2 Geometry of the fibred cusp ends

Recall from the introduction that the fibred cusp ends are identified with $\Gamma \setminus \mathbb {P}^1(F)$ and $\mathfrak {P}_{\Gamma }\subset \mathbb {P}^1(F)$ is a fixed subset of representatives that includes $[1:0]$ . Let us first describe the cusp end corresponding to $[1:0]\in \mathfrak {P}_{\Gamma }$ . Thus, let $B\subset \operatorname {SL}(2)$ be the standard Borel subgroup. Set

$$ \begin{align*}B_{\infty}= B(F\otimes_{\mathbb{R}}\mathbb{C})= B(\mathbb{R})^{r_1}\times B(\mathbb{C})^{r_2} \end{align*} $$

with

$$ \begin{align*}B(\mathbb{R})= \left\{ \left(\begin{array}{cc} \lambda & x \\ 0 & \lambda^{-1} \end{array} \right) \; | \; \lambda \in \mathbb{R}^*, x\in\mathbb{R}\right\} \end{align*} $$

and

$$ \begin{align*}B(\mathbb{C})= \left\{ \left(\begin{array}{cc} \mu & z \\ 0 & \mu^{-1} \end{array} \right) \; | \; \mu \in \mathbb{C}^*, z\in\mathbb{C}\right\}. \end{align*} $$

Let $\nu : B_{\infty }\to (\mathbb {R}^+)^*$ be defined by

(2.1) $$ \begin{align} \nu(b)= \left(\prod_{i=1}^{r_1}|\lambda_i|\right)\left( \prod_{j=1}^{r_2}|\mu_j|^2\right) \end{align} $$

for

$$ \begin{align*}b= \left( \left(\begin{array}{cc} \lambda_1 & x_1 \\ 0 & \lambda_1^{-1} \end{array} \right),\ldots, \left(\begin{array}{cc} \lambda_{r_1} & x_{r_1} \\ 0 & \lambda_{r_1}^{-1} \end{array} \right), \left(\begin{array}{cc} \mu_1 & z_1 \\ 0 & \mu_1^{-1} \end{array} \right),\ldots, \left(\begin{array}{cc} \mu_{r_2} & z_{r_2} \\ 0 & \mu_{r_2}^{-1} \end{array} \right)\right)\in B_{\infty}. \end{align*} $$

If

$$ \begin{align*}B_{\infty}(1)= \left\{ b\in B_{\infty} \; | \; \nu(b)=1 \right\}, \end{align*} $$

then

$$ \begin{align*}B_{\infty}(1)\cap K= B_{\infty}\cap K, \quad B_{\infty}(1)\cap \Gamma= B_{\infty}\cap \Gamma, \end{align*} $$

and

$$ \begin{align*}Y:= B_{\infty}\cap \Gamma \setminus B_{\infty}(1)/ B_{\infty}\cap K \end{align*} $$

is the cross-section of the cusp end associated to $[1:0]\in \mathfrak {P}_{\Gamma }$ , where $K=\operatorname {SO}(2)^{r_1}\times \operatorname {SU}(2)^{r_2}$ is the standard maximal compact subgroup of $G_{\infty }$ . In fact, by definition,

$$ \begin{align*}B_{\infty}\cap \Gamma\subset B_{\infty}\cap \operatorname{SL}(2,\mathcal{O}_{F})= \left\{ \iota\left( \left(\begin{array}{cc} \lambda & x_1 \\ 0 & \lambda^{-1} \end{array} \right)\right) \; | \; \lambda, \lambda^{-1}, x_1\in \mathcal{O}_{F} \right\}, \end{align*} $$

so $\lambda \in \mathcal {O}^*_{F}$ . If $N_{F/\mathbb {Q}}: F\to \mathbb {Q}$ is the norm defined by

$$ \begin{align*}N_{F/\mathbb{Q}}(k)= \left(\prod_{i=1}^{r_1} \sigma_i(k)\right) \left(\prod_{j=1}^{r_2} |\tau_j(k)|^2\right), \quad k\in F, \end{align*} $$

then since $\lambda $ is a unit, one has that

$$ \begin{align*}N_{F/\mathbb{Q}}(\lambda)=\pm 1, \end{align*} $$

so that

$$ \begin{align*}\nu\circ\iota \left( \left(\begin{array}{cc} \lambda & x_1 \\ 0 & \lambda^{-1} \end{array} \right)\right)= |N_{F/\mathbb{Q}}(\lambda)|=1, \end{align*} $$

confirming that $B_\infty \cap \Gamma = B_{\infty }(1)\cap \Gamma $ .

The other cusp ends admit a similar description. First, if $\Gamma =\operatorname {SL}(2,\mathcal {O}_{F})$ , the cusp ends are identified with the ideal classes $c_1,\ldots , c_h$ of F. If $\mathfrak {a}$ is a representative in the ideal class $c_{\nu }$ , pick $a,b\in \mathcal {O}_{F}$ such that $\mathfrak {a}$ is the ideal in $\mathcal {O}_{F}$ generated by a and b. Without loss of generality, we can assume that $a\ne 0$ . Then the cusp end associated to $c_{\nu }$ corresponds to the point

$$ \begin{align*}\eta:= [a:b] \in \mathbb{P}^1(F). \end{align*} $$

Since the element

$$ \begin{align*}\left( {\begin{matrix} a & 0 \\ b & a^{-1} \end{matrix} } \right) \in \operatorname{SL}(2,F) \end{align*} $$

sends $[1:0]$ onto $[a:b]$ , the parabolic subgroup $P_{\eta }$ associated to $\eta $ is given by

$$ \begin{align*}P_{\eta}= \left( {\begin{matrix} a & 0 \\ b & a^{-1} \end{matrix} } \right)^{-1} P_0 \left( {\begin{matrix} a & 0 \\ b & a^{-1} \end{matrix} } \right), \end{align*} $$

where

$$ \begin{align*}P_0= B(F)= \left\{ \left( {\begin{matrix} \lambda & z \\ 0 & \lambda^{-1} \end{matrix} } \right) \; | \; \lambda\in F^*, \; z\in F \right\} \end{align*} $$

is the parabolic subgroup of $[1:0]\in \mathbb {P}^1(F)$ . In particular, a computation shows that

$$ \begin{align*}\left( {\begin{matrix} a & 0 \\ b & a^{-1} \end{matrix} } \right) \Gamma\cap P_{\eta} \left( {\begin{matrix} a^{-1} & 0 \\ -b & a \end{matrix} } \right) = \left\{ \left( {\begin{matrix} \lambda& w\\ 0 & \lambda^{-1} \end{matrix} } \right) \; | \; \lambda\in \mathcal{O}_{F}^*, \; w\in \mathfrak{a}^{-2} \right\}, \end{align*} $$

where

$$ \begin{align*}\mathfrak{a}^{-2}= \left\{ w\in F \; | \; uw\in \mathcal{O}_{F} \; \forall u \in \mathfrak{a}^2 \right\}. \end{align*} $$

Thus, the cross-section of the cusp end associated to $\eta \in \mathbb {P}^1(F)$ is

(2.2) $$ \begin{align} Y_{\eta}:= \Gamma_{\mathfrak{a}}\setminus B_{\infty}(1)/B_{\infty}\cap K \end{align} $$

with

$$ \begin{align*}\Gamma_{\mathfrak{a}}:= \left\{ \iota\left( {\left( { \left( {\begin{matrix} \lambda & w \\ 0 & \lambda^{-1} \end{matrix} } \right)} \right)} \right) \; | \; \lambda\in \mathcal{O}_{F}^*, \; w\in \mathfrak{a}^{-2} \right\} \subset B_{\infty}(1). \end{align*} $$

More generally, if $\Gamma $ is a subgroup of $\operatorname {SL}(2,\mathcal {O}_{F})$ , the cusp ends are of the form

(2.3) $$ \begin{align} Y_{\eta}= \Gamma_{\eta} \setminus B_{\infty}(1) / B_{\infty}\cap K \end{align} $$

with $\Gamma _{\eta }$ a finite index subgroup of $\Gamma _{\mathfrak {a}}$ for some ideal $\mathfrak {a}$ representing an ideal class of F.

Denote the unipotent radical of B by N. Then $T=B/N$ is a split torus of dimension $r_1+r_2-1$ . Set

$$ \begin{align*}T_{\infty} := T(F\otimes_{\mathbb{Q}}\mathbb{R})= T(\mathbb{R})^{r_1}\times T(\mathbb{C})^{r_2} \end{align*} $$

with

$$ \begin{align*}T(\mathbb{R}):= \left\{ \left(\begin{array}{cc} \lambda & 0 \\ 0 & \lambda^{-1} \end{array} \right) \; | \; \lambda\in\mathbb{R}^* \right\} \quad\mbox{and} \quad T(\mathbb{C}):=\left\{ \left(\begin{array}{cc} \mu & 0 \\ 0 & \mu^{-1} \end{array} \right) \; | \; \mu\in\mathbb{C}^* \right\}. \end{align*} $$

Let $K_T$ (respectively $\Gamma _{\eta ,T}$ ) be the image of $K\cap B_{\infty }$ (respectively $\Gamma _{\eta }$ ) under the projection $p_{\infty }: B_{\infty }\to T_{\infty }$ . Then

(2.4) $$ \begin{align} K_T= (T(\mathbb{R})\cap \operatorname{SO}(2))^{r_1}\times (T(\mathbb{C})\cap\operatorname{SU}(2))^{r_2} \end{align} $$

with

$$ \begin{align*}T(\mathbb{R})\cap\operatorname{SO}(2)= \{\pm \operatorname{Id}\} \quad \mbox{and} \quad T(\mathbb{C})\cap\operatorname{SU}(2)= \left\{ \left( \begin{matrix} e^{i\theta} & 0 \\ 0 & e^{-i\theta} \end{matrix} \right) \; |\; \theta\in\mathbb{R} \right\}, \end{align*} $$

while

$$ \begin{align*}\Gamma_{\eta,T}\subset \left\{ \iota\left(\left(\begin{array}{cc} u & 0 \\ 0 & u^{-1} \end{array} \right)\right) \; | \; u\in \mathcal{O}_{F}^* \right\}. \end{align*} $$

In particular, (2.4) shows that

$$ \begin{align*}T_{\infty}/K_T\cong (((\mathbb{R}^+)^*)^{r_1+r_2}. \end{align*} $$

If we set

$$ \begin{align*}T_{\infty}(1):= \{ t\in T_{\infty} \; | \; \nu(t)=1\}, \end{align*} $$

this means $T_{\infty }(1)/K_T$ is identified with the kernel of the group homomorphism induced by $\nu $ ,

$$ \begin{align*}\begin{array}{llcl} \nu: & (((\mathbb{R}^+)^*)^{r_1+r_2} & \to & (\mathbb{R}^+)^* \\ & (\lambda_1,\ldots,\lambda_{r_1},\mu_1,\ldots,\mu_{r_2}) & \mapsto & \left( \prod_{i=1}^{r_1} \lambda_i\right) \left( \prod_{j=1}^{r_2} \mu_j^2\right), \end{array} \end{align*} $$

namely

$$ \begin{align*}T_{\infty}(1)/K_T\cong (((\mathbb{R}^+)^*)^{r_1+r_2-1}. \end{align*} $$

By the Dirichlet’s Unit Theorem (see for instance [Reference MarcusMar18, Theorem 38]),

$$ \begin{align*}\Gamma_{\eta,T}\setminus T_{\infty}(1) /K_T \cong \mathbb{T}^{r_1+r_2-1}. \end{align*} $$

is a real torus of dimension $r_1+r_2-1$ . The projection map $p_{\infty }: B_{\infty }\to T_{\infty }$ induces a projection

(2.5) $$ \begin{align} \phi_{\eta}: Y_{\eta}\to \Gamma_{\eta,T}\setminus T_{\infty}(1)/ K_T \cong \mathbb{T}^{r_1+r_2-1}. \end{align} $$

This map is a locally trivial fibration with fibre

$$ \begin{align*}\Gamma_{\eta}\cap N_{\infty}\setminus N_{\infty}\cong \mathbb{T}^{r_1+2r_2}=\mathbb{T}^{d_{F}}, \end{align*} $$

where

$$ \begin{align*}N_{\infty}= N(F\otimes_{\mathbb{Q}}\mathbb{R})= N(\mathbb{R})^{r_1}\times N(\mathbb{C})^{r_2} \end{align*} $$

with

$$ \begin{align*}N(\mathbb{R})= \left\{ \left(\begin{array}{cc} 1 & x \\ 0 & 1 \end{array} \right)\; | \; x\in\mathbb{R} \right\} \quad \mbox{and} \quad N(\mathbb{C})=\left\{ \left(\begin{array}{cc} 1 & z \\ 0 & 1 \end{array} \right)\; | \; z\in\mathbb{C} \right\}. \end{align*} $$

To describe the metric g in (1.7) in the cusp end corresponding to $\eta \in \mathfrak {P}_{\Gamma }$ , let us make the following change of variables with respect to the coordinates used in (1.7):

(2.6) $$ \begin{align} \log r= \frac{1}{d_{F}}\left(\sum_{i=1}^{r_1}\log y_i+ 2\sum_{j=1}^{r_2}\log t_j\right) , \quad u_i= \log y_i-\log y_1, \quad v_j= \log t_{j}-\log y_1, \end{align} $$

if the number field F admits at least one real embedding. If the number field F admits no real embedding, that is, if $r_1=0$ , then we define $\log r$ as before and set instead

$$ \begin{align*}v_j= \log t_j-\log t_1 \quad \forall j\in \{2,\ldots, r_2\}. \end{align*} $$

Using the conventions that $u_1=0$ if $r_1>0$ and $v_1=0$ if $r_1=0$ , we set

$$ \begin{align*}\mu:= \frac{1}{d_{F}}\left( \sum_{i=1}^{r_1} u_i + 2 \sum_{j=1}^{r_2} v_j \right), \end{align*} $$

so that

$$ \begin{align*}\log y_i=\log r +u_i-\mu \quad \mbox{and} \quad \log t_j= \log r+v_j -\mu. \end{align*} $$

Setting $\widetilde {u}_i=u_i-\mu $ and $\widetilde {v}_j=v_j-\mu $ , the metric $\widetilde {g}$ becomes

$$ \begin{align*}\widetilde{g}= d_{F}\frac{dr^2}{r^2}+ d_{F}g_{S_{\eta}}+ \frac{1}{r^2}\left( \sum_{i=1}^{r_1}e^{-2\widetilde{u}_i}dx_i^2+ 2 \sum_{j=1}^{r_2} e^{-2\widetilde{v}_j}|dz_j|^2 \right) \end{align*} $$

when $r_1>0$ , where $g_{S_{\eta }}$ is a flat metric on the base $S_{\eta }\cong \mathbb {T}^{r_1+r_2-1}$ of the fibred bundle (2.5) and the cusp is when $r\to \infty $ . Thus, the metric $g_{S_{\eta }}$ can be seen as a Euclidean metric in $u_2,\ldots , u_{r_1},v_1,\ldots , v_{r_2}$ (just in $v_2,\ldots , v_{r_2}$ if $r_1=0$ ), though not necessarily the canonical one.

To ease the comparison with [Reference Müller and RochonMR20], we will divide this metric by $d_{F}$ and let

(2.7) $$ \begin{align} g_{\operatorname{fc}}= \frac{dr^2}{r^2}+ g_{S_{\eta}}+ \frac{1}{d_{F}r^2}\left( \sum_{i=1}^{r_1}e^{-2\widetilde{u}_i}dx_i^2+ 2 \sum_{j=1}^{r_2} e^{-2\widetilde{v}_j}|dz_j|^2 \right) \end{align} $$

be the fibred cusp metric we will consider on X. For this metric, a local basis of orthonormal forms is given by

$$ \begin{align*}\frac{dr}{r}, \nu_1,\ldots, \nu_{r_1+r_2-1}, \frac{e^{-\widetilde{u}_1} dx_1}{\sqrt{d_{F}} r},\ldots, \frac{e^{-\widetilde{u}_{r_1}} dx_{r_1}}{\sqrt{d_{F}} r}, \frac{e^{-\widetilde{v}_1}dz_1}{\sqrt{d_{F}}r}, \frac{e^{-\widetilde{v}_1}d\overline{z}_1}{\sqrt{d_{F}}r},\ldots, \frac{e^{-\widetilde{v}_{r_2}}dz_{r_2}}{\sqrt{d_{F}}r}, \frac{e^{-\widetilde{v}_{r_2}}d\overline{z}_{r_2}}{\sqrt{d_{F}}r}, \end{align*} $$

where $\nu _1,\ldots , \nu _{r_1+r_2-1}$ is a basis of orthonormal parallel forms for $g_{S_{\eta }}$ .

