2.1 Basic properties of $\mathrm{SO}(3)$ -WRT representations

In this section, we will sketch a construction of the representation $\rho_p,$ and state some of its properties. Let p be an odd integer greater than or equal to 3, let M be a closed compact oriented 3-manifold, and let L be a framed link in M. In [Reference Blanchet, Habegger, Masbaum and VogelBHMV92], a topological invariant $Z_p(M,L)\in \mathbb{Q}[\zeta_p]$ of the pair (M,L) is defined. Roughly speaking, $Z_p(M,L)$ is the evaluation at a primitive pth root of unity $\zeta_p$ of a suitable linear combination of the colored Jones polynomials of the link $L_0 \cup L,$ where $L_0$ is a surgery presentation of M. Here ‘suitable’ means that the invariant thus defined is independent of the choice of surgery presentation $L_0$ for M; this is achieved by coloring the surgery presentation $L_0$ by the so-called Kirby color. We refer to [Reference Blanchet, Habegger, Masbaum and VogelBHMV92] for all details on this construction; we will use only some of the properties of this invariant $Z_p(M,L).$ First we note that as a consequence of the construction, the invariant $Z_p(M,L)$ satisfies the Kauffman relations in terms of the link L. Moreover, by a theorem of Masbaum and Roberts [Reference Masbaum and RobertsMR97], when p is prime then $Z_p(M,L) \in \mathbb{Z}[\zeta_p]$ .

Now let $\Sigma$ be a closed compact oriented surface. For two 3-manifolds M and M’ with a fixed identification of their respective boundaries $\partial M$ and $\partial M'$ with $\Sigma,$ we write $\langle M,M'\rangle= Z_p\big(M \underset{\Sigma}{\cup} \overline{M'}\big).$ We can do the same construction for manifolds containing framed links (M,L),(M’,L’). Moreover, we can extend $\langle,\rangle$ by bilinearity to get a sesquilinear form $\langle, \rangle$ on the $\mathbb{Q}[\zeta_p]$ -vector space $\mathcal{V}_p(\Sigma)$ formally spanned by 3-manifold M with a fixed isomorphism $\partial M=\Sigma,$ and containing a link L.

As examples of elements in $\mathcal{V}_p(\Sigma),$ consider a handlebody H with boundary $\Sigma,$ and such that H is itself a tubular neighborhood of a banded trivalent graph $\Gamma$ . A p-admissible coloring of the trivalent graph associates a non-negative even integer to each edge, with the additional conditions that colorings near a vertex satisfy triangle inequalities $c_1\leq c_2+c_3$ and also $c_1+c_2+c_3\leq 2p-4$ . Given a p-admissible coloring c of the trivalent graph $\Gamma,$ we get an element of $\mathcal{V}_p(\Sigma)$ as $(H,\Gamma(c)),$ where $\Gamma(c)$ is the linear combination of links obtained by cabling each edge e of $\Gamma$ by the Jones–Wenzl idempotent $f_{c_e}$ .

Theorem 2.1 [Reference Blanchet, Habegger, Masbaum and VogelBHMV92]. For any odd integer $p\geq 3$ , the vector space

\[V_p(\Sigma)=\mathcal{V}_p(\Sigma) / \mathrm{Ker} \langle,\rangle\]

is a $\mathbb{Q}[\zeta_p]$ -vector space of finite dimension d(g,p), on which $\mathrm{Mod}(\Sigma)$ has a natural projective-linear action by change of identification $\partial M=\Sigma$ .

Moreover, for any handlebody H with $\partial H=\Sigma,$ and any pants decomposition of $\Sigma$ by curves which bound disks in H, colored trivalent graphs in H with p-admissible colors form a basis of $V_p(\Sigma)$ .

An explicit formula for $\dim(V_p(\Sigma))$ may be found in [Reference Blanchet, Habegger, Masbaum and VogelBHMV92]. Now, if we suppose that p is prime, we can instead define $\mathcal{V}_p(\Sigma)$ to be the free $\mathbb{Z}[\zeta_p]$ -module spanned by pairs (M,L) where M is 3-manifold with a fixed identification $\partial M=\Sigma,$ and we get that $\mathcal{S}_p(\Sigma)=\mathcal{V}_p(\Sigma)/\mathrm{Ker} \langle,\rangle$ is a free $\mathbb{Z}[\zeta_p]$ -module and a lattice in $V_p(\Sigma)$ ; moreover, $\mathrm{Mod}(\Sigma)$ still admits a natural action on it.

The previous basis of $V_p(\Sigma)$ as a $\mathbb{Q}[\zeta_p]$ -vector space does not provide a basis of $\mathcal{S}_p(\Sigma)$ as a $\mathbb{Z}[\zeta_p]$ -module. In [Reference Gilmer and MasbaumGM07], Gilmer and Masbaum provide an integral basis for $\mathcal{S}_p(\Sigma).$ Unlike the previous basis, the integral basis can only be associated to some special pants decomposition of $\Sigma,$ the so-called lollipop tree decompositions. We will say that a pants decomposition of $\Sigma$ is a lollipop tree decomposition if it contains g non-separating curves and $2g-3$ separating curves.

Given a lollipop tree decomposition of $\Sigma,$ we associate elements $v(a,b)\in\mathcal{S}_p(\Sigma)$ to colorings of the edges of the trivalent graph $\Gamma$ associated to the decomposition as follows. We take H to be a handlebody with $\partial H=\Sigma$ , and such that the curve of the pants decomposition bounds disks in H. We let the internal edges of $\Gamma$ be colored by $(2a_i)_{1\leq i \leq 2g-3},$ and we assume that $2a_1,\ldots,2a_g$ are the colors of the edges that are adjacent to one-edge loops. The loop edges are colored by $a_i+b_i$ . We assume again that if $c_1,c_2,c_3$ are the colors around a trivalent vertex, then we have the p-admissibility conditions $c_1\leq c_2+c_3,$ and $c_1+c_2+c_3$ is even and less than or equal to $2p-4$ . Moreover, we ask for the colors $a_i+b_i$ to be at most $(p-3)/2,$ which Gilmer and Masbaum call a small coloring. We construct an element $\Gamma(a,b) \in \mathcal{S}_p(\Sigma)$ by embedding the trivalent graph $\Gamma$ in H, replacing the internal edges with $2a_i$ parallel copies with the Jones–Wenzl idempotent $f_{2a_i}$ inserted. However, for the loop edge colored by $a_i+b_i,$ we first cable the loop edge by the Jones–Wenzl idempotent $f_{a_i},$ then we add a copy of the loop colored by $((2+z)/h)^{b_i},$ where $h=1-\zeta_p.$ (Here we use the convention that a framed link colored by $z^n$ stands for n parallel copies of the link, and we can make sense of framed links colored by a polynomial in z by extending linearly.) Finally, we define $v(a,b)=h^{-\lfloor {(1/2)}(a_1+\cdots+a_g)\rfloor}\Gamma(a,b)$ .

In [Reference Gilmer and MasbaumGM14], Gilmer and Masbaum introduce a decomposition of $\mathcal{S}_p(\Sigma)$ as a sum of two $\mathbb{Z}[\zeta_p]$ -submodules. Let $v(a_i,b_i)$ be a lollipop tree basis of $\mathcal{S}_p(\Sigma),$ and assume (up to reindexing) that $2a_1,\ldots 2a_g$ are the colors of the edges that are adjacent to a one-edge loop. They define $\mathcal{S}_p^{\mathrm{odd}}(\Sigma)$ (respectively, $\mathcal{S}_p^{\mathrm{ev}}(\Sigma)$ ) to be the $\mathbb{Z}[\zeta_p]$ -submodule spanned by the basis vectors $v(a_i,b_i)$ such that $a_1+\cdots+a_g$ is odd (respectively, even).

