2.1 Motivic cohomology

We shall define motivic cohomology using Bloch's cycle complexes: see, for instance, [Reference BlochBlo86, Reference BlochBlo94, Reference LevineLev99, Reference LevineLev04]. This has a certain psychological advantage for us in that it allows us to think of our classes as coming from cycles. However, which theory of motivic cohomology is used does not matter in our final results, which concern smooth schemes over perfect fields.

Let $Y$ denote a quasi-projective scheme of finite type over a perfect field $F$. For nonnegative integers $j$ and $k$, let $z^k(Y,j)$ denote the group of codimension $k$ cycles in $Y \times \Delta ^j$ (the $F$-fiber product) that meet $Y \times \Phi$ for each face $\Phi$ of the algebraic $j$-simplex $\Delta ^j$ over $F$ properly. Via alternating sums of face maps, the $z^k(Y,\cdot )$ form a homological complex with $z^k(Y,j)$ in degree $j$. This is Bloch's cycle complex for $Y$; its homology groups are called higher Chow groups. These complexes admit pullbacks by flat maps and pushforwards by proper maps [Reference BlochBlo86, Proposition 1.3].

For any $i \in \mathbb {Z}$, we set

\[ H^i(Y,k) = H_{2k-i}(z^k(Y,\cdot)). \]

We also set $H^i(Y,k) = 0$ for negative integers $k$. If $Y$ is smooth, then $H^i(Y,k)$ is naturally isomorphic to the $i$th motivic cohomology group of $Y$ with $\mathbb {Z}(k)$-coefficients in the sense of Voevodsky [Reference VoevodskyVoe00] (see [Reference Mazza, Voevodsky and WeibelMVW06, Theorem 19.1]):Footnote ⁷

\[ H^i(Y, k) \cong H^i(Y, \mathbb{Z}(k)). \]

As such, we will refer to the groups $H^i(Y,k)$ themselves as motivic cohomology groups. (This is slightly nonstandard notation, which hopefully makes some of the typography easier to read.)

We briefly summarize a number of standard properties of these groups. To start with, as a consequence of Bloch's strong moving lemma [Reference BlochBlo94, Theorem 0.1], they admit arbitrary pullbacks (see [Reference BlochBlo86, Theorem 4.1]). They also satisfy:

• – if $Y = \coprod _{h=1}^t Y_h$ is a finite disjoint union of $F$-schemes, then $H^i(Y,k) \cong \bigoplus _{h=1}^t H^i(Y_h,k)$;

• – $H^i(Y,k) \cong H^i(Y \times \mathbb {A}^1,k)$ via pullback by the projection morphism $Y \times _F \mathbb {A}^1 \to Y$ (see [Reference BlochBlo86, Theorem 2.1]);

• – $H^0(Y, 0) \cong \mathbb {Z}$ if $Y$ is connected and $H^i(Y, 0) = 0$ for $i \neq 0$;

• – if $Y$ is smooth, then $H^1(Y,1)$ is naturally isomorphic to the group of global units on $Y$, and $H^2(Y,1)$ is naturally isomorphic to the Picard group of $Y$, while $H^i(Y,1) = 0$ for $i \notin \{1,2\}$ (see [Reference Mazza, Voevodsky and WeibelMVW06, Corollary 4.2]);

• – if $Y$ is smooth, then $H^i(Y,k) = 0$ for $i > k + \dim Y$ (see [Reference Mazza, Voevodsky and WeibelMVW06, Theorem 3.6]);

• – if $Y$ is a smooth variety over $F$, then $H^i(Y,k) = 0$ for $i > 2k$ (see [Reference Mazza, Voevodsky and WeibelMVW06, Theorem 19.3]);

• – if $f \colon X \rightarrow Y$ is a finite locally free morphism of quasi-projective $F$-schemes of finite type (so proper of relative dimension zero), then $f_* f^*$ is multiplication by the degree of $f$ (cf. [Sta24, Lemma 02RH]).

Suppose that $Y$ is equidimensional. Then, for any closed $F$-subscheme $\rho \colon Z \to Y$ of pure codimension $c$ and its complement $\iota \colon U \to Y$, there is an exact Gysin sequence

\[ \cdots \rightarrow H^i(Y, k) \xrightarrow{\iota^*} H^i(U, k) \xrightarrow{\partial} H^{i-2c+1}(Z, k-c) \xrightarrow{\rho_*} H^{i+1}(Y, k) \rightarrow \cdots. \]

We refer to the map $\partial$ as a residue map. It results from the distinguished triangle determined by the left exact sequence of complexes given by pushforward by $\iota$ and pullback by $\rho$ given by Bloch's moving lemma.

Motivic cohomology also has cup products

\[ \cup \colon H^i(Y,k) \times H^{i'}(Y,k') \to H^{i+i'}(Y,k+k'), \]

which can be constructed by pulling back an external product via the diagonal [Reference BlochBlo86, § 5]. There is then an isomorphism of graded rings

\[ \bigoplus_{i=0}^{\infty} K_i^M(F) \xrightarrow{\sim} \bigoplus_{i=0}^{\infty} H^i(F,i) \]

induced by the standard identifications of both sides with $\mathbb {Z}$ and $F^{\times }$ in degrees $0$ and $1$ (see [Reference Mazza, Voevodsky and WeibelMVW06, Theorem 5.1 and Lemma 5.6]). Recall that the canonical homomorphism $K_i^M(F) \to K_i(F)$ to the $i$th algebraic $K$-group of $F$ is an isomorphism for $i \le 2$, the case of $i=2$ being Matsumoto's theorem.

We will need to compare compositions of pushforwards and pullbacks. For instance, we shall often employ the following lemma in the case that the underlying schemes are spectra of fields and $i=k$, in which case the assertion is one of Milnor $K$-theory (see also [Reference RostRos96, Rule 1c, p. 329] for a direct formulation of this assertion, noting Theorem 1.4 therein).

Lemma 2.1.1 (Base change)

Suppose that

is a cartesian diagram of smooth, equidimensional quasi-projective schemes of finite type over $F$, with $\pi _Y$ flat and $f$ proper. Then $\pi _X$ is flat, $f'$ is proper, and

\[ (f')_* \pi_X^* = \pi_Y^* f_* \]

as morphisms $H^{i}(X, k) \rightarrow H^{i+2c}(Y', k)$, where $c = \dim Y - \dim X$ is the relative dimension of f.

Corollary 2.1.2 (Projection formula)

Let $f \colon X \to Y$ be a proper, relative dimension c morphism of smooth, equidimensional quasi-projective schemes of finite type over $F$, and let $\alpha \in H^i(X,k)$ and $\beta \in H^{i'}(Y,k')$. Then

\[ f_*(\alpha \cup f^*(\beta)) = f_*(\alpha) \cup \beta \in H^{i+i'+2c}(X,k+k'). \]

Proof. We need only apply Lemma 2.1.1 to the cartesian square

where $\Delta _X$ and $\Delta _Y$ are the diagonal embeddings of $X$ and $Y$, respectively.

We also have the following compatibility of residues with transfers and inclusions of fields.

Lemma 2.1.3 Let $E/F$ be a finite extension of fields. Then for $v$ a discrete valuation on $F$, one has

\[ \partial_{v} \circ \operatorname{N}_{E/F} = \sum_{w \mid v} \operatorname{N}_{k(w)/k(v)} \circ \partial_w \]

as morphisms $K_n^{M}(E) \rightarrow K_{n-1}^{M}(k(v))$ on Milnor $K$-theory; the sum on the right is over valuations $w$ on $E$ extending $v$, the symbols $\partial _v$ and $\partial _w$ are the residue maps on Milnor $K$-theory induced by the valuations $v$ and $w$, and $\operatorname {N}$ denotes transfer in Milnor $K$-theory.

Similarly, for each $w \mid v$ as above, we have

\[ \partial_w \circ \iota_{E/F} = e_{k(w)/k(v)} \cdot \iota_{k(w)/k(v)} \circ \partial_v \]

as morphisms $K_n^M(F) \rightarrow K_{n-1}^M(k(w))$, where $\iota$ denotes a map on Milnor $K$-theory induced by inclusions of fields, and $e_{k(w)/k(v)}$ is the ramification index.

2.2 Coniveau spectral sequences

Let us recall the coniveau spectral sequence for motivic cohomology. We refer to [Reference DégliseDeg08], which contains many of the details required to set this up. The primary role of this spectral sequence is that it provides complexes that compute motivic cohomology in our situations of interest, and these are also manifestly equivariant for the automorphism group of the ambient variety.

Continuity properties of motivic cohomology [Reference Mazza, Voevodsky and WeibelMVW06, Lemma 3.9] imply that for a finite- type smooth connected variety $Y$ over a field $F$ with function field $k(Y)$, we have

\[ H^p(k(Y),q) \cong \varinjlim_{U \subset Y} H^p(U, q), \]

where the limit is taken over open subvarieties $U$ of $Y$.Footnote ⁸

For $U$ as above and any irreducible divisor $D$ such that $D \cap U$ is nonempty, there is a residue homomorphism

\[ H^p(U - (D \cap U), q) \rightarrow H^{p-1}(D \cap U, q-1). \]

Consider the collection of open sets $U$ such that $D \cap U$ is smooth and nonempty. The collection of sets $U - (D \cap U)$ is cofinal in open sets on $Y$, and the collection of $D \cap U$ is cofinal in open sets on $D$. Therefore, the residue maps for $U$ in the collection induce a residue map

(2.1)\begin{equation} H^p(k(Y), q) \rightarrow H^{p-1}(k(D), q-1). \end{equation}

The latter map is determined by the field $k(Y)$ and the valuation $v$ on it which cuts out $D$ in $Y$ (see [Reference DégliseDeg08, Lemma 5.4.5]). When $p=q$, it is the residue in Milnor $K$-theory (see [Reference DégliseDeg08, Proposition 6.2.3]).