On the other hand, in terms of these coordinates and the bundle metric of [Reference Matsushima and MurakamiMM63], a local basis of orthonormal sections of the flat vector bundle $E_{m,n}$ when $|m|:=m_1+\cdots +m_{r_1}=1$ with $m_i=1$ for some fixed i and $n=0$ is given by

$$ \begin{align*}\begin{array}{l}e_{i,1}= \left( \begin{array}{cc} \lambda & t \\ 0 & \lambda^{-1}\end{array} \right)\left(\begin{array}{c} 1 \\ 0 \end{array} \right)= r^{\frac12}e^{\frac{\widetilde{u}_i}2 } \left( \begin{array}{c} 1 \\ 0 \end{array} \right), \\ e_{i,2}= \left( \begin{array}{cc} \lambda & t \\ 0 & \lambda^{-1}\end{array} \right)\left(\begin{array}{c} 0 \\ 1 \end{array} \right)= r^{-\frac12}e^{-\frac{\widetilde{u}_i}2} \left( \begin{array}{c} x_i \\ 1 \end{array} \right), \end{array} \quad \lambda= \sqrt{y_i}=r^{\frac12}e^{\frac{\widetilde{u}_i}2}, t= \frac{x_i}{\sqrt{y_i}}. \end{align*} $$

If instead $m=0$ and $|n|:= n_1+\overline {{n}}_1+\cdots + n_{r_2}+\overline {{n}}_{r_2}=1$ with $n_j=1$ for some fixed j, then a local basis of sections of $E_{m,n}$ is given by

$$ \begin{align*}\begin{array}{l}f_{j,1}= \left( \begin{array}{cc} \lambda & \zeta \\ 0 & \lambda^{-1}\end{array} \right)\left(\begin{array}{c} 1 \\ 0 \end{array} \right)= r^{\frac12}e^{\frac{\widetilde{v}_j}2} \left( \begin{array}{c} 1 \\ 0 \end{array} \right), \\ f_{j,2}= \left( \begin{array}{cc} \lambda & \zeta \\ 0 & \lambda^{-1}\end{array} \right)\left(\begin{array}{c} 0 \\ 1 \end{array} \right)= r^{-\frac12}e^{-\frac{\widetilde{v}_j}2} \left( \begin{array}{c} z_j \\ 1 \end{array} \right), \end{array} \quad \lambda= \sqrt{t_j}=r^{\frac12}e^{\frac{\widetilde{v}_j}{2}}, \zeta= \frac{z_j}{\sqrt{t_j}}. \end{align*} $$

Finally, if $m=0$ and $|n|:= n_1+\overline {{n}}_1+\cdots + n_{r_2}+\overline {{n}}_{r_2}=1$ with $\overline {{n}}_j=1$ for some fixed j, then a local basis of sections of $E_{m,n}$ is given by

$$ \begin{align*}\begin{array}{l}\overline{f}_{j,1}= \left( \begin{array}{cc} \lambda & \overline{\zeta} \\ 0 & \lambda^{-1}\end{array} \right)\left(\begin{array}{c} 1 \\ 0 \end{array} \right)= r^{\frac12}e^{\frac{\widetilde{v}_j}2} \left( \begin{array}{c} 1 \\ 0 \end{array} \right), \\ \overline{f}_{j,2}= \left( \begin{array}{cc} \lambda & \overline{\zeta} \\ 0 & \lambda^{-1}\end{array} \right)\left(\begin{array}{c} 0 \\ 1 \end{array} \right)= r^{-\frac12}e^{-\frac{\widetilde{v}_j}2} \left( \begin{array}{c} \overline{z}_j \\ 1 \end{array} \right), \end{array} \quad \lambda= \sqrt{t_j}=r^{\frac12}e^{\frac{\widetilde{v}_j}{2}}, \zeta= \frac{z_j}{\sqrt{t_j}}. \end{align*} $$

Notice in particular that $\overline {f}_{j,1}$ and $\overline {f}_{j,2}$ are precisely the complex conjugates of $f_{j,1}$ and $f_{j,2}$ respectively. Hence, more generally,

$$ \begin{align*}w_{k,l}:= \left(\bigotimes_{i=1}^{r_1}(e_{i,1}^{k_i}\otimes e_{i,2}^{m_i-k_i})\right) \otimes \left( \bigotimes_{j=1}^{r_2}(f_{j,1}^{l_j}\otimes \overline{f}_{j,1}^{\overline{l_j}}\otimes f_{j,2}^{n_j-l_j}\otimes \overline{f}_{j,2}^{\overline{{n}}_j-\overline{l}_j})\right), \end{align*} $$

for $k\le m\in \mathbb {N}^{r_1}_0$ and $l\le n\in \mathbb {N}^{2r_2}_0$ , form a local basis of orthonormal sections of $E_{m,n}$ . One computes that

$$ \begin{align*}de_{i,1}= \frac12\left( \frac{dr}{r}+ d\widetilde{u}_i \right)e_{i,1}, \quad de_{i,2}= \left( e^{-\widetilde{u}_i}\frac{dx_i}{r} \right)e_{i,1}- \frac{1}2 \left( \frac{dr}{r}+d\widetilde{u}_i \right)e_{i,2} \quad \end{align*} $$

and

$$ \begin{align*}df_{j,1}= \frac{1}2\left( \frac{dr}{r}+d\widetilde{v}_j \right)f_{j,1}, \quad df_{j,2}= \left(e^{-\widetilde{v}_j} \frac{dz_j}{r}\right)f_{j,1} -\frac{1}2 \left( \frac{dr}r+ d\widetilde{v}_j \right)f_{j,2}, \end{align*} $$

so that

(2.8) $$ \begin{align} & dw_{k,l}= \left[ \frac{(2|k|+2|l|-|m|-|n|)}2\frac{dr}r + \sum_{i=1}^{r_1}\frac{2k_i-m_i}2 d\widetilde{u}_i + \sum_{j=1}^{r_2} \frac{2(l_j+\overline{l}_j)-n_j-\overline{{n}}_j}2 d\widetilde{v}_j\right]w_{k,l} \\ & + \sum_{i=1}^{r_1}(m_i-k_i)\left(e^{-\widetilde{u}_i}\frac{dx_i}{r}\right)w_{k+1_i,l} \nonumber \\ & + \sum_{j=1}^{r_2}\left( { (n_j-l_j)\left( e^{-\widetilde{v}_j}\frac{dz_j}{r} \right)w_{k,l+1_j} + (\overline{{n}}_j-\overline{l}_j)\left( e^{-\widetilde{v}_j}\frac{d\overline{z}_j}{r} \right)w_{k,l+\overline{1}_j} } \right), \nonumber \end{align} $$

where $1_i$ is the $r_1$ -tuple with ith entry equal to one and the other entries equal to zero, $1_j$ is the $2r_2$ -tuple with $(2j-1)$ th entry equal to 1 and other entries equal to zero and $\overline {1}_j$ is the $2r_2$ -tuple with $(2j)$ th entry equal to 1 and other entries equal to zero.

Since the representation $\rho _{m,n}$ is self-dual, notice that the dual flat vector bundle $E_{m,n}^*$ is naturally isomorphic to $E_{m,n}$ as a flat vector bundle, as well as a hermitian vector bundle. This can be seen directly in terms of the local sections $w_{i,j}$ . The orthonormal basis of sections dual to $\{e_{i,1},e_{i,2}\}$ is given by

$$ \begin{align*}e^{1}_i= r^{-\frac12}e^{-\frac{\widetilde{u}_i}2} \left( \begin{array}{cc} 1 & -x_i \end{array} \right) \quad \mbox{and} \quad e^2_i= r^{\frac12}e^{\frac{\widetilde{u}_i}2}\left(\begin{array}{cc} 0 & 1 \end{array} \right), \end{align*} $$

while the local orthonormal basis of sections dual to $\{f_{j,1},f_{j,2}\}$ is given by

$$ \begin{align*}f^{1}_j= r^{-\frac12}e^{-\frac{\widetilde{v}_j}2} \left( \begin{array}{cc} 1 & -z_j \end{array} \right) \quad \mbox{and} \quad f^2_j= r^{\frac12}e^{\frac{\widetilde{v}_j}{2}}\left(\begin{array}{cc} 0 & 1 \end{array} \right) \end{align*} $$

with their complex conjugates $ \overline {f}^{1}_j$ and $\overline {f}^{2}_j$ giving the local orthonormal basis of sections dual to $\{\overline {f}_{j,1},\overline {f}_{j,2}\}$ . For these sections, one computes that

$$ \begin{align*}\begin{aligned} d e^1_i&= -\frac12\left( \frac{dr}r+d\widetilde{u}_i \right)e^1_i- \left( \frac{e^{-\widetilde{u}_i}dx_i}{r}\right)e^2_i, \\ de^2_i &= \frac12\left( {\frac{dr}r+ d\widetilde{u}_i} \right) e^2_i, \\ df^1_j &= -\frac12 \left( {\frac{dr}r+ d\widetilde{v}_j } \right)f^1_j -\left( {\frac{e^{-\widetilde{v}_j}dz_j}{r}} \right)f^2_j, \\ df^2_j &= \frac12 \left( {\frac{dr}r+d\widetilde{v}_j} \right)f^2_j, \end{aligned} \end{align*} $$

so that the natural isomorphisms of flat vector bundles $E_{1_i,0}\to E^*_{1_i,0}$ , $E_{0,1_j}\to (E_{0,1_j})^*$ and $E_{0,\overline {1}_j}\to (E_{0,\overline {1}_j})^*$ are induced by

$$ \begin{align*}e_{i,1}\mapsto e^2_i, \; e_{i,2}\mapsto -e^1_i, \quad f_{j,1}\mapsto f^2_j, \; f_{j,2}\mapsto -f^1_j \quad \mbox{and} \quad \quad \overline{f}_{j,1}\mapsto \overline{f}^2_j, \; \overline{f}_{j,2}\mapsto -\overline{f}^1_j. \end{align*} $$

3 Cusp degeneration and the Hodge-deRham operator

As in [Reference Müller and RochonMR20], X can be compactified to a manifold with boundary $\overline {X}$ by adding a copy of $Y_{\eta }$ at infinity for each cusp end $\eta \in \mathfrak {P}_{\Gamma }$ ,

$$ \begin{align*}\overline{X}= X \cup \left( { \sqcup_{\eta\in\mathfrak{P}_{\Gamma}} Y_{\eta} } \right), \quad \partial \overline{X}= \sqcup_{\eta\in\mathfrak{P}_{\Gamma}} Y_{\eta}. \end{align*} $$

On $\overline {X}$ , we can choose a boundary defining function x such that for each $\eta \in \mathfrak {P}_{\Gamma }$ , $x=\frac {1}{r}$ in the cusp end (2.7) corresponding to $\eta $ . If

$$ \begin{align*}M = \overline{X}\cup_{\partial \overline{X}}\overline{X} \end{align*} $$

is the double of $\overline {X}$ along $\partial X$ , then on M, one can consider a family of metrics $g_{{\varepsilon }}$ parametrized by ${\varepsilon }>0$ which in a tubular neighborhood $Y_{\eta }\times (-\delta ,\delta )_x$ of $Y_{\eta }$ in M takes the form

(3.1) $$ \begin{align} g_{\operatorname{fc},{\varepsilon}}=\frac{dx^2}{\rho^2}+ g_{S_{\eta}}+ \frac{\rho^2}{d_{F}}\left( \sum_{i=1}^{r_1} e^{-2\widetilde{u}_i}dx_i^2+ 2\sum_{j=1}^{r_2} e^{-2\widetilde{v}_j}|dz_j|^2 \right), \quad \rho:= \sqrt{x^2+{\varepsilon}^2} \end{align} $$

and which on $M\setminus \partial \overline {X}= X\sqcup X$ converges to $g_{\operatorname {fc}}$ on each copy of X as ${\varepsilon }\searrow 0$ .

Figure 1 The single surgery space $X_s$ .

For such a family of metrics, we can consider as in [Reference Albin, Rochon and SherARS21] the single surgery space of Mazzeo-Melrose [Reference Mazzeo and MelroseMM95]

(3.2) $$ \begin{align} X_s= [M\times [0,1]_{{\varepsilon}};\partial\overline{X}\times \{0\}] \end{align} $$

obtained by blowing up $\partial \overline {X}\times \{0\}$ inside $M\times [0,1]_{{\varepsilon }}$ in the sense of Melrose. It is a manifold with corners with natural blow-down map $\beta _s:X_s\to M\times [0,1]_{{\varepsilon }}$ . We denote by $\mathfrak {B}_{sm}:=\overline {\beta _s^{-1}(M\setminus \partial \overline {X})\times \{0\}}$ the lift of the old boundary hypersurface at ${\varepsilon }=0$ and by $\mathfrak {B}_{sb}:=\beta _s^{-1}(\partial \overline {X}\times \{0\})$ the new boundary hypersurface created by the blow-up.

There is a corresponding flat vector bundle $\widehat {E}_{m,n}$ corresponding to $E_{m,n}$ on each copy of X. We can equip $\widehat {E}_{m,n}$ with a bundle metric $h_{{\varepsilon }}$ depending on ${\varepsilon }>0$ and such that $h_{{\varepsilon }}\to h$ on each copy of X as ${\varepsilon }\searrow 0$ . To describe this metric near $Y_{\eta }$ , it suffices to give a local basis of orthonormal sections, which we take to be

$$ \begin{align*}\hat{w}_{k,l}:= \left(\bigotimes_{i=1}^{r_1}(\hat{e}_{i,1}^{k_i}\otimes \hat{e}_{i,2}^{m_i-k_i})\right) \otimes \left( \bigotimes_{j=1}^{r_2}(\hat{f}_{j,1}^{l_j}\otimes \hat{\overline{f}}_{j,1}^{\overline{l}_j}\otimes \hat{f}_{j,2}^{n_j-l_j}\otimes \hat{\overline{f}}_{j,2}^{\overline{{n}}_j-\overline{l}_j})\right) \end{align*} $$

for $k\le m\in \mathbb {N}^{r_1}_0$ and $ l\le n\in \mathbb {N}^{2r_2}_0$ with

$$ \begin{align*}\hat{e}_{i,1}:= \rho^{-\frac12}e^{\frac{\widetilde{u}_i}2} \left(\begin{array}{c} 1\\ 0 \end{array}\right), \; \hat{e}_{i,2}:= \rho^{\frac12}e^{-\frac{\widetilde{u}_i}2} \left( {\begin{array}{c} x_i\\1 \end{array}} \right), \; \hat{f}_{j,1}:= \rho^{-\frac12}e^{\frac{\widetilde{v}_j}2} \left( { \begin{array}{c} 1\\0 \end{array}} \right), \; \hat{f}_{j,2}:= \rho^{\frac12}e^{-\frac{\widetilde{v}_j}2} \left( { \begin{array}{c} z_j\\1 \end{array}} \right) \end{align*} $$

and $\hat {\overline {f}}_{j,1}, \hat {\overline {f}}_{j,2}$ the complex conjugates of $\hat {f}_{j,1}$ and $\hat {f}_{j,2}$ . In terms of these sections, notice that the following analog of (2.8) holds,

(3.3) $$ \begin{align} d\hat{w}_{k,l} &= \left[ \frac{(|m|+|n|-2|k|-2|l|)}2\frac{x}{\rho}\frac{dx}{\rho} + \sum_{i=1}^{r_1}\frac{2k_i-m_i}2 d\widetilde{u}_i + \sum_{j=1}^{r_2} \frac{2(l_j+\overline{l}_j)-n_j-\overline{{n}}_j}2 d\widetilde{v}_j\right]\hat{w}_{k,l} \nonumber\\ &\quad + \sum_{i=1}^{r_1}(m_i-k_i)\left(e^{-\widetilde{u}_i}\rho dx_i\right)\hat{w}_{k+1_i,l} \nonumber \\ &\quad +\sum_{j=1}^{r_2} \left( {(n_j-l_j)\left( e^{-\widetilde{v}_j}\rho dz_j \right)\hat{w}_{k,l+1_j} +(\overline{{n}}_j-\overline{l}_j)\left( e^{-\widetilde{v}_j}\rho d\overline{z}_j \right)\hat{w}_{k,l+\overline{1}_j} } \right). \end{align} $$