While this decomposition of $\mathcal{S}_p(\Sigma)$ depends on the choice of a lollipop tree basis of $\mathcal{S}_p(\Sigma),$ Gilmer and Masbaum show that the associated decomposition of the $\mathbb{Z}[\zeta_p]/(h)\simeq \mathbb{F}_p$ -module,

\[F_p(\Sigma):=\mathcal{S}_p(\Sigma)\underset{\mathbb{Z}[\zeta_p]}{\otimes}\mathbb{Z}[\zeta_p]/(h),\]

does not depend on such a choice.

Theorem 2.3 [Reference Gilmer and MasbaumGM14]. Let $\Sigma$ be a closed compact oriented surface and $p\geq 5$ be a prime. Then we have

\[\mathbb{Z}[\zeta_p][\rho_p(\mathrm{Mod}(\Sigma))] = \Bigg(\begin{array}{c@{\quad}c}\mathrm{End}(\mathcal{S}_p^{\mathrm{odd}}(\Sigma)) & \mathrm{Hom}(\mathcal{S}_p^{\mathrm{ev}}(\Sigma),\mathcal{S}_p^{\mathrm{odd}}(\Sigma))\\[5pt]h\mathrm{Hom}(\mathcal{S}_p^{\mathrm{odd}}(\Sigma),\mathcal{S}_p^{\mathrm{ev}}(\Sigma)) &\mathrm{End}(\mathcal{S}_p^{\mathrm{ev}}(\Sigma))\end{array}\Bigg)\]

where $h=1-\zeta_p$ .

In these results $\mathrm{Mod}(\Sigma)$ really means the central extension of $\mathrm{Mod}(\Sigma)$ that acts linearly rather than only projectively. In particular, we can talk here about Hom and End. The kernel of this extension is a cyclic group of order p and the elements of this kernel act as multiplication by a power of $\zeta_p$ .

This central extension of $\mathrm{Mod}(\Sigma)$ is trivial when restricted to the Torelli group as it is the pull-back of an extension of the symplectic group. The above theorem implies that the image of the Torelli and Johnson subgroups by $\rho_p$ have the following structure.

Proof. Part (i) is proved in [Reference Gilmer and MasbaumGM17, Lemma A1]. For part (ii), notice that it is trivially true for $f=t_{\alpha},$ a Dehn twist along a separating curve of the lollipop tree pants decomposition. Note that there is a Dehn twist of each possible genus among those. Conjugating with an element of $\mathrm{Mod}(\Sigma)$ and by Theorem 2.3, it is then also true for any separating Dehn twist, and therefore for any $f\in J_2(\Sigma)$ as $J_2(\Sigma)$ is generated by separating twists. Finally, part (iii) follows from parts (i) and (ii), as

$\rho([f,g])={\rm id}_{S_p(\Sigma)}+h\begin{pmatrix}0 & \quad AA_3-A_1A \\[3pt]0 & \quad 0\end{pmatrix}\ (\mathrm{mod}\, h^2).$

2.2 Computations of $\rho_p$ on bounding pairs and separating twists

The goal of this subsection is to prove that certain bounding pairs act non-trivially via $\rho_{p,1},$ and that the images of some specific family of separating twists by $\rho_{p,2}$ are linearly independent. These results are technical (based on skein theoretical arguments) but are crucial for the proofs of Theorem 1.2 and 1.3.

Let $p \ge 5$ be a prime and A be a 2pth primitive root of unity. Cutting the surface into smaller pieces is a standard trick used for computations in the setting of quantum representations.

Let B be a 3-ball, and let $\mathcal{P}$ be a set of four banded points on $\partial B$ , two of which are colored by 1 and two are colored by 2. Let $\gamma \subset \partial B$ be the following simple closed curve.

Let V be the skein module over $\mathbb{Q}[A]$ of B relative to $(\partial B, \mathcal{P})$ . This $\mathbb{Q}[A]$ vector space is two-dimensional with basis as follows.

Recall that for k a positive integer, $[k] = (A^{2k}-A^{-2k})/(A^{2}-A^{-2})$ .

Lemma 2.5. The action on V of the Dehn twist along $\gamma$ has the following matrix in the basis $(e_1,e_2)$ :

\[\begin{pmatrix}T_{1,1} & \quad T_{1,2} \\[3pt]T_{2,1} & \quad T_{2,2}\end{pmatrix} = \begin{pmatrix}-\dfrac{A^3+A^{19}+A^{11}}{[3]} & \quad -\dfrac{A^9(A^4+A^{-4})(A^2-A^{-2})}{A^2+A^{-2}} \\[15pt]A^9(A^4-A^{-4}) & \quad -\dfrac{A^7+A^{-1}+A^{15}}{[3]}\end{pmatrix}\!.\]

Proof. We note that since all bounding pairs of genus 1 are conjugated, it suffices to prove the proposition for one particular bounding pair. Let $\Gamma$ be the following graph.

Here $\Sigma$ is the boundary of a regular neighborhood of the graph, which is a handlebody of genus g, and the curves $c_1, c_2$ are on $\Sigma$ . We want to compute the action of $t_{c_1} t_{c_2}^{-1},$ which is a bounding pair of genus $1.$ We define the following two vectors associated to an admissible coloring of $\Gamma$ .

Here the three dots indicate that the remaining colors are 0. The triple

\[(h^{-1} u, h^{-1} v,h^{-2}(2+z_2)u)\]

is part of the integral basis. Here $z_2$ is the following curve.

Let W be the $\mathbb{Z}[\zeta_p]$ -module generated by $\{h^{-1} u, h^{-1} v,h^{-2}(2+z_2)u \}$ . It is easy to check that $\rho_p(t_{c_1})$ and $\rho_p(t_{c_2}^{-1})$ stabilize W. On the basis $(h^{-1} u, h^{-1} v,h^{-2}(2+z_2)u)$ of W, we compute

\[\rho_p(t_{c_1})_{|_W}=\begin{pmatrix}A^8 & \quad -2T_{1,2} & \quad 2h^{-1}(A^8-T_{1,1})\\[3pt]0 & \quad T_{2,2} & \quad h^{-1}T_{2,1} \\[3pt]0& \quad hT_{1,2} & \quad T_{1,1}\end{pmatrix},\quad\rho_p(t_{c_2})_{|_W}=\begin{pmatrix}1 & \quad 0 & \quad 2h^{-1}(1+A^3)\\[3pt]0 & \quad - A^3 & \quad 0 \\[3pt]0& \quad 0 & \quad -A^3\end{pmatrix}.\]

Recall that $A^2 = \zeta_p$ and $h=1-\zeta_p$ . We have that

$\rho_p(t_{c_1} t_{c_2}^{-1})_{|_W} \equiv \begin{pmatrix}1 & \quad 0 & \quad 0\\0 & \quad 1 & \quad -4 \\0& \quad 0 & \quad 1\end{pmatrix} \pmod h.$

Next we prove a linear independence result for the $\rho_{p,2}$ image of some separating twists. Recall that $\mathrm{Im}\rho_{p,2}|_{J_2(\Sigma)}$ has a natural $\mathbb{F}_p$ -vector space structure, obtained by considering the order 1 in h (see also Lemma 3.4.).

The separating Dehn twists $t_{\alpha_i}$ acts diagonally on the basis v(a,b), and we have

\[\rho_{p}(t_{\alpha_i})(v(a,b))=\zeta_p^{2a_i(a_i+1)}v(a,b),\]

by [Reference Blanchet, Habegger, Masbaum and VogelBHMV92, Remark 7.6(ii)]. Hence, we have

\[\rho_{p,2}(t_{\alpha_i})(v_j)=\begin{cases}(1+12h)v_i \ (\mathrm{mod}\, h^2), & \textrm{if }i=j,\\(1+4h)v_j \ (\mathrm{mod}\, h^2), & \textrm{otherwise}.\end{cases}\]

We conclude as the matrix

\[4\begin{pmatrix}3 & \quad 1 & \quad \cdots & \quad 1\\1 & \quad \ddots & \quad \ddots & \quad \vdots\\\vdots & \quad \ddots & \quad \ddots & \quad 1\\1 & \quad \cdots & \quad 1 & \quad 3\end{pmatrix}\]

has determinant $4^{2g-3}2^{2g-4}(2g-1),$ which is non-zero mod p unless p divides $2g-1.$ In the latter case its kernel is generated by ${}^t(1,1,\ldots,1)$ . However, in that case we also compute $\rho_{p,2}(t_{\alpha_1}\cdots t_{\alpha_{2g-3}})(w)=(1+4(2g-3)h)w \ (\mathrm{mod}\, h^2),$ hence the $\rho_{p,2}(t_{\alpha_i})$ are still linearly independent.