With these preliminaries in hand, we recall the coniveau spectral sequence for $n \ge 0$.

Theorem 2.2.1 There is a right half-plane spectral sequence with $E_1$-page

\[ E_1^{p,q} = \bigoplus_{x \in Y_p} H^{q-p}(k(x), n-p) \Rightarrow H^{p+q}(Y, n), \]

where $Y_p$ denotes the set of points of $Y$ of codimension $p$ and the differentials are residue maps.

The coniveau spectral sequence essentially carries the information of ‘all Gysin sequences at once’ and is a limit of spectral sequences attached to these Gysin sequences. We briefly explain its derivation: attached to a decreasing system $Z = (Z_p)_{p \in \mathbb {Z}}$ of closed $F$-subschemes of $Y$ with each $Z_p-Z_{p+1}$ smooth, $Z_p = Y$ for $p \le 0$, each $Z_p$ with $1 \le p \le n$ of pure codimension $p$, and $Z_p = \varnothing$ for $p > n$, we have Gysin sequences

(2.2)\begin{equation} \cdots \to H^i(Z_p,n-p) \to H^i(Z_p-Z_{p+1},n-p-1) \xrightarrow{\partial} H^{i-1}(Z_{p+1},n-p-1) \to \cdots \end{equation}

for $0 \le p \le n-1$. For $D^{p,q} = H^{q-p}(Z_p,n-p)$ and $E^{p,q} = H^q(Z_p-Z_{p+1},n)$, the exact couple $(D^{p,q},E^{p,q})$ determined by the exact sequences of (2.2) gives rise to a convergent right half-plane spectral sequence $E(Z)$ with $E_1$-page

\[ E_1^{p,q}(Z) = H^{q-p}(Z_p-Z_{p+1},n-p) \Rightarrow E^{p+q}(Z) = H^{p+q}(Y,n). \]

Note that the $q$th row of the $E_1$-page of this spectral sequence $E(Z)$ is a complex the form

\[ H^q(Y-Z_1,n) \xrightarrow{\partial} H^{q-1}(Z_1-Z_2,n-1) \xrightarrow{\partial} \cdots \xrightarrow{\partial} H^{q-n}(Z_n,0). \]

Our convention will be that $p$th term in this complex has homological degree the twist $n-p+1$.

If we have two collections $Z' = (Z'_p)_p$ and $Z = (Z_p)_p$ of closed subschemes as above with each $Z'_p$ a closed subscheme of $Z_p$, then we obtain morphisms $E^1_{p,q}(Z) \to E^1_{p,q}(Z')$ via composition $j^* \iota _*$ of pushforward and pullback along

\[ Z_p' - Z'_{p+1} \xrightarrow{\iota} Z_p - Z'_{p+1} \xleftarrow{j} Z_p -Z_{p+1} \]

(with $\iota$ a closed immersion and $j$ an open immersion). In particular, we can take direct limits of the spectral sequences over directed sets of such collections. If we use the collection of all $Z$, then we obtain the coniveau spectral sequence.

The row for $q = n$ in the $E_1$-page of the coniveau sequence is a homological complex $\mathsf {K}$ given in degrees $n$ through $0$ by

(2.3)\begin{equation} \mathsf{K} = \mathsf{K}^{(n)}(Y) \colon \quad K_n^M k(Y) \to \bigoplus_{x \in Y_1} K_{n-1}^M k(x) \to \cdots \to \bigoplus_{x \in Y_n} K_0^M k(x). \end{equation}

It follows from Lemmas 2.1.1 and 2.1.3 that pushforwards by proper maps and pullbacks by flat maps induce morphisms between these sequences via transfer maps and the maps induced by inclusions of fields, respectively, on Milnor $K$-theory. In this paper, we employ this complex for $n = 2$. So, let us describe this case in more detail.

Example 2.2.2 Suppose that $n=2$. Then the $E_1$-terms of the coniveau sequence in the range $0 \leq p \leq 2$ and $0 \leq q \leq 2$ look like this:

where the direct sums are over divisors $D$ and codimension $2$ points $x$.

Except possibly those with $p = 0$ and $q < 0$, all other terms vanish, recalling that the motivic cohomology $H^i(F,k)$ of a field $F$ vanishes when $i > k$. In particular, the spectral sequence degenerates, and the row

\[ \mathsf{K} = \mathsf{K}^{(2)}(Y) \colon \quad K_2 k(Y) \xrightarrow{\partial_2} \bigoplus_D K_1 k(D) \xrightarrow{\partial_1} \bigoplus_{x} \mathbb{Z} \]

is a complex in homological degrees $2$, $1$, and $0$ computing the cohomology groups $H^2(Y, 2)$, $H^3(Y, 2)$, and $H^4(Y, 2)$, respectively.

As noted after (2.1), the $D$-component of the map $\partial _2$ is given by the tame symbol in $K$-theory

(2.4)\begin{equation} \{f,g\} \mapsto (-1)^{v(f) v(g)} g^{v(f)} f^{-v(g)} \end{equation}

for the valuation $v$ attached to $D$. The map $\partial _1$ takes the divisor of $f \in k(D)^{\times }$ (i.e., yielding the order of vanishing at $f$ in each $K_0k(x) \cong \mathbb {Z}$ for $x \in D$), which we interpret in the sense of intersection theory if $D$ is not smooth.

Remark 2.2.3 Suppose that $n \le 2$. For any (connected) open subscheme $U$ of $Y$, the maps

\[ H^n(U,n) \rightarrow H^n(k(Y), n) \cong K_n^M k(Y) \]

are injective, as follows for $n = 2$ from the form of the coniveau spectral sequence for $U$ in Example 2.2.2, noting that $k(U) = k(Y)$ (and for $n \le 1$ more easily). Accordingly, we will say that a class in $K_n^M k(Y)$ is defined on $U$ if it lies in the image of the morphism $H^n(U,n) \rightarrow H^n(k(Y),n)$. Given a class $\kappa \in K_n^Mk(Y)$ defined on $U$ and a closed point $x \in U$, it is then meaningful to specialize $\kappa$ to $x$ via pullback, producing a class in $K_n^M k(x)$.

2.3 Trace maps

Let $G$ be a smooth, connected commutative group scheme over our base smooth variety $Y$ over $F$. Let $U$ be a nonempty open $F$-subscheme of a closed $F$-subscheme of $G$ of pure codimension. Multiplication by any positive integer $m$ defines a morphism $m \colon m^{-1}U \to U$. Pushforward by the finite map given by multiplication by $m$ on $G$ induces a map

\[ H^i(m^{-1}U,k) \to H^i(U,k). \]

If $m^{-1}U$ is a subscheme of $U$, then precomposing the pushforward by $m$ with pullback under inclusion gives a morphism

\[ [m]_* \colon H^i(U,k) \to H^i(U,k) \]

denoted by the same symbol, which we refer to as a trace map for $m$. (The reader might compare with [Reference Kings and RösslerKR17, Definition 2.1.1].)

In the remainder of this paper, we will frequently be interested in the ‘fixed parts’ of motivic cohomology groups, comprising all elements fixed by all (but finitely many) trace maps $[p]_*$ for $p$ prime not equal to the characteristic of $F$. This frequently isolates a subspace of elements of geometric significance.

Example 2.3.1 Let $z$ denote the coordinate function on $G = \mathbb {G}_m$ over $F$. Choose $m$ not divisible by the characteristic of $F$, and suppose that $U$ open in $G$ satisfies $m^{-1}U \subset U$. The map $[m]_*$ on $H^1(U,1) \subset F(z)^{\times }$ is characterized by the property that for $f \in H^1(U,1)$ and $\alpha \in U(F) \subseteq F^{\times }$,

\[ ([m]_*f)(\alpha) = \prod_{\beta^m = \alpha} f(\beta), \]

with the product taken over $m$th roots of $\alpha$ inside an algebraic closure of $F$. The pullback $[m]^*$ is given more simply by

\[ ([m]^*f)(\alpha) = f(\alpha^m). \]

In particular, for $U = \mathbb {G}_m-\{1\}$, the norm map $[m]_*$ fixes $1-z$ in $H^1(\mathbb {G}_m-\{1\},1)$, as follows from the calculation

(2.5)\begin{align} \prod_{i=0}^{m-1} (1-\zeta_m^i z^{1/m}) = 1-z, \end{align}

where $\zeta _m$ denotes a primitive $m$th root of unity. (In fact, $1-z$ is $[m]_*$-fixed even for $m$ divisible by $\operatorname {char} F$.)

Example 2.3.2 Take two smooth connected commutative group schemes $G_1$ and $G_2$ over $F$, and set $G =G_1 \times G_2$. For $\nu _j \in H^{i_j}(G_j,k_j)$ with $i_j \in \mathbb {Z}$ and $k_j \ge 0$, define the exterior product $\nu _1 \boxtimes \nu _2 \in H^{i_1+i_2}(G, k_1+k_2)$ as the cup product $\pi _1^* \nu _1 \cup \pi _2^* \nu _2$, with $\pi _j \colon G \to G_j$ the projection maps. We then have

(2.6)\begin{equation} [m]_* (\nu_1 \boxtimes \nu_2) = [m]_* \nu_1 \boxtimes [m]_* \nu_2. \end{equation}

(To verify this from basic properties, factor the multiplication-by-$m$ map $[m]$ as a product of corresponding maps $[m]_1$ and $[m]_2$ in the first and second coordinates. Then (2.6) follows from the equality

\[ [m]_{1*} (\pi_1^* \nu_1 \cup \pi_2^* \nu_2) = [m]_{1*} (\pi_1^* \nu_1 \cup [m]_1^* \pi_2^* \nu_2) = [m]_{1*} \pi_1^* \nu_1 \cup \pi_2^* \nu_2 = \pi_1^* [m]_{*} \nu_1 \cup \pi_2^* \nu_2, \]

where the middle equality is the projection formula of Lemma 2.1.2, together with the analogous assertion with the roles of first and second variables switched.)