Let $\eth _{\operatorname {fc},{\varepsilon }}=d+d^*$ be the Hodge-deRham operator associated to $(M,\widehat {E}_{m,n}, g_{\operatorname {fc},{\varepsilon }},h_{{\varepsilon }})$ , where d is the exterior differential and $d^*$ is its formal adjoint with respect to $g_{\operatorname {fc},{\varepsilon }}$ and $h_{{\varepsilon }}$ . In a tubular neighborhood $Y_{\eta }\times (-\delta ,\delta )_x$ of $Y_{\eta }$ in M, we can describe $\eth _{\operatorname {fc},{\varepsilon }}$ in terms of the decomposition of forms

(3.4) $$ \begin{align} \omega= \omega_0 + \frac{dx}{\rho}\wedge \omega_1 \end{align} $$

with $\omega _0$ and $\omega _1$ forms not involving $\frac {dx}{\rho }$ . To do so, we need to introduce a weight operator W defined on forms by the number operator in the fibres of the fibre bundle $Y_{\eta }\to S_{\eta }$ ,

(3.5) $$ \begin{align} \begin{aligned} & W\left(\frac{dx}{\rho}\right)= W(\nu_q)=0, \quad q\in \{1,\ldots, r_1+r_2-1\}, \\ & W(\omega)=\omega \quad \mbox{for} \quad \omega\in\left\{ e^{-\widetilde{u}_i}\rho dx_i, e^{-\widetilde{v}_j}\rho dz_j, e^{-\widetilde{v}_j}\rho d\overline{z}_j \right\}, \quad i\le r_1, \; j\le r_2, \end{aligned} \end{align} $$

and on sections of $\widehat {E}_{m,n}$ by

(3.6) $$ \begin{align} W(\hat{w}_{k,l})= \left( {\frac{|m|+|n|}2-|k|-|l|} \right)\hat{w}_{k,l}. \end{align} $$

In terms of this weight and the decomposition (3.4), we have that

(3.7) $$ \begin{align} \eth_{\operatorname{fc},{\varepsilon}}= \left( {\begin{array}{cc} \rho^{-1}\eth_{Y_{\eta}/S_{\eta}}+ \eth_{S_{\eta}} & \frac{x}{\rho}(W-d_{F})-\rho\frac{\partial}{\partial x} \\ \rho\frac{\partial}{\partial x}+ \frac{x}{\rho}W & -\rho^{-1}\eth_{Y_{\eta}/S_{\eta}}- \eth_{S_{\eta}} \end{array}} \right) \end{align} $$

For some vertical and horizontal operators $\eth _{Y_{\eta }/S_{\eta }}$ and $\eth _{S_{\eta }}$ with respect to the fibre bundle $\phi _{\eta }:Y_{\eta }\to S_{\eta }$ of (2.5). Compared to [Reference Albin, Rochon and SherARS21, § 2.2], notice that there is no curvature term, which is due to the fact that the Riemannian submersion $\phi _{\eta }:Y_{\eta } \to S_{\eta }$ has trivial curvature in the sense of [Reference Berline, Getzler and VergneBGV04, § 10.1].

Let $D_{v,\eta }$ be the vertical family of [Reference Albin, Rochon and SherARS21] associated to the family of Hodge-deRham operators $\eth _{\operatorname {fc},{\varepsilon }}$ at $Y_{\eta }$ , that is,

$$ \begin{align*}D_{v,\eta}= \left( {\begin{array}{cc} \eth_{Y_{\eta}/S_{\eta}}& 0 \\ 0 & -\eth_{Y_{\eta}/S_{\eta}}\end{array}} \right). \end{align*} $$

This is a family of operators acting fibrewise on the fibres of the fibre bundle

$$ \begin{align*}\phi_{\eta}: Y_{\eta}\to S_{\eta}. \end{align*} $$

As in [Reference Müller and RochonMR20], in each fibre, it corresponds to the Hodge-deRham operator with respect to the induced fibrewise metric and trivial flat vector bundle of rank $\operatorname {rank} E_{m,n}$ , that is the trivial vector bundle $E_{m,n}$ with flat structure obtained by declaring the sections $\hat {w}_{i,j}$ to be flat. Its kernel is naturally a vector bundle over $S_{\eta }$ with local sections given by

$$ \begin{align*}\Lambda^*\langle \frac{dx}{\rho}, \nu_q, e^{-\widetilde{u}_i}\rho dx_i, e^{-\widetilde{v}_j}\rho dz_j, e^{-\widetilde{v}_j}\rho d\overline{z}_j \; | \; q\le r_1+r_2-1, i\le r_1, j\le r_2 \rangle\otimes \langle \hat{w}_{k,l}\; | \; k\le m, \; l\le n\rangle. \end{align*} $$

The corresponding horizontal operator $D_{b,\eta }$ of [Reference Albin, Rochon and SherARS21] is obtained by letting

(3.8) $$ \begin{align} \rho^{\frac{d_{F}}2}\eth_{\operatorname{fc},{\varepsilon}}\rho^{-\frac{d_F}2}= \left( {\begin{array}{cc} \rho^{-1}\eth_{Y_{\eta}/S_{\eta}}+ \eth_{S_{\eta}} & \frac{x}{\rho}\left(W-\frac{d_{F}}{2}\right)-\rho\frac{\partial}{\partial x} \\ \rho\frac{\partial}{\partial x}+ \frac{x}{\rho}\left(W-\frac{d_{F}}2 \right) & -\rho^{-1}\eth_{Y_{\eta}/S_{\eta}}- \eth_{S_{\eta}} \end{array}} \right) \end{align} $$

act on such sections extended smoothly off the boundary hypersurface $\mathfrak {B}_{sb}$ of $X_s$ (defined in (3.2)) and then restricting back to $\mathfrak {B}_{sb}$ . A computation shows that in terms of the decomposition (3.4),

(3.9) $$ \begin{align} D_{b,\eta}=\left( { \begin{array}{cc} \eth_{S_{\eta}} & -D(\frac{d_{F}}2-W) \\D(W-\frac{d_{F}}2) & -\eth_{S_{\eta}} \end{array} } \right), \end{align} $$

where

$$ \begin{align*}D(a)=\langle X\rangle^{-a} \langle X\rangle \frac{\partial}{\partial X} \langle X\rangle^a= \langle X\rangle \frac{\partial}{\partial X}+ \frac{aX}{\langle X\rangle}, \quad a\in \mathbb{R}, \quad X= \frac{x}{{\varepsilon}}, \; \langle X\rangle= \sqrt{1+X^2}, \end{align*} $$

and $\eth _{S_{\eta }}$ is the Hodge-deRham operator on $S_{\eta }$ (with metric induced by $g_{\operatorname {fc}}$ ) acting on the flat vector bundle generated by the space of sections

(3.10) $$ \begin{align} \mathcal{C}^*= \Lambda^*\langle e^{-\widetilde{u}_i}\rho dx_i, e^{-\widetilde{v}_j}\rho dz_j, e^{-\widetilde{v}_j}\rho d\overline{z}_j \; | \; i\le r_1, j\le r_2 \rangle\otimes \langle \hat{w}_{k,l}\; | \; k\le m, \; l\le n\rangle. \end{align} $$

Let $\eth _{\mathcal {C}}$ be the natural (Kostant type) Hodge-deRham operator associated with the finite dimensional complex (3.10) with differential $d_{\mathcal {C}}$ defined by

(3.11) $$ \begin{align} d_{\mathcal{C}} \hat{w}_{k,l}&= \sum_{i=1}^{r_1}(m_i-k_i)\left(e^{-\widetilde{u}_i}\rho dx_i\right)\hat{w}_{k+1_i,l} \\ &\quad + \sum_{j=1}^{r_2} \left( { (n_j-l_j)\left( e^{-\widetilde{v}_j}\rho dz_j \right)\hat{w}_{k,l+1_j} + (\overline{{n}}_j-\overline{l}_j)\left( e^{-\widetilde{v}_j}\rho d\overline{z}_j \right)\hat{w}_{k,l+\overline{1}_j} } \right)\nonumber \end{align} $$

with natural metric induced by $g_{\operatorname {fc},{\varepsilon }}$ and $h_{{\varepsilon }}$ . By Hodge theory, the kernel of $\eth _{\mathcal {C}}$ corresponds to the cohomology of the complex $\mathcal {C}^*$ . It admits in fact the following explicit description.

Lemma 3.1. The kernel of $\eth _{\mathcal {C}}$ is given by

(3.12) $$ \begin{align} & \mathcal{H}^*(\mathcal{C})=\left(\bigwedge_{i=1}^{r_{1}} \langle \hat{e}^{m_i}_{i,1}, \hat{e}_{i,2}^{m_i}e^{-\widetilde{u}_i}\rho dx_i \rangle\right) \wedge \\ & \left( \bigwedge_{j=1}^{r_2} \langle \hat{f}_{j,1}^{n_j}\otimes\hat{\overline{f}}^{\overline{{n}}_j}_{j,1}, \hat{f}^{n_j}_{j,1}\otimes\hat{\overline{f}}^{\overline{{n}}_j}_{j,2}e^{-\widetilde{v}_j}\rho d\overline{z}_j, \hat{f}_{j,2}^{n_j}\otimes\hat{\overline{f}}^{\overline{{n}}_j}_{j,1}e^{-\widetilde{v}_j}\rho dz_j, \hat{f}^{n_j}_{j,2}\otimes\hat{\overline{f}}^{\overline{{n}}_j}_{j,2}e^{-2\widetilde{v}_j}\rho^2dz_j\wedge d\overline{z}_j \rangle \right).\nonumber \end{align} $$

Proof. Notice first that the complex $\mathcal {C}^*$ admits the decomposition into subcomplexes

(3.13) $$ \begin{align} \mathcal{C}^*= \left( \bigwedge_{i=1}^{r_1} \mathcal{C}_i^* \right)\wedge \left( \bigwedge_{j=1}^{r_2} \mathcal{D}^*_j \right) \wedge \left( \bigwedge_{j=1}^{r_2} \overline{\mathcal{D}}^*_j \right), \end{align} $$

where

$$ \begin{align*}\mathcal{C}_i^*=\langle \hat{e}^{k_i}_{i,1}\otimes \hat{e}^{m_i-k_i}_{i,2}, \hat{e}^{k_i}_{i,1}\otimes \hat{e}^{m_i-k_i}_{i,2} e^{-\widetilde{u}_i}\rho dx_i \; | \; k_i\in \{0,1,\ldots, m_i\} \rangle \end{align*} $$

is the complex with differential given by

$$ \begin{align*}\begin{aligned} & d_{\mathcal{C}_i}(\hat{e}^{k_i}_{i,1}\otimes \hat{e}^{m_i-k_i}_{i,2} )= (m_i-k_i)e^{k_i+1}_{i,1}\otimes e^{m_i-k_i-1}_{i,2} e^{-\widetilde{u}_i}\rho dx_i, \\ & d_{\mathcal{C}_i}(\hat{e}^{k_i}_{i,1}\otimes \hat{e}^{m_i-k_i}_{i,2} e^{-\widetilde{u}_i}\rho dx_i) = 0, \end{aligned} \end{align*} $$

while

$$ \begin{align*}\mathcal{D}_j^*=\langle \hat{f}^{l_j}_{j,1}\otimes \hat{f}^{n_j-l_j}_{j,2} \; | \; l_j\in \{0,1,\ldots, n_j\} \rangle \otimes \langle 1, dz_j \rangle \end{align*} $$

is the complex with differential given by

$$ \begin{align*}d_{\mathcal{D}_j} (\hat{f}^{l_j}_{j,1}\otimes \hat{f}^{n_j-l_j}_{j,2}\wedge \omega)= (n_j-l_j) \hat{f}^{l_j+1}_{j,1}\otimes \hat{f}^{n_j-l_j-1}_{j,2} e^{-\widetilde{v}_j}\rho dz_j\wedge \omega \end{align*} $$

for $\omega \in \langle 1, dz_j \rangle $ and $\overline {\mathcal {D}}^{*}_j$ is its complex conjugate. There are corresponding Hodge-deRham operators $\eth _{\mathcal {C}_i}$ and $\eth _{\mathcal {D}_j\wedge \overline {\mathcal {D}}_j}$ . A direct computation shows that their kernels are respectively given by

$$ \begin{align*}\mathcal{H}^*(\mathcal{C}_i)= \langle \hat{e}^{m_i}_{i,1}, \hat{e}_{i,2}^{m_i}e^{-\widetilde{u}_i}\rho dx_i \rangle, \end{align*} $$

$$ \begin{align*}\mathcal{H}^*(\mathcal{D}_j\wedge \overline{\mathcal{D}}_j)= \langle \hat{f}_{j,1}^{n_j}, \hat{f}_{j,2}^{n_j}e^{-\widetilde{v}_j}\rho dz_j \rangle \wedge \langle \hat{\overline{f}}_{j,1}^{n_j}, \hat{\overline{f}}_{j,2}^{n_j}e^{-\widetilde{v}_j}\rho d\overline{z}_j \rangle. \end{align*} $$

Since

$$ \begin{align*}\eth_{\mathcal{C}}= \sum_{i=1}^{r_1}\eth_{\mathcal{C}_i}+ \sum_{j=1}^{r_2} \eth_{\mathcal{D}_j\wedge \overline{\mathcal{D}}_j} \end{align*} $$

and the various Hodge-deRham operators in this sum anti-commute, the result follows. In terms of cohomology, this is just the Künneth formula for the decomposition (3.13).

The differential $d_{S_{\eta }}$ of the complex

(3.14) $$ \begin{align} \Omega^*(S_{\eta};\ker\eth_{Y_{\eta}/S_{\eta}}) \end{align} $$

is then of the form

(3.15) $$ \begin{align} d_{S_{\eta}}= \widetilde{d}_{S_{\eta}}+ d_{\mathcal{C}} \end{align} $$

with $\widetilde {d}_{S_{\eta }}$ also a differential, namely

(3.16) $$ \begin{align} \begin{aligned} & \widetilde{d}_{S_{\eta}} \hat{w}_{k,l}= \left[ \sum_{i=1}^{r_1}\frac{2k_i-m_i}2 d\widetilde{u}_i + \sum_{j=1}^{r_2} \frac{2(l_j+\overline{l}_j)-n_j-\overline{{n}}_j}2 d\widetilde{v}_j\right]\hat{w}_{k,l}, \\ & \widetilde{d}_{S_{\eta}} (e^{-\widetilde{u}_i}\rho dx_i) = -d\widetilde{u_i}\wedge (e^{-\widetilde{u}_i}\rho dx_i), \quad i\le r_1, \\ &\widetilde{d}_{S_{\eta}} (e^{-\widetilde{v}_j}\rho dz_j) = -d\widetilde{v}_j\wedge (e^{-\widetilde{v}_j}\rho dz_j), \quad j\le r_2, \\ &\widetilde{d}_{S_{\eta}} (e^{-\widetilde{v}_j}\rho d\overline{z}_j) = -d\widetilde{v}_j\wedge (e^{-\widetilde{v}_j}\rho d\overline{z}_j), \quad j\le r_2, \end{aligned} \end{align} $$

and

$$ \begin{align*}\widetilde{d}_{S_{\eta}} ( \omega \wedge \nu)= (d\omega) \wedge \nu + (-1)^q \omega\wedge \widetilde{d}_{S_{\eta}}\nu \quad \mbox{for} \quad \omega\in\Omega^q(S_{\eta}), \; \nu\in \mathcal{C}^*. \end{align*} $$

Using the natural metric induced by $g_{\operatorname {fc},{\varepsilon }}$ and $h_{{\varepsilon }}$ , the differential $\widetilde {d}_{S_{\eta }}$ has a formal adjoint $d^*_{S_{\eta }}$ . In particular, there is a corresponding Hodge-deRham operator $\widetilde {\eth }_{S_{\eta }}$ and we see from (3.15) that

$$ \begin{align*}\eth_{S_{\eta}}= \widetilde{\eth}_{S_{\eta}}+ \eth_{\mathcal{C}}. \end{align*} $$

Since, $d_{S_{\eta }}=\widetilde {d}_{S_{\eta }}+d_{\mathcal {C}}$ is a differential, we see that the differentials $\widetilde {d}_{S_{\eta }}$ and $d_{\mathcal {C}}$ anti-commute. Correspondingly, their formal adjoints $\widetilde {d}_{S_{\eta }}^*$ and $d_{\mathcal {C}}^*$ anti-commute. In fact, writing the formal adjoints in terms of the corresponding Hodge star operators we see that $\widetilde {\eth }_{S_{\eta }}$ and $\eth _{\mathcal {C}}$ anti-commutes. Hence, we see that

$$ \begin{align*}\eth_{S_{\eta}}^2= \widetilde{\eth}_{S_{\eta}}^2+ \eth_{\mathcal{C}}^2. \end{align*} $$

This implies that an element in the kernel of $\eth _{S_{\eta }}$ will be in the kernels of $\widetilde {\eth }_{S_{\eta }}$ and $\eth _{\mathcal {C}}$ and vice-versa. Combined with Lemma 3.1, this can be used to obtain the following description of $\ker \eth _{S_{\eta }}$ .