Recall that if $\Sigma$ is a surface of genus at least 2 with one boundary component, we denote by $\hat{\Sigma}$ the surface obtained by gluing a disk on the boundary of $\Sigma$ . We also denote by $\rho_p^s$ the representation of $\pi_1(\hat{\Sigma})$ obtained by restriction of $\rho_p$ where the boundary color is 2.

Proof. If $\gamma$ is a loop in $\pi_1(\hat{\Sigma})$ and $\varphi \in \mathrm{Mod}(\Sigma)$ then

\[\rho_p(\varphi) \rho_p(\gamma) \rho_p(\varphi)^{-1} = \rho_p(\varphi(\gamma)).\]

Moreover, the action of $\mathrm{Mod}(\Sigma)$ is transitive on the set of non-separating simple loops in $\pi_1(\hat{\Sigma})$ . Therefore it is enough to find a simple loop $\gamma \in\pi_1(\hat{\Sigma})$ such that $\rho_{p,1}^s(\gamma) \neq 1$ . Let $\Gamma$ be the following graph.

As before, $\hat{\Sigma}$ is the boundary of a regular neighborhood of the graph; the univalent vertex (marked by a dot) is attached to the banded point colored by 2 on $\hat{\Sigma}$ . Also the loop $\gamma$ lies on $\hat{\Sigma}$ . Let $c_1$ and $c_2$ be the following two curves on $\Sigma$ .

It is known that the loop $\gamma$ is $t_{c_1} t_{c_2}^{-1}$ when viewed as an element of $\mathrm{Mod}(\Sigma)$ . We will omit the computations, as they are almost identical to the ones done for the proof of Lemma 2.6, but they show that $\rho_{p,1}(t_{c_1}t_{c_2}^{-1})$ is not trivial.

Proof. We will use the same notation as in the proof of Lemma 2.6. Let $\alpha$ be the meridian curve of the base edge of the second lollipop of the graph $\Gamma$ . Let also $f=t_{c_1}t_{c_2}^{-1},$ the bounding pair considered in the proof of Lemma 2.6. We have that the $\mathbb{Z}[\zeta_p]$ -submodule W of $\mathcal{S}_p(\Sigma)$ spanned by $h^{-1}u,h^{-1}v,h^{-2}(2+z_2)u$ is invariant not only by $f=t_{c_1}t_{c_2}^{-1}$ but also by $t_{\alpha}$ . Indeed, in the basis $\lbrace h^{-1}u,h^{-1}v,h^{-2}(2+z_2)u \rbrace,$ we have

$\rho_p(t_{\alpha})|_W=\begin{pmatrix}1 & \quad 0 & \quad 0 \\0 & \quad A^8 & \quad 0 \\0 & \quad 0 & \quad 1\end{pmatrix},$

and thus

\[\rho_{p,2}(t_{\alpha})|_W=\begin{pmatrix}1 & \quad 0 & \quad 0 \\0 & \quad 1-4h & \quad 0 \\0 & \quad 0 & \quad 1\end{pmatrix}\ (\mathrm{mod}\, h^2).\]

Note that since $\rho_p(t_\alpha)={\rm Id} \mod h,$ it suffices to know $\rho_p(f)$ mod h to compute $\rho_p(ft_{\alpha}f^{-1})$ mod $h^2$ . We get

$\rho_{p,2}(ft_{\alpha}f^{-1})=\begin{pmatrix}1 & \quad\! 0 & \quad\! 0 \\0 & \quad\! 1 & \quad\! -4 \\0 & \quad\! 0 & \quad\! 1\end{pmatrix}\begin{pmatrix}1 & \quad\! 0 & \quad\! 0 \\0 & \quad\! 1-4h & \quad\! 0 \\0 & \quad\! 0 & \quad\! 1\end{pmatrix}\begin{pmatrix}1 & \quad\! 0 & \quad\! 0 \\0 & \quad\! 1 & \quad\! 4 \\0 & \quad\! 0 & \quad\! 1\end{pmatrix}=\begin{pmatrix}1 & \quad\! 0 & \quad\! 0 \\0 & \quad\! 1-4h & \quad\! -16h \\0 & \quad\! 0 & \quad\! 1\end{pmatrix}\neq \rho_{p,2}(t_{\alpha}).$

We will call a subsurface $S\subset \Sigma$ essential if no boundary component of S bounds a disk.

For $f \in \mathrm{Mod}(\Sigma)$ and S a subsurface of $\Sigma,$ we say that the support of f is included in S if there is a representative of f which is the identity on $\Sigma \setminus S$ . The mapping class f can then be seen as a mapping class of the surface S’ obtained from S by filling boundary components with disks. The surface S’ may have smaller genus than $\Sigma.$ We will write $\rho_p$ for the $\mathrm{SO}(3)$ -WRT quantum representation of $\mathrm{Mod}(\Sigma)$ or of $\mathrm{Mod}(S'),$ indifferently.

1. (i) $\rho_p(f')\notin \mathrm{Ker} \rho_p \Longrightarrow \rho_p(f) \notin \mathrm{Ker} \rho_p$ ;

2. (ii) let J be an ideal of $\mathbb{Z}[\zeta_p]$ ; furthermore, if $\partial S$ consists only of separating curves in $\Sigma,$ then

   \[\rho_p(f') \neq {\rm Id} \mod J \Longrightarrow \rho_p(f) \neq {\rm Id} \mod J.\]

2.3 Proof of Theorem 1.3

In this short subsection we give a proof of Theorem 1.3. Let $\Sigma$ be a surface of genus at least 2 with one boundary component and let $p \ge 5$ be prime. Suppose that the boundary of $\Sigma$ is colored by 2. We want to understand the kernel of $\rho^s_{p,1}:\pi_1(\hat{\Sigma}) \to \mathrm{PGL}_{d}(\mathbb{F}_p)$ .

Recall that any Dehn twist along a separating curve is trivial via $\rho_{p,1}$ . Now any separating simple loop $\delta \in \pi_1(\hat{\Sigma})$ , when viewed in $\mathrm{Mod}(\Sigma)$ , can be written as $t_{c_1} t_{c_2}^{-1}$ , where $c_1,c_2$ are separating curves. Therefore the image by $\rho^s_{p,1}$ of any separating simple loop is trivial.

By [Reference PutmanPut07, Lemma A.1], the group generated by separating simple loops on $\hat{\Sigma}$ is $[\pi_1(\hat{\Sigma}),\pi_1(\hat{\Sigma})]$ , so $\rho_{p,1}^s$ factors through $\pi_1(\hat{\Sigma})/[\pi_1(\hat{\Sigma}),\pi_1(\hat{\Sigma})]$ . Moreover, $\rho_{p,1}^s$ kills the pth powers of each generator of $\pi_1(\hat{\Sigma})$ , so $\rho_{p,1}^s$ induces a map:

\[\bar{\rho}_{p,1}^s : H_1(\hat{\Sigma},\mathbb{F}_p) \to \mathrm{PGL}_{d}(\mathbb{F}_p).\]

Let $I_p$ be the kernel of $\bar{\rho}_{p,1}^s$ . To conclude, we need to prove that $I_p=0$ . The map $\bar{\rho}_{p,1}^s$ is equivariant with respect to the mapping class group in the sense that if $[\gamma]$ is the homology class of the loop $\gamma$ then

\[\bar{\rho}_{p,1}^s(\varphi_*[\gamma]) = \rho_{p,1}^s(\varphi) \bar{\rho}_{p,1}^s([\gamma])\rho_{p,1}^s(\varphi)^{-1}\]

for any $\varphi \in \mathrm{Mod}(\Sigma)$ . This property implies that $I_p$ is a $\mathbb{F}_p$ -subspace of $H_1(\hat{\Sigma},\mathbb{F}_p)$ invariant under $\mathrm{Mod}(\Sigma)$ . The action of $\mathrm{Mod}(\Sigma)$ on $H_1(\hat{\Sigma},\mathbb{F}_p)$ is irreducible and by Lemma 2.8 the map $\rho_{p,1}^s$ is not trivial, therefore $I_p = 0$ .