The maps $[m]_*$ commute with each other, with pullback to open subschemes, with pushforward by inclusion of closed subschemes, and with residue maps in Gysin sequences (see [Reference Kings and RösslerKR17, § 2.1]). They also induce a self-map of the $E_1$-page of the coniveau spectral sequence of Theorem 2.2.1.

Remark 2.3.3 For $x \in G_p$ (i.e., a codimension $p$ point) and $y \in G_p$ with $my = x$, the trace map

\[ [m]_* \colon \bigoplus_{y \in G_p} K_{n-p}^M k(y) \to \bigoplus_{x \in G_p} K_{n-p}^M k(x) \]

is the sum of norm maps associated to the induced inclusions $k(x) \hookrightarrow k(y)$ for $x, y \in G_p$ with $my = x$. By [Reference HesselholtHes05, Lemma 14], this is compatible with residues, and so differentials on the $E_1$-page of the coniveau sequence. In particular, we have trace maps on our complexes $\mathsf {K}^{(n)}(G)$ of (2.3).

2.4 Powers of commutative group schemes

If we start with a smooth, equidimensional quasi-projective scheme $\mathscr {Y}$ of finite type over $Y$, then instead of taking a limit of motivic cohomology groups over all open subvarieties of $\mathscr {Y}$, it is natural to use only those subvarieties which are themselves defined over $Y$. As in § 2.2, we have a coniveau-type spectral sequence for this limit. We are actually interested in only very special cases with finer structure. Correspondingly, we consider here complements of much smaller collections of closed subsets defined over $Y$ and limits thereof.

Now let us fix $n \ge 1$ and let $\mathscr {Y} = G^n$ be the $n$th power of a smooth, connected commutative group scheme $G$ of relative dimension $1$ over $Y$, such as $\mathbb {G}_{m/Y}$ or a smooth family of elliptic curves over $Y$. We use throughout the convention that the monoid

\[ \Delta = M_n(\mathbb{Z}) \cap \mathrm{GL}_n(\mathbb{Q}) \]

of integral matrices of nonzero determinant acts by right multiplication on $G^n$. For example, if $n = 2$ and $\big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big ) \in \Delta$, then for any $g_1, g_2 \in G$, we have

(2.7)\begin{equation} (g_1,g_2) \cdot \bigg(\begin{matrix} a & b\\c & d \end{matrix}\bigg) = (g_1^ag_2^c,g_1^bg_2^d). \end{equation}

This being a right action, the monoid $\Delta$ then acts on the left on the motivic cohomology groups $H^i(G^n,k)$ by pullback.

Even better, $\Delta$ acts on the left on the complex $\mathsf {K} = \mathsf {K}^{(n)}(\mathscr {Y})$ of (2.3), also by pullback. That is, if $x \in \mathscr {Y}_q$ is a codimension $q$ point of $\mathscr {Y} = G^n$, $\delta \in \Delta$, and $y \in \mathscr {Y}_q$ is such that $y \cdot \gamma = x$, then pullback yields a map $\gamma ^* \colon k(x) \to k(y)$ of residue fields, and this induces $\gamma ^* \colon K_{n-q}^M k(x) \to K_{n-q}^M k(y)$. The pullback map on $\mathsf {K}_{n-q}$ is the sum of these maps, and the residue maps are clearly equivariant for this action.

Now let us focus on the case $n = 2$ of interest to us. We consider divisors of the form

(2.8)\begin{equation} S_{\alpha} = S_{i,j} = \ker\Big(G^2 \xrightarrow{z_1^jz_2^j} G \Big) \end{equation}

for nonzero $\alpha = (i,j) \in \mathbb {Z}^2$. Then $S_{i,j}$ is connected if and only if $i$ and $j$ are relatively prime.

Take a finite indexing set $I \subset \mathbb {Z}^2 - \{(0,0)\}$ with at least two elements and containing at most one representative of each element of $\mathbb {P}^1(\mathbb {Q})$. We set

\[ S_I = \bigcup_{\alpha \in I} S_{\alpha} \quad \mathrm{and} \quad U_I = G^2 - S_I, \]

where we regard $S_I$ as a closed subscheme of $G^2$ with its reduced scheme structure and $U_I$ as an open subscheme of $\mathbb {G}_m^2$. We then consider the union of pairwise intersections

\[ T_I = \bigcup_{\substack{\alpha, \beta \in I \\ \alpha \neq \beta}} (S_{\alpha} \cap S_{\beta}), \]

which is a finite subgroup scheme of $G^2$ by our choice of $I$. Let

\[ S_I^{\circ} = S_I - T_I \]

so that $S_I^{\circ }$ is the disjoint union of the smooth subschemes $S_{\alpha }^{\circ } = S_{\alpha } \cap S_I^{\circ }$ for $\alpha \in I$.

This fits into the setting above for $n = 2$ with $Z_1 = S_I$ and $Z_2 = T_I$, so $Z_0-Z_1 = U_I$ and $Z_1-Z_2 = S_I^{\circ }$. We obtain a spectral sequence having the following terms in degrees $(p,q)$ with $0 \le p \le 2$ and $0 \le q \le 2$:

This spectral sequence maps to the coniveau sequence detailed in Example 2.2.2 (with $Y$ replaced by $G^2$). It follows from Remark 2.2.3 that each of the complexes

\[ \mathsf{K}_I \colon \quad H^2(U_I,2) \to H^1(S_I^{\circ},1) \to H^0(T_I,0) \]

injects quasi-isomorphically into the big complex

\[ \mathsf{K} \colon \quad K_2(k(G^2)) \to \bigoplus_D K_1k(D) \to \bigoplus_x K_0k(x). \]

If we order our indexing sets by $I \le I'$ if $U_{I'} \subseteq U_I$, then the limit complex $\varinjlim _I \mathsf {K}_I$ is also a quasi-isomorphic subcomplex of $\mathsf {K}$. The constructions in this paper can all be carried out using this complex, which is just large enough to allow for the definition of Hecke actions on $\mathrm {GL}_2(\mathbb {Z})$-cocycles, in that it is preserved under the pullback action of the monoid ${\Delta = M_2(\mathbb {Z}) \cap \mathrm {GL}_2(\mathbb {Q})}$.

3.1 Motivic cohomology of $\mathbb {G}_m^r$

Let $z$ denote the coordinate function on the multiplicative group $\mathbb {G}_m$ over a field $F$, normalized so that the value of $z$ at the identity element is $1$. The motivic cohomology of $\mathbb {G}_m$ involves classes directly constructed from $-z$, together with classes pulled back from the motivic cohomology of $\operatorname {Spec} F$ itself. We extend this description to powers of $\mathbb {G}_m$.

First, we construct suspension isomorphisms in motivic cohomology. Recall the definition of exterior product from Example 2.3.2.

Proposition 3.1.1 Let $Y$ denote an equidimensional quasi-projective scheme of finite type over $F$. There is a natural isomorphism

\[ H^i(Y, k) \oplus H^{i-1}(Y, k-1) \xrightarrow{\sim} H^i(\mathbb{G}_m \times Y,k), \]

where the map on the first summand is pullback under projection to the first factor and the map on the second summand is left exterior product with $-z$, considered as a class in $H^1(\mathbb {G}_m, 1)$.

The reason for choosing $-z$, as opposed to $z$, will be made clear in Lemma 4.1.1. This result is well known and corresponds to the ‘fundamental theorem’ of algebraic $K$-theory (proved for $K_0$ and $K_1$ by Bass and in general by Quillen).

Proof. Consider the canonical embedding $\iota \colon \mathbb {G}_m \times Y \hookrightarrow \mathbb {A}^1 \times Y$ given by the usual embedding in the first coordinate and the identity in the second. The Gysin sequence has the form

\[ \cdots \to H^i(\mathbb{A}^1 \times Y,k) \to H^i(\mathbb{G}_m \times Y,k) \xrightarrow{\partial} H^{i-1}(Y,k-1) \to \cdots. \]

As noted in § 2.1, the pullback $H^i(Y,k) \to H^i(\mathbb {A}^1 \times Y,k)$ by the projection map is an isomorphism. Thus, it suffices to show that $\partial$ is split by a map on the right-hand summand in the theorem, and this follows from $\partial (-z \boxtimes x) = \partial (-z) \boxtimes x = x$ for $x \in H^{i-1}(Y,k-1)$.

Corollary 3.1.2 Let $Y$ denote an equidimensional quasi-projective scheme of finite type over a field $F$, and let $r \ge 1$. There is a natural isomorphism

\[ H^i(\mathbb{G}_m^r,k) \cong \bigoplus_{j=0}^{\min(k,r)} H^{i-j}(F,k-j)^{\binom{r}{j}}. \]

Since $H^i(F,k) = 0$ for $i > k$, we obtain in particular the following corollary.

3.2 Symbols in the complex computing motivic cohomology

Recall from Example 2.2.2 that the coniveau spectral sequence gives rise to a homological complex $\mathsf {K}$ with nonzero terms in degrees $2$, $1$, and $0$ given by

(3.1)\begin{equation} \mathsf{K} \colon \quad K_2 k(\mathbb{G}_m^2) \rightarrow \bigoplus_{D} K_1 k(D) \rightarrow \bigoplus_{x} K_0 k(x), \end{equation}

the sums being taken over irreducible divisors and closed points, respectively. This complex computes the cohomology groups $H^*(\mathbb {G}_m^2, 2)$ in degrees $2$ to $4$ from left to right. Therefore, by Corollary 3.1.3, the sequence is exact in the middle and at the right, and its homology at the left is $H^2(\mathbb {G}_m^2,2)$. Let $\bar {\mathsf {K}}_2$ be the quotient of $\mathsf {K}_2$ by the image of $H^2(\mathbb {G}_m^2, 2)$ so that we get a short exact sequence

\[ 0 \rightarrow \bar{\mathsf{K}}_2 \rightarrow \mathsf{K}_1 \rightarrow \mathsf{K}_0 \rightarrow 0. \]

We shall denote the boundary maps in this complex by the generic symbol $\partial$.

