (for all primes ∤ M).

(Dual) Manin symbols
and M ∤ d.The restriction

Algebraic K-theory and motivic cohomology
Fix an algebraic closure . We have a canonical group isomorphism If A is a commutative ring, let K 2 (A) be the second K-group of A, as defined by Quillen.For any x, y ∈ A × , there is an element {x, y} of K 2 (A), called the Steinberg symbol of x and y.It is bilinear in x and y and has the property that if x + y = 1, then {x, y} = 1.
There is an action of Gal(Q(ζ M )/ Q) (and in particular of the complex conjugation) on 1  2 ] on which the complex conjugation acts trivially.Note that K M is a (Z /M Z) × / ± 1-module.

Sharifi's ϖ M map and summary of known results
The map Z[S 0 M ] → K M given by ), and thus induces a map Let us note that our map ϖ M is, in the notation of [9, Proposition 4.
As Sharifi and Venkatesh mention right after [9,Theorem 4.3.6], it is expected that the conjecture holds without restricting ϖ M to H 1 (X 0 (M), Z).
This conjecture has a history of partial results: [4,5] and most recently [9].Let us recall the main results of Sharifi-Venkatesh and Fukaya-Kato on this conjecture.
is annihilated by the Hecke operators T − ⟨ ⟩ − 1 for primes not dividing M and by the Hecke operators U − 1 for primes | M.
We refer to [4, Theorem 5.2.3 (1)] for this result.Let us note that Fukaya and Kato actually consider (the p-ordinary part of) H 1 (Y 1 (M), Z p ) instead of H 1 (X 1 (M), C M , Z p ).These two groups are canonically isomorphic, but the isomorphism transfers dual Hecke operators (i.e., T * , U * or ⟨x⟩ −1 ) to usual Hecke operators (i.e., T , U or ⟨x⟩).

Our main results
Another important aspect of Sharifi's theory is the way in which the maps ϖ M relate with each other when varying M.This has been studied under some assumptions in [4,11].If p is a prime, there are two degeneracy maps π 1 , π 2 ∶ X 1 (M p) → X 1 (M) given on the upper half-plane by π 1 ∶ z ↦ z and π 2 ∶ z ↦ pz.On the K-side, there is a norm map Norm ∶ K M p → K M .Our main result is the following.Theorem 1. 4 Let p ≥ 2 be a prime number, and let M ≥ 4. Let C ⊂ C 0 M p be a subset of cusps which are all in the same orbit under the action of Ker((Z /M p Z) × → (Z /M Z) × ) (the action being given by diamond operators as recalled above).
(i) Assume that p divides M. We have a commutative diagram (ii) Assume that p does not divide M. We have a commutative diagram The étale Chern class map (cf.[10]) provides an isomorphism ) and K M is identified with the fixed part by the complex conjugation in Fukaya and Kato's map actually takes values in K M (by construction).They also do not need to restrict to the subset C of C 0 M p .Their techniques rely on p-adic Hodge theory.(ii) Similarly, Theorem 1.4(ii) has been proved (for the absolute homology) by Scott in [11,Theorem 7] after tensoring by Z for a prime ≠ p dividing M (Scott's p is our and vice versa).Scott relies on the techniques of Fukaya and Kato.(iii) Thus, the main novelty of our result is to work with Z coefficients.This is because we rely instead on the motivic techniques of Sharifi and Venkatesh.(iv) It would be interesting to allow less restrictive conditions on C, and in particular replace H 1 (X 1 (M), Z) in the bottom line of our diagrams by a relative homology group.We were actually able to improve slightly our result when C contains the cusp ∞ (cf.diagrams (5.7) and (5.11)).We were not able to go beyond these results because the techniques of Sharifi and Venkatesh essentially deal with the absolute homology of modular curves.(v) The techniques of Sharifi and Venkatesh, combined with the result of Section 4 actually show that the restriction of in the same diamond orbit as ∞.This is a slight improvement on Theorem 1.2 (which holds for the restriction of ϖ M to H 1 (X 1 (M), Z)).
By combining Theorem 1.4 and the results of Fukaya and Kato, one gets the following result. 1 6 ] obtained by restricting ϖ M to H 1 (X 1 (M), Z) and inverting 6 is annihilated by the Hecke operator U − 1 for all primes dividing M. Here, U is the classical Hecke operator of index , corresponding to the double coset of ( 1 0 0 ).
Remark 1.7 (i) As mentioned above, Theorem 1.6 goes beyond Theorem 1.2.Our result thus completes the proof of Conjecture 1.1 for the absolute homology, after inverting 6. (ii) Fukaya and Kato proved Theorem 1.6 after tensoring with Z p for a prime p ≥ 5 dividing M (cf.Theorem 1.3).Our trick is to use Theorem 1.3 after adding p to the level, and then descend using Theorem 1.4(ii).The reason we have to invert 6 is that Fukaya and Kato assume that p ∤ 6 (note that 2 is inverted anyway in the definition of ϖ M ).It would be nice to be able to avoid inverting 3 in our result.
The plan of this paper is as follows: in Section 2, we recall some basic facts and notation about various kinds of homology and cohomology groups.In Section 3, we recall some constructions of Sharifi and Venkatesh.In Section 4, we explain how to use the cocycle of Sharifi and Venkatesh to produce a map on a certain relative homology group.Finally, in Section 5, we prove Theorems 1.4 and 1.6.