2.4 Structure of the abelianization of the Torelli and Johnson subgroups

For G a group, let $\mathrm{Ab}(G)=G/[G,G]$ be its abelianization, which we can consider as a $\mathbb{Z}$ -module, and let $\mathrm{Ab}_{\mathbb{Q}}(G)=\mathrm{Ab}(G) \underset{\mathbb{Z}}{\otimes}\mathbb{Q}$ .

We will first describe the abelianization of the Torelli group $J_1(\Sigma)$ of a closed surface of genus $g \geq 3,$ as described in the following theorem of Johnson [Reference JohnsonJoh85]. Note that $\mathrm{Mod}(\Sigma)$ acts on $J_1(\Sigma)$ by conjugation. This action induces a $\mathrm{Mod}(\Sigma)$ action on $\mathrm{Ab}(J_1(\Sigma)),$ which factors through an $\mathrm{Sp}_{2g}(\mathbb{Z})$ -action.

We will write $\omega$ for the intersection form on $\Sigma$ , and by an abuse of notation, for its dual which is a $\mathrm{Sp}_{2g}(\mathbb{Z})$ -invariant vector in $\Lambda^2 H_1(\Sigma,\mathbb{Z})$ . An explicit formula for $\omega$ is $\omega=a_1\wedge b_1 +\cdots +a_g\wedge b_g,$ where $a_1,b_1,\ldots,a_g,b_g$ is any symplectic basis of $H_1(\Sigma,\mathbb{Z})$ . Similarly, $\omega \wedge H_1(\Sigma,\mathbb{Z})$ is a subrepresentation of $\Lambda^3 H_1(\Sigma,\mathbb{Z})$ . The contraction map $\kappa$ on $\Lambda^3 H_1(\Sigma,\mathbb{Z})$ is also a surjective morphism of $\mathrm{Sp}_{2g}(\mathbb{Z})$ -representation:

\[\begin{array}{rccl}\kappa : & \Lambda^3 H_1(\Sigma,\mathbb{Z}) & \longrightarrow & H_1(\Sigma,\mathbb{Z})\\& a\wedge b \wedge c & \longmapsto & \omega(a,b)c+ \omega(b,c)a + \omega(c,a)b.\end{array}\]

We note that $\kappa$ maps an element $\omega \wedge c\in \omega \wedge H_1(\Sigma,\mathbb{Z})$ to $(g-1)c,$ hence $\kappa$ induces a map

\[\widetilde{\kappa}:\Lambda^3 H_1(\Sigma,\mathbb{Z})/(\omega\wedge H_1(\Sigma,\mathbb{Z}))\longrightarrow H_1(\Sigma,\mathbb{Z}/(g-1)\mathbb{Z}).\]

Theorem 2.11 [Reference JohnsonJoh85]. For any closed surface of genus $g\geq 3,$ we have an isomorphism of $\mathrm{Sp}_{2g}(\mathbb{Z})$ -modules:

\[{\mathrm{Ab}}(J_1(\Sigma))\simeq \Lambda^3 H_1(\Sigma,\mathbb{Z}) /(\omega \wedge H_1(\Sigma,\mathbb{Z})) \bigoplus T,\]

where T is a $\mathrm{Sp}_{2g}(\mathbb{Z})$ -module which as an abelian group is a finite-rank 2-torsion group.

We note that the map

\[\tau_1 : {\mathrm{Ab}}(J_1(\Sigma))\longrightarrow \Lambda^3 H_1(\Sigma,\mathbb{Z})/(\omega \wedge H_1(\Sigma,\mathbb{Z}))\]

induced by the above isomorphism is also known as the first Johnson isomorphism, and the right hand-side is also isomorphic to $J_1(\Sigma)/J_2(\Sigma)$ .

Moreover, the result of Johnson actually explicitly describes the $\mathrm{Sp}_{2g}(\mathbb{Z})$ -module structure of the 2-torsion group, but we will not need it here; we will only use that it is a 2-torsion group.

Next, we want to describe the abelianization of the Johnson subgroup $J_2(\Sigma).$ As previously, the action of $\mathrm{Mod}(\Sigma)$ by conjugation on $J_2(\Sigma)$ induces a $\mathcal{M}=\mathrm{Mod}(\Sigma)/J_2(\Sigma)$ module structure on $\mathrm{Ab}(J_2(\Sigma))$ . Since $\mathcal{M}\simeq \mathrm{Im} \tau_1 \rtimes \mathrm{Sp}_{2g}(\mathbb{Z}),$ we also get a $\mathrm{Sp}_{2g}(\mathbb{Z})$ -module structure on $\mathrm{Ab}(J_2(\Sigma)),$ albeit a non-canonical one.

At the time of writing, only the rational abelianization of $J_2(\Sigma)$ is known. It was first computed by Dimca, Hain and Papadima [Reference Dimca, Hain and PapadimaDHP14] after Dimca and Papadima showed that it was of finite rank [Reference Dimca and PapadimaDP13]. The description that we will use here comes from the work of Morita, Sakasai and Suzuki [Reference Morita, Sakasai and SuzukiMSS20].

Theorem 2.12 [Reference Dimca, Hain and PapadimaDHP14, Reference Morita, Sakasai and SuzukiMSS20]. For any closed surface of genus $g\geq 6,$ we have an isomorphism of $\mathrm{Sp}_{2g}(\mathbb{Z})$ -modules:

\[{\mathrm{Ab}}_{\mathbb{Q}}(J_2(\Sigma)) \simeq \mathbb{Q} \oplus [2^2] \oplus [31^2],\]

where $[2^2]$ and $[31^2]$ stands for the $\mathrm{Sp}_{2g}(\mathbb{Q})$ representations associated to the Young diagrams $[2^2]$ and $[31^2]$ . Moreover, those representations are absolutely irreducible representations of $\mathrm{Sp}_{2g}(\mathbb{Z})$ .

We recall that a representation over $\mathbb{Q}$ is absolutely irreducible if it is irreducible over $\overline{\mathbb{Q}}$ . We note that, just like the $\mathrm{Sp}_{2g}(\mathbb{Z})$ -module structure, this splitting is not canonical. We will be interested also in the $\mathcal{M}$ -module structure of $\mathrm{Ab}_{\mathbb{Q}}(J_2(\Sigma))$ .

1. (1) The summand $\mathbb{Q}$ in the above isomorphism is the rational span of the core of the Casson invariant, which by [Reference MoritaMor91, Theorem∼5.7] is a morphism

   \[d:J_2(\Sigma)\longrightarrow \mathbb{Z}\]
   which is invariant by the action of conjugation by $\mathrm{Mod}(\Sigma).$ In particular, d vanishes on $[J_1(\Sigma),J_2(\Sigma)]$ .

2. (2) The summand $[2^2]$ is the rational image of the second Johnson homomorphism, which is the map

   \[\tau_2: J_2(\Sigma) \longrightarrow (J_2(\Sigma)/J_3(\Sigma)) \otimes \mathbb{Q}.\]
   Therefore, it vanishes on $[J_1(\Sigma),J_2(\Sigma)],$ as $[J_1(\Sigma),J_2(\Sigma)]\subset J_3(\Sigma)$ .