The monoid $\Delta = \mathrm {GL}_2(\mathbb {Q}) \cap M_2(\mathbb {Z})$ acts on the right on $\mathbb {G}_m^2$ by the formula of (2.7). As explained in § 2.4, the complex $\mathsf {K}$ is correspondingly endowed with a left $\Delta$-action via pullback. This action descends to an action on $\bar {\mathsf {K}}$. For now, we use only the induced action of the group $\mathrm {GL}_2(\mathbb {Z})$; we will employ the full $\Delta$-action in § 3.4.

Let us define special elements

\[ e \in \mathsf{K}_0, \quad \langle a, c \rangle \in \mathsf{K}_1, \quad \langle \gamma \rangle \in \mathsf{K}_2 \]

attached to a primitive vector $(a,c) \in \mathbb {Z}^2$ or to a matrix $\gamma \in \mathrm {GL}_2(\mathbb {Z})$ that satisfy

(3.2)\begin{equation} \partial \bigg\langle \!\bigg(\begin{matrix} a & b\\c & d \end{matrix}\bigg) \!\bigg\rangle = \begin{cases} \langle a,c \rangle -\langle -b, -d\rangle & \text{if } \det \gamma=1, \\ \langle -a,-c \rangle -\langle b, d\rangle & \text{if } \det \gamma=-1 \end{cases} \quad \text{and} \quad \partial \langle a,c \rangle = e. \end{equation}

These are:

• – the $\mathrm {GL}_2(\mathbb {Z})$-fixed class $e \in \mathsf {K}_0$ of the element $1 \in \mathbb {Z}$ supported at the identity of $\mathbb {G}_m$;

• – for a primitive vector $(a,c) \in \mathbb {Z}^2$ and the torus $S_{a,c} = \ker (\mathbb {G}_m^2 \xrightarrow {(x,y) \mapsto x^ay^c} \mathbb {G}_m)$ of (2.8), the image $\langle a,c \rangle \in \mathsf {K}_1$ of the invertible function

  \[ 1-z_1^b z_2^d \in \mathscr{O}(S_{a,c}-\{1\})^{\times} \hookrightarrow K_1( \mathbb{Q}(S_{a,c})), \]
  where $\big (\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\big ) \in \mathrm {SL}_2(\mathbb {Z})$ extends $(a,c)$ – this is independent of the choice of $(b,d)$, since another choice simply alters the function $z_1^b z_2^d$ by a multiple of $z_1^a z_2^c$, which is $1$ on $S_{a,c}$;

• – for $\gamma = \big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big ) \in \mathrm {GL}_2(\mathbb {Z})$, and its columns $v_1= (a,c)$ and $v_2= (b,d)$, the Steinberg symbol

  \[ \langle \gamma \rangle = \langle v_1, v_2 \rangle = \{1-z_1^az_2^c, 1-z_1^b z_2^d\} \in \mathsf{K}_2. \]
  Note that $\langle \gamma \rangle = \gamma ^* \langle \big (\begin{smallmatrix} 1 & 0\\0 & 1 \end{smallmatrix}\big ) \rangle$ is the image of $(1-z_1^az_2^c) \cup (1-z_1^bz_2^d) \in H^2(\mathbb {G}_m^2 - S_{a,c} \cup S_{b,d},2)$.

The special elements of the form $e$, $\langle a,c \rangle$ for $(a,c) \in \mathbb {Z}^2$ primitive, and $\langle \gamma \rangle$ for $\gamma \in \mathrm {GL}_2(\mathbb {Z})$ together span a subcomplex $\mathsf {Symb}$ of $\mathsf {K}$ that we refer to as the symbol complex. We will return to it in §§ 4 and 5. That these symbols satisfy (3.2) follows directly from the description in Example 2.2.2 of the residue maps in $\mathsf {K}$ in terms of tame symbols (2.4) and divisors.

Remark 3.2.1 We note for later use that, for $\gamma = \big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big ) \in \mathrm {GL}_2(\mathbb {Z})$, the pullback $\gamma ^* \langle 0,1 \rangle$ is supported on the torus $S_{b,d}$ which is the kernel of $(z_1, z_2) \mapsto z_1^b z_2^d$. On this divisor, it is given by $\gamma ^* (1-z_1^{-1}) = 1- z_1^{-a} z_2^{-c}$. Thus

(3.3)\begin{equation} \gamma^* \langle 0,1 \rangle = \begin{cases} \langle b,d \rangle & \text{if } \det(\gamma)=1,\\ \langle -b, -d \rangle & \text{if } \det(\gamma)=-1. \end{cases} \end{equation}

3.3 The cocycle

Pulling back the complex $\bar {\mathsf {K}}_2 \rightarrow \mathsf {K}_1 \rightarrow \mathsf {K}_0$ to the cyclic subgroup generated by $e \in \mathsf {K}_0$, we get an extension of $\mathbb {Z}$ by $\bar {\mathsf {K}}_2$, and so an extension class in

\[ \operatorname{Ext}^1_{\mathbb{Z}[\mathrm{GL}_2(\mathbb{Z})]}(\mathbb{Z},\bar{\mathsf{K}}_2) = H^1(\mathrm{GL}_2(\mathbb{Z}), \bar{\mathsf{K}}_2). \]

We shall describe a cocycle representing this class more explicitly in Proposition 3.3.1 below. We then give an explicit recipe for $\Theta _{\gamma }$ as a sum of symbols $\langle \rho \rangle$ with $\rho \in \mathrm {SL}_2(\mathbb {Z})$ and show that it lies in parabolic cohomology. Because $H^2(\mathrm {GL}_2(\mathbb {Z}),\mathbb {Z})$ is torsion, a multiple of $\Theta$ can actually be lifted to $\mathsf {K}_2$; in § 5, we sketch how to do this explicitly.

Proposition 3.3.1 There is a $1$-cocycle

\[ \Theta \colon \mathrm{GL}_2(\mathbb{Z}) \to \bar{\mathsf{K}}_2, \quad \gamma \mapsto \Theta_{\gamma}, \]

uniquely characterized by the property that

\[ \partial \Theta_{\gamma} = (\gamma^* - 1) \langle 0,1 \rangle. \]

Proof. Since $(\gamma ^*-1)\langle 0,1 \rangle$ has trivial boundary $e-e = 0$, it is the boundary of a unique $\Theta _{\gamma } \in \bar {\mathsf {K}}_2$. Since pullback is a left action, we have

\[ \partial\Theta_{\gamma \gamma'} = (\gamma^*(\gamma')^*-1)\langle 0,1 \rangle = \gamma^*((\gamma')^*-1)\langle 0,1 \rangle + (\gamma^*-1)\langle 0,1 \rangle = \partial(\gamma^*\Theta_{\gamma'} + \Theta_{\gamma}), \]

for $\gamma, \gamma ' \in \mathrm {GL}_2(\mathbb {Z})$. That $\Theta$ is a cocycle therefore follows by the exactness of $\bar {\mathsf {K}}$.

Next, we give an explicit recipe for values of $\Theta$ in terms of our special symbols in $\mathsf {K}_2$ using a standard variant of the Euclidean algorithm, analogous to writing a geodesic between cusps on the modular curve as a sum of Manin symbols. We make the latter analogy precise in § 4.3.

Given $\gamma = \big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big ) \in \mathrm {GL}_2(\mathbb {Z})$, the Euclidean algorithm allows us to find a sequence $(v_i)_{i=0}^k$ in $\mathbb {Z}^2$ for some $k \ge 0$ with $v_0 = (0,1)$ and $v_k = \det (\gamma ) (b,d)$ and such that the $v_i = (b_i,d_i)$ satisfy $\det \big (\begin{smallmatrix} b_{i-1} & b_i \\ d_{i-1} & d_i \end{smallmatrix}\big ) = 1$ for all $1 \le i \le k$. We call such a sequence $(v_i)_{i=0}^k$ a connecting sequence for $\gamma$.

Proposition 3.3.2 Let $\gamma = \big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big ) \in \mathrm {GL}_2(\mathbb {Z})$, and choose a connecting sequence $(v_i)_{i=0}^k$ for $\gamma$. Then we have the following equality in $\bar {\mathsf {K}}_2$:

\[ \Theta_{\gamma} = \sum_{i=1}^k \langle v_i, -v_{i-1} \rangle. \]

Recall from § 3.2 that $\langle v_i, -v_{i-1} \rangle$ is the symbol associated to the matrix with first column $v_i$ and second column $-v_{i-1}$.

Proof. By (3.3) and (3.2) we have

\[ \partial \bigg( \sum_{i=1}^k \langle v_i, -v_{i-1} \rangle \bigg) = \sum_{i=1}^k (\langle v_i \rangle - \langle v_{i-1} \rangle) = \langle v_k \rangle - \langle v_0 \rangle = \gamma^* \langle 0,1 \rangle - \langle 0,1 \rangle. \]

Since $\Theta _{\gamma }$ and $\sum _{i=1}^k \langle v_i,-v_{i-1} \rangle$ have the same boundary, they are equal in $\bar {\mathsf {K}}_2$.

Example 3.3.3 Take $\gamma = \big (\begin{smallmatrix} -1 & 0\\0 & -1 \end{smallmatrix}\big )$. Then $v_0 = (0,1)$, $v_1 = (-1,0)$, and $v_2 = (0,-1)$ form a connecting sequence for $\gamma$, so

\[ \Theta_{\gamma} = \langle (-1,0),(0,-1) \rangle + \langle (0,-1),(1,0) \rangle = \{ -z_1^{-1}, 1-z_2^{-1} \}, \]

which equals $-\{ -z_1, 1-z_2 \}$ in $\bar {\mathsf {K}}_2$.