Lemma 3.2. If $r_1\ge 1$ , the kernel of $\eth _{S_{\eta }}$ is trivial unless $m_1=\cdots =m_{r_1}$ and $n_j\in \{2m_1-\overline {{n}}_j,2m_1+2+\overline {{n}}_j, \overline {{n}}_j-2m_1-2\}$ for all $j\in \{1,\ldots ,r_2\}$ . In this latter case, the kernel is given by

(3.17) $$ \begin{align} & \langle w_{m,l}\left( \bigwedge_{j\in J} e^{-\widetilde{v}_j}\rho dz_j\right)\wedge \left( \bigwedge_{j\in \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right) , \\ & w_{0,n-l} \left(\bigwedge_{i=1}^{r_1} e^{-\widetilde{u}_i}\rho dx_i\right)\wedge \left( \bigwedge_{j\notin J} e^{-\widetilde{v}_j}\rho dz_j \right)\wedge \left( \bigwedge_{j\notin \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right)\rangle\wedge \mathcal{H}^*(S_{\eta}), \nonumber \end{align} $$

where $J= \{ j\in \{1,\ldots , r_2\}\; | \; n_j= \overline {{n}}_j-2m_1-2\}$ , $\overline {J}= \{ j\in \{1,\ldots , r_2\}\; | \; n_j= 2m_1+2+\overline {{n}}_j\}$ ,

$$ \begin{align*}l_j= \left\{ \begin{array}{ll} 0, & j\in J, \\ n_j, & j\notin J, \end{array}\right. \quad \overline{l}_j= \left\{ \begin{array}{ll} 0, & j\in \overline{J}, \\ \overline{n}_j, & j\notin \overline{J}, \end{array}\right. \end{align*} $$

and $\mathcal {H}^*(S_{\eta })$ , which can be identified with the cohomology ring of $S_{\eta }\cong \mathbb {T}^{r_1+r_2-1}$ , is the finite dimensional exterior algebra generated by $d\widetilde {u}_2,\ldots , d\widetilde {u}_{r_1}, d\widetilde {v}_1,\ldots , d\widetilde {v}_{r_2}$ .

Proof. First, in the special case where $r_1=1$ and $r_2=0$ , notice that $S_{\eta }$ is a point and the result is a direct consequence of Lemma 3.1. Otherwise, since we assume $r_1\ge 1$ , we must have $\dim S_{\eta }\ge 1$ and by the discussion above, $\ker \eth _{S_{\eta }}$ corresponds to the kernel of $\widetilde {\eth }_{S_\eta }$ acting on $\Omega ^*(S_{\eta };\ker \eth _{\mathcal {C}})$ . In terms of the differential $\widetilde {d}_{S_{\eta }}$ and the induced metric, we see from Lemma 3.1 that $\ker \eth _{\mathcal {C}}$ splits orthogonally into a direct sum of flat line bundles on $S_{\eta }$ . It thus suffices to compute the kernel of $\widetilde {\eth }_{S_{\eta }}$ when acting on sections of each of these flat line bundles. To describe the corresponding differentials, it is convenient to use, instead of $u_2,\ldots , u_{r_1}, v_1,\ldots , v_{r_2}$ , the local coordinates $\widetilde {u}_2,\ldots , \widetilde {u}_{r_1}, \widetilde {v}_1,\ldots , \widetilde {v}_{r_2}$ on $S_{\eta }$ . This is possible, since as readily checked, the corresponding linear change of coordinates has non-vanishing Jacobian. In terms of these coordinates, notice that

$$ \begin{align*}\mu =\frac{1}{d_{F}}\left(\sum_{i=2}^{r_1}u_i+ 2\sum_{j=1}^{r_2} v_j \right)= \sum_{i=2}^{r_1}\widetilde{u}_i +2 \sum_{j=1}^{r_2} \widetilde{v}_j, \end{align*} $$

so the differential $\widetilde {d}_{S_{\eta }}$ is given by

$$ \begin{align*}\widetilde{d}_{S_{\eta}} \hat{w}_{k,l}=\left[ \sum_{i=2}^{r_1} \frac{2\widetilde{k_i}-\widetilde{m}_i}2d\widetilde{u}_i + \sum_{j=1}^{r_2}\frac{2(\widetilde{l}_j+\widetilde{\overline{l}}_j)-\widetilde{n}_j-\widetilde{\overline{{n}}}_j}2 d\widetilde{v}_j \right] \hat{w}_{k,l} \end{align*} $$

with $\widetilde {k}_i=k_i-k_1$ , $\widetilde {m}_i=m_i-m_1$ , $\widetilde {l}_j=l_j-k_1$ , $\widetilde {\overline {l}}_j=\overline {l}_j-k_1$ , $\widetilde {n}_j=n_j-m_1$ and $\widetilde {\overline {{n}}}_j=\overline {{n}}_j-m_1$ , while

$$ \begin{align*}\widetilde{d}_{S_{\eta}}(e^{-\widetilde{u}_1}\rho dx_1)= d\mu\wedge e^{-\widetilde{u}_1}\rho dx_1= \left( \sum_{i=2}^{r_1}d\widetilde{u}_i+ 2\sum_{j=1}^{r_2} d\widetilde{v}_j\right)\wedge e^{-\widetilde{u}_1}\rho dx_1 \end{align*} $$

and otherwise is described as in (3.16). Now, by Lemma 3.1, assuming that $m\in \mathbb {N}^{r_1}$ and $n\in \mathbb {N}^{2r_2}$ for the moment, the flat line bundles of the decomposition of $\ker \eth _{\mathcal {C}}$ over $S_{\eta }$ are spanned by sections of the form

$$ \begin{align*}\sigma= \hat{w}_{k,l} \left( \bigwedge_{i, k_i=0} e^{-\widetilde{u}_i}\rho dx_i\right)\wedge \left( \bigwedge_{j,l_j=0} e^{-\widetilde{v}_i}\rho dz_j\right)\wedge \left( \bigwedge_{j,\overline{l}_j=0} e^{-\widetilde{v}_i}\rho d\overline{z}_j\right) \end{align*} $$

for k and l such that $k_i\in \{0,m_i\}$ for all i and $l_j\in \{0,n_j\}$ , $\overline {l}_j\in \{0,\overline {{n}}_j\}$ for all j. In particular, we compute that

(3.18) $$ \begin{align} \widetilde{d}_{S_{\eta}}\sigma&= \left[ \sum_{i=2}^{r_1} \left( \frac{2\widetilde{k}_i-\widetilde{m}_i}2-\delta_{k_i0}+\delta_{k_10} \right)d\widetilde{u}_i \right. \\ &\quad \left.+ \sum_{j=1}^{r_2}\left( \frac{2(\widetilde{l}_j+\widetilde{\overline{l}}_j)-\widetilde{n}_j-\widetilde{\overline{{n}}}_j}2+ 2\delta_{k_10}-\delta_{l_j0}-\delta_{\overline{l}_j0} \right)d\widetilde{v}_j \right]\wedge \sigma.\nonumber \end{align} $$

Using the Künneth formula in terms of the decomposition induced by the coordinates $\widetilde {u}_2,\ldots ,\widetilde {u}_{r_1}, \widetilde {v}_1,\ldots ,\widetilde {v}_{r_2}$ , we see that the cohomology is trivial unless $d\sigma =0$ , that is, unless the line bundle is trivial as a flat vector bundle, in which case the cohomology is isomorphic to $\mathcal {H}^*(S_{\eta })$ . A careful inspection of (3.18) then shows that the only way $d\sigma =0$ is if

$$ \begin{align*}\sigma= w_{m,l}\left( \bigwedge_{j\in J} e^{-\widetilde{v}_j}\rho dz_j\right)\wedge \left( \bigwedge_{j\in \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right) \end{align*} $$

or

$$ \begin{align*}\sigma= w_{0,n-l} \left(\bigwedge_{i=1}^{r_1} e^{-\widetilde{u}_i}\rho dx_i\right)\wedge \left( \bigwedge_{j\notin J} e^{-\widetilde{v}_j}\rho dz_j \right)\wedge \left( \bigwedge_{j\notin \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right) \end{align*} $$

with $m,n$ ,l, J and $\overline {J}$ as in the statement of the lemma, yielding the result. When we allow $m\in \mathbb {N}^{r_1}_0$ and $n\in \mathbb {N}^{r_2}_0$ , the formula (3.18) must be modified appropriately, but again the same conclusion can be reached.

When $r_1=0$ , the kernel of $\eth _{S_{\eta }}$ can be computed as follows.

Lemma 3.3. If $r_1=0$ , suppose that $\overline {{n}}_1=\cdots \overline {{n}}_{r_2}=0$ and assume without loss of generality that $n_1\ge \cdots \ge n_{r_2}\ge 0$ . If $n_1>0$ , then the kernel of $\eth _{S_{\eta }}$ is trivial unless $n_j\in \{n_1, n_1-2\}$ for all j, in which case it is given by

(3.19) $$ \begin{align} \langle w_{0,n}(e^{-\widetilde{v}}\rho d\overline{z})^{\alpha}, w_{0,0}\left( \bigwedge_{j=1}^{r_2} e^{-\widetilde{v}_j}\rho dz_j \right)\wedge (e^{-\widetilde{v}}\rho d\overline{z})^{\beta} \rangle \wedge \mathcal{H}^*(S_{\eta}), \end{align} $$

where $\alpha ,\beta \in \{0,1\}^{r_2}$ are such that

$$ \begin{align*}\begin{aligned} n_j=n_1 \;&\Longrightarrow \; \alpha_j=\alpha_1, \; \beta_j= \beta_1, \\ n_j= n_1-2 & \Longrightarrow \; \alpha_j=0, \alpha_1=1, \beta_j=1, \beta_1=0. \end{aligned} \end{align*} $$

In particular, if $n_j=n_1-2$ for some j, then $\alpha $ and $\beta $ are uniquely determined by these conditions, while otherwise $\alpha ,\beta \in \{ (0,\ldots ,0), (1,\ldots ,1)\}$ . If instead $n_1=\cdots =n_{r_2}=0$ , then the kernel is of the form

(3.20) $$ \begin{align} \langle (\rho dz)^\alpha\wedge (\rho d\overline{z})^\beta \; | \; \alpha_j+\beta_j= \alpha_1+\beta_1 \; \forall j \rangle \wedge \mathcal{H}^*(S_{\eta}). \end{align} $$

with

(3.21) $$ \begin{align} (e^{-\widetilde{v}}\rho d\overline{z})^\beta=(e^{-\widetilde{v}_1}\rho d\overline{z}_1)^{\beta_1}\wedge\cdots \wedge (e^{\widetilde{v}_{r_2}}\rho d\overline{z})^{\beta_{r_2}}_{r_2} \end{align} $$

and similarly for $(e^{-\widetilde {v}}\rho dz)^\alpha $ .

Proof. If $r_2=1$ , then $S_{\eta }$ is a point and the result is a direct consequence of Lemma 3.1. Otherwise, we can still follow the strategy of the proof of Lemma 3.2. We will only highlight the changes needed. First, we can use the local coordinates $\widetilde {v}_2,\ldots , \widetilde {v}_{r_2}$ on $S_{\eta }$ instead of $v_2,\ldots , v_{r_2}$ . In terms of these coordinates,

$$ \begin{align*}\mu= \frac{2}{d_{F}}\sum_{j=2}^{r_2}v_j= \sum_{j=2}^{r_2}\widetilde{v}_j \end{align*} $$

and the differential $\widetilde {d}_{S_{\eta }}$ takes the form

$$ \begin{align*}\widetilde{d}_{S_{\eta}} \hat{w}_{0,l}= \left[ \sum_{j=2}^{r_2} \left( {\frac{2\widetilde{l}_j-\widetilde{n}_j}2} \right)d\widetilde{v}_j \right] \hat{w}_{0,l} \end{align*} $$

with now $\widetilde {l}_j=l_j-l_1$ and $\widetilde {n}_j=n_j-n_1$ , while

$$ \begin{align*}\begin{aligned} \widetilde{d}_{S_{\eta}} (e^{-\widetilde{v}_1}\rho dz_1)&= d\mu\wedge e^{-\widetilde{v}_1}\rho dz_1= \left( {\sum_{j=2}^{r_2}d\widetilde{v}_j} \right)\wedge e^{-\widetilde{v}_1}\rho dz_1, \\ \widetilde{d}_{S_{\eta}} (e^{-\widetilde{v}_1}\rho d\overline{z}_1)&= d\mu\wedge e^{-\widetilde{v}_1}\rho d\overline{z}_1= \left( {\sum_{j=2}^{r_2}d\widetilde{v}_j} \right)\wedge e^{-\widetilde{v}_1}\rho d\overline{z}_1, \end{aligned} \end{align*} $$

and otherwise is as in (3.16). Now, $\ker \eth _{\mathcal {C}}$ still splits into flat line bundles. Assuming that $n\in \mathbb {N}^{r_2}$ , it is spanned by sections of the form

$$ \begin{align*}\sigma= \hat{w}_{0,l}\left( {\bigwedge_{j, l_j=0}e^{-\widetilde{v}_j}\rho dz_j} \right)\wedge (e^{-\widetilde{v}}\rho d\overline{z})^{\beta} \end{align*} $$

for l such that $l_j\in \{0,n_j\}$ for all j and with $d\overline {z}^{\beta }$ as in (3.21). We compute in this case that

(3.22) $$ \begin{align} \widetilde{d}_{S_{\eta}} \sigma= \left[ \sum_{j=2}^{r_2} \left( {\frac{2\widetilde{l}_j-\widetilde{n}_j}2 +\delta_{l_10} -\delta_{l_j0}+\delta_{\beta_11}-\delta_{\beta_j1} } \right)d\widetilde{v}_j \right] \wedge \sigma. \end{align} $$

Again, to have non-trivial kernel, we must have that $\widetilde {d}_S\sigma =0$ . Looking at (3.22), we see that the kernel must be of the claimed form (3.19). If more generally $n\in \mathbb {N}_0^{r_2}$ , formula (3.22) must be suitably modified, but again the same conclusion can be reached, except in the case where $n_1=0$ , that is, when $n=(0,\ldots ,0)$ and $E_{0,n}$ is a trivial flat line bundle, in which case the kernel is instead described by (3.20).

These results can be used to study the operator $D_{b,\eta }$ . To see this, we need the following lemma.

Using Lemma 3.5, we see that the operator $D_{b,\eta }$ of (3.9) is such that

(3.23) $$ \begin{align} \begin{aligned} D_{b,\eta}^2 &= \left( {\begin{array}{cc} \eth_{S_{\eta}}^2- D(\frac{d_{F}}2-W)D(W-\frac{d_F}2) & 0 \\ 0 & \eth_{S_{\eta}}^2- D(W-\frac{d_{F}}2)D(\frac{d_{F}}2-W) \end{array}} \right) \\ &=\left( {\begin{array}{cc} \eth_{S_{\eta}}^2+ D(W-\frac{d_{F}}2)^*D(W-\frac{d_{F}}2) & 0 \\ 0 & \eth_{S_{\eta}}^2+D(\frac{d_{F}}2-W)^*D(\frac{d_{F}}2-W) \end{array}} \right). \end{aligned} \end{align} $$

Hence, as in [Reference Müller and RochonMR20], $D_{b,\eta }$ can possibly be non-Fredholm or have a non-trivial $L^2$ -kernel only if $\eth _{S_{\eta }}$ has a non-trivial kernel.