3. (3) The summand $[31^2]$ is isomorphic to $(J_3/J_4)_{\mathbb{Q}}:=(J_3(\Sigma)/J_4(\Sigma))\otimes \mathbb{Q}$ as a $\mathrm{Sp}_{2g}(\mathbb{Z})$ -representation. The projection from ${\mathrm{Ab}}(J_2(\Sigma))$ to this summand is non-canonical, and depends on a choice of splitting of the exact sequence

   \[1\longrightarrow \mathrm{Im}\tau_1 \longrightarrow \mathcal{M}\longrightarrow \mathrm{Sp}_{2g}(\mathbb{Z})\longrightarrow 1,\]
   which induces a splitting of the exact sequence
   \[1\longrightarrow (J_3/J_4)_{\mathbb{Q}} \longrightarrow (J_2/J_4)_{\mathbb{Q}} \longrightarrow (J_2/J_3)_{\mathbb{Q}}\longrightarrow 1,\]
   and thus an isomorphism $(J_2/J_4)_{\mathbb{Q}}\simeq (J_2/J_3)_{\mathbb{Q}} \times (J_3/J_4)_{\mathbb{Q}}.$ We remark that $J_2(\Sigma)/J_4(\Sigma)$ is abelian since $[J_2(\Sigma),J_2(\Sigma)]\subset J_4(\Sigma)$ .

The map ${\mathrm{Ab}}(J_2(\Sigma))\longrightarrow [2^2]\oplus[31^2]$ is the so-called refined second Johnson homomorphism, which was first introduced in [Reference MoritaMor99] and [Reference Morita, Sakasai and SuzukiMSS15]. This morphism vanishes on $J_4(\Sigma),$ as noted in the second equation of [Reference Morita, Sakasai and SuzukiMSS20] between Equations (9) and (10), and as follows from our description. Finally, it can be noted that the map ${\mathrm{Ab}}(J_2(\Sigma))\longrightarrow [31^2]$ does not vanish on $[J_1(\Sigma),J_2(\Sigma)]$ (as we shall also see from our computations).

2.5 Irreducibility of some modular representations of $\mathrm{Sp}_{2g}(\mathbb{Z})$

In this section, we gather some results about representations of $\mathrm{Sp}_{2g}(\mathbb{Z})$ that are necessary for the proofs of our main theorems. We recall that we can view the group $\mathrm{Sp}_{2g}(\mathbb{Z})$ as the image of the homology representation of a surface of genus g. To stay coherent with the other sections of the paper, we will write $H_1(\Sigma_g,\mathbb{Z})$ for the fundamental representation of $\mathrm{Sp}_{2g}(\mathbb{Z})$ . Also let $\overline{\kappa}$ be the reduction mod p of $\kappa$ .

Since $\kappa(\omega\wedge h)=(g-1)h$ for any $h\in H_1(\Sigma,\mathbb{Z}),$ if p divides $g-1$ then $\omega \wedge H_1(\Sigma,\mathbb{F}_p) \subset \mathrm{Ker} \overline{\kappa},$ and $\mathrm{Ker} \overline{\kappa} /(\omega \wedge H_1(\Sigma,\mathbb{F}_p))$ is a non-trivial subrepresentation of $\Lambda^3 H_1(\Sigma,\mathbb{F}_p)/(\omega \wedge H_1(\Sigma,\mathbb{F}_p))$ .

If, on the other hand, p does not divide $g-1,$ then $\mathrm{Ker} \overline{\kappa}$ is a direct summand of $\omega\wedge H_1(\Sigma,\mathbb{F}_p)$ in $\Lambda^3H_1(\Sigma,\mathbb{F}_p)$ and thus isomorphic to $V=\Lambda^3H_1(\Sigma,\mathbb{F}_p)/(\omega \wedge H_1(\Sigma,\mathbb{F}_p))$ .

Moreover, if p divides $g-1,$ then the only subrepresentations of V are $\lbrace 0 \rbrace,$ V and $\mathrm{Ker} \overline{\kappa} / (\omega \wedge H_1(\Sigma,\mathbb{F}_p))$ .

We note that the above proposition is part of the results of [Reference Premet and SuprunenkoPS83] on the composition factors of Weyl modules for $\mathrm{Sp}_{2g}(\mathbb{Z})$ (see also [Reference FoulleFou05, Theorem 1.1]). However, for the convenience of the reader, we include an elementary proof.

Throughout the proof, we write K to mean $\mathrm{Ker} \overline{\kappa}\simeq V$ if p does not divide $g-1,$ or $K=\mathrm{Ker} \overline{\kappa} / (\omega \wedge H_1(\Sigma,\mathbb{F}_p))$ if p divides $g-1.$

Indeed, since $\mathrm{Sp}_{2g}(\mathbb{Z})$ acts transitively on basis vectors $c_i\wedge c_j\wedge c_k$ with $c_i,c_j,c_k$ generating an isotropic subspace of $H_1(\Sigma,\mathbb{F}_p),$ the subspace W would contain all such vectors. Since a generating set of K consists of those vectors and the vectors $(a_i\wedge b_i)\wedge c_l-(a_j\wedge b_j)\wedge c_l$ where $c_l \notin \lbrace a_i,b_i,a_j,b_j\rbrace,$ it suffices to show that the latter vectors are also in W. However,

\begin{align*}& t_{a_i\,+\,a_j}(b_i\wedge b_j\wedge c_l)=(b_i+a_i+a_j)\wedge (b_j+a_i+a_j)\wedge c_l\\&\quad =b_i\wedge b_j\wedge c_l + b_i\wedge a_j\wedge c_l + a_i\wedge b_j \wedge c_l +(a_j\wedge b_j \wedge c_l-a_i\wedge b_i \wedge c_ls).\end{align*}

Hence W should also contain the vectors $a_j\wedge b_j \wedge c_l -a_i\wedge b_i \wedge c_l,$ and thus all of K.

Indeed, Claim 2 will follow from Claim 1 if we show that we can realize the projection $\pi$ from V onto $\mathrm{Span}(c_i\wedge c_j \wedge c_k)$ as an element of $\mathbb{F}_p[\mathrm{Sp}_{2g}(\mathbb{F}_p)].$ For $\lambda$ a generator of $\mathbb{F}_p^*,$ let $\phi_{\lambda}^l\in \mathrm{Sp}_{2g}(\mathbb{F}_p)$ such that $\phi_{\lambda}^l(c_l)=\lambda c_l$ and $\phi_{\lambda}(c_l')=\lambda^{-1}c_l',$ and $\phi_{\lambda}(c_m)=c_m$ for the other basis vectors of $H_1(\Sigma,\mathbb{F}_p).$ Note that the endomorphism of V induced by $\phi^l_{\lambda}$ (which we will abusively also denote by $\phi^l_{\lambda}$ ) acts diagonally in the basis $c_i\wedge c_j \wedge c_k$ of V. Moreover, the eigenvalue is $\lambda$ if there is one $c_l$ and no $c_l'$ among $c_i,c_j,c_k,$ it is 1 if there is either no $c_l$ and no $c_l'$ among $c_i,c_j,c_k$ or if $c_l$ and $c_l'$ both appear, and it is $\lambda^{-1}$ if only $c_l'$ appears. Then the endomorphism of V defined by

\[\pi^l=-\underset{\lambda \in \mathbb{F}_p^*}{\sum} \lambda^{-1}\phi_{\lambda}^l\]

is the projection on the subspace spanned by basis vectors with one component equal to $c_l$ and no $c_l'$ component. The above follows from the classical identity that $\sum_{\lambda \in\mathbb{F}_p^*} \lambda^k=0$ if $p-1$ does not divide k, and $=-1$ otherwise.

Moreover, $\pi=\pi^i \pi^j \pi^k$ is the projection on $\mathrm{Span}(c_i\wedge c_j \wedge c_k)$ .

