1 Introduction and notation
Sharifi [Reference Sharifi8] 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 [Reference Fukaya and Kazuya4] and more recently by Sharifi and Venkatesh [Reference Sharifi and Venkatesh9].
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 [Reference Williams11]. 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 [Reference Lecouturier and Wang6]. We now set up some notation and describe our results in details.
1.1 Homology of modular curves and Hecke operators
Let
$M\geq 4$
be an integer. Let
$$ \begin{align*}\Gamma_1(M) = \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{\mathrm{SL}}_2(\operatorname{\mathrm{\mathbf{Z}}}) \text{ such that } a-1 \equiv c \equiv 0 \text{ (modulo } M\text{)}\right\},\end{align*} $$
and denote by
$X_1(M)$
the compact modular curve (over
$\operatorname {\mathrm {\mathbf {C}}}$
) of level
$\Gamma _1(M)$
. Let
$$ \begin{align*}C_M = \Gamma_1(M) \backslash \mathbf{P}^1(\operatorname{\mathrm{\mathbf{Q}}})\end{align*} $$
be the set of cusps of
$X_1(M)$
, and let
$C_M^0$
be those cusps in
$C_M$
of the form
$\Gamma _1(M) \cdot \frac {a}{b}$
with
$\gcd (a,b)=1$
and
$a \not \equiv 0 \text { (modulo } M \text {)}$
(in the case
$b=0$
, we have the cusp
${\Gamma _1(M)\cdot \infty }$
).
Let
$H_1(X_1(M), C_M, \operatorname {\mathrm {\mathbf {Z}}})$
be the singular homology of
$X_1(M)$
relative to
$C_M$
. If
$\alpha $
and
$\beta $
are in
$\mathbb {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
, let
$$ \begin{align*}\{\alpha, \beta\} \in H_1(X_1(M), C_M, \operatorname{\mathrm{\mathbf{Z}}})\end{align*} $$
be the class of the hyperbolic geodesic from
$\alpha $
to
$\beta $
in
$X_1(M)$
. The group
$H_1(X_1(M), C_M, \operatorname {\mathrm {\mathbf {Z}}})$
carries an action of various Hecke operators, which we now recall.
If
$\ell $
is a prime number, the
$\ell $
th Hecke operator
$T_{\ell }$
is the double coset operator
$\Gamma _1(M) \begin {pmatrix} 1 & 0 \\ 0 & \ell \end {pmatrix} \Gamma _1(M)$
. As usual, we denote
$T_{\ell }$
by
$U_{\ell }$
is
$\ell $
divides M. The Atkin–Lehner involution
$W_M$
is the involution of
$H_1(X_1(M), C_M, \operatorname {\mathrm {\mathbf {Z}}})$
induced by the map
$z \mapsto -\frac {1}{Mz}$
of the upper half-plane.
For any
$x \in (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
, we denote by
$\langle x \rangle $
the corresponding diamond operator, which is the automorphism of
$X_1(M)$
induced by the action of any matrix
$\begin {pmatrix} a & b \\ c & d \end {pmatrix} \in \Gamma _0(M)$
such that
$$ \begin{align*}d \equiv x \text{ (modulo }M\text{)} \text{ .}\end{align*} $$
This gives by functoriality an action of
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
on
$H_1(X_1(M), C_M, \operatorname {\mathrm {\mathbf {Z}}})$
. Note that diamond operators act on the set of cusps
$C_M$
and that this action preserves
$C_M^0$
. We say that two cusps c and
$c'$
are in the same diamond orbit if there exists
$x \in (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
such that
$\langle x \rangle \cdot c = c'$
.
There are also dual Hecke operators: if T is one of the operators defined above, we let
$T^* = W_M^{-1} T W_M$
. As is well known (cf. [Reference Diamond and Shurman1, Theorem 5.5.3]), we have
$\langle x \rangle ^* = \langle x \rangle ^{-1}$
(for all
$x \in (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
) and
$T_{\ell }^* = \langle \ell \rangle ^{-1} T_{\ell }$
(for all primes
$\ell \nmid M$
).
1.2 (Dual) Manin symbols
Let
$\xi _M : \operatorname {\mathrm {\mathbf {Z}}}[\Gamma _1(M) \backslash \operatorname {\mathrm {SL}}_2(\operatorname {\mathrm {\mathbf {Z}}})] \rightarrow H_1(X_1(M), C_M, \operatorname {\mathrm {\mathbf {Z}}})$
be the (modified) Manin map: it sends a coset
$\Gamma _1(M) \cdot \begin {pmatrix} a & b \\ c & d \end {pmatrix}$
to
$\{-\frac {d}{Mb}, -\frac {c}{Ma}\}$
(it is the usual Manin map sending
$\Gamma _1(M) \cdot \begin {pmatrix} a & b \\ c & d \end {pmatrix}$
to
$\{\frac {b}{d}, \frac {a}{c}\}$
composed with the Atkin–Lehner involution
$W_M$
). Manin showed that
$W_M \circ \xi _M$
is surjective, and therefore
$\xi _M$
is surjective (cf. [Reference Manin7, Section 1.6]).
Let
$S_M^0 \subset \Gamma _1(M) \backslash \operatorname {\mathrm {SL}}_2(\operatorname {\mathrm {\mathbf {Z}}})$
be the subset consisting of
$\Gamma _1(M) \cdot \begin {pmatrix} a & b \\ c & d \end {pmatrix}$
with
$M \nmid c$
and
$M \nmid d$
. The restriction
$$ \begin{align*}\xi_M^0 : \operatorname{\mathrm{\mathbf{Z}}}[S_M^0] \rightarrow H_1(X_1(M), C_M^0, \operatorname{\mathrm{\mathbf{Z}}})\end{align*} $$
is surjective (cf. [Reference Fukaya, Kato and Sharifi3, Section 2.1.3]).
1.3 Algebraic K-theory and motivic cohomology
Fix an algebraic closure
$\overline {\operatorname {\mathrm {\mathbf {Q}}}}$
of
$\operatorname {\mathrm {\mathbf {Q}}}$
. For any integer
$M \geq 4$
, choose a primitive Mth root of unity
$\zeta _M \in \overline {\operatorname {\mathrm {\mathbf {Q}}}}$
such that for all
$M' \mid M$
, we have
$\zeta _{M'} = \zeta _M^{M/M'}$
.
We have a canonical group isomorphism
$$ \begin{align*}(\operatorname{\mathrm{\mathbf{Z}}}/M\operatorname{\mathrm{\mathbf{Z}}})^{\times} \xrightarrow{\sim} \operatorname{\mathrm{Gal}}(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M)/\operatorname{\mathrm{\mathbf{Q}}})\end{align*} $$
sending
$a \in (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
to the Galois automorphism characterized by
$\zeta _M \mapsto \zeta _M^a$
. The complex conjugation of
$ \operatorname {\mathrm {Gal}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M)/\operatorname {\mathrm {\mathbf {Q}}})$
corresponds to the class of
$-1$
in
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
under that isomorphism.
If A is a commutative ring, let
$K_2(A)$
be the second K-group of A, as defined by Quillen. For any
$x,y \in A^{\times }$
, 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
$\operatorname {\mathrm {Gal}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M)/\operatorname {\mathrm {\mathbf {Q}}})$
(and in particular of the complex conjugation) on
$K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M, \frac {1}{M}])$
. We denote by
$\mathcal {K}_M$
the largest quotient of
$K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M, \frac {1}{M}]) \otimes \operatorname {\mathrm {\mathbf {Z}}}[\frac {1}{2}]$
on which the complex conjugation acts trivially. Note that
$\mathcal {K}_M$
is a
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
-module.
1.4 Sharifi’s
$\varpi _M$
map and summary of known results
The map
$\operatorname {\mathrm {\mathbf {Z}}}[S_M^0] \rightarrow \mathcal {K}_M$
given by
$$ \begin{align*}\Gamma_1(M) \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \{ 1 - \zeta_M^c, 1-\zeta_M^d \}\end{align*} $$
factors through
$\xi _M^0$
(cf. [Reference Fukaya, Kato and Sharifi3, Section 2.1.4]), and thus induces a map
$$ \begin{align} \varpi_M : H_1(X_1(M), C_M^0, \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow \mathcal{K}_M \text{ .} \end{align} $$
Let us note that our map
$\varpi _M$
is, in the notation of [Reference Sharifi and Venkatesh9, Proposition 4.3.2], equal to
$\Pi _M^{\circ }\circ W_M$
. Sharifi made the following conjecture.