Proposition 3.6. The operators $D_{b,\eta }$ and $D_{b,\eta }^2$ are Fredholm as b-operators for the b-density $\frac {dX}{\langle X\rangle }$ for $X\in \mathbb {R}$ if $r_1>0$ or if $r_1=0$ , $n\ne 0$ and $\overline {{n}}_1=\cdots =\overline {{n}}_2=0$ . When $r_1> 0$ the $L^2$ -kernel of $D_{b,\eta }$ (and $D_{b,\eta }^2$ ) is trivial unless $m_1=\cdots =m_{r_1}$ and

$$ \begin{align*}n_j\in\{2m_1-\overline{{n}}_j,2m_1+2+\overline{{n}}_j, \overline{{n}}_j-2m_1-2\}\end{align*} $$

for all $j\in \{1,\ldots ,r_2\}$ , in which case it is given by

(3.24) $$ \begin{align} & \langle w_{m,l}\left( \bigwedge_{j\in J} e^{-\widetilde{v}_j}\rho d z_j\right)\wedge \left( \bigwedge_{j\in \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right)\wedge\langle X\rangle^{-\frac{d_{F}}2-\frac{|m|-|n|}2-|l|+|J|+|\overline{J}|}\frac{dX}{\langle X\rangle}, \nonumber \\ & w_{0,n-l} \left(\bigwedge_{i=1}^{r_1} e^{-\widetilde{u}_i}\rho dx_i\right)\wedge \left( \bigwedge_{j\notin J} e^{-\widetilde{v}_j}\rho dz_j \right)\wedge \left( \bigwedge_{j\notin \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right) \langle X\rangle^{-\frac{d_{F}}2-\frac{|m|+|n|}2+|l|+|J|+|\overline{J}|} \rangle\wedge \mathcal{H}^*(S_{\eta}) \end{align} $$

with $J, \overline {J}$ and l as in Lemma 3.2. If instead $r_1=0$ , but $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ , $n_1\ge 1$ and we assume without loss of generality that $n_1\ge \cdots \ge n_{r_2}$ , then the $L^2$ -kernel of $D_{b,\eta }$ is trivial unless $n_{j}\in \{n_1, n_1-2\}$ for all j, in which case it is given by

(3.25) $$ \begin{align} \langle w_{0,n}(e^{-\widetilde{v}}\rho d\overline{z})^{\alpha}\wedge \langle X\rangle^{-\frac{d_{F}}2-\frac{|n|}2+|\alpha|}\frac{dX}{\langle X \rangle}, w_{0,0}\left( \bigwedge_{j=1}^{r_2} e^{-\widetilde{v}_j}\rho dz_j \right)\wedge (e^{-\widetilde{v}}\rho d\overline{z})^{\beta}\langle X\rangle^{-\frac{|n|}2-|\beta|} \rangle \wedge \mathcal{H}^*(S_{\eta}), \end{align} $$

with $\alpha ,\beta \in \{0,1\}^{r_2}$ such that

$$ \begin{align*}\begin{aligned} n_j=n_1 \;&\Longrightarrow \; \alpha_j=\alpha_1, \; \beta_j= \beta_1, \\ n_j= n_1-2 & \Longrightarrow \; \alpha_j=0, \alpha_1=1, \beta_j=1, \beta_1=0. \end{aligned} \end{align*} $$

Proof. By [Reference MelroseMel93], it suffices to check that the indicial family of $I(D^2_{b,\eta },\lambda )$ is invertible for all $\lambda \in \mathbb {R}$ . As shown in [Reference Albin, Rochon and SherARS18, (2.12)],

$$ \begin{align*}I(D(a),\lambda)= \pm(a-i\lambda) \end{align*} $$

at $X=\pm \infty $ , hence

$$ \begin{align*}I(D_{b,\eta}^2,\lambda)= \left( {\begin{array}{cc} \eth_{S_{\eta}}^2 + (W-\frac{d_{F}}2)^2+\lambda^2 & 0 \\ 0 & \eth_{S_{\eta}}^2 + (W-\frac{d_{F}}2)^2+ \lambda^2 \end{array}} \right) \end{align*} $$

at both ends. Clearly, this is invertible for $\lambda \ne 0$ , while at $\lambda =0$ , it is invertible provided $W-\frac {d_{F}}2\ne 0$ when acting on $\ker \eth _{S_{\eta }}$ . By the explicit description of $\ker \eth _{S_{\eta }}$ given in Lemmas 3.2 and 3.3 and the definition of W in (3.5) and (3.6), this is indeed the case unless $r_1=0$ and $n=0$ . To describe the $L^2$ -kernel of $D_{b,\eta }$ and $D_{b,\eta }^2$ , notice from (3.23) that it corresponds to the kernel of

$$ \begin{align*}\left( { \begin{array}{cc} 0 & -D(\frac{d_{F}}2-W) \\D(W-\frac{d_{F}}2) & 0 \end{array} } \right) \end{align*} $$

acting on sections of the trivial vector bundle $\ker \eth _{S_{\eta }}\oplus \ker \eth _{S_{\eta }}$ over $\mathbb {R}$ . Since the $L^2$ -kernel of $D(a)$ is non-trivial and spanned by $\langle X \rangle ^{-a}$ if and only if $a>0$ , the description of the $L^2$ -kernel of $D_{b,\eta }$ and $D_{b,\eta }^2$ therefore follows from Lemmas 3.2 and 3.3 and the definition of W.

When $D_{b,\eta }$ is Fredholm, we can apply the uniform construction of the resolvent of [Reference Albin, Rochon and SherARS21, Theorem 4.5] to $\eth _{\operatorname {fc},{\varepsilon }}$ as ${\varepsilon }\searrow 0$ .

Theorem 3.7. Suppose that either $r_1\ne 0$ or $r_1=0$ with $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ and $n\ne 0$ . Then the family of operators $D_{\operatorname {fc},{\varepsilon }}:=\rho ^{\frac {d_{F}}2}\eth _{\operatorname {fc},{\varepsilon }}\rho ^{-\frac {d_{F}}2}$ (using b-densities) has finitely many small eigenvalues, that is, there are finitely many eigenvalues of $D_{\operatorname {fc},{\varepsilon }}$ tending to $0$ as ${\varepsilon }\searrow 0$ . Furthermore, the projection $\Pi _{\operatorname {small}}$ on the eigenspace of small eigenvalues is a polyhomogeneous operator of order $-\infty $ in the surgery calculus of [Reference Mazzeo and MelroseMM95] and

(3.26) $$ \begin{align} \operatorname{rank} \Pi_{\operatorname{small}}= 2 \dim\ker_{L^2} \eth_{\operatorname{fc}}+ \sum_{\eta\in\mathfrak{P}_{\Gamma}}\dim\ker_{L^2_b} D_{b,\eta}. \end{align} $$

4 Cusp degeneration of analytic torsion

Let $T(M,\widehat {E}_{m,n},g_{\operatorname {fc},{\varepsilon }},h_{{\varepsilon }})$ be the analytic torsion of $(M,\widehat {E}_{m,n},g_{\operatorname {fc},{\varepsilon }},h_{{\varepsilon }})$ . In this section, we will study the limiting behaviour of $T(M;\widehat {E}_{m,n}, g_{\operatorname {fc},{\varepsilon }},h_{{\varepsilon }})$ as ${\varepsilon } \searrow 0$ . To do so, we need first to give a better description of the small eigenvalues occurring in Theorem 3.7. First, on the fibred cusp end $(T,\infty )_r\times Y_{\eta }$ , we can compute $L^2$ -cohomology as follows. As in [Reference Müller and RochonMR20], we can use polyhomogeneous forms, so the $L^2$ -cohomology can be computed using the complex

(4.1) $$ \begin{align} L^2\mathcal{A}^{\prime}_{\operatorname{phg}}&\Omega^q((T,\infty)_r\times Y_{\eta};E_{m,n})= \nonumber\\ &\{\nu \in L^2\mathcal{A}_{\operatorname{phg}}\Omega^q((T,\infty)_r\times Y_{\eta};E_{m,n})\; | \; d\nu\in L^2\mathcal{A}_{\operatorname{phg}}\Omega^{q+1}((T,\infty)_r\times Y_{\eta};E_{m,n}) \}, \end{align} $$

where $L^2\mathcal {A}_{\operatorname {phg}}\Omega ^q((T,\infty )_r\times Y_{\eta };E_{m,n})$ is the space of smooth $L^2$ -forms admitting a polyhomogeneous expansion in $x=\frac 1r$ in the sense of [Reference MelroseMel93]. The differential of this complex decomposes as

$$ \begin{align*}d= d_r + d_{Y_{\eta}} \end{align*} $$

with $d_r$ the differential in the $(T,\infty )_r$ factor and $d_{Y_{\eta }}$ the differential in the $Y_{\eta }$ factor. Using that the natural connection of the fibred bundle $\phi _{\eta }: Y_{\eta }\to S_{\eta }$ is flat, we see that the differential $d_{Y_{\eta }}$ further decomposes as

$$ \begin{align*}d_{Y_{\eta}}= d_{\mathcal{C}} +\widetilde{d}_{S_{\eta}}+d_{Y_{\eta}/S_{\eta}} \end{align*} $$

where $d_{\mathcal {C}}$ is the differential of the complex (3.10) with $\rho =1$ , $d_{Y_{\eta }/S_{\eta }}$ is corresponding to the vertical differential of the smooth coefficients in front of the basis generating the complex $\mathcal {C}$ and $\widetilde {d}_{S_{\eta }}$ is the remaining horizontal differential. Using this decomposition, we can in particular induce a double complex out of (4.1) with differentials $d_A:=d_{Y_{\eta }/S_{\eta }}$ and $d_B:= d_r+\widetilde {d}_{S_{\eta }}+d_{\mathcal {C}}$ with bi-degree given by $(W, N_r+N_{S_{\eta }}+N_{Y_{\eta }/S_{\eta }}-W)$ , where W is the weight operator defined earlier on and $N_r$ , $N_{S_{\eta }}$ and $N_{Y_{\eta }/S_{\eta }}$ are the number operators in the factors $(T,\infty )_r$ the base and the fibres of the fibre bundle $Y_{\eta }\to S_{\eta }$ . The first page of the corresponding spectral sequence is

$$ \begin{align*}E_1= L^2\mathcal{A}^{\prime}_{\operatorname{phg}}\Omega^*((T,\infty)_r\times S_{\eta}; \ker D_{v,\eta}) \end{align*} $$

with $d_1=d_B=d_r+\widetilde {d}_{S_{\eta }}+d_{\mathcal {C}}$ . In fact, this spectral sequence degenerates at the second page, which is just the cohomology of $(E_1,d_1)$ . We can again decompose this differential by

$$ \begin{align*}d_1= d_{1,A}+ d_{1,B} \quad \mbox{with} \; d_{1,A}= \widetilde{d}_{S_{\eta}}+d_{\mathcal{C}} \quad \mbox{and} \quad d_{1,B}=d_r, \end{align*} $$

inducing on $(E_1,d_1)$ a structure of double complex with bi-degree $(N_{S_{\eta }}+N_{Y_{\eta }/S_{\eta }},N_r)$ . The corresponding spectral sequence has first page

$$ \begin{align*}E_1'= L^2\mathcal{A}_{\operatorname{phg}}'\Omega((T,\infty)_r, \ker\eth_{S_{\eta}}) \end{align*} $$

with differential $d_1'=d_{1,B}=d_r$ , where a basis for $\ker \eth _{S_{\eta }}$ is given by Lemmas 3.2 and 3.3. This spectral sequence degenerates at the second page $E_1"$ so that the $L^2$ -cohomology of the fibred cusp end is identified with $E_1"$ , that is, with the cohomology of the complex

$$ \begin{align*}L^2\mathcal{A}_{\operatorname{phg}}'\Omega^*((T,\infty)_r;\ker\eth_{S_{\eta}}) \end{align*} $$

with differential $d_1'=d_r$ . By [Reference Hausel, Hunsicker and MazzeoHHM04, p.501], we deduce the following.

Proposition 4.1. When $r_1>0$ , the $L^2$ -cohomology of $((T,\infty )_r\times Y_{\eta }; E_{m,n})$ is trivial unless $m_1=\cdots =m_{r_1}$ and $n_j\in \{2m_1-\overline {{n}}_j,2m_1+2+\overline {{n}}_j, \overline {{n}}_j-2m_1-2\}$ for all $j\in \{1,\ldots , r_2\}$ , in which case it is given by

$$ \begin{align*}H^*_{(2)}((T,\infty)_r\times Y_{\eta};E_{m,n})\cong \langle \overline{w}_{m,l}\left( {\bigwedge_{j\in J}e^{-\widetilde{v}_j}dz_j} \right)\wedge \left( {\bigwedge_{j\in \overline{J}}e^{-\widetilde{v}_j}d\overline{z}_j} \right)\rangle\wedge \mathcal{H}^*(S_{\eta}) \end{align*} $$

with $J, \overline {J}$ and l as in Lemma 3.2 and $\overline {w}_{k,l}$ is $w_{k,l}$ with $r=1$ . If instead $r_1=0$ , but $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ and $n_1\ge 1$ , then assuming without loss of generality that $n_1\ge \cdots \ge n_{r_2}$ , the $L^2$ -cohomology of $((T,\infty )_r\times Y, E_{m,n})$ is trivial unless $n_j\in \{n_1, n_1-2\}$ for all j, in which case it is given by

$$ \begin{align*}H^*_{(2)}((T,\infty)_r\times Y_{\eta};E_{m,n})\cong \langle \overline{w}_{0,n} (e^{-\widetilde{v}}d\overline{z})^{\alpha}\rangle \wedge \mathcal{H}^*(S_{\eta}) \end{align*} $$

with $\alpha \in \{0,1\}^{r_2}$ such that

$$ \begin{align*}n_j=n_1\; \Longrightarrow \; \alpha_j=\alpha_1, \quad n_j=n_1-2 \; \Longrightarrow \; \alpha_j=0, \; \alpha_1=1. \end{align*} $$

If we forget about the factor of $(T,\infty )_r$ , notice that the above spectral sequence argument also shows that

(4.2) $$ \begin{align} H^*(Y_{\eta};E_{m,n})\cong \ker\eth_{S_{\eta}}, \end{align} $$

so Lemmas 3.2 and 3.3 give an explicit description of $H^*(Y_{\eta };E_{m,n})$ . It also induces a natural orthogonal decomposition

(4.3) $$ \begin{align} H^*(Y_{\eta};E_{m,n})= H_+^*(Y_{\eta};E_{m,n})\oplus H_-^*(Y_{\eta};E_{m,n}), \end{align} $$

when the cohomology is not trivial, where

(4.4) $$ \begin{align} \begin{aligned} H_-^*(Y_{\eta};E_{m,n})&= \overline{w}_{0,n-l} \left(\bigwedge_{i=1}^{r_1} e^{-\widetilde{u}_i}\rho dx_i\right)\wedge \left( \bigwedge_{j\notin J} e^{-\widetilde{v}_j}\rho dz_j \right)\wedge \left( \bigwedge_{j\notin \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right)\wedge \mathcal{H}^*(S_{\eta}), \\ H_+^*(Y_{\eta};E_{m,n})&= \overline{w}_{m,l}\left( \bigwedge_{j\in J} e^{-\widetilde{v}_j}\rho dz_j\right)\wedge \left( \bigwedge_{j\in \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right)\wedge\mathcal{H}^*(S_{\eta}) \end{aligned} \end{align} $$

in the setting of Lemma 3.2, while in the setting of (3.19),

(4.5) $$ \begin{align} \begin{aligned} H_-^*(Y_{\eta};E_{m,n})&= \overline{w}_{0,0}\left( \bigwedge_{j=1}^{r_2} e^{-\widetilde{v}_j}\rho dz_j \right)\wedge (e^{-\widetilde{v}}\rho d\overline{z})^{\beta} \wedge \mathcal{H}^*(S_{\eta}), \\ H_+^*(Y_{\eta};E_{m,n})&= \overline{w}_{0,n}(e^{-\widetilde{v}}\rho d\overline{z})^{\alpha} \wedge \mathcal{H}^*(S_{\eta}). \end{aligned} \end{align} $$

In particular, notice that Proposition 3.6 implies that

(4.6) $$ \begin{align} \ker_{L^2_b}D_{b,\eta}\cong H^*_+(Y_{\eta};E_{m,n})\oplus H^*_-(Y_{\eta};E_{m,n}). \end{align} $$

The decomposition (4.3) also implies that

(4.7) $$ \begin{align} H^*(\partial \overline{X}; E_{m,n})= H^*_+(\partial\overline{X};E_{m,n})\oplus H^*_-(\partial\overline{X};E_{m,n}) \end{align} $$

with

$$ \begin{align*}H^*_{\pm}(\partial\overline{X};E_{m,n})= \bigoplus_{\eta\in\mathfrak{P}_{\Gamma}} H_{\pm}^*(Y_{\eta};E_{m,n}). \end{align*} $$

Let $\operatorname {pr}_{\pm }: H^*(\partial \overline {X};E_{m,n})\to H_{\pm }^*(\partial \overline {X};E_{m,n})$ be the induced projections. Lemmas 3.2 and 3.3 yields the following description of these spaces.