Let $W\subset K$ be a non-trivial subrepresentation, and let $v\neq 0 \in W$ . Let us assume that v has no non-zero coefficient along any $c_i\wedge c_j\wedge c_k$ with $\omega(c_i,c_j)=\omega(c_j,c_k)=\omega(c_k,c_i)=0,$ since otherwise Claim 2 applies. Up to applying an element of $\mathrm{Sp}_{2g}(\mathbb{Z})$ that permutes the basis $c_i$ of $H_1(\Sigma,\mathbb{F}_p),$ we may assume that v has a non-zero coefficient along a vector basis of the form $a_i\wedge b_i \wedge a_g$ where $i<g.$ Then, applying the projection $\pi^{2g-1}$ as in the proof of Claim 2, we can assume that

\[v=\lambda_1 a_1\wedge b_1 \wedge a_g + \lambda_2 a_2\wedge b_2 \wedge a_g + \cdots +\lambda_{g-1} a_{g-1} \wedge b_{g-1} \wedge a_g,\]

for some coefficients $\lambda_i \in \mathbb{F}_p$ . Since v is not in $\omega \wedge H_1(\Sigma,\mathbb{F}_p),$ we can also assume that the $\lambda_i$ are not all equal. Without loss of generality, we assume that $\lambda_1\neq \lambda_2$ .

Then we have

\[t_{a_1+a_2}(v)-v=(\lambda_1-\lambda_2)a_1\wedge a_2\wedge a_g,\]

and therefore we can conclude by Claim 2.

Since $H_1(\Sigma,\mathbb{F}_p)$ is an irreducible representation, without loss of generality, we can assume that W contains a vector v such that $\overline{\kappa}(v) = a_g$ . We claim that we can assume that v is of the form

\[v=\lambda_1 a_1\wedge b_1\wedge a_g +\cdots + \lambda_{g-1} a_{g-1}\wedge b_{g-1}\wedge a_g.\]

Let us introduce maps

\[\tilde{\pi}^l= -\underset{\lambda \in \mathbb{F}_p^*}{\sum} \phi_{\lambda}^l\]

where the maps $\phi_{\lambda}^l$ were introduced in the proof of Claim 2. Then $\tilde{\pi}^l$ is the projection on the subspace spanned by basis vectors $c_i\wedge c_j\wedge c_k$ where either none of $c_i,c_j,c_k$ is $a_l$ or $b_l,$ or one is $a_l$ and another one is $b_l$ . Therefore we claim that $\tilde{\pi}^1 \cdots \tilde{\pi}^{g-1} \pi^g (v)$ is the form required and is in W. We conclude similarly as in the proof of Claim 3: since $\overline{\kappa}(v)=a_g$ we have $\lambda_1+\cdots +\lambda_{g-1}=1,$ therefore they are not all equal. Assume without loss of generality that $\lambda_1\neq \lambda_2$ . then $t_{a_1+a_2}(v)-v=(\lambda_1-\lambda_2)a_1\wedge a_2\wedge a_g,$ therefore W contains K by Claim 1, and thus contains all of V.

Lemma 2.15. For any $g\geq 3$ and for any odd prime p, the representation

\[\mathrm{Sp}_{2g}(\mathbb{F}_p) \longrightarrow \mathrm{GL}(\Lambda^3 H_1(\Sigma,\mathbb{F}_p)/(\omega \wedge H_1(\Sigma,\mathbb{F}_p)))\]

is faithful.

Now, it is well known that $\mathrm{PSp}_{2g}(\mathbb{F}_p)$ is a simple group for any $g\geq 2$ and any prime $p\geq 3,$ and that the only normal subgroups of $\mathrm{Sp}_{2g}(\mathbb{F}_p)$ are the trivial subgroup, $\mathrm{Sp}_{2g}(\mathbb{F}_p)$ and $Z(\mathrm{Sp}_{2g}(\mathbb{F}_p))=\lbrace \pm \mathrm{id} \rbrace$ ; see, for example, [Reference DieudonnéDie71]. Since the kernel does not contain $-\mathrm{id},$ it must be trivial.

2.6 Simply intersecting pairs and the contraction map

Proposition 2.14 shows the importance of understanding the kernel of the contraction map

\[\widetilde{\kappa}: \Lambda^3 H_1(\Sigma,\mathbb{Z})/(\omega \wedge H_1(\Sigma,\mathbb{Z}))\rightarrow H_1(\Sigma,\mathbb{Z}/(g-1)\mathbb{Z}).\]

In this section, we will introduce some elements of the Torelli subgroup $J_1(\Sigma)$ whose images are in this kernel.

The work of Childers [Reference ChildersChi12] shows that SIP maps are in the kernel of the contraction map. More precisely, if $[t_{\alpha},t_{\beta}]$ is a SIP map then a regular neighborhood of $\alpha\cup \beta$ is a four-holed sphere with some curves x,y,z,w as boundary components. Then [Reference ChildersChi12, Main Result 2] states that

\[\tau_1([t_{\alpha},t_{\beta}])=[x]\wedge [y]\wedge [z],\]

where $\tau_1: J_1(\Sigma) \rightarrow \Lambda^3 H_1(\Sigma,\mathbb{Z})/(\omega \wedge H_1(\Sigma,\mathbb{Z}))$ is the first Johnson homomorphism.

From this we get the following lemma.

2.7 Casson invariant and quantum invariant

For M an integral homology 3-sphere, let $\lambda(M) \in \mathbb{Z}$ be its Casson invariant. An important theorem by Murakami relates quantum invariants with Casson invariants. Recall that for $p\ge 5$ a prime, $\zeta_p$ denotes a pth primitive root of unity and $h=1-\zeta_p$ .

Theorem 2.19 [Reference MurakamiMur95]. If M is an integral homology 3-sphere, then

\[Z_p(M) = Z_p(S^3) (1+6h \lambda(M))\mod h^2\]

Now let $\Sigma$ be a surface of genus at least 6. We choose two handlebodies $H_1$ and $H_2$ with boundary $\Sigma$ such that $H_1 \underset{\mathrm{Id}}{\cup} \bar{H_2} = S^3$ . If $\varphi\in J_1(\Sigma)$ , the 3-manifold $H_1 \underset{\varphi}{\cup} \bar{H_2}$ is an integral homology 3-sphere. In [Reference MoritaMor89], Morita showed that the map

\[\lambda_{H_1,H_2}: \varphi \in J_2(\Sigma) \mapsto \lambda(H_1 \underset{\varphi}{\cup} \bar{H_2})\in \mathbb{Z}\]

is a group homomorphism. Moreover, he proved that when restricted to $J_3(\Sigma)$ , the map $\lambda_{H_1,H_2}$ does not depend on the choice of the handlebodies $H_1$ and $H_2$ . For $\varphi\in J_3(\Sigma)$ , we will simply write $\lambda(\varphi)$ for $\lambda_{H_1,H_2}(\varphi)$ . The map $\lambda : \varphi \in J_3(\Sigma) \mapsto \lambda(\varphi) \in \mathbb{Z}$ is invariant by conjugation under $\mathrm{Mod}(\Sigma)$ and is proportional to the so-called core of the Casson invariant. We will also need the following theorem by Hain.

Although we will not need it here, we note that for the case $k=4$ , Faes [Reference FaesFae22] showed that $\lambda(J_4) = \mathbb{Z}$ .

3.1 The h-adic expansion of quantum representations

In this section, we fix a prime $p\geq 5.$ As in the previous section, let $h=1-\zeta_p\in\mathbb{Z}[\zeta_p].$ We recall that h is a prime in $\mathbb{Z}[\zeta_p],$ and that p is equal to $xh^{p-1}$ for some unit $x\in \mathbb{Z}[\zeta_p].$ We denote by $\rho_{p,k}$ the representation $\rho_p$ reduced modulo $h^k,$ whose coefficients then belong to $\mathbb{Z}[\zeta_p]/(h^k)$ .