Background and notation regarding homology and cohomology
Let be a torsion-free finite index subgroup of SL 2 (Z) (e.g., = 1 (M) for M ≥ 3).Let Y = /h be the open modular curve of level , where h is the upper half-plane.We denote by X the corresponding compactified modular curve: we have X = Y ∪ C where C = /P 1 (Q) is the set of cusps of X.
We denote by H 1 (X, C, Z) the first singular homology group of X relative to C. We have the following exact sequence coming from the long exact sequence for the pair (X, C): where the map Z[C] → Z is the degree map.
The Poincaré duality yields a perfect bilinear pairing H 1 (X, C, Z) × H 1 (Y , Z) → Z (also called the intersection pairing, due to its interpretation in terms of intersection number of cycles).Under this duality, (2.1) becomes where the map The first cohomology group H 1 (G, T) can be computed as the abelian group of 1-cocycles c ∶ G → T modulo the cocycles of the form c(g) = gx − x for some x ∈ T (independent of g).
Similarly, using the projective resolution of Z as a Z[G]-module in terms of inhomogeneous chains, one can compute the first homology group H 1 (G, T) as Finally, let us recall that since is torsion-free, it is isomorphic to the fundamental group of Y, and therefore we have canonical group isomorphisms

Reminders from the work of Sharifi and Venkatesh
Sharifi and Venkatesh constructed a 1-cocycle Here, Let us recall a characterization of Θ.Let where D runs through all the irreducible divisors of G 2 m .There is a divisor map where v is the valuation coming from D (cf.[9, equation (2.6)]).The map ∂ induces an embedding (for every M ≥ 4).Here, the action of 0 (M) on ) is given as follows: we have a surjective group homomorphism given by and The idea is to "evaluate" Θ(γ) at This does not make sense in general because f (1, ζ M ) or g(1, ζ M ) may not be well defined (f or g may have a pole or zero at (1, ζ M )).
The idea of Sharifi and Venkatesh is to prove that Θ(γ) is actually a combination of Steinberg symbols which can be evaluated at (1, ζ M ).They make this precise by using motivic cohomology groups.We refer to [9, Section 2.1] for the precise definition and results they are using regarding motivic cohomology groups.In particular, if U ⊂ G 2  m is an open subset, there is a motivic cohomology group H 2 (U , 2) (which is an abelian group).As explained in [9, Remark 2.2.3], the functorial map . By [9, Corollary 4.2.5],Θ M (γ) ∶= s * M (Θ(γ)) actually belongs to the subgroup We therefore have a 1-cocycle By [9, Proposition 4.2.1], the cocycle Θ M is parabolic.This means that if c ∈ P 1 (Q) and c ⊂ 0 (M) is the stabilizer of c, then the restriction of Θ M to c is a coboundary, i.e., of the form γ