We will use a perhaps slightly nonstandard notion of parabolic cohomology for $\mathrm {GL}_2(\mathbb {Z})$, consistent with our use of right actions of $\mathrm {GL}_2(\mathbb {Z})$ on group schemes. That is, we define the $\mathrm {GL}_2(\mathbb {Z})$-parabolic cohomology group $H^1_P(\mathrm {GL}_2(\mathbb {Z}),M)$ for a $\mathbb {Z}[\mathrm {GL}_2(\mathbb {Z})]$-module $M$ to be the intersection of the kernels of the restriction maps from $H^1(\mathrm {GL}_2(\mathbb {Z}),M) \to H^1(P,M)$, where $P$ runs over all stabilizers of nonzero elements of $\mathbb {Z}^2$ under the right action of $\mathrm {GL}_2(\mathbb {Z})$. We say that a $1$-cocycle $\mathrm {GL}_2(\mathbb {Z}) \to M$ is parabolic if its class lies in the parabolic cohomology group $H^1_P(\mathrm {GL}_2(\mathbb {Z}),M)$.

Proof. Since the right action of $\mathrm {GL}_2(\mathbb {Z})$ on the set of relatively prime pairs of integers is transitive, it is enough to verify triviality upon restriction to the stabilizer $P_{\infty } = \{ \big (\begin{smallmatrix} 1 & 0 \\ c & \pm 1 \end{smallmatrix}\big ) \mid c \in \mathbb {Z}\}$ of $(1,0) \in \mathbb {Z}^2$. If we take $\gamma \in P_{\infty }$, then by (3.3),

\[ \partial\Theta_{\gamma} = (\gamma^*-1)\langle 0,1 \rangle = \langle 0,1 \rangle - \langle 0,1 \rangle = 0, \]

so $\Theta _{\gamma } = 0$ in $\bar {\mathsf {K}}_2$.

Remark 3.3.5 The parabolic cocycle $\Theta$ is also integral in a sense we shall now describe. For this, we employ the theory of motivic cohomology over schemes over Dedekind domains (see the work of Levine [Reference LevineLev99], Geisser [Reference GeisserGei04], and Spitzweck [Reference SpitzweckSpi18]), which we denote as over fields. Take the direct limit of second motivic cohomology groups $\mathsf {K}_{2/\mathbb {Z}} = \varinjlim _U H^2(U,2)$, where $U$ runs over the open $\mathbb {Z}$-subschemes of $\mathbb {G}_m^2{}_{/\mathbb {Z}}$ that are complements of unions of kernels of morphisms $\mathbb {G}_m^2{}_{/\mathbb {Z}} \to \mathbb {G}_m^2{}_{/\mathbb {Z}}$ with $(z_1,z_2) \mapsto z_1^az_2^c$ for some primitive $(a,c) \in \mathbb {Z}^2-\{0\}$. There is a canonical injection $\mathsf {K}_{2/\mathbb {Z}} \hookrightarrow \mathsf {K}_2$, under which the inverse image of $H^2(\mathbb {G}_m^2,2)$ is $H^2(\mathbb {G}_m^2{}_{/\mathbb {Z}},2)$. The statement is then that

\[ \text{$\Theta$ takes values in $\mathsf{K}_{2/\mathbb{Z}}/H^2(\mathbb{G}_m^2{}_{/\mathbb{Z}},2)$}. \]

This can be seen directly from the explicit formula of Proposition 3.3.2 or without recourse to this formula using Gysin sequences and Lemma 4.1.2 over finite fields (supposing an expected compatibility of pushforwards and residues as in Lemma 2.1.3 that we did not endeavor to check).

3.4 Hecke actions

We now turn to the action of Hecke operators on the class of our $1$-cocycle $\Theta$. To set the stage, suppose that $\Delta$ is a submonoid of $M_2(\mathbb {Z}) \cap \mathrm {GL}_2(\mathbb {Q})$ and $\Gamma$ is a finite index subgroup of $\mathrm {GL}_2(\mathbb {Z}) \cap \Delta$. We recall the explicit formulas for the action of Hecke operators of double cosets for $\Gamma \backslash \Delta / \Gamma$ on the cohomology $H^1(\Gamma, M)$ for any $\mathbb {Z}[\Delta ]$-module $M$.

For $g \in \Delta$, write

(3.4)\begin{equation} \Gamma g \Gamma = \coprod_{j=1}^t g_j\Gamma. \end{equation}

For $\gamma \in \Gamma$, there exist a permutation $\sigma \in S_t$ (the permutation group on $t$ letters) and elements $\gamma _j \in \Gamma$ such that $\gamma g_j = g_{\sigma (j)}\gamma _j$ for $1 \le j \le t$. For a $1$-cocycle $\theta \colon \Gamma \to M$ and $\gamma \in \Gamma$, we set

(3.5)\begin{equation} T(g)\theta(\gamma) = \sum_{j=1}^t g_{\sigma(j)}\theta(\gamma_j), \end{equation}

which in general depends on the chosen coset representatives $g_j$. The following lemma is well known and verified simply by writing out the definitions.

Returning to our case of interest, we again take

\[ \Delta = M_2(\mathbb{Z}) \cap \mathrm{GL}_2(\mathbb{Q}) \]

and $\Gamma = \mathrm {GL}_2(\mathbb {Z})$. The monoid $\Delta$ acts on the right on $\mathbb {G}_m^2(Y)$ for any smooth $\mathbb {Q}$-scheme $Y$ by formula (2.7), and this right action on $\mathbb {G}_m^2$ induces a left pullback action on $\mathsf {K}$ as in § 2.4.

For example, the pullback by $\gamma \in \Delta$ of an invertible regular function $f$ on $S_{i,j}-\{1\}$ is the function on $S_{ai+bj,ci+dj} = S_{i,j}\gamma ^{-1}$ defined by

\[ (\gamma^*f)(z_1,z_2) = f((z_1,z_2)\gamma) = f(z_1^a z_2^c, z_1^b z_2^d), \]

and the divisor of $\gamma ^*f$ is the pullback of the divisor of $f$. This action descends to an action on $\bar {\mathsf {K}}$ also, since $M_2(\mathbb {Z}) \cap \mathrm {GL}_2(\mathbb {Q})$ acts by pullback on $H^2(\mathbb {G}_m^2,2)$, compatibly with its morphism to $\mathsf {K}_2$. In the case of the matrix $\big (\begin{smallmatrix} \ell & 0 \\ 0 & \ell \end{smallmatrix}\big )$, we denote this action on $\mathsf {K}$ more succinctly by $[\ell ]^*$.

For a prime $\ell$, let $T_{\ell } = T(g)$ for $g = \big (\begin{smallmatrix} \ell \\ & 1 \end{smallmatrix}\big )$, where $T(g)$ is as in (3.5) above for the coset representativesFootnote ⁹

(3.6)\begin{equation} g_j = \begin{pmatrix} \ell & j \\ & 1 \end{pmatrix} \text{ for } 0 \le j \le \ell - 1 \quad \text{ and } \quad g_{\ell} = \begin{pmatrix} 1 \\ & \ell \end{pmatrix} \end{equation}

in $\Gamma g \Gamma = \coprod _{j=0}^{\ell } g_j \Gamma$. By Lemma 3.4.1, this gives an action of $T_{\ell }$ on cohomology independent of the latter decomposition. Let us also define an endomorphism of the complex $\mathsf {K}$ by the rule

\[ T_{\ell}^{\mathsf{K}} = \sum_{j=0}^{\ell} g_j^* \colon \mathsf{K} \to \mathsf{K}. \]

This depends upon the choice of $g_j$ and appears primarily as a computational aid.

We can now compute the action of $T_{\ell }$ on the class of $\Theta$ from the action of $T_{\ell }^K$ on $e$ of § 3.2.

Lemma 3.4.3 For each prime $\ell$, we have an equality

\[ T_{\ell}^{\mathsf{K}} e = (\ell + [\ell]^*)e \]

of elements of $\mathsf {K}_0$.

In the following, note that $[\ell ]^*$ acts on the cocycle $\Theta$ through its action on $\bar {\mathsf {K}}_2$.

Proof. As the residue of $\langle 0,1 \rangle$ is $e$, Lemma 3.4.3 and the $\Delta$-equivariance of the boundary maps in $\mathsf {K}$ imply that $(T_{\ell }^{\mathsf {K}} - \ell - [\ell ]^*)\langle 0,1 \rangle$ has zero residue. Accordingly there exists $\psi \in \mathsf {K}_2$ so that

(3.7)\begin{equation} \partial \psi = (T_{\ell}^{\mathsf{K}} - \ell - [\ell]^*)\langle 0,1 \rangle. \end{equation}

For $\gamma \in \mathrm {GL}_2(\mathbb {Z})$, let $\sigma$ be a permutation of $\{0,\ldots,\ell \}$, and let $\gamma _j \in \mathrm {GL}_2(\mathbb {Z})$ for $0 \le j \le \ell$ be such that $\gamma g_j = g_{\sigma (j)} \gamma _j$. We then have

(3.8)\begin{align} (\gamma^*-1)T_{\ell}^{\mathsf{K}} \langle 0,1 \rangle &=(\gamma^* -1) \sum_{j=0}^{\ell} g_j^* \langle 0,1 \rangle = \sum_{j=0}^{\ell} g_{\sigma(j)}^*(\gamma_j^*-1) \langle 0,1 \rangle\nonumber\\ &= \partial \bigg( \sum_{j=0}^{\ell} g_{\sigma(j)}^* \Theta_{\gamma_j} \bigg) = \partial (T_{\ell}\Theta)_{\gamma}, \end{align}

where the last equality follows by definition (3.5) of our Hecke action on $\mathsf {K}$. Comparing (3.7) and (3.8), we see that

\[ \partial (\gamma^*-1)\psi = (\gamma^*-1)(T_{\ell}^K-\ell-[\ell]^*)\langle 0,1 \rangle = \partial((T_{\ell}-\ell-[\ell]^*)\Theta)_{\gamma}. \]

It follows that $((T_{\ell }-\ell -[\ell ]^*)\Theta )_{\gamma }$ and $(\gamma ^*-1)\psi$ coincide in $\bar {\mathsf {K}}_2$, and therefore the cocycle $(T_{\ell }-\ell -[\ell ]^*)\Theta$ is the coboundary of $\psi$.