1. (i) for all $k,l \geq 1,$ one has $[N_k,N_l] \subset N_{k+l}$ ;

2. (ii) for all $k \geq 1,$ one has $N_k^{p} \subset N_{k+p-1},$ where $N_k^p$ is the subgroup of $\mathrm{Mod}(\Sigma)$ generated by pth powers of elements of $N_k.$

Proof. We warn the reader that in the following proof, we consider all equalities to hold up to a power of $\zeta_p$ . An element $f\in \mathrm{Mod} (\Sigma)$ is in $N_k$ if and only if $\rho_{p}(f)=\mathrm{id}_{\mathcal{S}_p(\Sigma)} + h^k u,$ where $u\in\mathrm{End}_{\mathbb{Z}[\zeta_p]}(\mathcal{S}_p(\Sigma))$ . Let $f\in N_k$ and $g\in N_l$ , and write $\rho_p(f)={\rm id}_{\mathcal{S}_p(\Sigma)} + hu$ and $\rho_p(g)=\mathrm{id} +h^lv$ . Then

\[\rho(f)^{-1}=\mathrm{id}_{\mathcal{S}_p(\Sigma)} -h^ku +h^{2k}u^2 -\cdots \mod h^{k+l}.\]

One can write the same formula for $\rho(g)^{-1}$ modulo $h^{k+l}$ . Then a direct computation shows that $\rho([f,g])={\rm id}_{\mathcal{S}_p(\Sigma)}$ modulo $h^{k+l}$ .

As for point (ii), we recall that p is equal to $h^{p-1}$ up to a unit in $\mathbb{Z}[\zeta_p]$ . Let us assume again that $f\in N_k$ and $\rho_p(f)={\rm id} +hu$ . Then we have

\[\rho_p(f)^p={\rm id} +\binom{p}{1} h^ku + \binom{p}{2} h^{2k}u^2 + \cdots + h^{pk}u^{p}.\]

Note that p divides all binomial coefficients $\binom{p}{j}$ with $1\leq j \leq p-1,$ so all terms $h^{jk}\binom{p}{j} u^j$ with $1\leq j \leq p-1$ are zero modulo $h^{k+p-1}.$ Since $pk\geq k+p-1,$ the last term is also zero modulo $h^{k+p-1}$ and we get the second claim.

3.2 Analysis of $\mathrm{Ker}(\rho_{p,1})$

In this section, we will study the representation $\rho_{p,1}$ restricted to the Torelli group $J_1(\Sigma)$ and give a proof of Theorem 1.2. First, note that since we restrict $\rho_{p,1}$ to $J_1(\Sigma),$ we can consider it a linear representation instead of just a projective representation. For G a group, we write ${\mathrm{Ab}}(G)$ for its abelianization, and ${\mathrm{Ab}}_p(G)$ for its mod p abelianization, ${\mathrm{Ab}}_p(G)=G/\langle [G,G],G^p\rangle$ .

Lemma 3.2. The morphism

\[\rho_{p,1}: J_1(\Sigma) \longrightarrow \mathrm{Aut}(\mathcal{S}_p(\Sigma)) \mod h\]

factors through ${\mathrm{Ab}}_p(J_1(\Sigma))$ .

We can put a $\mathrm{Mod}(\Sigma)$ -module structure on $\rho_{p,1}(J_1(\Sigma))$ by defining for $f\in\mathrm{Mod}(\Sigma),\ g \in J_1(\Sigma)$ ,

\[f\cdot \rho_{p,1}(g) = \rho_{p,1}(fgf^{-1}).\]

However, since $\rho_{p,1}$ is abelian on $J_1(\Sigma),$ this induces a $\mathrm{Mod}(\Sigma)/J_1(\Sigma)\simeq \mathrm{Sp}_{2g}(\mathbb{Z})$ -representation structure on $\rho_{p,1}(J_1(\Sigma))$ .

Now let us treat the case where p divides $g-1$ . We still have that $\rho_{p,1}$ induces a surjective morphism of $\mathrm{Sp}_{2g}(\mathbb{Z})$ -representation ${\mathrm{Ab}}_p(J_1(\Sigma)) \longrightarrow\rho_{p,1}(J_1(\Sigma))$ ; however, since ${\mathrm{Ab}}_p(J_1(\Sigma)) \simeq \Lambda^3H_1(\Sigma,\mathbb{F}_p)/(\omega \wedge H_1(\Sigma,\mathbb{F}_p)),$ the kernel might not be trivial. Since the kernel of this map is a $\mathrm{Sp}_{2g}(\mathbb{Z})$ -subrepresentation by Proposition 2.14, we only have to exclude the possibility that $\mathrm{Ker}\rho_{p,1} = \mathrm{Ker} \overline{\kappa}/(\omega \wedge H_1(\Sigma,\mathbb{F}_p))$ .

For this, let us introduce S, a subsurface of $\Sigma$ of genus $g'=3$ with one boundary component, which is a separating curve in $\Sigma$ . By Lemma 2.18, there is a SIP map, $[t_a,t_b],$ with support on S, and such that $\tau_1([t_a,t_b])$ is a non-zero element of $\mathrm{Ker}\overline{\kappa}$ for any prime $p\geq 5$ . However, since p does not divide $g'-1=2$ , we have that the mapping class induced by $[t_a,t_b]$ on $\hat{S}$ is not in $\mathrm{Ker} \rho_{p,1}$ . By Lemma 2.10, since the boundary of S is separating in $\Sigma,$ the mapping class $[t_a,t_b]$ , as an element of $\mathrm{Mod}(\Sigma),$ is also not in $\mathrm{Ker} \rho_{p,1}$ . Hence $\mathrm{Ker}\rho_{p,1}$ does not contain $\mathrm{Ker} \overline{\kappa},$ and induces an isomorphism ${\mathrm{Ab}}_p(J_1(\Sigma)) \simeq \rho_{p,1}(J_1(\Sigma))$ .

Proof of Theorem 1.2. First, let us note that the inclusion $[J_1(\Sigma),J_1(\Sigma)]T_p\subset \mathrm{Ker} \rho_{p,1}$ is clear, since $T_p\subset \mathrm{Ker} \rho_p$ and $[J_1(\Sigma),J_1(\Sigma)]\subset J_2(\Sigma)\subset \mathrm{Ker}\rho_{p,1}$ as noted before. Let $f \in \mathrm{Ker} \rho_{p,1}$ . Then f acts trivially on $\rho_{p,1}(J_1(\Sigma))$ by conjugation. By Proposition 3.3, we have that $\rho_{p,1}(J_1(\Sigma)) \simeq \Lambda^3 H_1(\Sigma,\mathbb{F}_p)/(\omega \wedge H_1(\Sigma,\mathbb{F}_p)),$ and by Lemma 2.15, the group $\mathrm{Sp}_{2g}(\mathbb{F}_p)$ acts faithfully on $\Lambda^3H_1(\Sigma,\mathbb{F}_p)/(\omega \wedge H_1(\Sigma,\mathbb{F}_p))$ . Therefore the image of f in $\mathrm{Sp}_{2g}(\mathbb{F}_p)$ is trivial, that is, f is in the Torelli mod p subgroup. Since $g\geq 2,$ the Torelli mod p subgroup is the subgroup $J_1(\Sigma)T_p$ (this is a consequence of the classical fact that the p-congruence subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z})$ is generated by pth powers of transvections). Now, since the subgroup $T_p$ generated by pth powers of Dehn twists is contained in $\mathrm{Ker} \rho_p,$ we must have $f=f_1 f_2$ with $f_2\in T_p$ and $f_1 \in J_1(\Sigma) \cap \mathrm{Ker} \rho_{p,1}=\mathrm{Ker}\rho_{p,1}|_{J_1(\Sigma)}$ . However, by Proposition 3.3, this kernel is the same as the kernel of the mod p abelianization of the Torelli group. Since the Torelli group is generated by bounding pairs, whose pth powers are in $T_p,$ this is the same as the subgroup generated by $[J_1(\Sigma),J_1(\Sigma)]$ and p-powers of bounding pairs, and therefore $f\in[J_1(\Sigma),J_1(\Sigma)]T_p$ .

3.3 Analysis of $\mathrm{Ker}(\rho_{p,2})$

In this section, we will analyze the kernel of $\rho_{p,2}$ restricted to $J_2(\Sigma),$ and deduce Theorem 1.3.