Conjecture 1.1 (Sharifi)
The restriction of
$\varpi _M$
to
$H_1(X_0(M), \operatorname {\mathrm {\mathbf {Z}}})$
is annihilated by the Hecke operators
$T_{\ell }-\ell \langle \ell \rangle - 1$
for primes
$\ell $
not dividing M and by the Hecke operators
$U_{\ell }-1$
for primes
$\ell \mid M$
.
This is equivalent to [Reference Sharifi and Venkatesh9, Conjecture 4.3.5(a)], where the authors use dual Hecke operators but use
$\Pi _M^{\circ } = \varpi _M \circ W_M$
instead of
$\varpi _M$
. As Sharifi and Venkatesh mention right after [Reference Sharifi and Venkatesh9, Theorem 4.3.6], it is expected that the conjecture holds without restricting
$\varpi _M$
to
$H_1(X_0(M), \operatorname {\mathrm {\mathbf {Z}}})$
.
This conjecture has a history of partial results: [Reference Fukaya and Kazuya4, Reference Lecouturier and Wang5] and most recently [Reference Sharifi and Venkatesh9]. Let us recall the main results of Sharifi–Venkatesh and Fukaya–Kato on this conjecture.
Theorem 1.2 (Sharifi–Venkatesh)
The restriction of
$\varpi _M$
to
$H_1(X_0(M), \operatorname {\mathrm {\mathbf {Z}}})$
is annihilated by the Hecke operators
$T_{\ell }-\ell \langle \ell \rangle - 1$
for primes
$\ell $
not dividing M.
This follows from [Reference Sharifi and Venkatesh9, Theorem 4.3.7]. Therefore, to prove Conjecture 1.1, it only remains to consider the Hecke operators
$U_{\ell }-1$
for primes
$\ell \mid M$
. Fukaya–Kato do get a result including
$U_{\ell }-1$
, but they have to tensor with
$\operatorname {\mathrm {\mathbf {Z}}}_p$
where
$p\geq 5$
is a prime dividing M.
Theorem 1.3 (Fukaya–Kato)
Let
$p\geq 5$
be a prime dividing M. The map
$$ \begin{align*}\varpi_M \otimes 1 : H_1(X_1(M),C_M^0, \operatorname{\mathrm{\mathbf{Z}}}_p) \rightarrow \mathcal{K}_M \otimes \operatorname{\mathrm{\mathbf{Z}}}_p\end{align*} $$
is annihilated by the Hecke operators
$T_{\ell }-\ell \langle \ell \rangle - 1$
for primes
$\ell $
not dividing M and by the Hecke operators
$U_{\ell }-1$
for primes
$\ell \mid M$
.
We refer to [Reference Fukaya and Kazuya4, Theorem 5.2.3(1)] for this result. Let us note that Fukaya and Kato actually consider (the p-ordinary part of)
$H^1(Y_1(M), \operatorname {\mathrm {\mathbf {Z}}}_p)$
instead of
$H_1(X_1(M), C_M, \operatorname {\mathrm {\mathbf {Z}}}_p)$
. These two groups are canonically isomorphic, but the isomorphism transfers dual Hecke operators (i.e.,
$T_{\ell }^*$
,
$U_{\ell }^*$
or
$\langle x \rangle ^{-1}$
) to usual Hecke operators (i.e.,
$T_{\ell }$
,
$U_{\ell }$
or
$\langle x \rangle $
).
1.5 Our main results
Another important aspect of Sharifi’s theory is the way in which the maps
$\varpi _M$
relate with each other when varying M. This has been studied under some assumptions in [Reference Fukaya and Kazuya4, Reference Williams11]. If p is a prime, there are two degeneracy maps
$\pi _1, \pi _2 : X_1(Mp) \rightarrow X_1(M)$
given on the upper half-plane by
$\pi _1 : z\mapsto z$
and
$\pi _2 : z \mapsto pz$
. On the K-side, there is a norm map
$\operatorname {\mathrm {Norm}} : \mathcal {K}_{Mp} \rightarrow \mathcal {K}_M$
. Our main result is the following.
Theorem 1.4 Let
$p\geq 2$
be a prime number, and let
$M \geq 4$
. Let
$C \subset C_{Mp}^0$
be a subset of cusps which are all in the same orbit under the action of
$\operatorname {\mathrm {Ker}}((\operatorname {\mathrm {\mathbf {Z}}}/Mp\operatorname {\mathrm {\mathbf {Z}}})^{\times } \rightarrow (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times })$
(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
Here,
$\langle p \rangle $
is the pth diamond operator, induced by the action of a matrix
$\begin {pmatrix} a & b \\ c & d \end {pmatrix} \in \Gamma _0(M)$
with
$d \equiv p \text { (modulo }M\text {)}$
on
$X_1(M)$
.
Remark 1.5
-
(i) Theorem 1.4(i) has been proved by Fukaya and Kato in [Reference Fukaya and Kazuya4, Theorem 5.2.3(2)] after tensoring by
$\operatorname {\mathrm {\mathbf {Z}}}_p$
for
$p\geq 5$
. They use
$H^2(G_{\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M)}, \operatorname {\mathrm {\mathbf {Z}}}_p(2))$
instead of
$\mathcal {K}_M \otimes \operatorname {\mathrm {\mathbf {Z}}}_p$
. The étale Chern class map (cf. [Reference Tate10]) provides an isomorphism
$$ \begin{align*}K_2\left(\operatorname{\mathrm{\mathbf{Z}}}\left[\zeta_M, \frac{1}{M}\right]\right) \otimes \operatorname{\mathrm{\mathbf{Z}}}_p \simeq H^2_{\text{\'et}}\left(\operatorname{\mathrm{\mathbf{Z}}}\left[\zeta_M, \frac{1}{M}\right], \operatorname{\mathrm{\mathbf{Z}}}_p(2)\right) \text{ .}\end{align*} $$
Since
$H^2_{\mathrm {\'et}}(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M, \frac {1}{M}], \operatorname {\mathrm {\mathbf {Z}}}_p(2))$
is a subgroup
$H^2(G_{\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M)}, \operatorname {\mathrm {\mathbf {Z}}}_p(2))$
and
$\mathcal {K}_M$
is identified with the fixed part by the complex conjugation in
$K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M, \frac {1}{M}]) \otimes \operatorname {\mathrm {\mathbf {Z}}}_p$
, we get a canonical embedding
$$ \begin{align*}\mathcal{K}_M \otimes \operatorname{\mathrm{\mathbf{Z}}}_p \hookrightarrow H^2(G_{\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M)}, \operatorname{\mathrm{\mathbf{Z}}}_p(2)) \text{ .}\end{align*} $$
Fukaya and Kato’s map actually takes values in
$\mathcal {K}_M$
(by construction). They also do not need to restrict to the subset C of
$C_{Mp}^0$
. Their techniques rely on p-adic Hodge theory. -
(ii) Similarly, Theorem 1.4(ii) has been proved (for the absolute homology) by Scott in [Reference Williams11, Theorem 7] after tensoring by
$\operatorname {\mathrm {\mathbf {Z}}}_{\ell }$
for a prime
$\ell \neq p$
dividing M (Scott’s p is our
$\ell $
and vice versa). Scott relies on the techniques of Fukaya and Kato. -
(iii) Thus, the main novelty of our result is to work with
$\operatorname {\mathrm {\mathbf {Z}}}$
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
$H_1(X_1(M), \operatorname {\mathrm {\mathbf {Z}}})$
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
$\infty $
(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
$\varpi _M$
to
$H_1(X_1(M), C_{\infty }, \operatorname {\mathrm {\mathbf {Z}}})$
is annihilated by
$T_{\ell }-\ell \langle \ell \rangle - 1$
for primes
$\ell $
not dividing M, where
$C_{\infty }$
are the cusps of
$X_1(M)$
in the same diamond orbit as
$\infty $
. This is a slight improvement on Theorem 1.2 (which holds for the restriction of
$\varpi _M$
to
$H_1(X_1(M), \operatorname {\mathrm {\mathbf {Z}}})$
).