4.1 Fixed parts via suspension

4.1.1 Fixed parts of the cohomology of $\mathbb {G}_m^2$

The results of § 3.1 imply that the motivic cohomology group $H^2(\mathbb {G}_m^2,2)$ breaks up as a direct sum of motivic cohomology classes that are $\mathbb {Z}$-multiples of $(-z_1) \cup (-z_2)$ and sums of classes pulled back via one of the two projection maps. We want to be able to ‘ignore’ the latter classes, and to kill them we will use trace maps.

We work in this subsection over a base field $F$. For $r \ge 1$, we define the fixed part of the motivic cohomology group $H^i(\mathbb {G}_m^r,k)$ as

(4.1)\begin{equation} H^i(\mathbb{G}_m^r,k)^{(0)} = \{ \alpha \in H^i(\mathbb{G}_m^r,k) \mid ([p]_*-1)\alpha = 0 \text{ for all primes } p \neq \operatorname{char} F \}. \end{equation}

In general, if $F$ has zero or sufficiently large characteristic, then this fixed part is the direct summand of $H^i(\mathbb {G}_m^r,k)$ given by $H^i(F,k)$ in the decomposition of Corollary 3.1.2. However, we shall only be interested in these groups in very specific cases: in particular, let us study them for $i = k \le r \in \{1,2\}$.

Lemma 4.1.1 The element $-z \in H^1(\mathbb {G}_m,1)$ generates $H^1(\mathbb {G}_m,1)^{(0)}$.

Proof. The group $H^1(\mathbb {G}_m, 1)$ consists of the invertible functions on $\mathbb {G}_{m/F}$. As such, each element is uniquely of the form $\eta (-z)^k$ for $\eta \in F^{\times }, k \in \mathbb {Z}$.

Suppose that such a class is $[m]_*$-fixed for $m$ prime to $\operatorname {char} F$. The global unit $-z$ is $[m]_*$-fixed for $m$ prime to $\operatorname {char} F$ since

(4.2)\begin{equation} [m]_*(-z) = \prod_{i=0}^{m-1} (-\zeta_m^i z^{1/m}) = - z. \end{equation}

Thus, we also have $[m]_* \eta = \eta ^m$ for such $m$.

If $\operatorname {char} F \neq 2$, then $\eta ^2 =\eta$ and so $\eta =1$; in characteristic $2$ the equality $\eta ^3=\eta$ implies the same conclusion. So $\eta =1$, and the claim follows.

Let us turn to $\mathbb {G}_m^2$ over $F$, on which we let $z_i$ denote the $i$th coordinate function.

Lemma 4.1.2 The group $H^1(\mathbb {G}_m^2,1)^{(0)}$ is trivial, and if $F$ has characteristic not $2$ or $3$ or is a finite field, then $H^2(\mathbb {G}_m^2, 2)^{(0)}$ is $\mathbb {Z}$-free of rank $1$, generated by $(-z_1) \cup (-z_2)$.

Proof. Let us use $\nu _j$ to denote the class of $-z_j$ in $H^1(\mathbb {G}_m^2,1)$. As in Corollary 3.1.2, we have an isomorphism

(4.3)\begin{equation} H^2(F, 2) \oplus H^1(F, 1) \oplus H^1(F, 1) \oplus H^0(F, 0) \xrightarrow{\sim} H^2(\mathbb{G}_m^2,2), \end{equation}

where the maps are given by pullback and (left) cup product with $1$, $\nu _1$, $\nu _2$, and $\nu _1 \cup \nu _2$, respectively.

Let $m \ge 1$ with $\operatorname {char} F \nmid m$. By Example 2.3.2 and (4.2), the class $\nu _1 \cup \nu _2$ is $[m]_*$-fixed. Any class $\eta$ that is pulled back from $\operatorname {Spec} F$ satisfies $[m]^*\eta = \eta$, so for such $\eta$ and any $\alpha$ among $1$, $\nu _1$, $\nu _2$, and $\nu _1 \cup \nu _2$, Corollary 2.1.2 yields

\[ [m]_*(\alpha \cup \eta) = [m]_*(\alpha \cup [m]^*\eta) = ([m]_*\alpha) \cup \eta, \]

which tells us that the trace maps preserve the summands in (4.3). Thus, we need only consider the summands individually. So, let us suppose that $\alpha \cup \eta$ is $[m]_*$-fixed.

1. (i) If $\alpha = 1$, then $[m]_*\eta = m^2\eta$. So, if $\eta$ is $[m]_*$-fixed, then $(m^2-1)\eta = 0$. If $\operatorname {char} F \notin \{2,3\}$, then since this is true for $m = 2$ and $m = 3$, we have $\eta = 0$. In the case $i = k = 1$, note that if $\operatorname {char} F = 2$, then $8\eta = 0$ implies $\eta = 0$, and if $\operatorname {char} F = 3$, then $3\eta = 0$ implies $\eta = 0$. If $i = k = 2$ and $F$ is a finite field, then $H^2(F,2) = 0$, so $\eta = 0$.

2. (ii) If $\alpha = \nu _j$, then we have

   \[ [m]_*(\nu_j \cup \eta) = m(\nu_j \cup \eta), \]
   so if $\nu _j \cup \eta$ is $[m]_*$-fixed, then it is $(m-1)$-torsion. If $i = k = 1$, then $H^0(F,0) \cong \mathbb {Z}$, so $\nu _j \cup \eta = 0$. In general, so long as $\operatorname {char} F \neq 2$, then $\nu _j \cup \eta$ is trivial taking $m = 2$. If $i = k = 2$ and $\operatorname {char} F = 2$, then by taking $m=3$, we see that $\nu _j \cup \eta$ is $2$-torsion in $F^{\times }$, so trivial.

3. (iii) If $\alpha = \nu _1 \cup \nu _2$, then it is indeed $[m]_*$-fixed.

4.1.2 Fixed parts of complexes

We return to the consideration of $\mathbb {G}_m^2$ over $\mathbb {Q}$. As explained in Remark 2.3.3, the trace maps $[m]_*$ act on the complex $\mathsf {K}$ as well. We define the fixed complex $\mathsf {K}^{(0)}$ in exactly the same way as (4.1).

Lemma 4.1.3 The symbols defined in § 3.2 all lie in the fixed part of $\mathsf {K}$:

\[ e \in \mathsf{K}_0^{(0)}, \quad \langle a,c \rangle \in \mathsf{K}_1^{(0)}, \quad \langle\gamma\rangle \in \mathsf{K}_2^{(0)}. \]

Proof. First, note that $[m]_*e = e$ for all $m$. From (2.5), we see that $\langle a,c \rangle \in \mathsf {K}_1^{(0)}$. Finally, since $[m]_*$ on $\mathsf {K}_2$ is given by the ‘product of the pushforwards by $m$ in the first and second variable’ by (2.6), it fixes $\langle (1,0),(0,1) \rangle$. We then note that $\gamma ^*$ and $[m]_*$ commute.

Proposition 4.1.4 The cocycle $\Theta$ lifts to a cocycle valued in $\mathsf {K}_2/\langle \{-z_1, -z_2\} \rangle$.

Proof. Indeed, Proposition 3.3.2 and Lemma 4.1.3 imply that for each $\gamma \in \mathrm {GL}_2(\mathbb {Z})$, the cocycle $\Theta _{\gamma }$ is valued in the image of $\mathsf {K}_2^{(0)} \rightarrow \bar {\mathsf {K}}_2$. By Lemma 4.1.2, that image is isomorphic to $\mathsf {K}_2^{(0)}/ \langle \{-z_1, -z_2\} \rangle$. Thus $\Theta$ lifts to that group, and a fortiori to $\mathsf {K}_2/ \langle \{-z_1, -z_2\} \rangle$.

In the remainder of this section, we will implicitly regard $\Theta$ as valued in $\mathsf {K}_2/\langle \{-z_1, -z_2\} \rangle$.

Remark 4.1.5 The symbol complex $\mathsf {Symb}$ defined in § 3.2 is contained in the fixed part under trace maps of the limit complex $\varinjlim _I \mathsf {K}_I \subset \mathsf {K}$ of § 2.4. In fact, it is the fixed part of a certain motivic subcomplex $\mathsf {k}$ of $\varinjlim _I \mathsf {K}_I$ that we refer to as the small complex.

To make this precise, for $\gamma = \big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big ) \in \mathrm {SL}_2(\mathbb {Z})$, let us set $\mathsf {k}_{\gamma } = \mathsf {K}_I$ for $I = \{(a,c),(b,d)\}$. The complex $\mathsf {k}_{\gamma }$ has the form

\[ H^2(\mathbb{G}_m^2 - S_{a,c} \cup S_{b,d},2) \to H^1(S_{a,c}-\{1\},1) \oplus H^1(S_{b,d}-\{1\},1) \to H^0(\{1\},0), \]

where $S_{a,c}$ and $S_{b,d}$ are the rank $1$ tori of (2.8), and the last term is identified with $\mathbb {Z}$. Using Gysin sequences and the results of § 3.1, it is not hard to see that $\mathsf {k}_{\gamma }^{(0)}$ is canonically a direct summand of $\mathsf {k}_{\gamma }$ such that

• – $\mathsf {k}_{\gamma,2}^{(0)} \cong \mathbb {Z}^4$ is generated by $\langle \pm (a,c), \pm (b,d) \rangle$,

• – $\mathsf {k}_{\gamma,1}^{(0)} \cong \mathbb {Z}^4$ is generated by $\langle \pm (a,c) \rangle$ and $\langle \pm (b,d) \rangle$, and

• – $\mathsf {k}_{\gamma,0}^{(0)} \cong \mathbb {Z}$ is generated by $e$.