Now, as in the previous section, we put a $\mathrm{Mod}(\Sigma)$ -module structure on $\rho_{p,2}(J_2(\Sigma))$ by defining for $f\in \mathrm{Mod}(\Sigma),\ g \in J_2(\Sigma)$ ,

\[f \cdot \rho_{p,2}(g) = \rho_{p,2}(fgf^{-1}).\]

Again, since $\rho_{p,2}|_{J_2(\Sigma)}$ is abelian, this actually induces an $\mathcal{M}=\mathrm{Mod}(\Sigma)/J_2(\Sigma)$ -module structure. Note that $\mathcal{M}\simeq (\Lambda^3H_1(\Sigma,\mathbb{Z})/(\omega \wedge H_1(\Sigma,\mathbb{Z}))) \rtimes \mathrm{Sp}_{2g}(\mathbb{Z}),$ hence we get also a $\mathrm{Sp}_{2g}(\mathbb{Z})$ -module structure on $\rho_{p,2}(J_2(\Sigma))$ .

Proposition 3.5. Let $\Sigma$ be a closed surface of genus $g\geq 6.$ For all large enough primes p, the map

\[\rho_{p,2}:{\mathrm{Ab}}_p(J_2(\Sigma)) \longrightarrow \rho_{p,2}(J_2(\Sigma))\]

is an isomorphism of $\mathrm{Mod}(\Sigma)/J_2(\Sigma)$ modules.

By Theorem 2.12, the rational abelianization ${\mathrm{Ab}}_{\mathbb{Q}}(J_2(\Sigma))$ of $J_2(\Sigma)$ is a sum of three absolutely irreducible $\mathrm{Sp}_{2g}(\mathbb{Z})$ -representations. Moreover, $\mathrm{Ab}_{\mathbb{Q}}(J_2(\Sigma))\simeq \mathrm{Ab}^f(J_2(\Sigma))\underset{\mathbb{Z}}{\otimes} \mathbb{Q}$ . By Lemma 2.16, we get that $\mathrm{Ab}_p(J_2(\Sigma))$ is also a sum of three irreducible representations of $\mathrm{Sp}_{2g}(\mathbb{Z})$ over $\mathbb{F}_p,$ whenever p is large enough. Then $\rho_{p,2}(J_2(\Sigma))$ is a quotient $\mathrm{Sp}_{2g}(\mathbb{Z})$ -representation of $\mathrm{Ab}_p(J_2(\Sigma)).$ Note that $\rho_{p,2}(J_2(\Sigma))$ is also a quotient $\mathcal{M}$ -representation of $\mathrm{Ab}_p(J_2(\Sigma)),$ where $\mathcal{M}=\mathrm{Mod}(\Sigma)/J_2(\Sigma)$ .

We have to show that the three irreducible $\mathrm{Sp}_{2g}(\mathbb{Z})$ -summands in $\mathrm{Ab}_p(J_2(\Sigma))$ survive in $\rho_{p,2}(J_2(\Sigma))$ .

First, by Theorem 2.20, we can find $\varphi \in J_4(\Sigma)$ with $\lambda(\varphi) \neq0$ . In particular, by Remark 2.13, $\varphi$ projects to a non-zero element in the $\mathbb{Q}$ -summand of $\mathrm{Ab}_{\mathbb{Q}}(J_2(\Sigma))$ . Hence, for p big enough, $\lambda(\varphi) \neq 0$ in $\mathbb{F}_p$ and $\varphi$ projects to a non-zero element of the $\mathbb{F}_p$ -summand of $\mathrm{Ab}_p(J_2(\Sigma))$ . Now let $H_1$ and $H_2$ be two handlebodies with boundary $\Sigma$ such that $H_1 \underset{\mathrm{Id}}{\cup} \bar{H_2} = S^3$ . By Theorem 2.19,

\[Z_p(H_1 \underset{\varphi}{\cup} \bar{H_2}) =Z_p(S^3) (1+6h \lambda(\varphi))\mod h^2,\]

which implies that

(1) \begin{equation}Z_p(H_1 \underset{\varphi}{\cup} \bar{H_2})\neq Z_p(S^3) \mod h^2.\end{equation}

On the other hand, by the topological quantum field theory axioms, $Z_p(H_1 \underset{\varphi}{\cup} \bar{H_2}) = \langle Z_p(H_1),\rho_p(\varphi) Z_p(H_2) \rangle$ and $Z_p(S^3) = \langle Z_p(H_1), Z_p(H_2) \rangle$ . As the vectors $Z_p(H_1)$ and $Z_p(H_2)$ belong to the lattice $\mathcal{S}_p(\Sigma)$ , Equation (1) implies that $\rho_{p,2}(\varphi)$ is not trivial. Hence the $\mathbb{F}_p$ -summand of $\mathrm{Ab}_p(J_2(\Sigma))$ survives in $\rho_{p,2}(J_2(\Sigma))$ .

Next, by Lemma 2.9, $\rho_{p,2}(J_2(\Sigma))$ is not invariant under the $J_1(\Sigma)$ -action. However, the first and second summands of $\mathrm{Ab}_p(J_2(\Sigma))$ are both invariant under the $J_1(\Sigma)$ -action, hence the last summand $[31^2]$ has to survive in $\rho_{p,2}(J_2(\Sigma))$ .

Finally, assume that the factor $[2^2]$ is killed in $\rho_{p,2}(J_2(\Sigma))$ . Then we claim that $\rho_{p,2}([J_1(\Sigma),J_2(\Sigma)])$ would have codimension 1 in $\rho_{p,2}(J_2(\Sigma))$ . Indeed, $\mathrm{Ab}_p([J_1(\Sigma),J_2(\Sigma)])$ is a sub- $Sp_{2g}(\mathbb{Z})$ -representation of $Ab_p(J_2(\Sigma))$ since $[J_1(\Sigma),J_2(\Sigma)]$ is stable under $\mathrm{Mod}(\Sigma)$ -conjugation, and we have shown that it is non-trivial (since it is not killed by $\rho_{p,2}$ ). However, by Remark 2.13, $Ab_p([J_1(\Sigma),J_2(\Sigma)])$ lies entirely in $[31^2]$ .

Now we appeal to Lemma 2.7. The images of the separating Dehn twists of the lollipop tree decomposition of $\Sigma$ span an $\mathbb{F}_p$ subspace of $\rho_{p,2}(J_2(\Sigma))$ of dimension $2g-3,$ which has trivial intersection with $\rho_{p,2}([J_1(\Sigma),J_2(\Sigma)]),$ since by Corollary 2.4(iii), $\rho_{p,2}([J_1(\Sigma),J_2(\Sigma)])$ consists of matrices whose diagonal is $1\ (\mathrm{mod}\, h^2)$ . Therefore, $\rho_{p,2}([J_1(\Sigma),J_2(\Sigma)])$ has codimension greater than 1 in $\rho_{p,2}(J_2(\Sigma)),$ and hence the factor $[2^2]$ must survive.

Proof of Theorem 1.4. The inclusion $[J_2(\Sigma),J_2(\Sigma)]T_p \subset \ker \rho_{p,2}$ is clear, since $T_p\subset\ker \rho_p,$ and $\rho_{p,2}$ restricted to $J_2(\Sigma)$ has abelian image by Lemma 3.4.

By Theorem 1.1, we have that $\mathrm{Ker} \rho_{p,1} \subset [J_1(\Sigma),J_1(\Sigma)]T_p\subset J_2(\Sigma)T_p.$ Since $T_p\subset \mathrm{Ker} \rho_p,$ we only need to describe which elements of $J_2(\Sigma)$ are in $\mathrm{Ker} \rho_{p,2}.$ However, by Proposition 3.5, for p large enough, we have that $f\in J_2(\Sigma)$ is in $\mathrm{Ker} \rho_{p,2}$ if and only if f is in the kernel of the mod p abelianization morphism. Since $J_2(\Sigma)$ is generated by separating twists, the kernel of the mod p abelianization is generated by commutators in $J_2(\Sigma)$ and pth powers of separating Dehn twists, both of which are in $[J_2(\Sigma),J_2(\Sigma)]T_p$ .