From cocycles to relative homology
In this section, we explain how the cocycle Θ M defined in (3.2) gives rise to a group homomorphism where C 0 is the set of cusps of X 1 (M) which are in the same diamond orbit as the cusp 1 (M) ⋅ 0.
Recall that we have denoted by 1  2 ] on which the complex conjugation acts trivially.Note that since the Steinberg symbol {−1, −ζ M } has order dividing 2, its image in K M is trivial.Furthermore, the action of (Z /M Z) × on K M factors through (Z /M Z) × / ± 1 (by definition).
It suffices to prove that the map sending (g, γ) to g ⋅ {0, γ0} is an isomorphism.Note that φ is well defined since (we have used the fact that γ c ∈ 1 , so the diamond operator ⟨γ c ⟩ is trivial).
Since M > 3, the group 1 is torsion-free and we have . By Shapiro's lemma for group homology, we have Using the description of group homology in terms of inhomogeneous chains (cf. (2.3)), one gets a short exact sequence for g ∈ G, γ, γ ′ ∈ 0 (M).The last map Z[G] → Z is the augmentation (degree) map (note that J is indeed in the kernel of ∂).

E. Lecouturier and J. Wang
As in (2.2), we have an exact sequence where C M is the set of cusps of Y 1 (M).Here, the map Z[C M ] sends a cusp c to the homology class of a small loop around c in Y 1 (M).
Under the isomorphism ) and the embedding Thus, we have an exact sequence where I ′ is the subgroup of Z[G × 0 ] generated by the elements and by the Let us apply Proposition 4.1 to T = K M and u ∶ 0 (M) → T induced by Θ M .Since u is parabolic and 1 (M) acts trivially on T, the condition that u vanishes on parabolic elements of 1 (M) is satisfied.Therefore, we get a

Proofs of the theorems
We start with the following lemma (we thank Venkatesh for explaining this to us).
where α * is the trace map induced by α.
(note that f is induced by the right action of the matrix α of Lemma 5.1).
Let U = U γ be as in Section 3 and Consider the following Cartesian diagram of schemes: where s M is given by the closed point (1, ζ M ) ∈ U, and X makes the diagram Cartesian by definition.