By combining Theorem 1.4 and the results of Fukaya and Kato, one gets the following result.
Theorem 1.6 Let
$M \geq 4$
. The map
$H_1(X_1(M), \operatorname {\mathrm {\mathbf {Z}}}[\frac {1}{6}]) \rightarrow \mathcal {K}_M \otimes \operatorname {\mathrm {\mathbf {Z}}}[\frac {1}{6}]$
obtained by restricting
$\varpi _M$
to
$H_1(X_1(M), \operatorname {\mathrm {\mathbf {Z}}})$
and inverting
$6$
is annihilated by the Hecke operator
$U_{\ell }-1$
for all primes
$\ell $
dividing M. Here,
$U_{\ell }$
is the classical Hecke operator of index
$\ell $
, corresponding to the double coset of
$\begin {pmatrix} 1 & 0 \\ 0 & \ell \end {pmatrix}$
.
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
$\operatorname {\mathrm {\mathbf {Z}}}_p$
for a prime
$p\geq 5$
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
$p \nmid 6$
(note that
$2$
is inverted anyway in the definition of
$\varpi _M$
). 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.
2 Background and notation regarding homology and cohomology
Let
$\Gamma $
be a torsion-free finite index subgroup of
$\operatorname {\mathrm {SL}}_2(\operatorname {\mathrm {\mathbf {Z}}})$
(e.g.,
$\Gamma = \Gamma _1(M)$
for
$M \geq 3$
). Let
$Y = \Gamma \backslash \mathfrak {h}$
be the open modular curve of level
$\Gamma $
, where
$\mathfrak {h}$
is the upper half-plane. We denote by X the corresponding compactified modular curve: we have
$X = Y \cup C$
where
$C = \Gamma \backslash \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
is the set of cusps of X.
We denote by
$H_1(X, C, \operatorname {\mathrm {\mathbf {Z}}})$
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
$(X,C)$
:
$$ \begin{align} 0 \rightarrow H_1(X, \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H_1(X, C, \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow \operatorname{\mathrm{\mathbf{Z}}}[C] \rightarrow \operatorname{\mathrm{\mathbf{Z}}} \rightarrow 0 \text{ ,} \end{align} $$
where the map
$\operatorname {\mathrm {\mathbf {Z}}}[C] \rightarrow \operatorname {\mathrm {\mathbf {Z}}}$
is the degree map.
The Poincaré duality yields a perfect bilinear pairing
$H_1(X, C, \operatorname {\mathrm {\mathbf {Z}}}) \times H_1(Y, \operatorname {\mathrm {\mathbf {Z}}}) \rightarrow \operatorname {\mathrm {\mathbf {Z}}}$
(also called the intersection pairing, due to its interpretation in terms of intersection number of cycles). Under this duality, (2.1) becomes
$$ \begin{align} 0 \rightarrow \operatorname{\mathrm{\mathbf{Z}}} \rightarrow \operatorname{\mathrm{\mathbf{Z}}}[C] \rightarrow H_1(Y \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H_1(X, \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow 0 \text{ ,} \end{align} $$
where the map
$\operatorname {\mathrm {\mathbf {Z}}}[C] \rightarrow H_1(Y \operatorname {\mathrm {\mathbf {Z}}})$
sends
$c \in C$
to the class of a little oriented circle around c.
If G is a group and T is a left G-module, we say that
$c : G \rightarrow T$
is a
$1$
-cocycle if for all
$g,g' \in G$
we have
$c(gg') = c(g)+g\cdot c(g')$
. The first cohomology group
$H^1(G,T)$
can be computed as the abelian group of
$1$
-cocycles
$c : G \rightarrow T$
modulo the cocycles of the form
$c(g) = gx-x$
for some
$x \in T$
(independent of g).
Similarly, using the projective resolution of
$\operatorname {\mathrm {\mathbf {Z}}}$
as a
$\operatorname {\mathrm {\mathbf {Z}}}[G]$
-module in terms of inhomogeneous chains, one can compute the first homology group
$H_1(G,T)$
as
$$ \begin{align} H_1(G,T) = Z/B, \end{align} $$
where
$Z \subset \operatorname {\mathrm {\mathbf {Z}}}[G] \otimes _{\operatorname {\mathrm {\mathbf {Z}}}} T$
is the kernel of the map sending
$[g] \otimes x$
to
$g^{-1}x-x$
(for
$g \in G$
and
$x \in T$
) and B is generated by the elements of the form
$[gg'] \otimes x - [g] \otimes x - [g'] \otimes g^{-1}x$
for
$g,g' \in G$
and
$x \in T$
.
Finally, let us recall that since
$\Gamma $
is torsion-free, it is isomorphic to the fundamental group of Y, and therefore we have canonical group isomorphisms
$$ \begin{align} H_1(\Gamma, \operatorname{\mathrm{\mathbf{Z}}}) \simeq \Gamma^{ab} \simeq H_1(Y, \operatorname{\mathrm{\mathbf{Z}}}) \text{ .} \end{align} $$
3 Reminders from the work of Sharifi and Venkatesh
Sharifi and Venkatesh constructed a
$1$
-cocycle
$$ \begin{align*}\Theta : \operatorname{\mathrm{SL}}_2(\operatorname{\mathrm{\mathbf{Z}}}) \rightarrow {\mathbf{K}}_2/\operatorname{\mathrm{\mathbf{Z}}}\cdot \{ -z_1, -z_2 \} \text{ .}\end{align*} $$
Here,
$$ \begin{align*}{\mathbf{K}}_2 := K_2(\operatorname{\mathrm{\mathbf{Q}}}(\mathbf{G}_m^2)) = K_2(\operatorname{\mathrm{\mathbf{Q}}}(z_1, z_2))\end{align*} $$
carries a left action of
$\operatorname {\mathrm {SL}}_2(\operatorname {\mathrm {\mathbf {Z}}})$
induced by the natural right action of
$\operatorname {\mathrm {SL}}_2(\operatorname {\mathrm {\mathbf {Z}}})$
on
$\mathbf {G}_m^2$
given by
$$ \begin{align*}(z_1, z_2) \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \left(z_1^az_2^c, z_1^bz_2^d\right) \text{ .}\end{align*} $$
Furthermore,
$\operatorname {\mathrm {\mathbf {Z}}}\cdot \{ -z_1, -z_2 \}$
is the subgroup of
$ {\mathbf {K}}_2$
generated by the Steinberg symbol of
$-z_1$
and
$-z_2$
. The cocycle
$\Theta $
actually takes values in
$ {\mathbf {K}}_2^{(0)}/ \{ -z_1, -z_2 \}$
, where
$ {\mathbf {K}}_2^{(0)}$
is the subgroup of
$ {\mathbf {K}}_2$
fixed by the pushforward
$[m]_*$
of the multiplication by m map for all
$m \in \mathbf {N}$
(cf. [Reference Sharifi and Venkatesh9, Section 4.1.2]).
Let us recall a characterization of
$\Theta $
. Let
$$ \begin{align*}{\mathbf{K}}_1 = \bigoplus_D K_1(\operatorname{\mathrm{\mathbf{Q}}}(D)) = \bigoplus_D \operatorname{\mathrm{\mathbf{Q}}}(D)^{\times} \text{ ,}\end{align*} $$
where D runs through all the irreducible divisors of
$\mathbf {G}_m^2$
. There is a divisor map
$\partial : {\mathbf {K}}_2 \rightarrow {\mathbf {K}}_1$
sending a Steinberg symbol
$\{ f, g\}$
to the element of
$K_1$
whose component in D is
$$ \begin{align*}(-1)^{v(f)v(g)}g^{v(f)}f^{-v(g)} \text{ ,}\end{align*} $$
where v is the valuation coming from D (cf. [Reference Sharifi and Venkatesh9, equation (2.6)]). The map
$\partial $
induces an embedding
$$ \begin{align*}\partial : {\mathbf{K}}_2^{(0)}/\operatorname{\mathrm{\mathbf{Z}}}\cdot \{ -z_1, -z_2 \} \hookrightarrow {\mathbf{K}}_1\end{align*} $$
(cf. [Reference Sharifi and Venkatesh9, Section 3.2 and Lemma 4.1.2]). As in [Reference Sharifi and Venkatesh9, Section 3.2], for any
$a, c \in \operatorname {\mathrm {\mathbf {Z}}}$
with
$\gcd (a,c)=1$
, there is a special element
$$ \begin{align} \langle a , c \rangle \in {\mathbf{K}}_1, \end{align} $$
which is supported on the divisor
$D : 1-z_1^az_2^c=0$
and is given there by the function
$1-z_1^bz_2^d$
for any
$b, d\in \operatorname {\mathrm {\mathbf {Z}}}$
such that
$ad-bc=1$
(this is independent of the choice of b and d).