The homology of $\mathsf {k}_{\gamma }^{(0)}$ is then concentrated in degree $2$, being isomorphic to $H^2(\mathbb {G}_m^2,2)^{(0)} \cong \mathbb {Z}$.

Define the small complex $\mathsf {k} \subset \mathsf {K}$ to be the span of the $\mathsf {k}_{\gamma }$ for $\gamma \in \mathrm {SL}_2(\mathbb {Z})$. One may verify that the small complex is, like its subcomplexes $\mathsf {k}_{\gamma }$, a quasi-isomorphic subcomplex of $\mathsf {K}$, and from our description of each $\mathsf {k}_I^{(0)}$, we see that $\mathsf {k}^{(0)}$ is precisely the symbol complex $\mathsf {Symb}$. In fact, $\mathsf {Symb} = \mathsf {k}^{(0)}$ is a $\mathbb {Z}[\mathrm {GL}_2(\mathbb {Z})]$-direct summand of $\mathsf {k}$ with homology $H^2(\mathbb {G}_m^2,2)^{(0)}$ in degree $2$. In particular, by Proposition 3.3.2, our cocycle $\Theta$ takes values in $\bar {\mathsf {k}}_2^{(0)} = \,\mathsf {k}_2^{(0)}/\langle (-z_1) \cup (-z_2) \rangle$.

4.2 Specialization at an $N$-torsion point

Fix a positive integer $N \ge 2$. Let us fix the notation for the congruence subgroups of $\mathrm {GL}_2(\mathbb {Z})$ that we shall use from this point forward. That is, we set

(4.4)\begin{gather} \tilde{\Gamma}_0(N) =\bigg\{ \bigg(\begin{matrix} a & b\\c & d \end{matrix}\bigg) \in \mathrm{GL}_2(\mathbb{Z}) \mid N \mid c \bigg\}, \end{gather}

(4.5)\begin{gather}\tilde{\Gamma}_1(N) = \bigg\{ \bigg(\begin{matrix} a & b\\c & d \end{matrix}\bigg) \in \tilde{\Gamma}_0(N) \mid d \equiv 1 \bmod N \bigg\}. \end{gather}

We denote by $\Gamma _0(N)$ and $\Gamma _1(N)$ their respective intersections with $\mathrm {SL}_2(\mathbb {Z})$, as usual. Setting $\Delta = M_2(\mathbb {Z}) \cap \mathrm {GL}_2(\mathbb {Q})$, we also have associated monoids

(4.6)\begin{gather} \Delta_0(N) = \bigg\{ \bigg(\begin{matrix} a & b\\c & d \end{matrix}\bigg) \in \Delta \mid (d,N) = 1 \text{ and } N \mid c \bigg\}, \end{gather}

(4.7)\begin{gather}\Delta_1(N) = \bigg\{ \bigg(\begin{matrix} a & b\\c & d \end{matrix}\bigg) \in \Delta_0(N) \mid d \equiv 1 \bmod N \bigg\}. \end{gather}

In this section, we specialize our cocycle $\Theta$ at the $N$-torsion point

(4.8)\begin{equation} s \colon \operatorname{Spec} \mathbb{Q}(\mu_N) \rightarrow \mathbb{G}_m^2 \end{equation}

with value $(1,\zeta _N) \in \mathbb {G}_m(\mathbb {Q}(\mu _N))^2$ to obtain a cocycle

\begin{align*} \Theta_N \colon \tilde{\Gamma}_0(N) \to K_2(\mathbb{Q}(\mu_N))/\langle \{-1,-\zeta_N\} \rangle. \end{align*}

4.2.1 The specialized cocycle

We turn to the specialization of $\Theta$ at the $\mathbb {Q}(\mu _N)$-point $s$ of (4.8), which is given by the map $z_1 \mapsto 1$ and $z_2 \mapsto \zeta _N$ on coordinate rings. The stabilizer of $s$ in $\mathrm {GL}_2(\mathbb {Z})$ under its right action on $\mathbb {G}_m(\mathbb {Q}(\mu _N))^2$ is the congruence subgroup $\tilde {\Gamma }_1(N)$ of (4.5). For $j \in (\mathbb {Z}/N\mathbb {Z})^{\times }$, let $\sigma _j \in \operatorname {Gal}(\mathbb {Q}(\mu _N)/\mathbb {Q})$ be such that $\sigma _j(\zeta _N) = \zeta _N^j$. Via the isomorphism

(4.9)\begin{equation} \tilde{\Gamma}_0(N)/\tilde{\Gamma}_1(N) \xrightarrow{\sim} (\mathbb{Z}/N\mathbb{Z})^{\times}, \quad \bigg(\begin{matrix} a & b \\ c & d \end{matrix}\bigg) \mapsto d \end{equation}

and its composite with $d \mapsto \sigma _d$, we may consider any $\mathbb {Z}[\operatorname {Gal}(\mathbb {Q}(\mu _N)/\mathbb {Q})]$-module as a $\mathbb {Z}[\tilde {\Gamma }_0(N)]$-module. In particular, we let $\tilde {\Gamma }_0(N)$ act on $K_2(\mathbb {Q}(\mu _N))$ in this fashion.

Note that $s^*$ is not well defined on the whole of $\mathsf {K}_2$: there is no field map $\mathbb {Q}(\mathbb {G}_m^2) \to \mathbb {Q}(\mu _N)$. In order to pull back the values of our cocycle via $s$, we show that for $\gamma = \big (\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\big ) \in \tilde {\Gamma }_0(N)$, any lift to $\mathsf {K}_2$ of $\Theta _{\gamma }$ lies in a sufficiently small subgroup of $\mathsf {K}_2$ upon which $s^*$ can be defined.

For this, let

\[ U_{\gamma} = \mathbb{G}_m^2 - S_{b,d} \cup S_{0,1}, \]

which is to say the complement in $\mathbb {G}_m^2$ of the subtori that are the kernels of $(z_1, z_2) \mapsto z_1^b z_2^d$ and $(z_1,z_2) \mapsto z_2$. Since $(1,\zeta _N) \in U_{\gamma }$, the specialization map

\[ s^* \colon H^2(U_{\gamma},2) \to H^2(\mathbb{Q}(\mu_N),2) \cong K_2(\mathbb{Q}(\mu_N)) \]

is well defined. The residue of $\Theta _{\gamma }$ in $\mathsf {K}_1$ is $\langle \det (\gamma )(b,d) \rangle - \langle 0,1 \rangle$. Therefore, any lift of $\Theta _{\gamma }$ to $\mathsf {K}_2$ is defined on $U_{\gamma }$ in the sense of Remark 2.2.3.

Proposition 4.1.4 provides a canonical lift of $\Theta$ to a cocycle valued in $\mathsf {K}_2/\langle \{-z_1, -z_2\} \rangle$, which we also denote by $\Theta$. The value $\Theta _{\gamma }$ lies inside

\[ \bar{\mathsf{K}}_2(N) :=\varinjlim_{s \in U}\, H^2(U,2)/\langle (-z_1) \cup (-z_2) \rangle, \]

where the limit runs over the open $\mathbb {Q}$-subschemes $U$ of the $\mathbb {Q}$-scheme $\mathbb {G}_m^2$ containing all multiples of $s$ by an element of $(\mathbb {Z}/N\mathbb {Z})^{\times }$. Specialization at $s$ now defines a morphism

(4.10)\begin{equation} s^* \colon \bar{\mathsf{K}}_2(N) \rightarrow K_2(\mathbb{Q}(\mu_N))/Z_N \end{equation}

where $Z_N$ is the $\tilde {\Gamma }_0(N)$-stable subgroup of $K_2(\mathbb {Q}(\mu _N))$ generated by the specialization $\{-1,-\zeta _N\}$ of the symbol $\{-z_1,-z_2\} \in K_2(\mathbb {Q}(\mathbb {G}_m^2))$ under $s^*$. It therefore makes sense to speak of

\[ \Theta_{N,\gamma} := s^*\Theta_{\gamma} \]

as an element of $K_2(\mathbb {Q}(\mu _N))/Z_N$, Note that

\[ \{-1,-\zeta_N\} = \begin{cases} \{-1,-1\} & \text{if } N \text{ is odd}, \\ 0 & \text{if } N \text{ is even}, \end{cases} \]

so $Z_N$ is a group of order dividing $2$.

Proposition 4.2.1 The map

\begin{align*} \Theta_N \colon \tilde{\Gamma}_0(N) \to K_2(\mathbb{Q}(\mu_N))/Z_N, \quad \gamma \mapsto \Theta_{N,\gamma} \end{align*}

is a parabolic cocycle.

Proof. That $\Theta _N$ is a cocycle follows from if we can show that $s^*$ as in (4.10) is a homomorphism of $\tilde {\Gamma }_0(N)$-modules. Here, note that $\tilde {\Gamma }_0(N)$ acts on $K_2(\mathbb {Q}(\mu _N))$ as in (4.9). Write $\gamma = \big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big )$ and note that the two maps $\operatorname {Spec} \mathbb {Q}(\mu _N) \rightarrow \mathbb {G}_m^2$ defined by $\gamma \circ s$ and $s \circ \sigma _d$ coincide, since viewed on coordinate rings both send $z_1$ to $1 = \zeta _N^c$ and $z_2$ to $\zeta _N^d$. In particular, $\tilde {\Gamma }_0(N)$ acts on $\mathsf {K}_2(N)$, since for any $\mathbb {Q}$-subscheme $U$ of $\mathbb {G}_m^2$ with $s \in U(\mathbb {Q}(\mu _N))$, the composition $\gamma \circ s = s \circ \sigma _d$ is also a $\mathbb {Q}(\mu _N)$-point of $U$. This implies that $s^* \circ \gamma ^* = \sigma _d \circ s^*$ on $\bar {\mathsf {K}}_2(N)$.