Lemma 5.2
We have a natural isomorphism of schemes over Spec(Q): where the maps ) are given by T ↦ ζ M and T ↦ t p , respectively.Under this isomorphism, the map X → Spec(Q(ζ M )) is the projection onto the first factor.The compositum map X → U ) is given by the compositum of the projection Proof Let Y be such that the following diagram is Cartesian: We claim that there is a natural isomorphism Y ≃ X.To prove that, it is enough to prove that we have a commutative diagram It suffices to prove that the image of Y (which we view as a closed subscheme of G 2 m ) is contained in U ′ .This follows from the fact that f To conclude the proof of Lemma 5.2, note that there is a commutative diagram T=t p z1=1 z2=t f z1=1 z2=T ∎ Lemma 5.2 yields a more concrete description of X: we have This latter isomorphism can be rewritten more simply in a way which depends on whether p divides M or not.
After applying the Atkin-Lehner involution W M p and W M to the two lines of (5.6), we get a commutative diagram where ) in the same orbit as ∞.We have used the facts that ϖ M p = ΘMp ○ W M p and ϖ M = ΘM ○ W M .This follows from [9,Proposition 4.3.3],where the authors use usual Manin symbols (whereas our map ϖ M uses Manin symbols twisted by the Atkin-Lehner involution).
Note that ϖ M p and ϖ M are anti-equivariant for the actions of (Z /M p Z) × / ± 1 and (Z /M Z) × / ± 1, respectively.This means that for any x ∈ H 1 (X 1 (M p), C ′ ∞ , Z) and g ∈ (Z /M p Z) × / ± 1, we have ϖ M p (g ⋅ x) = g −1 ⋅ ϖ M p (x) (5.8) (and similarly for ϖ M ).Indeed, we have W M p ○ ⟨g⟩ = ⟨g −1 ⟩ ○ W M p .This could also have been checked easily directly on the definition of ϖ M p and ϖ M in terms of dual Manin symbols.Let us note that (5.8) is true independently on whether p divides M or not.Now, let C be a subset of cusps of X 1 (M p) as in Theorem 1.4.If C ⊂ C ′ ∞ , then Theorem 1.4 follows from (5.7) (we just restrict ϖ M p to H 1 (X 1 (M p), C, Z) ⊂ H 1 (X 1 (M p), C ′ ∞ , Z)).Let us explain how to deduce the general case from this special case.
Fix c ∈ P 1 (Q) such that 1 (M p) ⋅ c ∈ C.An element of H 1 (X 1 (M p), C, Z) is of the form {c, γc} for some γ ∈ 0 (M p).The assumption that all the elements of C are in the same diamond orbit under Ker((Z /M p Z) × → (Z /M Z) × ) means that we can actually choose γ in 0 (M p) ∩ 1 (M).
We This concludes the proof of Theorem 1.4 in the case p | M.

The case p ∤ M
Assume now that p does not divide M. Note that in this case we have (1, ζ M ) ∈ U ′ .By (5.3), there is an isomorphism such that: • The map X → U ′ is given by the two inclusions Applying the functor H 2 (⋅, 2) to (5.2) (cf.[9, Lemma 2.1.1]),we get the following commutative diagram:

1 (
M).As usual, we denote T by U is divides M. The Atkin-Lehner involution W M is the involution of H 1 (X 1 (M), C M , Z) induced by the map z ↦ − 1 Mz of the upper half-plane.For any x ∈ (Z /M Z) × , we denote by ⟨x⟩ the corresponding diamond operator, which is the automorphism of X 1 (M) induced by the action of any matrix ( a b c d ) ∈ 0 (M) such that d ≡ x (modulo M) .
3.2], equal to Π ○M ○ W M .Sharifi made the following conjecture.Conjecture 1.1 (Sharifi) The restriction of ϖ M to H 1 (X 0 (M), Z) is annihilated by the Hecke operators T − ⟨ ⟩ − 1 for primes not dividing M and by the Hecke operators U − 1 for primes | M.This is equivalent to [9, Conjecture 4.3.5(a)],where the authors use dual Hecke operators but use Π

Theorem 1 . 2 (
Sharifi-Venkatesh) The restriction of ϖ M to H 1 (X 0 (M), Z) is annihilated by the Hecke operators T − ⟨ ⟩ − 1 for primes not dividing M.This follows from [9, Theorem 4.3.7].Therefore, to prove Conjecture 1.1, it only remains to consider the Hecke operators U − 1 for primes | M. Fukaya-Kato do get a result including U − 1, but they have to tensor with Z p where p ≥ 5 is a prime dividing M. Theorem 1.3 (Fukaya-Kato) Let p ≥ 5 be a prime dividing M. The map

[ 9 ,
Section 3.2 and Lemma 4.1.2]).As in [9, Section 3.2], for any a, c ∈ Z with gcd(a, c) = 1, there is a special element ⟨a, c⟩ ∈ K 1 , (3.1) which is supported on the divisor D ∶ 1 − z a 1 z c 2 = 0 and is given there by the function 1 − z b 1 z d 2 for any b, d ∈ Z such that ad − bc = 1 (this is independent of the choice of b and d).