For any
$\gamma = \begin {pmatrix} a & b \\ c & d \end {pmatrix} \in \operatorname {\mathrm {SL}}_2(\operatorname {\mathrm {\mathbf {Z}}})$
,
$\Theta (\gamma )$
is characterized by the equality
$$ \begin{align*}\partial(\Theta(\gamma)) = \langle b, d \rangle - \langle 0, 1 \rangle\end{align*} $$
in
$K_1$
(cf. [Reference Sharifi and Venkatesh9, Proposition 3.3.1]). As in the proof of [Reference Sharifi and Venkatesh9, Proposition 3.3.4], one sees that
$\Theta (\gamma )=0$
if
$\gamma (0)=0$
, i.e., if
$\gamma = \begin {pmatrix} 1 & 0 \\ m & 1 \end {pmatrix}$
for some
$m \in \operatorname {\mathrm {\mathbf {Z}}}$
(this follows from the injectivity of
$\partial : {\mathbf {K}}_2^{(0)}/\operatorname {\mathrm {\mathbf {Z}}}\cdot \{ -z_1, -z_2 \} \hookrightarrow {\mathbf {K}}_1$
).
Finally, let us recall how Sharifi and Venkatesh (cf. [Reference Sharifi and Venkatesh9, Section 4.2.1]) specialize
$\Theta $
to a cocycle
$$ \begin{align*}\Theta_M : \Gamma_0(M) \rightarrow K_2\left(\operatorname{\mathrm{\mathbf{Z}}}\left[\zeta_M, \frac{1}{M}\right]\right)/\operatorname{\mathrm{\mathbf{Z}}}\cdot \{-1, -\zeta_M\}\end{align*} $$
(for every
$M \geq 4$
). Here, the action of
$ \Gamma _0(M)$
on
$K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M, \frac {1}{M}])$
is given as follows: we have a surjective group homomorphism
$$ \begin{align*}\Gamma_0(M) \rightarrow (\operatorname{\mathrm{\mathbf{Z}}}/M\operatorname{\mathrm{\mathbf{Z}}})^{\times}\end{align*} $$
given by
$$ \begin{align*}\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto d \text{ (modulo }M\text{)} \text{ ,}\end{align*} $$
and
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
acts on
$K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M, \frac {1}{M}])$
via our identification
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times } \simeq \operatorname {\mathrm {Gal}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M)/\operatorname {\mathrm {\mathbf {Q}}})$
.
The idea is to “evaluate”
$\Theta (\gamma )$
at
$(z_1, z_2) = (1, \zeta _M)$
. To “evaluate” at
$(z_1, z_2) = (1, \zeta _M)$
, a naive idea would be to send a Steinberg symbol
$\{f, g\} \in {\mathbf {K}}_2$
to
$$ \begin{align*}\{f(1, \zeta_M), g(1, \zeta_M)\} \in K_2(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M)) \text{ .}\end{align*} $$
This does not make sense in general because
$f(1, \zeta _M)$
or
$g(1, \zeta _M)$
may not be well defined (f or g may have a pole or zero at
$(1, \zeta _M)$
).
The idea of Sharifi and Venkatesh is to prove that
$\Theta (\gamma )$
is actually a combination of Steinberg symbols which can be evaluated at
$(1, \zeta _M)$
. They make this precise by using motivic cohomology groups. We refer to [Reference Sharifi and Venkatesh9, Section 2.1] for the precise definition and results they are using regarding motivic cohomology groups. In particular, if
${U \subset \mathbf {G}_m^2}$
is an open subset, there is a motivic cohomology group
$H^2(U, 2)$
(which is an abelian group). As explained in [Reference Sharifi and Venkatesh9, Remark 2.2.3], the functorial map
$H^2(U, 2) \rightarrow H^2(\operatorname {\mathrm {\mathbf {Q}}}(\mathbf {G}_m^2), 2)$
is injective and
$H^2(\operatorname {\mathrm {\mathbf {Q}}}(\mathbf {G}_m^2), 2)$
is canonically identified with
$ {\mathbf {K}}_2$
.
As noted in [Reference Sharifi and Venkatesh9, Section 4.2.1], for all
$\gamma = \begin {pmatrix} a & b \\ c & d \end {pmatrix} \in \Gamma _0(M)$
, the element
$$ \begin{align*}\Theta(\gamma) \in {\mathbf{K}}_2/\operatorname{\mathrm{\mathbf{Z}}}\cdot \{ -z_1, -z_2 \}\end{align*} $$
lies in the image of
$H^2(U_{\gamma }, 2)/\{-z_1, -z_2\}$
, where
$U_{\gamma }$
is the open subset of
$\mathbf {G}_m^2$
, which is the complement of
$\{z_1^bz_2^d = 1\} \cup \{z_2=1\}$
. Since
$(1, \zeta _M) \in U_{\gamma }$
, there is a functorial map
$$ \begin{align*}s_M^* : H^2(U_{\gamma}, 2) \rightarrow H^2(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M), 2) \simeq K_2(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M)) \text{ .}\end{align*} $$
By [Reference Sharifi and Venkatesh9, Corollary 4.2.5],
$\Theta _M(\gamma ) := s_M^*(\Theta (\gamma ))$
actually belongs to the subgroup
$$ \begin{align*}K_2\left(\operatorname{\mathrm{\mathbf{Z}}}\left[\zeta_M, \frac{1}{M}\right]\right)/\operatorname{\mathrm{\mathbf{Z}}}\cdot \{-1, -\zeta_M\}\end{align*} $$
of
$K_2(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M))/\operatorname {\mathrm {\mathbf {Z}}}\cdot \{-1, -\zeta _M\}$
.
We therefore have a
$1$
-cocycle
$$ \begin{align} \Theta_M : \Gamma_0(M) \rightarrow K_2(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M))/\operatorname{\mathrm{\mathbf{Z}}}\cdot \{-1, -\zeta_M\} \text{ .} \end{align} $$
By [Reference Sharifi and Venkatesh9, Proposition 4.2.1], the cocycle
$\Theta _M$
is parabolic. This means that if
$c \in \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
and
$\Gamma _c \subset \Gamma _0(M)$
is the stabilizer of c, then the restriction of
$\Theta _M$
to
$\Gamma _c$
is a coboundary, i.e., of the form
$\gamma \mapsto \gamma \cdot x -x$
for some
$x \in K_2(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M))/\operatorname {\mathrm {\mathbf {Z}}}\cdot \{-1, -\zeta _M\}$
(depending on c a priori).
4 From cocycles to relative homology
In this section, we explain how the cocycle
$\Theta _M$
defined in (3.2) gives rise to a group homomorphism
$$ \begin{align*}\tilde{\Theta}_M : H_1(X_1(M), C_0, \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow \mathcal{K}_M,\end{align*} $$
where
$C_0$
is the set of cusps of
$X_1(M)$
which are in the same diamond orbit as the cusp
$\Gamma _1(M) \cdot 0$
.
Recall that we have denoted by
$\mathcal {K}_M$
the largest quotient of
$K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M, \frac {1}{M}]) \otimes \operatorname {\mathrm {\mathbf {Z}}}[\frac {1}{2}]$
on which the complex conjugation acts trivially. Note that since the Steinberg symbol
$\{-1, -\zeta _M\}$
has order dividing
$2$
, its image in
$\mathcal {K}_M$
is trivial. Furthermore, the action of
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
on
$\mathcal {K}_M$
factors through
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
(by definition).