Lemma 4.2. When $r_1>0$ , the cohomology of $(\partial \overline {X},E_{m,n})$ is trivial unless $m_1=\cdots =m_{r_1}$ and $n_j\in \{2m_1-\overline {{n}}_j,2m_1+2+\overline {{n}}_j, \overline {{n}}_j-2m_1-2\}$ for all $j\in \{1,\ldots ,r_2\}$ , in which case it is given by

$$ \begin{align*}\begin{aligned} H_-^*(\partial \overline{X};E_{m,n}) &\cong \bigoplus_{\eta\in\mathfrak{P}_{\Gamma}} \langle \overline{w}_{0,n-l}\left( {\bigwedge_{i=1}^{r_1}e^{-\widetilde{u}_i}dx_i} \right)\wedge \left( {\bigwedge_{j\notin J}e^{-\widetilde{v}_j}dz_j} \right)\wedge \left( {\bigwedge_{j\notin \overline{J}}e^{-\widetilde{v}_j}d\overline{z}_j} \right)\rangle\wedge \mathcal{H}^*(S_{\eta}) \\ H_+^*(\partial \overline{X};E_{m,n}) &\cong \bigoplus_{\eta\in\mathfrak{P}_{\Gamma}} \langle \overline{w}_{m,l}\left( \bigwedge_{j\in J} e^{-\widetilde{v}_j}\rho dz_j\right) \wedge \left( \bigwedge_{j\in \overline{J}} e^{-\widetilde{v}_j}\rho d\overline{z}_j\right) \rangle \wedge \mathcal{H}^*(S_{\eta}), \end{aligned} \end{align*} $$

for $J=\{ j\in \{1,\ldots , r_2\} \; | \; n_j=\overline {{n}}_j+2m_1+2\}$ and $\overline {J}= \{ j\in \{1,\ldots , r_2\}\; | \; n_j= 2m_1+2+\overline {{n}}_j\}$ . If instead $r_1=0$ , but $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ , $n_1\ge 1$ and we assume without loss of generality that $n_1\ge \cdots n_{r_2}$ , then the cohomology of $(\partial \overline {X}, E_{m,n})$ is trivial unless $n_j\in \{n_1,n_1-2\}$ for all j, in which case it is given by

$$ \begin{align*}\begin{aligned} H^*_-(\partial\overline{X};E_{m,n})&\cong \bigoplus_{\eta\in\mathfrak{P}_{\Gamma}} \langle \overline{w}_{0,0}\left( {\bigwedge_{j=1}^{r_2}e^{-\widetilde{v}_j}dz_j} \right) \wedge (e^{-\widetilde{v}}d\overline{z})^{\beta} \rangle\wedge \mathcal{H}^*(S_{\eta}), \\ H_+^*(\partial \overline{X};E_{m,n})&= \bigoplus_{\eta\in\mathfrak{P}_{\Gamma}} \langle \overline{w}_{0,n}(e^{-\widetilde{v}}\rho d\overline{z})^{\alpha} \rangle \wedge \mathcal{H}^*(S_{\eta}). \end{aligned} \end{align*} $$

where $\alpha , \beta \in \{0,1\}^{r_2}$ is such that

$$ \begin{align*}\begin{aligned} n_j=n_1 \;&\Longrightarrow \; \alpha_j=\alpha_1, \quad \beta_j= \beta_1, \\ n_j= n_1-2 & \Longrightarrow \; \alpha_j=0, \alpha_1=1, \beta_j=1, \beta_1=0. \end{aligned} \end{align*} $$

We can use this information to compute the cohomology group $H^*(\overline {X};E_{m,n})$ using known results and the Mayer-Vietoris long exact sequence in $L^2$ -cohomology

(4.8)

This yields the following.

Theorem 4.3. Suppose that $r_1>0$ or that $r_1=0$ with $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ and $n\ne 0$ . Then there is an isomorphism

(4.9) $$ \begin{align} H^*(\overline{X}; E_{m,n})\cong H^*_{(2)}(X;E_{m,n}) \oplus H^*_-(\partial\overline{X};E_{m,n}). \end{align} $$

Furthermore, if $n_j\ne \overline {{n}}_j$ for some $j\in \{1,\ldots ,r_2\}$ , then

(4.10) $$ \begin{align} H^*_{(2)}(X;E_{m,n})= \{0\} \end{align} $$

and this simplifies to

(4.11) $$ \begin{align} H^*(\overline{X};E_{m,n})\cong H^*_-(\partial\overline{X};E_{m,n}). \end{align} $$

Proof. By a result of Harder [Reference HarderHar75, Remark (3), p.159], we know that the image of the restriction map

$$ \begin{align*}\iota_{\partial\overline{X}}^*: H^*(\overline{X};E_{m,n})\to H^*(\partial \overline{X};E_{m,n}) \end{align*} $$

is identified with $H^*_-(\partial \overline {X}; E_{m,n})$ via the projection $\operatorname {pr}_-: H^*(\partial \overline {X};E_{m,n})\to H^*_-(\partial X;E_{m,n})$ ,

(4.12) $$ \begin{align} \iota_{\partial \overline{X}}^* H^*(\overline{X};E_{m,n})\cong H_-^*(\partial\overline{X};E_{m,n}). \end{align} $$

On the other hand, Proposition 4.1 and Lemma 4.2 show that the restriction map also induces an isomorphism

(4.13) $$ \begin{align} H^*_{(2)}((T,\infty)_r\times Y_{\eta}; E_{m,n})\cong H^*_+(\partial\overline{X};E_{m,n}). \end{align} $$

By the decomposition (4.7), this means that the boundary homomorphism of the long exact sequence (4.8) is trivial, yielding the isomorphism (4.9). To prove (4.10), let $\tau \colon G_\infty \to \operatorname {GL}(V_\tau )$ be an irreducible finite dimensional representation. Let $E_\tau \to X$ be the flat vector bundle which is associated to $\tau |_\Gamma $ . Equip $E_\tau $ with the Hermitian fibre metric defined in [Reference Matsushima and MurakamiMM63]. Let

$$\begin{align*}\Delta_{q}(\tau)\colon \Lambda^q(X,E_\tau)\to \Lambda^q(X,E_\tau) \end{align*}$$

be the Laplacian acting in the space of $E_\tau $ -valued q-forms. Let be the subspace of $L^2\Lambda ^q(X,E_\tau )$ which is spanned by the square integrable eigenforms of $\Delta _q(\tau )$ . Let be the subspace of $L^2(\Gamma \backslash G_\infty )$ , which is spanned by the irreducible subrepresentations of the right regular representation of $G_\infty $ on $L^2(\Gamma \backslash G_\infty )$ . Then we have

(4.14)

Let be the restriction of $\Delta _q(\tau )$ to . Now assume that $\tau \not \cong \tau \circ \vartheta $ where $\vartheta $ is the standard Cartan involution of $G_{\infty }$ with respect to $K_{\infty }$ . Using (4.14) it follows as in the proof of [Reference Bergeron and VenkateshBV13, Lemma 4.1] that there exists $c>0$ such that

(4.15)

for all $q=0,...,n$ . In particular, it follows that

(4.16) $$ \begin{align} H^\ast_{(2)}(X,E_\tau)= \mathcal{H}^\ast_{(2)}(X,E_\tau)=0, \end{align} $$

where the right hand side denotes the space of square integrable harmonic forms. Now assume that $r_2>0$ and that there exists $j\in \{1,...,r_2\}$ with $n_j\neq \bar n_j$ . Then $\varrho _{m,n}$ is not self-conjugate, that is, $\varrho _{m,n}\ncong \varrho _{m,n}\circ \vartheta $ , so (4.10) follows from (4.16).

Remark 4.4. Comparing (4.8) with the long exact sequence in cohomology for the pair $(\overline {X},\partial \overline {X})$ , we can also deduce from (4.12) that

$$ \begin{align*}H^*_{(2)}(X;E_{m,n})\cong \mathrm{Im} \left( {H^*_c(X;E_{m,n})\to H^*(X;E_{m,n})} \right). \end{align*} $$

Using the Mayer-Vietoris long exact sequence in cohomology

(4.17)

and Proposition 3.6, we deduce the following.

Lemma 4.5. If $r_1>0$ or $r_1=0$ with $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ and $n\ne 0$ , then

$$ \begin{align*}\begin{aligned} H^*(M;\widehat{E}_{m,n})&\cong H^*_{(2)}(X;E_{m,n})\bigoplus H^*_{(2)}(X;E_{m,n}) \bigoplus \left( {\bigoplus_{\eta\in\mathfrak{P}_{\Gamma}}\ker_{L^2_b}D_{b,\eta}} \right), \\ &\cong \ker_{L^2} \eth_{\operatorname{fc},0} \bigoplus \left( {\bigoplus_{\eta\in\mathfrak{P}_{\Gamma}}\ker_{L^2_b}D_{b,\eta}} \right), \end{aligned} \end{align*} $$

where $\eth _{\operatorname {fc},0}$ is the operator corresponding to $\eth _{\operatorname {fc}}$ on each copy of X in $M\setminus \partial \overline {X}$ .

We deduce from this lemma and Theorem 3.7 that the eigenspaces associated to small eigenvalues are cohomological, that is, all the small eigenvalues are zero. As in [Reference Müller and RochonMR20], this allows us to apply [Reference Albin, Rochon and SherARS21, Corollary 11.3]. However, one important hypothesis in this corollary is that the base $S_{\eta }$ of the fibre bundle $\phi _{\eta }: Y_{\eta }\to S_{\eta }$ has to be even dimensional to ensure that some terms in the short time asymptotic expansion of the trace of the heat kernel vanish. Our setting, on the other hand, is very special within the framework considered in [Reference Albin, Rochon and SherARS21, Theorem 11.2]. This will allow us to establish directly the vanishing of these terms in the asymptotic expansion of the trace of the heat kernel without assuming that $S_{\eta }$ is even dimensional.

The first useful special feature of our setting is that the family of metrics $g_{\operatorname {fc},{\varepsilon }}$ is not just asymptotic to the model (3.1). It is in fact exactly given by (3.1) for $\rho =\sqrt {x^2+{\varepsilon }^2}$ sufficiently small. Moreover, as already observed, the fibre bundle $\phi _{\eta }: Y_{\eta }\to S_{\eta }$ has trivial curvature in the sense of [Reference Berline, Getzler and VergneBGV04]. Consequently, there is no curvature term in the asymptotic model (3.7) of the corresponding Hodge-deRham operator in the limit $\rho \to 0$ and this operator is in fact exactly given by (3.7) in a small region near $\rho =0$ .

To study the asymptotic behaviour of the heat kernel near $\rho =0$ in the limit $t\to 0$ , this means that we can use the model operator (3.8), namely

(4.18) $$ \begin{align} \widehat{D}_{\operatorname{fc},{\varepsilon}}= \left( {\begin{array}{cc} \rho^{-1}\eth_{Y_{\eta}/S_{\eta}}+ \eth_{S_{\eta}} & \frac{x}{\rho}\left(W-\frac{d_{F}}{2}\right)-\rho\frac{\partial}{\partial x} \\ \rho\frac{\partial}{\partial x}+ \frac{x}{\rho}\left(W-\frac{d_{F}}2 \right) & -\rho^{-1}\eth_{Y_{\eta}/S_{\eta}}- \eth_{S_{\eta}} \end{array}} \right) \end{align} $$

seen as defined on $Y_{\eta }\times \mathbb {R}_x$ for ${\varepsilon }$ small. Since $\eth _{S_{\eta }}$ commutes with W and anti-commutes with $\eth _{Y_{\eta }/S_{\eta }}$ , we see that

(4.19) $$ \begin{align} \widehat{D}_{\operatorname{fc},{\varepsilon}}^2= \widehat{D}_{c,{\varepsilon}}^2 + \eth^2_{S_{\eta}}, \end{align} $$

where

$$ \begin{align*}\widehat{D}_{c,{\varepsilon}}:= \left( {\begin{array}{cc} \rho^{-1}\eth_{Y_{\eta}/S_{\eta}} & \frac{x}{\rho}\left(W-\frac{d_{F}}{2}\right)-\rho\frac{\partial}{\partial x} \\ \rho\frac{\partial}{\partial x}+ \frac{x}{\rho}\left(W-\frac{d_{F}}2 \right) & -\rho^{-1}\eth_{Y_{\eta}/S_{\eta}} \end{array}} \right) \end{align*} $$

is an operator anti-commuting with $\eth _{S_{\eta }}$ . The corresponding heat kernel is therefore given by

$$ \begin{align*}e^{-t\widehat{D}^2_{\operatorname{fc},{\varepsilon}}}= e^{-t\widehat{D}^2_{c,{\varepsilon}}}e^{-t\eth^2_{S_{\eta}}}. \end{align*} $$

For analytic torsion, we are in fact interested in the weighted version

$$ \begin{align*}(-1)^N N e^{-t\widehat{D}_{\operatorname{fc},{\varepsilon}}^2}= (-1)^N N \left( {e^{-t \widehat{D}^2_{c,{\varepsilon}}}e^{-t \eth_{S_{\eta}}^2}} \right), \end{align*} $$

where N is the number operator multiplying a form (of pure degree) by its degree. This can be rewritten

(4.20) $$ \begin{align} (-1)^N N e^{-t\widehat{D}^2_{\operatorname{fc},{\varepsilon}}}= (-1)^{N_x+ N_{Y_{\eta}/S_{\eta}}+ N_{S_{\eta}}}(N_x+N_{Y_{\eta}/S_{\eta}}+N_{S_{\eta}})\left( {e^{-t\widehat{D}^2_{c,{\varepsilon}}}e^{-t\eth_{S_{\eta}}^2}} \right), \end{align} $$

where $N_x$ is the number operator in the factor $\mathbb {R}_x$ , while as before $N_{Y_{\eta }/S_{\eta }}$ and $N_{S_{\eta }}$ are the number operators in the fibres and the base of the fibre bundle $\phi _{\eta }: Y_{\eta }\to S_{\eta }$ . To understand this operator, it suffices to understand it when acting on each of the eigenspaces of $(N_x+ N_{Y_{\eta }/S_{\eta }})$ and W.

We can also decompose in terms of the eigenbundles of $\eth _{Y_{\eta }/S_\eta }$ . Let us first see what happens in a given fibre $\phi _{\eta }^{-1}(s)$ of the fibre bundle (2.5). There, the restriction $(\eth _{Y_\eta /S_\eta })^2_s$ of $\eth _{Y_\eta /S_\eta }^2$ is the Hodge Laplacian associated to the trivial flat Hermitian vector bundle $\phi _{\eta }^{-1}(s)\times \mathbb {C}^{\operatorname {rank} E_{m,n}}$ on a flat torus. In this case,

$$ \begin{align*}\dim_{\mathbb{C}} \ker (\eth_{Y_\eta/S_\eta})^2_s= \dim_{\mathbb{C}} \ker (\eth_{Y_\eta/S_\eta})_s= 2^{d_F}\operatorname{rank} E_{m,n} \end{align*} $$

and $(\eth _{Y_\eta /S_\eta })^2_s$ is canonically identified with $2^{d_F}\operatorname {rank} E_{m,n}$ copies of the scalar Laplacian $\Delta _{\phi _{\eta }^{-1}(s)}$ on $\phi _{\eta }^{-1}(s)$ . Hence, the eigenspace of $(\eth _{Y_\eta /S_\eta })^2_s$ associated to the eigenvalue $\nu $ is canonically of the form

(4.21) $$ \begin{align} \ker((\eth_{Y_\eta/S_\eta})^2_s-\nu)\cong \ker(\Delta_{\phi_\eta^{-1}(s)}-\nu)\otimes \ker(\eth_{Y_\eta/S_\eta})_s. \end{align} $$

The torus $\phi _{\eta }^{-1}(s)$ can be taken to be $(\Gamma _{\eta }\cap N_\infty )\setminus N_\infty $ , but notice that the flat metric on it depends on s. In other words, the fibres of $\phi _\eta : Y_\eta \to S_\eta $ are not isometric with their induced metrics. Nevertheless, under the identification $\phi _{\eta }^{-1}(s)\cong (\Gamma _{\eta }\cap N_\infty )\setminus N_\infty $ , notice that for $t\ne s$ , $\ker (\Delta _{\phi _\eta ^{-1}(s)}-\nu )$ , can also be seen as an eigenspace of $\Delta _{\phi _\eta ^{-1}(t)}$ but possibly with a different eigenvalue. These eigenspaces can be combined to form a flat vector bundle on $S_\eta $ as follows.