The proof of Proposition 3.3.4 argued that $\Theta$ is trivial on a lower-triangular parabolic $P_{\infty }$ in $\mathrm {GL}_2(\mathbb {Z})$. An arbitrary parabolic $Q$ of $\tilde {\Gamma }_0(N)$ has the form $Q = \mu P_{\infty } \mu ^{-1} \cap \tilde {\Gamma }_0(N)$ for some $\mu \in \mathrm {SL}_2(\mathbb {Z})$. For $\gamma \in P_{\infty }$, we have $\Theta _{\mu \gamma \mu ^{-1}} = (1-(\mu \gamma \mu ^{-1})^*)\Theta _{\mu }$. So long as $\Theta _{\mu } \in \mathsf {K}_2(N)$, we then have that $\Theta |_Q \colon Q \to \mathsf {K}_2(N)$ is a coboundary, and for this it suffices that $\mu = \big (\begin{smallmatrix} a' & b'\\c' & d' \end{smallmatrix}\big )$ satisfies $N \nmid d'$. On the other hand, the set of $\mu$ with $N \mid d'$ is exactly the coset $\tilde {\Gamma }_0(N)\big (\begin{smallmatrix} 0 & 1\\1 & 0 \end{smallmatrix}\big )$, and in this case $Q = \mu P_{\infty } \mu ^{-1}$. We may then suppose $\mu = \big (\begin{smallmatrix} 0 & 1\\1 & 0 \end{smallmatrix}\big )$, for which $Q = \{ \big (\begin{smallmatrix} \pm 1 & n \\ 0 & 1 \end{smallmatrix}\big ) \mid n \in \mathbb {Z}\}$. Since $\Theta _N$ lifts to a map taking the value $\{1-\zeta _N,1-\zeta _N^{-1}\} = 0$ on $\big (\begin{smallmatrix} 1 & -1 \\ 0 & 1 \end{smallmatrix}\big )$ and the value $\{-1,-\zeta _N\} \in Z_N$ on $\big (\begin{smallmatrix} -1 & 0 \\ 0 & 1 \end{smallmatrix}\big )$, it is trivial on $Q$, and we have parabolicity.

We need to modify the notion of connecting sequence of § 3.3 to adapt it to the level $N$ structure. Specifically, let us refer to a connecting sequence $(b_i,d_i)_{i=0}^k$ for $\gamma \in \tilde {\Gamma }_0(N)$ with the property that $N \nmid d_i$ for all $0 \le i \le k$ as an $N$-connecting sequence for $\gamma$. Note that an $N$-connecting sequence always exists, as we verify with a little fiddling.

Lemma 4.2.2 Given a primitive vector $(b,d) \in \mathbb {Z}^2$ with $N \nmid d$, there exists a sequence $(v_i)_{i=0}^k$ in $\mathbb {Z}^2$ with $v_i \wedge v_{i+1} = 1$ for $0 \le i < k$ such that $v_0 = (0,1)$, $v_k = (b,d)$, and $v_i = (b_i,d_i)$ with $N \nmid d_i$ for all $0 \le i \le k$.

Proof. Choose any connecting sequence $(v_i)_{i=0}^k$ with $v_k = (b,d)$. Suppose that $v_i$ has second coordinate divisible by $N$. Then neither $v_{i-1}$ nor $v_{i+1}$ does. We will insert another sequence between $v_{i-1}$ and $v_{i+1}$, no element of which has second coordinate divisible by $N$. For $v, w \in \mathbb {Z}^2$, consider $v \wedge w$ as an integer via the identification of $\bigwedge ^2 \mathbb {Z}^2$ with $\mathbb {Z}$ using the basis vector $(1,0) \wedge (0,1)$. Note that $v_{i-1} \wedge v_i = v_i \wedge v_{i+1} = 1$ for $1 \le i \le k-1$. Write $t = v_{i-1} \wedge v_{i+1}$. We suppose that $t \geq 1$, the other case being easier. The sequence with $v_{i-1},v_i,v_{i+1}$ replaced by $v_{i-1}, v_{i+1}+(1-t) v_i, \ldots, v_{i+1} - v_i, v_{i+1}$ has nearly the desired properties since

\[ v_{i-1} \wedge (v_{i+1} + (1-t)v_i) = (v_{i+1}-(j-t)v_i) \wedge (v_{i+1} - (j+1-t)v_i) = 1 \]

for $1 \le j \leq t-1$. However, the last pair of adjacent vectors $x = v_{i+1}-v_i$ and $y = v_{i+1}$ satisfies $x \wedge y = -1$, rather than $1$. To remedy this, we replace $x,y$ by the sequence $x, -y, -x, y$.

Remark 4.2.3 As in Remark 4.1.5, Lemma 4.2.2 tells us that $\Theta$ restricted to $\tilde {\Gamma }_0(N)$ takes values in (a quotient of) the degree $2$ term of the subcomplex of the small complex $\mathsf {k}$ spanned by the $\mathsf {k}_{\gamma }$ for those $\gamma = \big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big ) \in \mathrm {SL}_2(\mathbb {Z})$ such that $N \nmid c$ and $N \nmid d$.

We then have the following explicit formula for our cocycle.

Proposition 4.2.4 Let $\gamma = \big (\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\big ) \in \tilde {\Gamma }_0(N)$, and let $(b_i,d_i)_{i=0}^k$ be an $N$-connecting sequence for $\gamma$. Then

\[ \Theta_{N,\gamma} = \sum_{i=1}^k \{ 1- \zeta_N^{d_i}, 1-\zeta_N^{-d_{i-1}} \}. \]

Proof. By Proposition 3.3.2, we need only note that

\[ s^*\langle (b_i,d_i),(-b_{i-1},-d_{i-1}) \rangle = \{1-\zeta_N^{d_i},1-\zeta_N^{-d_{i-1}}\} \]

for $1 \le i \le k$.

Since $1-\zeta _N^c$ is an $N$-unit for all $c \not \equiv 0 \bmod N$ and the map $K_2(\mathbb {Z}[\mu _N,{1}/{N}]) \to K_2(\mathbb {Q}(\mu _N))$ is an injection, we have the following corollary.

Corollary 4.2.5 The cocycle $\Theta _N$ takes values in $K_2(\mathbb {Z}[\mu _N,{1}/{N}])/Z_N$.

Remark 4.2.6 One could use Remark 3.3.5 to avoid the explicit formula in proving this corollary (assuming the same expected property of integral motivic cohomology), since pullback by $(1,\zeta _N)$ defines a morphism $\mathsf {K}_{2/\mathbb {Z}}(N) \to K_2(\mathbb {Z}[\mu _N,{1}/{N}])$, where $\mathsf {K}_{2/\mathbb {Z}}(N) = \mathsf {K}_{2/\mathbb {Z}} \cap \mathsf {K}_2(N)$.

In fact, we can do slightly better.

Lemma 4.2.7 If $N$ is divisible by two distinct primes, then $\Theta _N$ takes values in $K_2(\mathbb {Z}[\mu _N])/Z_N$. Otherwise, its restriction to $\Gamma _1(N)$ does.

Proof. Fix a prime $\ell$ dividing $N$. Let $F_{\ell }$ denote the residue field at a prime of $\mathbb {Q}(\mu _N)$ over $\ell$, and consider the tame symbol map

\[ \delta_{\ell} \colon K_2\bigg(\mathbb{Z}\bigg[\mu_N,\frac{1}{N}\bigg]\bigg)\bigg/Z_N \to F_{\ell}^{\times} \]

of (2.4). The common kernel of the maps $\delta _{\ell }$ is $K_2(\mathbb {Z}[\mu _N])/Z_N$. Thus, it suffices to see that $\delta _{\ell } \circ \Theta _N$ is trivial on the congruence subgroups of interest.

Suppose first that $N$ is divisible by two distinct primes. For any prime $p \mid N$ with $p \neq \ell$, we have that $\tilde {\Gamma }_0(N) \subset \tilde {\Gamma }_0(p)$, so there exists a $p$-connecting sequence $(b_i,d_i)_{i=0}^k$ for any $\gamma \in \tilde {\Gamma }_0(N)$. But then each $1-\zeta _N^{d_i}$ is a unit locally at primes over $\ell$, so $\delta _{\ell }(\{1-\zeta _N^{d_i},1-\zeta _N^{-d_{i-1}}\})$ vanishes. By Proposition 4.2.4, we then have $\delta _{\ell }(\Theta _{N,\gamma }) = 1$, independent of $\ell$.

Next, suppose that $N$ is a power of a prime $\ell$. Given $\gamma = \big (\begin{smallmatrix} a & b\\c & d \end{smallmatrix}\big ) \in \tilde {\Gamma }_0(N)$, there exists an $\ell$-connecting sequence $(b_i,d_i)_{i=0}^k$ for $\gamma$. Then each $1-\zeta _N^{d_i}$ has valuation $1$ at $(1-\zeta _N)$, so

\[ \delta_{\ell}(\{1-\zeta_N^{d_i},1-\zeta_N^{-d_{i-1}}\}) = -\frac{1-\zeta_N^{-d_{i-1}}}{1-\zeta_N^{d_i}} \bmod (1-\zeta_N), \]

which reduces to ${d_{i-1}}/{d_i}$ in $\mathbb {F}_{\ell }^{\times }$. Proposition 4.2.4 then yields that $\delta _{\ell }(\Theta _{\gamma }) = \det (\gamma ) d^{-1} \bmod \ell$, which is trivial if $\gamma \in \Gamma _1(N)$.