Recall that we have a group homomorphism
$\Gamma _0(M) \rightarrow (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
given by
$\gamma \mapsto \langle \gamma \rangle $
, where if
$\gamma = \begin {pmatrix} a & b \\ c & d \end {pmatrix}$
, we let
$\langle \gamma \rangle = d \text { (modulo }M\text {)}$
. Therefore, a (left)
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
-module can be considered naturally as a (left)
$\Gamma _0(M)$
-module. In particular, this is the case of
$H_1(X_1(M), C_0, \operatorname {\mathrm {\mathbf {Z}}})$
, on which
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
acts via diamond operators.
There is a map
$$ \begin{align*}f : \Gamma_0(M) \rightarrow H_1(X_1(M), C_0, \operatorname{\mathrm{\mathbf{Z}}})\end{align*} $$
given by
$\gamma \mapsto \{0, \gamma 0\}$
. The map f is a
$1$
-cocycle, since for all
$\gamma , \gamma ' \in \Gamma _0(M)$
, we have
$$ \begin{align*} f(\gamma \gamma') &= \{0, \gamma\gamma' 0\} \\& = \{0, \gamma 0\} + \{\gamma0, \gamma \gamma' 0\} \\& = f(\gamma) + \langle \gamma \rangle f(\gamma') \text{ .} \end{align*} $$
We shall need the following result, which allows us to transfer a
$1$
-cocycle to a map on homology.
Proposition 4.1 Let T be a (left)
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
-module (where
$M>3$
). Let
$u : \Gamma _0(M) \rightarrow T$
be a
$1$
-cocycle satisfying
$u(\gamma )=0$
for any
$\gamma \in \Gamma _1(M)$
such that there exists
$c \in \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
with
$\gamma c = c$
. Then u factors through the map
$f : \Gamma _0(M) \rightarrow H_1(X_1(M), C_0, \operatorname {\mathrm {\mathbf {Z}}})$
, thus inducing a morphism of left
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
-modules
$\tilde {u} : H_1(X_1(M), C_0, \operatorname {\mathrm {\mathbf {Z}}}) \rightarrow T$
.
Proof For notational simplicity, let
$G = (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times } / \pm 1$
,
$\Gamma _0 = \Gamma _0(M)/\pm 1$
, and
$\Gamma _1 = \Gamma _1(M) \subset \Gamma _0$
. Recall that if
$\gamma \in \Gamma _0$
, we let
$\langle \gamma \rangle \in G$
be the class of the lower-right corner of
$\gamma $
. For any
$c \in \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
, let
$\gamma _c \in \Gamma _1$
be a generator of the stabilizer of c in
$\Gamma _1$
.
By assumption, we have
$u(\gamma _c) = 0$
for all
$c \in \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
. Thus, u induces a group homomorphism
$u' : \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]/I \rightarrow T$
given by
$u'(g, \gamma ) = g\cdot u(\gamma )$
, where I is the subgroup of
$ \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]$
generated by the elements
$(g, \gamma \gamma ') - (g, \gamma ) - (g\langle \gamma \rangle , \gamma ')$
and by the
$(1, \gamma _c) - (1, 1)$
for all
$g \in G$
,
$\gamma , \gamma ' \in \Gamma _0(M)$
and
$c \in \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
.
It suffices to prove that the map
$$ \begin{align*}\varphi : \operatorname{\mathrm{\mathbf{Z}}}[G \times \Gamma_0]/I \rightarrow H_1(X_1(M), C_0, \operatorname{\mathrm{\mathbf{Z}}})\end{align*} $$
sending
$(g, \gamma )$
to
$g\cdot \{0, \gamma 0 \}$
is an isomorphism. Note that
$\varphi $
is well defined since
$$ \begin{align*} \{0, \gamma_c 0\}&= \{0, c\} + \{c, \gamma_c c\} + \{\gamma_c c, \gamma_c 0\} \\&= (1-\langle \gamma_c \rangle) \cdot \{ 0, c\} \\& = 0 \end{align*} $$
(we have used the fact that
$\gamma _c \in \Gamma _1$
, so the diamond operator
$\langle \gamma _c \rangle $
is trivial).
Let us first prove that
$\varphi $
is surjective. Any element of
$H_1(X_1(M), C_0, \operatorname {\mathrm {\mathbf {Z}}})$
is a combination of modular symbols of the form
$\{\alpha , \beta \}$
where
$\alpha , \beta \in \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
project onto
$C_0$
in
$X_1(M)$
. This latter condition means that
$\alpha = \gamma 0$
and
$\beta = \gamma ' 0$
for some
$\gamma , \gamma ' \in \Gamma _0(M)$
. Note that
$$ \begin{align*} \{\alpha, \beta\} &= \{\alpha, 0\} + \{0, \beta\} \\&= \{0, \gamma' 0\} - \{0, \gamma 0\} \text{ ,} \end{align*} $$
so
$H_1(X_1(M), C_0, \operatorname {\mathrm {\mathbf {Z}}})$
is generated by the elements of the form
$\{0, \gamma 0\}$
for
$\gamma \in \Gamma _0(M)$
. We have thus proved that
$\varphi $
is surjective. To prove that
$\varphi $
is injective, it is enough to show that
$ \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]/I$
is a free
$\operatorname {\mathrm {\mathbf {Z}}}$
-module of the same rank as
$H_1(X_1(M), C_0, \operatorname {\mathrm {\mathbf {Z}}})$
.
Since
$M>3$
, the group
$\Gamma _1$
is torsion-free and we have
$H_1(\Gamma _1, \operatorname {\mathrm {\mathbf {Z}}}) \simeq H_1(Y_1(M), \operatorname {\mathrm {\mathbf {Z}}})$
(cf. (2.4)). By Shapiro’s lemma for group homology, we have
$H_1(\Gamma _1, \operatorname {\mathrm {\mathbf {Z}}}) \kern1.2pt{\simeq}\kern1.2pt H_1(\Gamma _0, \operatorname {\mathrm {\mathbf {Z}}}[G])$
. Using the description of group homology in terms of inhomogeneous chains (cf. (2.3)), one gets a short exact sequence
$$ \begin{align*}0 \rightarrow H_1(\Gamma_0, \operatorname{\mathrm{\mathbf{Z}}}[G]) \rightarrow \operatorname{\mathrm{\mathbf{Z}}}[G \times \Gamma_0]/J \xrightarrow{\partial} \operatorname{\mathrm{\mathbf{Z}}}[G] \rightarrow \operatorname{\mathrm{\mathbf{Z}}} \rightarrow 0,\end{align*} $$
where
$\partial (g, \gamma ) = g\cdot \langle \gamma \rangle ^{-1} - g$
and J is the subgroup of
$\operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]$
generated by
$$ \begin{align*}(g, \gamma\gamma') - (g, \gamma) - (g\langle \gamma \rangle^{-1}, \gamma')\end{align*} $$
for
$g \in G$
,
$\gamma , \gamma ' \in \Gamma _0(M)$
. The last map
$\operatorname {\mathrm {\mathbf {Z}}}[G] \rightarrow \operatorname {\mathrm {\mathbf {Z}}}$
is the augmentation (degree) map (note that J is indeed in the kernel of
$\partial $
).
As in (2.2), we have an exact sequence
$$ \begin{align*}0 \rightarrow \operatorname{\mathrm{\mathbf{Z}}} \rightarrow \operatorname{\mathrm{\mathbf{Z}}}[C_M] \rightarrow H_1(Y_1(M), \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow 0,\end{align*} $$
where
$C_M$
is the set of cusps of
$Y_1(M)$
. Here, the map
$ \operatorname {\mathrm {\mathbf {Z}}}[C_M]$
sends a cusp c to the homology class of a small loop around c in
$Y_1(M)$
.
Under the isomorphism
$H_1(Y_1(M), \operatorname {\mathrm {\mathbf {Z}}}) \simeq H_1(\Gamma _0, \operatorname {\mathrm {\mathbf {Z}}}[G])$
and the embedding
$H_1(\Gamma _0, \operatorname {\mathrm {\mathbf {Z}}}[G]) \hookrightarrow \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]/J$
described above, the map
$\operatorname {\mathrm {\mathbf {Z}}}[C_M] \rightarrow H_1(\Gamma _0, \operatorname {\mathrm {\mathbf {Z}}}[G])$
sends a cusp c to the class of
$(1,\gamma _c) - (1,1)$
in
$\operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]/J$
.