Denote by $1$ the element of $S_\eta = \Gamma _{\eta ,T} \setminus T_{\infty }(1)/K_T$ corresponding to taking

$$ \begin{align*}\lambda_1=\cdots= \lambda_{r_1}=\mu_1=\cdots=\mu_{r_2}=1. \end{align*} $$

The fundamental group $\Gamma _{\eta ,T}$ of $S_\eta $ acts on $\phi _\eta ^{-1}(s)$ in such a way that

$$ \begin{align*}Y_\eta= \Gamma_{\eta,T}\setminus ( (T_{\infty}(1)/K_T)\times \phi_\eta^{-1}(1) ) \end{align*} $$

with respect to the diagonal action of $\Gamma _{\eta ,T}$ on $ (T_{\infty }(1)/K_T)\times \phi _\eta ^{-1}(1)$ . The action of $\Gamma _{\eta ,T}$ on $\phi _{\eta }^{-1}(1)$ also induces an action on $\ker (\Delta _{\phi _\eta ^{-1}(1)}-\nu )$ , so for $\nu $ an eigenvalue of $\Delta _{\phi _\eta ^{-1}(1)}$ , we can consider the flat vector bundle

(4.22) $$ \begin{align} H_\nu:= \Gamma_{\eta,T} \setminus ((T_{\infty}(1)/K_T)\times \ker(\Delta_{\phi_{\eta}^{-1}(1)}-\nu)) \end{align} $$

on $S_{\eta }$ , where the quotient is taken with respect to the diagonal action of $\Gamma _{\eta ,T}$ on $(T_{\infty }(1)/K_T)\times \ker (\Delta _{\phi _{\eta }^{-1}(1)}-\nu )$ . For $s\in S_\eta $ , the fibre of $H_\nu $ above s corresponds to an eigenspace of $\Delta _{\phi _\eta ^{-1}(s)}$ with eigenvalue $\nu (s)$ possibly different from $\nu $ .

Together with the canonical identification (4.21), we see that the flat vector bundles $H_\nu $ as $\nu $ varies yield the decomposition

(4.23) $$ \begin{align} L^2(Y_\eta; \Lambda^*(T^*(Y_\eta/S_\eta))\otimes E_{m,n})= \bigoplus_{\nu\in \operatorname{Spec}(\Delta_{\phi_\eta^{-1}(1)})} L^2(S_\eta; H_\nu\otimes \ker\eth_{Y_\eta/S_\eta}). \end{align} $$

Correspondingly, the restriction of $\eth _{S_{\eta }}$ acting on sections of the vector bundle $H_\nu \otimes \ker (\eth _{Y_{\eta }/S_{\eta }})$ is given by

$$ \begin{align*}\eth_{S_{\eta},\nu}= \eth_{\mathcal{C}}+ \widetilde{\eth}_{S_{\eta},\nu} \quad \mbox{with} \quad \eth_{S_{\eta},\nu}^2= \eth^2_{\mathcal{C}}+ \widetilde{\eth}_{S_{\eta},\nu}^2, \end{align*} $$

where $\eth _{\mathcal {C}}$ acts trivially on $H_\nu $ and $\eth _{S_{\eta },\nu }$ is a version of $\eth _{S_{\eta }}$ twisted by the flat vector bundle $H_{\nu }$ with Hermitian structure $h_\nu $ . Hence, to understand the pointwise trace of (4.20), it suffices to understand the sum

(4.24) $$ \begin{align} \sum_{q=0}^{r_1+r_2-1} (-1)^{p+q}(p+q)\operatorname{tr}\left( { (e^{-t\widetilde{\eth}_{S_{\eta},\nu}^2})_{\Omega^q(S_{\eta};L\otimes H_{\nu})} } \right), \end{align} $$

where $L\to S_{\eta }$ is a flat line bundle on $S_{\eta }$ , seen as a subbundle of $\mathcal {C}^*\to S_{\eta }$ , which corresponds to a common eigenspace of $\eth _{\mathcal {C}}$ and W contained in $\{ \omega \; | \; (N_x+ N_{Y_{\eta }/S_{\eta }})\omega =p\omega \}$ . But by (3.16), in a trivialization of such a flat line bundle L,

(4.25) $$ \begin{align} \widetilde{d}_{S_{\eta},\nu}= d_{\nu}+a_L \operatorname{Id}_{\nu} \end{align} $$

for some parallel 1-form $a_L\in \Omega ^1(S_{\eta })$ , where $\operatorname {Id}_{\nu }$ is the identity section of $H_{\nu }$ and $d_{\nu }$ is the differential on $\Omega ^*(S_{\eta };H_{\nu })$ , so

(4.26) $$ \begin{align} \operatorname{tr}\left( { e^{-t\widetilde{\eth}_{S_{\eta},\nu}^2}|_{\Omega^q(S_{\eta};L\otimes H_{\nu})} } \right)= \operatorname{tr}(e^{-t\Delta_{\nu}}) e^{-t |a_L|^2} \left( {\begin{matrix} r_1+r_2-1 \\ q \end{matrix}} \right), \end{align} $$

where $\Delta _{\nu }$ is the scalar Laplacian of $(S_{\eta },g_{S_{\eta }}, H_{\nu },h_\nu )$ and $|a_L|$ is the pointwise norm of $a_L$ with respect to the metric $g_{S_{\eta }}$ . Hence the sum (4.24) will vanish provided we can show that

(4.27) $$ \begin{align} \sum_{q=0}^{r_1+r_2-1} (-1)^{p+q}(p+q) \left( {\begin{matrix} r_1+r_2-1 \\ q \end{matrix}} \right) \end{align} $$

vanishes. When $r_1+r_2-1>1$ , the vanishing of (4.27) is a consequence of the binomial theorem, since

$$ \begin{align*}\begin{aligned} \sum_{q=0}^{r_1+r_2-1}(-1)^{p+q}(p+q)\left( {\begin{matrix} r_1+r_2-1 \\ q \end{matrix}} \right)&= (-1)^pp\sum_{q=0}^{r_1+r_2-1}(-1)^q \left( {\begin{matrix} r_1+r_2-1 \\ q \end{matrix}} \right) \\ & \quad+ (-1)^p \sum_{q=0}^{r_1+r_2-1} (-1)^qq \left( {\begin{matrix} r_1+r_2-1 \\ q \end{matrix}} \right) \\ &= (-1)^pp(1-1)^{r_1+r_2-1} \\ & \quad +(-1)^p \sum_{q=1}^{r_1+r_2-1} (-1)^qq\left( {\begin{matrix} r_1+r_2-1 \\ q \end{matrix}} \right) \\ &= 0 +(-1)^p \sum_{q=1}^{r_1+r_2-1}(-1)^q(r_1+r_2-1)\left( {\begin{matrix} r_1+r_2-2 \\ q-1 \end{matrix}} \right) \\ &= (-1)^p (r_1+r_2-1) \sum_{q=1}^{r_1+r_2-1} (-1)^q \left( {\begin{matrix} r_1+r_2-2 \\ q-1 \end{matrix}} \right) \\ &= (-1)^p(r_1+r_2-1) \sum_{q=0}^{r_1+r_2-2}(-1)^{q+1} \left( {\begin{matrix} r_1+r_2-2 \\ q \end{matrix}} \right) \\ &= (-1)^{p+1} (r_1+r_2-1)(1-1)^{r_1+r_2-2}=0. \end{aligned} \end{align*} $$

If $r_1+r_2-1=1$ , this argument does not show that (4.27) vanishes. However, we should notice that the flat line bundles $L\to S_{\eta }$ arising in (4.24) come in pairs using the Hodge star operator of $\mathcal {C}^*$ and $\mathbb {R}_x$ . Namely, if $L\to S_{\eta }$ is such that

(4.28) $$ \begin{align} \left(W-\frac{d_{F}}2 \right)(\sigma)=\nu \sigma \quad (N_x+ N_{Y_{\eta}/S_{\eta}})\sigma= p \sigma \end{align} $$

for each of its sections, then there is a dual line bundle $L^*$ also arising in the decomposition (4.24) such that

(4.29) $$ \begin{align} \left(W-\frac{d_{F}}2 \right)(\sigma)=-\nu \sigma \quad (N_x+ N_{Y_{\eta}/S_{\eta}})\sigma= (d_{F}+1-p) \sigma. \end{align} $$

If the flat connection of $L\otimes H_{\nu }$ is locally given by (4.25), then the one of $L^*\otimes H_{\nu }$ is given by

(4.30) $$ \begin{align} d_{\nu}-a_L\operatorname{Id}_{\nu} \end{align} $$

since $L^*$ is the dual of L. Alternatively this can be seen directly from the description of $\widetilde {d}_{S_{\eta }}$ in (3.16), see also (3.18) and (3.22). In particular, from (4.26), we see that

(4.31) $$ \begin{align} \operatorname{tr} \left( {e^{-t\widetilde{\eth}_{S_{\eta}}^2}|_{\Omega^q(S_{\eta};L^*\otimes H_{\nu})}} \right)= \operatorname{tr} \left( {e^{-t\widetilde{\eth}_{S_{\eta}}^2}|_{\Omega^q(S_{\eta};L\otimes H_{\nu})}} \right). \end{align} $$

However, since (4.28) is replaced by (4.29) for sections of $L^*$ , we see from (3.23) that the contribution (4.27) coming from L, namely

(4.32) $$ \begin{align} \sum_{q=0}^{1} (-1)^{p+q}(p+q) \left( {\begin{matrix} r_1+r_2-1 \\ q \end{matrix}} \right)= (-1)^{p}p+ (-1)^{p+1}(p+1)=(-1)^{p+1} \end{align} $$

since $r_1+r_2-1=1$ , becomes the contribution

(4.33) $$ \begin{align} \sum_{q=0}^{1} (-1)^{d_{F}+1-p+q}(d_{F}+1-p+q) \left( {\begin{matrix} r_1+r_2-1 \\ q \end{matrix}} \right)= (-1)^{d_{F}+1-p+1}= (-1)^{d_{F}+p} \end{align} $$

for $L^*$ . Therefore, if $d_{F}$ is even, we see that (4.33) exactly cancels (4.32).

To summarize the discussion so far, provided $r_1+ r_2-1>1$ or that $r_1+r_2-1=1$ and $d_{F}$ is even (i.e. $r_1=2$ and $r_2=0$ or $r_1=0$ and $r_2=2$ ), we see that the pointwise trace of (4.20) identically vanishes,

(4.34) $$ \begin{align} \operatorname{tr} \left( { (-1)^N N e^{-t\widehat{D}^2_{\operatorname{fc},{\varepsilon}}}} \right)=0. \end{align} $$

This implies the following result for the heat kernel of $D^2_{\operatorname {fc},{\varepsilon }}$ .

Theorem 4.7. Suppose that $r_1>0$ or that $r_1=0$ with $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ and $n\ne 0$ . Suppose also that $r_1+r_2-1>1$ or that $(r_1,r_2)\in \{(2,0),(0,2)\}$ . Then as ${\varepsilon } \searrow 0$ , the logarithm of the analytic torsion of $(M, \widehat {E}_{m,n}, g_{\operatorname {fc},{\varepsilon }}, h_{{\varepsilon }})$ has a polyhomogeneous expansion and its finite part is given by

$$ \begin{align*}\operatorname*{\mathrm{\operatorname{FP}}}_{{\varepsilon}=0} \log T(M;\widehat{E}_{m,n},g_{\operatorname{fc},{\varepsilon}}, h_{{\varepsilon}})= 2\log T(X;E_{m,n},g_{\operatorname{fc}}, h), \end{align*} $$

where $T(X;E_{m,n},g_{\operatorname {fc}}, h)$ is the analytic torsion of $(X;E_{m,n},g_{\operatorname {fc}}, h)$ .

Proof. By the discussion above, in a region near $\rho =0$ , the heat kernel of $D_{\operatorname {fc}, {\varepsilon }}^2$ has the same short time asymptotic expansion as the heat kernel of the model (4.19). In particular, we see from (4.34) that

$$ \begin{align*}\operatorname{tr} \left( { (-1)^N N e^{-tD^2_{\operatorname{fc},{\varepsilon}}}} \right) \end{align*} $$

has a trivial short time expansion near $\rho =0$ , in particular at the front face $\mathfrak {B}_{tff}$ of the surgery heat space of [Reference Albin, Rochon and SherARS21]. Thus, by [Reference Albin, Rochon and SherARS21, Theorem 11.2], $ T(M;\widehat {E}_{m,n},g_{\operatorname {fc},{\varepsilon }}, h_{{\varepsilon }})$ has a polyhomogeneous expansion as ${\varepsilon }\searrow 0$ with finite part given by

$$ \begin{align*}\operatorname*{\mathrm{\operatorname{FP}}}_{{\varepsilon}=0} \log T(M;\widehat{E}_{m,n},g_{\operatorname{fc},{\varepsilon}}, h_{{\varepsilon}})= 2\log T(X;E_{m,n},g_{\operatorname{fc}}, h) +\sum_{\eta\in\mathfrak{P}_{\Gamma}} \log T(D_{b,\eta}^2), \end{align*} $$

where $T(D_{b,\eta }^2)$ is the analytic torsion defined by

$$ \begin{align*}\log T(D^2_{b,\eta})= \frac12 \zeta_{D^2_{b,\eta}}'(0) \end{align*} $$

with

$$ \begin{align*}\zeta_{D^2_{b,\eta}}(s):= \frac{1}{\Gamma(s)}\int_{0}^{\infty} (t^s) {}^{R}\operatorname{Tr}\left( {(-1)^N N \left( {e^{-tD_{b,\eta}^2}-\Pi_{\ker_{L^2}D_{b,\eta}^2}} \right)} \right) \frac{dt}{t}, \quad \operatorname{Re}(s)>\frac{r_1+r_2}2, \end{align*} $$

admitting a meromorphic extension to the complex plane which is holomorphic near $s=0$ . Here, N is the number operator giving the form degree, but also including the vertical degree of the forms $e^{-\widetilde {u}_i}\rho dx_i$ , $e^{-\widetilde {v}_j}\rho dz_j$ and $e^{-\widetilde {v}_j}\rho d\overline {z}_j$ seen as sections of $\ker D_{v,\eta }$ when multiplied by $\hat {w}_{k,l}$ for some k and l. The vanishing of (4.34) however implies that

$$ \begin{align*}\zeta_{D^2_{b,\eta}}(s)=0. \quad \forall s>\frac{r_1+r_2}2, \end{align*} $$

so $\log T(D^2_{b,\eta })=0$ and the result follows.

5 Small eigenvalues

When $\ker _{L^2} D_{b,\eta }\ne 0$ , there are small eigenvalues. However, assuming that $r_1>0$ or that $r_1=0$ with $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ and $n\ne 0$ , all these small eigenvalues are zero by Lemma 4.5, which will allow us to obtain a formula relating analytic torsion and Reidemeister torsion. We will in fact closely follow the argument in [Reference Müller and RochonMR20].