Thus, we have an exact sequence
$$ \begin{align*}0 \rightarrow H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow \operatorname{\mathrm{\mathbf{Z}}}[G \times \Gamma_0]/I' \xrightarrow{\partial} \operatorname{\mathrm{\mathbf{Z}}}[G] \rightarrow \operatorname{\mathrm{\mathbf{Z}}} \rightarrow 0 \text{ ,}\end{align*} $$
where
$I'$
is the subgroup of
$ \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]$
generated by the elements
$$ \begin{align*}(g, \gamma\gamma') - (g, \gamma) - (g\langle \gamma \rangle^{-1}, \gamma')\end{align*} $$
and by the
$(1, \gamma _c) - (1, 1)$
for all
$g \in G$
,
$\gamma , \gamma ' \in \Gamma _0(M)$
and
$c \in \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
.
The involution
$G \rightarrow G$
given by
$g \mapsto g^{-1}$
induces an isomorphism
$ \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]/I' \xrightarrow {\sim } \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]/I$
. This shows that
$ \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]/I$
is a free
$\operatorname {\mathrm {\mathbf {Z}}}$
-module of rank
$\operatorname {\mathrm {rk}}_{\operatorname {\mathrm {\mathbf {Z}}}} H_1(X_1(M), \operatorname {\mathrm {\mathbf {Z}}}) + \#G -1$
. We have
$\# G = \# C_0$
, and the exact sequence (cf. (2.1))
$$ \begin{align*}0 \rightarrow H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H_1(X_1(M), C_0, \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow \operatorname{\mathrm{\mathbf{Z}}}[C_0] \rightarrow \operatorname{\mathrm{\mathbf{Z}}} \rightarrow 0\end{align*} $$
shows that
$\operatorname {\mathrm {rk}}_{\operatorname {\mathrm {\mathbf {Z}}}} \operatorname {\mathrm {\mathbf {Z}}}[G \times \Gamma _0]/I = \operatorname {\mathrm {rk}}_{\operatorname {\mathrm {\mathbf {Z}}}} H_1(X_1(M), C_0, \operatorname {\mathrm {\mathbf {Z}}})$
, as wanted.
Let us apply Proposition 4.1 to
$T = \mathcal {K}_M$
and
$u : \Gamma _0(M) \rightarrow T$
induced by
$\Theta _M$
. Since u is parabolic and
$\Gamma _1(M)$
acts trivially on T, the condition that u vanishes on parabolic elements of
$\Gamma _1(M)$
is satisfied. Therefore, we get a
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
-equivariant homomorphism
$$ \begin{align*}\tilde{\Theta}_M : H_1(X_1(M), C_0, \operatorname{\mathrm{\mathbf{Z}}}) \rightarrow \mathcal{K}_M \text{ .}\end{align*} $$
5 Proofs of the theorems
We start with the following lemma (we thank Venkatesh for explaining this to us).
Lemma 5.1 Let
$M \geq 4$
and
$p\geq 2$
be a prime. Let
$\alpha = \begin {pmatrix} 1 & 0 \\ 0 & p \end {pmatrix}$
. Let
$\phi _p : \Gamma _0(Mp) \rightarrow \Gamma _0(M)$
be the group homomorphism sending
$\begin {pmatrix} a & b \\ c & d \end {pmatrix}$
to
$\begin {pmatrix} a & pb \\ c/p & d \end {pmatrix}$
. We have a commutative diagram
where
$\alpha _*$
is the trace map induced by
$\alpha $
.
Proof Since
$\partial : {\mathbf {K}}_2^{(0)}/\operatorname {\mathrm {\mathbf {Z}}} \cdot \{-z_1, -z_2\} \hookrightarrow {\mathbf {K}}_1$
is injective, it suffices to prove that the following diagram is commutative:
In other words, if
$\gamma = \begin {pmatrix} a & b \\ c & d \end {pmatrix} \in \Gamma _0(Mp)$
, then it suffices to check that
$$ \begin{align} \alpha_*(\langle b, d \rangle - \langle 0, 1 \rangle) = \langle pb, d \rangle - \langle 0, 1 \rangle \end{align} $$
(cf. (3.1) for the definition of the symbol
$\langle b , d \rangle $
).
As in [Reference Sharifi and Venkatesh9, equation (3.2)], we have
$\langle b , d \rangle = \gamma ^* \langle 0, 1 \rangle $
where
$\gamma ^* : {\mathbf {K}}_1 \rightarrow {\mathbf {K}}_1$
is the pullback induced by the right action of
$\gamma $
on
$\mathbf {G}_m^2$
. Note that we have
$\gamma ^* = (\gamma ^{-1})_*$
. Thus, we have
$$ \begin{align*}\alpha_* \langle b, d \rangle &= \alpha_* (\gamma^{-1})_* \langle 0, 1 \rangle = (\gamma^{-1}\cdot \alpha)_* \langle 0, 1 \rangle = (\alpha^{-1} \cdot \gamma^{-1}\cdot\alpha)_* \alpha_* \langle 0, 1 \rangle \\&=(\alpha^{-1} \cdot \gamma \cdot\alpha)^* \alpha_* \langle 0, 1 \rangle \text{ .}\end{align*} $$
Let us prove that
$\alpha _*\langle 0 , 1 \rangle = \langle 0, 1 \rangle $
. By definition,
$\langle 0 , 1 \rangle $
is the function
$1-z_1^{-1}$
on the divisor
$z_2=1$
of
$\mathbf {G}_m^2$
. Furthermore, right multiplication by
$\alpha $
on
$\mathbf {G}_m^2$
is given by
$(z_1, z_2) \mapsto (z_1, z_2^p)$
. Thus, the divisor
$z_2=1$
is mapped to itself by right multiplication by
$\alpha $
. Therefore (cf. [Reference Sharifi and Venkatesh9, Remark 2.3.3]),
$\alpha _*\langle 0 , 1 \rangle $
is the norm of
$1-z_1^{-1}$
via the identity map
$K_1(\operatorname {\mathrm {\mathbf {Q}}}(z_1)) \rightarrow K_1(\operatorname {\mathrm {\mathbf {Q}}}(z_1))$
, on the divisor
$z_2=1$
. This proves that
$\alpha _*\langle 0 , 1 \rangle = \langle 0, 1 \rangle $
.
Since
$\alpha ^{-1} \cdot \gamma \cdot \alpha = \begin {pmatrix} a & pb \\ c/p & d \end {pmatrix}$
, we get
$\alpha _* \langle b, d \rangle = \langle pb, d \rangle $
. This proves (5.1), and concludes the proof of Lemma 5.1.
We are now ready to prove Theorem 1.4. Let
$$ \begin{align*}\gamma' \in \Gamma_0(Mp),\end{align*} $$
and let
$$ \begin{align*}\gamma = \phi_p(\gamma') \in \Gamma_0(M)\end{align*} $$
be as in Lemma 5.1. Let
$f : \mathbf {G}_m^2 \rightarrow \mathbf {G}_m^2$
given by
$$ \begin{align*}f : (z_1, z_2) \mapsto (z_1, z_2^p)\end{align*} $$
(note that f is induced by the right action of the matrix
$\alpha $
of Lemma 5.1).
Let
$U = U_{\gamma }$
be as in Section 3 and
$$ \begin{align*}U' = f^{-1}(U) \cap U_{\gamma'} \text{ .}\end{align*} $$
Both U and
$U'$
are open subschemes of
$\mathbf {G}_m^2$
, and we have
$(1, \zeta _M) \in U$
and
$(1, \zeta \cdot \zeta _{Mp}) \in U'$
for all pth root of unity
$\zeta $
.
Consider the following Cartesian diagram of schemes:
where
$s_M$
is given by the closed point
$(1, \zeta _M) \in U$
, and X makes the diagram Cartesian by definition.
Lemma 5.2 We have a natural isomorphism of schemes over
$\operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}})$
:
$$ \begin{align*} X \simeq \operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M)) \times_{\operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}[T, T^{-1}])} \operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}[t, t^{-1}]), \end{align*} $$
where the maps
$\operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M)) \rightarrow \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}[T, T^{-1}])$
and
$\operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}[t, t^{-1}]) \rightarrow \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}[T, T^{-1}])$
are given by
$T \mapsto \zeta _M$
and
$T \mapsto t^p$
, respectively.
Under this isomorphism, the map
$X \rightarrow \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _M))$
is the projection onto the first factor. The compositum map
$X \rightarrow U' \hookrightarrow \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}[z_1, z_2, z_1^{-1}, z_2^{-1}])$
is given by the compositum of the projection
$$ \begin{align*} \operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M)) \times_{\operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}[T, T^{-1}])} \operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}[t, t^{-1}]) \rightarrow \operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}[t, t^{-1}]) \end{align*} $$
and of the map
$ \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}[t, t^{-1}]) \rightarrow \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}[z_1, z_2, z_1^{-1}, z_2^{-1}])$
defined by
$z_1 \mapsto 1$
and
${z_2 \mapsto t}$
.
Proof Let Y be such that the following diagram is Cartesian:
We claim that there is a natural isomorphism
$Y \simeq X$
. 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
$\mathbf {G}_m^2$
) is contained in
$U'$
. This follows from the fact that
$f^{-1}(1, \zeta _M) \subset U'$
.
To conclude the proof of Lemma 5.2, note that there is a commutative diagram
Lemma 5.2 yields a more concrete description of X: we have
$$ \begin{align} X \simeq \operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_M)[t]/(t^p-\zeta_M)) \text{ .} \end{align} $$
This latter isomorphism can be rewritten more simply in a way which depends on whether p divides M or not.
5.1 The case
$p \mid M$
Assume first that p divides M. Then
$X \simeq \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _{Mp}))$
and the map
$X \rightarrow U'$
comes from the point
$(1, \zeta _{Mp}) \in U'$
. Applying the functor
$H^2( \cdot , 2)$
to (5.2) (cf. [Reference Sharifi and Venkatesh9, Lemma 2.1.1]), we get a commutative diagram
and hence a commutative diagram
As explained in Section 3, we have
$$ \begin{align*} \Theta(\gamma') \in H^2(U_{\gamma'}, 2) /\operatorname{\mathrm{\mathbf{Z}}} \cdot \{-z_1, -z_2\} \end{align*} $$
and
$$ \begin{align*} \Theta(\gamma) \in H^2(U_{\gamma}, 2)/\operatorname{\mathrm{\mathbf{Z}}} \cdot \{-z_1, -z_2\} \text{ .} \end{align*} $$
Since
$U'$
is an open subset of
$U_{\gamma '}$
, we have a functorial embedding
$$ \begin{align*} H^2(U_{\gamma'}, 2) /\operatorname{\mathrm{\mathbf{Z}}} \cdot \{-z_1, -z_2\} \hookrightarrow H^2(U', 2) /\operatorname{\mathrm{\mathbf{Z}}} \cdot \{-z_1, -z_2\} \text{ .} \end{align*} $$
Thus, we have
$$ \begin{align*} \Theta(\gamma') \in H^2(U', 2) /\operatorname{\mathrm{\mathbf{Z}}} \cdot \{-z_1, -z_2\} \text{ .} \end{align*} $$
By Lemma 5.1, we have
$$ \begin{align*} f_* \Theta(\gamma') = \Theta(\gamma) \text{ .} \end{align*} $$
Therefore, using (5.4), we get
$s_{M}^*(\Theta (\gamma )) = \operatorname {\mathrm {Norm}}(s_{Mp}^*(\Theta (\gamma ')))$
, i.e.,
$$ \begin{align} \operatorname{\mathrm{Norm}}(\Theta_{Mp}(\gamma')) = \Theta_M(\gamma) \text{ .} \end{align} $$
By Proposition 4.1, equation (5.5) yields the following commutative diagram:
where
$C_0'$
(resp.
$C_0$
) is the set of cusps of
$X_1(Mp)$
(resp.
$X_1(M)$
) in the same diamond orbit as
$0$
. The top horizontal map (resp. bottom horizontal map) is equivariant for the action of
$(\operatorname {\mathrm {\mathbf {Z}}}/Mp\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
(resp.
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
).
After applying the Atkin–Lehner involution
$W_{Mp}$
and
$W_M$
to the two lines of (5.6), we get a commutative diagram
where
$C_{\infty }'$
(resp.
$C_{\infty }$
) is the set of cusps of
$X_1(Mp)$
(resp.
$X_1(M)$
) in the same orbit as
$\infty $
. We have used the facts that
$\varpi _{Mp} = \tilde {\Theta }_{Mp} \circ W_{Mp}$
and
$\varpi _{M} = \tilde {\Theta }_{M} \circ W_{M}$
. This follows from [Reference Sharifi and Venkatesh9, Proposition 4.3.3], where the authors use usual Manin symbols (whereas our map
$\varpi _M$
uses Manin symbols twisted by the Atkin–Lehner involution).
Note that
$\varpi _{Mp}$
and
$\varpi _{M}$
are anti-equivariant for the actions of
$(\operatorname {\mathrm {\mathbf {Z}}}/Mp\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
and
$(\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
, respectively. This means that for any
$x \in H_1(X_1(Mp), C_{\infty }', \operatorname {\mathrm {\mathbf {Z}}})$
and
$g \in (\operatorname {\mathrm {\mathbf {Z}}}/Mp\operatorname {\mathrm {\mathbf {Z}}})^{\times }/\pm 1$
, we have
$$ \begin{align} \varpi_{Mp}(g\cdot x) = g^{-1} \cdot \varpi_{Mp}(x) \end{align} $$
(and similarly for
$\varpi _M$
). Indeed, we have
$W_{Mp} \circ \langle g \rangle = \langle g^{-1} \rangle \circ W_{Mp}$
. This could also have been checked easily directly on the definition of
$\varpi _{Mp}$
and
$\varpi _{M}$
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
$X_1(Mp)$
as in Theorem 1.4. If
$C \subset C_{\infty }'$
, then Theorem 1.4 follows from (5.7) (we just restrict
$\varpi _{Mp}$
to
$H_1(X_1(Mp), C, \operatorname {\mathrm {\mathbf {Z}}}) \subset H_1(X_1(Mp), C_{\infty }', \operatorname {\mathrm {\mathbf {Z}}})$
). Let us explain how to deduce the general case from this special case.
Fix
$c \in \mathbf {P}^1(\operatorname {\mathrm {\mathbf {Q}}})$
such that
$\Gamma _1(Mp)\cdot c \in C$
. An element of
$H_1(X_1(Mp), C, \operatorname {\mathrm {\mathbf {Z}}})$
is of the form
$\{c, \gamma c\}$
for some
$\gamma \in \Gamma _0(Mp)$
. The assumption that all the elements of C are in the same diamond orbit under
$\operatorname {\mathrm {Ker}}((\operatorname {\mathrm {\mathbf {Z}}}/Mp\operatorname {\mathrm {\mathbf {Z}}})^{\times } \rightarrow (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times })$
means that we can actually choose
$\gamma $
in
$\Gamma _0(Mp) \cap \Gamma _1(M)$
.
We have
$$ \begin{align*} \{c, \gamma c\} &= \{c, \infty\} + \{\infty, \gamma \infty\} + \{\gamma \infty, \gamma c\} \\& = \{\infty, \gamma \infty\} + (\langle \gamma \rangle - 1)\cdot \{\infty, c \} \text{ ,} \end{align*} $$
where
$\langle \gamma \rangle $
is the diamond operator associated with
$\gamma $
. Since
$\gamma \in \Gamma _0(Mp) \cap \Gamma _1(M)$
, we have
$ \pi _2\left ( (\langle \gamma \rangle - 1)\cdot \{\infty , c \}\right ) = 0$
. Thus, we have
$$ \begin{align*}\pi_2(\{c, \gamma c \}) = \pi_2(\{\infty, \gamma \infty \}) \text{ .}\end{align*} $$
We also have
$$ \begin{align*} \operatorname{\mathrm{Norm}}\left( \varpi_{Mp}((\langle \gamma \rangle - 1)\{\infty, c \} ) \right) = 0 \end{align*} $$
in
$ \mathcal {K}_M$
since by (5.8) we have
$$ \begin{align*} \varpi_{Mp}((\langle \gamma \rangle - 1)\{\infty, c \} ) \in (\langle \gamma \rangle^{-1} - 1)\cdot \mathcal{K}_{Mp} \text{ .} \end{align*} $$
Thus, we have
$$ \begin{align*} \operatorname{\mathrm{Norm}}(\varpi_{Mp}( \{c, \gamma c\} )) &= \operatorname{\mathrm{Norm}}(\varpi_{Mp}( \{\infty, \gamma \infty\} )) \\& = \varpi_M(\pi_2( \{\infty, \gamma \infty\} )) \\& =\varpi_M(\pi_2( \{c, \gamma c\} )) \text{ .} \end{align*} $$
This concludes the proof of Theorem 1.4 in the case
$p \mid M$
.
5.2 The case
$p \nmid M$
Assume now that p does not divide M. Note that in this case we have
$(1, \zeta _M) \in U'$
. By (5.3), there is an isomorphism
$$ \begin{align*} X\simeq \operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_{Mp})) \sqcup \operatorname{\mathrm{Spec}}(\operatorname{\mathrm{\mathbf{Q}}}(\zeta_{M})) \end{align*} $$
such that:
-
• The map
$X \rightarrow U'$
is given by the two inclusions
$(1, \zeta _M^{p^*}) \in U'$
and
$(1, \zeta _{Mp}) \in U'$
, where
$p^* \in (\operatorname {\mathrm {\mathbf {Z}}}/M\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
is the inverse of p modulo M. -
• The map
$X \rightarrow \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _{M}))$
is given by the canonical map
$\operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _{Mp})) \rightarrow \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _{M}))$
and the identity map
$\operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _{M})) \rightarrow \operatorname {\mathrm {Spec}}(\operatorname {\mathrm {\mathbf {Q}}}(\zeta _{M}))$
.
Applying the functor
$H^2( \cdot , 2)$
to (5.2) (cf. [Reference Sharifi and Venkatesh9, Lemma 2.1.1]), we get the following commutative diagram:
Combining Lemma 5.1 and (5.9), we get
$$ \begin{align} \operatorname{\mathrm{Norm}}(\Theta_{Mp}(\gamma')) + (\sigma_p^{-1})^*(\Theta_M(\gamma')) = \Theta_M(\gamma) \text{ .} \end{align} $$
By (5.8), we have
$$ \begin{align*} \varpi_{Mp} \circ \langle p \rangle =(\sigma_p^{-1})^* \circ \varpi_{Mp} \text{ .} \end{align*} $$
As in (5.7), we then get a commutative diagram
(note that both
$\pi _1$
and
$\langle p \rangle \pi _2$
send
$C_{\infty }'$
to
$C_{\infty }$
, so that this diagram makes sense). An argument identical to the one when
$p\mid M$
shows that the diagram of Theorem 1.4 commutes. This concludes the proof of Theorem 1.4.
5.3 Proof of Theorem 1.6
Let us now prove Theorem 1.6. Let
$M \geq 4$
and
$p \geq 5$
be a prime. One needs to prove that
$$ \begin{align*} \varpi_M \otimes \operatorname{\mathrm{\mathbf{Z}}}_p : H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}_p) \rightarrow \mathcal{K}_M \otimes \operatorname{\mathrm{\mathbf{Z}}}_p \end{align*} $$
is annihilated by the operator
$U_{\ell }-1$
for any prime
$\ell \mid M$
. If
$p \mid M$
, then this is a result of Fukaya and Kato (cf. Theorem 1.3). Therefore, we shall assume in what follows that p does not divide M.
By Theorem 1.4(ii), we have a commutative diagram
By the result of Fukaya and Kato, one knows that
$\varpi _{Mp} \otimes \operatorname {\mathrm {\mathbf {Z}}}_p$
is annihilated by the Hecke operator
$U_{\ell }-1$
. Since
$\pi _1-\langle p \rangle \pi _2$
commutes with the action of
$U_{\ell }-1$
on both sides, it suffices to prove that
$$ \begin{align*} \pi_1-\langle p \rangle \pi_2 : H_1(X_1(Mp), \operatorname{\mathrm{\mathbf{Z}}}_p) \rightarrow H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}_p) \end{align*} $$
is surjective. Note that
$$ \begin{align*} H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}_p) \otimes_{\operatorname{\mathrm{\mathbf{Z}}}} \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}} \simeq H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \end{align*} $$
by the Universal Coefficient Theorem (as
$H_0(X, \operatorname {\mathrm {\mathbf {Z}}})$
is torsion-free).
By the Nakayama lemma, the surjectivity of our map
$ H_1(X_1(Mp), \operatorname {\mathrm {\mathbf {Z}}}_p) \rightarrow H_1(X_1(M), \operatorname {\mathrm {\mathbf {Z}}}_p)$
is equivalent to the surjectivity of the map
$$ \begin{align*} \pi_1-\langle p \rangle \pi_2 : H_1(X_1(Mp), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \text{ .} \end{align*} $$
By the Poincaré duality, it suffices to prove that
$$ \begin{align*} \pi_1^*- \pi_2^* \circ \langle p \rangle^{-1} : H_1(X_1(M), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H_1(X_1(Mp), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \end{align*} $$
is injective. Note that
$H_1(X_1(M), \operatorname {\mathrm {\mathbf {Z}}}/p\operatorname {\mathrm {\mathbf {Z}}})$
is canonically isomorphic to the parabolic cohomology
$H^1_p(\Gamma _1(M), \operatorname {\mathrm {\mathbf {Z}}}/p\operatorname {\mathrm {\mathbf {Z}}})$
, i.e., the subgroup of
$H^1(\Gamma _1(M), \operatorname {\mathrm {\mathbf {Z}}}/p\operatorname {\mathrm {\mathbf {Z}}})$
consisting of classes of cocycles which are coboundaries when restricted to stabilizers of cusps.
Thus, it is enough for us to prove that the map
$$ \begin{align} \pi_1^*- \pi_2^* \circ \langle p \rangle^{-1} : H^1(\Gamma_1(M), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H^1(\Gamma_1(Mp), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \end{align} $$
is injective. By [Reference Edixhoven and Khare2, Lemma 1], the map
$$ \begin{align*} H^1(\Gamma_1(M), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}})^2 \xrightarrow{\pi_1^*+ \pi_2^*} H^1(\Gamma_1(M)\cap \Gamma_0(p), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \end{align*} $$
is injective.
By the inflation-restriction exact sequence, we have an exact sequence
$$ \begin{align*} 0 \rightarrow H^1((\operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}})^{\times}, \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H^1(\Gamma_1(M) \cap \Gamma_0(p), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H^1(\Gamma_1(Mp), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \text {.} \end{align*} $$
Since
$(\operatorname {\mathrm {\mathbf {Z}}}/p\operatorname {\mathrm {\mathbf {Z}}})^{\times }$
has order prime to p, we have
$H^1((\operatorname {\mathrm {\mathbf {Z}}}/p\operatorname {\mathrm {\mathbf {Z}}})^{\times }, \operatorname {\mathrm {\mathbf {Z}}}/p\operatorname {\mathrm {\mathbf {Z}}})=0$
, so the map
$$ \begin{align*} H^1(\Gamma_1(M) \cap \Gamma_0(p), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \rightarrow H^1(\Gamma_1(Mp), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \end{align*} $$
is injective. We then conclude that the map
$$ \begin{align*} H^1(\Gamma_1(M), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}})^2 \xrightarrow{\pi_1^*+ \pi_2^*} H^1(\Gamma_1(Mp), \operatorname{\mathrm{\mathbf{Z}}}/p\operatorname{\mathrm{\mathbf{Z}}}) \end{align*} $$
is injective. This proves the injectivity of (5.13), and thus concludes the proof of Theorem 1.6.
Acknowledgment
The first author would like to thank Akshay Venkatesh for a useful discussion concerning this work, especially regarding Lemma 5.1. We would like to thank the anonymous referee for providing detailed and useful comments, which helped to greatly improve the exposition.
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