First, Lemma 4.5 can be used to compute the fibred cusp degeneration of Reidemeister torsion using the long exact sequence in cohomology (4.17). This requires choosing suitable bases for the spaces involved in this long exact sequence. When the cohomology on the boundary is not trivial, we can pick an orthonormal basis

(5.1) $$ \begin{align} \mu_{\partial \overline{X}}^q=( \mu_{+}^q, \mu_{-}^q) \end{align} $$

for $H^q(\partial \overline {X};E_{m,n})$ , compatible with the orthogonal decomposition (4.7) with $\mu _{\pm }^q$ an orthonormal basis (with each element of pure degree) for $H^q_{\pm }(\partial \overline {X};E_{m,n})$ . Without loss of generality, we can assume that $\mu _{+}$ is Poincaré dual to $\mu _{-}$ . On $H^q(\overline {X};E_{m,n})$ , we can then just take the basis $\mu _X^q=(\mu _{X,(2)}^q,\mu _{X,\operatorname {inf}}^q)$ compatible with the decomposition (4.9) and such that

(5.2) $$ \begin{align} \operatorname{pr}_-\circ\iota^*_{\partial\overline{X}} \mu_X^q= \operatorname{pr}_-\circ\iota^*_{\partial\overline{X}} \mu_{X,\operatorname{inf}}^q= \mu_{-}^q, \end{align} $$

where $\mu _{X,(2)}^q$ is an orthonormal basis of $H^q_{(2)}(X;E_{m,n})\cong \ker _{L^2}\eth _{\operatorname {fc}}$ . On $H^q(M;\widehat {E}_{m,n})$ , there is a decomposition

(5.3) $$ \begin{align} H^q(M;\widehat{E}_{m,n})=\Im \partial_{q-1}\oplus \Im i_q \end{align} $$

in terms of the long exact sequence (4.17), so we can choose a basis $\mu _M^q$ compatible with this decomposition and the previous choices of bases, namely

(5.4) $$ \begin{align} \mu_M^q= (\partial_*(\mu_{+}^{q-1}), \mu_{M,i}^q). \end{align} $$

More precisely, if $\mu _X^q=\{\nu _1,\ldots ,\nu _K\}$ with $\mu ^q_{X,(2)}=\{\nu _1,\ldots ,\nu _{K'}\}$ , then $\mu _{M,i}^q$ will be chosen such that

$$ \begin{align*}i_q(\mu_{M,i}^q)=\{(\nu_1,0), (0,\nu_1), \ldots (\nu_{K'},0), (0,\nu_{K'}), (\nu_{K'+1},\nu_{K'+1}),\ldots, (\nu_K,\nu_K) \}. \end{align*} $$

Now, on $H^q(\overline {X};E_{m,n})\oplus H^q(\overline {X};E_{m,n})$ , we can consider, instead of $\mu _{X}^q\oplus \mu _{X}^q,$ the equivalent basis

(5.5) $$ \begin{align} \Bigg\{(\nu_1,0), (0,\nu_1), \ldots (\nu_{K'},0), (0,\nu_{K'}), \left( {\frac{\nu_{K'+1}}{\sqrt{2}},\frac{\nu_{K'+1}}{\sqrt{2}}} \right),&\left( {\frac{\nu_{K'+1}}{\sqrt{2}},-\frac{\nu_{K'+1}}{\sqrt{2}}} \right),\ldots, \\ & \left( {\frac{\nu_K}{\sqrt{2}},\frac{\nu_K}{\sqrt{2}}} \right),\left( {\frac{\nu_K}{\sqrt{2}},-\frac{\nu_K}{\sqrt{2}}} \right) \Bigg\} \nonumber \end{align} $$

obtained by an orthonormal transformation. It is such that

$$ \begin{align*}\ker j_q= \operatorname{span}\left\{(\nu_1,0), (0,\nu_1), \ldots (\nu_{K'},0), (0,\nu_{K'}), \left( {\frac{\nu_{K'+1}}{\sqrt{2}},\frac{\nu_{K'+1}}{\sqrt{2}}} \right),\ldots, \left( {\frac{\nu_K}{\sqrt{2}},\frac{\nu_K}{\sqrt{2}}} \right) \right\}, \end{align*} $$

$$ \begin{align*}j_q \left\{ \left( {\frac{\nu_{K'+1}}{\sqrt{2}},-\frac{\nu_{K'+1}}{\sqrt{2}}} \right),\ldots, \left( {\frac{\nu_K}{\sqrt{2}},-\frac{\nu_K}{\sqrt{2}}} \right) \right\}= \sqrt{2}\iota_{\partial\overline{X}}^*(\mu_X) \end{align*} $$

and

$$ \begin{align*}i_q(\mu_M)= \sqrt{2} \left\{ \left( {\frac{\nu_{K'+1}}{\sqrt{2}},\frac{\nu_{K'+1}}{\sqrt{2}}} \right),\ldots, \left( {\frac{\nu_K}{\sqrt{2}},\frac{\nu_K}{\sqrt{2}}} \right) \right\}. \end{align*} $$

In particular, in terms of these bases, we see that

$$ \begin{align*}|{\operatorname{det}}(j_q)_{\perp}|= |{\operatorname{det}}(i_q)_{\perp}|. \end{align*} $$

Using the formula of Milnor applied to the long exact sequence (4.17) therefore yields the following.

Theorem 5.1. If $r_1>0$ or $r_1=0$ with $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ and $n\ne 0$ , then

$$ \begin{align*}\tau(M;\widehat{E}_{m,n},\mu_M)= \frac{\tau(X;E_{m,n},\mu_X)^2}{\tau(\partial\overline{X};E_{m,n},\mu_{\partial\overline{X}})}= \tau(X;E_{m,n},\mu_X)^2. \end{align*} $$

Proof. The result would follow from the discussion above if instead of $\mu _{\partial \overline {X}}$ , we would have chosen the basis

$$ \begin{align*}\mu_{\partial\overline{X}}'=(\mu_{+},\iota_{\partial\overline{X}}^*\mu_X). \end{align*} $$

However, since the change of basis from $\mu _{\partial \overline {X}}$ to $\mu _{\partial \overline {X}}'$ is not orthonormal, but has determinant $1$ , we can simply replace $\mu _{\partial \overline {X}}'$ by $\mu _{\partial \overline {X}}$ to obtain the result. Since we assume that the basis $\mu _{\partial \overline {X}}$ is self-dual, we also know from [Reference MüllerMül93, Corollary 1.9] that

$$ \begin{align*}\tau(\partial\overline{X};E_{m,n},\mu_{\partial\overline{X}})=1. \end{align*} $$

To relate analytic torsion and Reidemeister torsion on X when $\dim X$ is odd, that is, when $r_2$ is odd, the idea is to start with the formula of [Reference MüllerMül93]

(5.6) $$ \begin{align} \log\tau(M;\widehat{E}_{m,n},\mu_M)=\log T(M,\widehat{E}_{m,n},g_{\operatorname{fc},{\varepsilon}},h_{{\varepsilon}})-\log\left( {\prod_q [\mu^q_M|\omega_{{\varepsilon}}^q]^{(-1)^q}} \right) \end{align} $$

for ${\varepsilon }>0$ , where $\omega _{{\varepsilon }}^q$ is an orthonormal basis of harmonic forms in degree q with respect to $g_{\operatorname {fc},{\varepsilon }}$ and $h_{{\varepsilon }}$ , while $[\mu ^q_M|\omega _{{\varepsilon }}^q]= |{\operatorname {det}} W^q|$ with $W^q$ the matrix describing the change of bases

$$ \begin{align*}(\mu^q_M)_i= \sum_jW^q_{ij}(\omega_{{\varepsilon}}^q)_j. \end{align*} $$

Taking the finite part at ${\varepsilon }=0$ of the formula (5.6) shall lead to a formula on X. First, assuming that $r_1+r_2-1>1$ , we know by Theorem 4.7 that

(5.7) $$ \begin{align} \operatorname*{\mathrm{\operatorname{FP}}}_{{\varepsilon}=0} \log T(M,\widehat{E}_{m,n},g_{\operatorname{fc},{\varepsilon}},h_{{\varepsilon}})= 2\log T(X,E_{m,n},g_{\operatorname{fc}},h). \end{align} $$

By Theorem 5.1, we can express $\tau (M,E_{m,n},\mu _M)$ in terms of the torsion on $\overline {X}$ . So, what remains to be done is to take the finite part of

$$ \begin{align*}\log\left( {\prod_q [\mu^q_M|\omega_{{\varepsilon}}^q]^{(-1)^q}} \right) \end{align*} $$

as ${\varepsilon }\searrow 0$ .

By the definition of $c_b$ in [Reference Müller and RochonMR20, (4.16)] and our definition of $\mu _{\partial \overline {X}}$ in (5.1), we see that an orthonormal basis of $\bigoplus _{\eta \in \mathfrak {P}_{\Gamma }}\ker _{L^2_b} D_{b,\eta }$ is given by

(5.8) $$ \begin{align} \left( { \frac{1}{\sqrt{c_{\frac{d_{F}}{2}-W}}} \langle X\rangle^{W-\frac{d_{F}}2} \rho^W \mu_{+} \frac{dX}{\langle X \rangle }, \frac{1}{\sqrt{c_{W-\frac{d_{F}}2}}} \langle X\rangle^{\frac{d_{F}}2 -W} \rho^W \mu_{-} } \right). \end{align} $$

As in [Reference Müller and RochonMR20], this can be extended to a polyhomogeneous basis of harmonic forms on the single surgery space $(X_s,\widehat {E}_{m,n})$ of (3.2) with respect to $D_{\operatorname {fc},{\varepsilon }}:= \rho ^{\frac {d_{F}}2}\eth _{\operatorname {fc},{\varepsilon }}\rho ^{-\frac {d_{F}}2}$ . Let $(\omega _{{\varepsilon },(2)},\omega _{{\varepsilon },+},\omega _{{\varepsilon },-})$ be a corresponding orthonormal basis of harmonic forms with respect to $\eth _{\operatorname {fc},{\varepsilon }}$ , where $\omega _{{\varepsilon },(2)}$ corresponds in the limit ${\varepsilon }\searrow 0$ to an orthonormal basis of $\ker _{L^2_b} D_{\operatorname {fc},0}$ with $D_{\operatorname {fc},0}= \rho ^{\frac {d_{F}}2}\eth _{\operatorname {fc},0}\rho ^{-\frac {d_{F}}2}$ , $\omega _{{\varepsilon },+}$ corresponds to

$$ \begin{align*}\frac{1}{\sqrt{c_{\frac{d_{F}}{2}-W}}} \langle X\rangle^{W-\frac{d_{F}}2} \rho^W \mu_{+} \frac{dX}{\langle X \rangle } \end{align*} $$

and $\omega _{{\varepsilon },-}$ corresponds to

$$ \begin{align*}\frac{1}{\sqrt{c_{W-\frac{d_{F}}2}}} \langle X\rangle^{\frac{d_{F}}2 -W} \rho^W \mu_{-}. \end{align*} $$

If ${\varepsilon }^{\mu }w$ for $\mu>0$ is a higher order term in the expansion of $\omega _{{\varepsilon }}\in \omega _{{\varepsilon },+}$ as ${\varepsilon }\searrow 0$ , then since $D_{\operatorname {fc},{\varepsilon }}$ commutes with multiplication by ${\varepsilon }$ , we see that $\rho ^{\frac {d_{F}}2}w\in \ker D_{\operatorname {fc},{\varepsilon }}$ , so in particular its restriction $\rho ^{\frac {d_{F}}2}w_{sb}$ to $\mathfrak {B}_{sb}$ , the front face of the single surgery space $X_s$ considered in [Reference Albin, Rochon and SherARS21, Reference Müller and RochonMR20], is in $\bigoplus _{\eta \in \mathfrak {P}_{\Gamma }}D_{b,\eta }$ . As in [Reference Müller and RochonMR20], we can check that cohomologically, these terms are negligible, so that in the limit ${\varepsilon }\searrow 0$ ,

(5.9) $$ \begin{align} \begin{aligned} \omega_{{\varepsilon},+} &\sim \frac{1}{\sqrt{c_{\frac{d_{F}}2-W}}} (\langle X\rangle\rho)^{W-\frac{d_{F}}2}\mu_{+}\frac{dx}{\rho} \\ &= {\varepsilon}^{W-\frac{d_{F}}2}\langle X\rangle^{2W-d_{F}} \frac{1}{\sqrt{c_{\frac{d_{F}}2-W}}}\mu_{+} \frac{dX}{\langle X\rangle }. \end{aligned} \end{align} $$

Since

$$ \begin{align*}\int_{-\infty}^{\infty} \frac{{\varepsilon}^{W-\frac{d_{F}}2}}{\sqrt{c_{\frac{d_{F}}{2}-W}}} \langle X\rangle^{2W-d_{F}} \frac{dX}{\langle X\rangle}= {\varepsilon}^{W-\frac{d_{F}}2}\sqrt{c_{\frac{d_{F}}{2}-W}} \end{align*} $$

when acting on the negative eigenspaces of $W-\frac {d_{F}}2$ , we see that cohomologically,

(5.10) $$ \begin{align} [\omega_{{\varepsilon},+}]\sim {\varepsilon}^{W-\frac{d_{F}}2}\sqrt{c_{\frac{d_{F}}{2}-W}} \partial_*[\mu_{+}] \end{align} $$

as ${\varepsilon }\searrow 0$ . On the other hand, if ${\varepsilon }^{\mu }w$ for some $\mu>0$ is a higher order term in the expansion of $\omega _{{\varepsilon }}\in \omega _{{\varepsilon },-}$ , then again $\rho ^{\frac {d_{F}}2}w\in \ker D_{\operatorname {fc},{\varepsilon }}$ , so its restriction $\rho ^{\frac {d_{F}}2}w_{sb}$ to $\mathfrak {B}_{sb}$ is in $\bigoplus _{\eta \in \mathfrak {P}_{\Gamma }}D_{b,\eta }$ . As in [Reference Müller and RochonMR20], though such a term is negligible in $L^2$ -norm, it could still give a contribution in cohomology. More precisely, if

$$ \begin{align*}\rho^{\frac{d_{F}}2}w_{sb}= \langle X\rangle^{\frac{d_{F}}2-W}\rho^W \psi \end{align*} $$

for some $\psi \in \operatorname {span} \langle \mu _{\partial \overline {X},m,n}\rangle $ , then cohomologically

$$ \begin{align*}\begin{aligned} [{\varepsilon}^{\mu}\iota_Y^*w] &\sim {\varepsilon}^{\mu}\iota_Y^*\left[ \left( {\frac{\langle X\rangle}{\rho}} \right)^{\frac{d_{F}}2-W} \psi \right] \\ &= {\varepsilon}^{\mu+W-\frac{d_{F}}2} [\psi]. \end{aligned} \end{align*} $$

As in [Reference Müller and RochonMR20, (6.18) and (6.19)] and keeping in mind [Reference Müller and RochonMR20, Remark 6.1], we expect in general a non-trivial contribution in cohomology from some lower order terms and we can deduce that

(5.11) $$ \begin{align} [\iota_{\partial\overline{X}}^*\omega_{{\varepsilon},-}]\sim \frac{{\varepsilon}^{\hat{W}-\frac{d_{F}}2}}{\sqrt{c_{\hat{W}-\frac{d_{F}}2}}} [\iota_{\partial\overline{X}}^*\mu_X] \quad \mbox{as} \; {\varepsilon}\searrow 0, \end{align} $$

where the weight operator $\hat {W}$ is defined by

$$ \begin{align*}\hat{W}\Xi= \nu \Xi \end{align*} $$

if $\Xi \in \iota _{\partial \overline {X}}^*\mu _X$ is such that

$$ \begin{align*}\Xi= \hat{\Xi}\quad \mod \operatorname{span} \langle \mu_{+}\rangle \end{align*} $$

with $\hat {\Xi }\in \mu _{\partial \overline {X},0,0}$ such that $W\hat {\Xi }=\nu \hat {\Xi }$ .

Combining (5.10) and (5.11) yields the following.

Lemma 5.2. When $r_1+r_2-1>0$ ,

(5.12) $$ \begin{align} \operatorname*{\mathrm{\operatorname{FP}}}_{{\varepsilon}=0} \log\left( {\prod_q [\mu^q_M|\omega^q]^{(-1)^q}} \right)=0 \end{align} $$

in the limit ${\varepsilon }\searrow 0$ .

Proof. From (5.10) and (5.11), we see that

$$ \begin{align*}\operatorname*{\mathrm{\operatorname{FP}}}_{{\varepsilon}=0} \log\left( {\prod_q [\mu^q_M|\omega^q]^{(-1)^q}} \right)= \sum_{\eta\in\mathfrak{P}_{\Gamma}}B_{\eta}\chi(S_{\eta}), \end{align*} $$

where $\chi \left ( {S_{\eta } } \right )$ is the Euler characteristic of $S_{\eta }$ and $B_{\eta }$ is a certain sum of terms. Our assumption that $\dim S_{\eta }=r_1+r_2-1>0$ ensures that $\chi \left ( {S_{\eta } } \right )=0$ , from which the result follows.

This yields our main result.

Theorem 5.3. Suppose that $r_1>0$ or that $r_1=0$ with $\overline {{n}}_1=\cdots =\overline {{n}}_{r_2}=0$ and $n\ne 0$ . If $r_2$ is odd (i.e. $\dim X$ is odd) and $r_1+r_2-1>1$ , then

$$ \begin{align*}\log T(X,E_{m,n},g_{\operatorname{fc}},h)= \log\tau(\overline{X},E_{m,n},\mu_X). \end{align*} $$

In many cases, the long exact sequence (4.17) is trivial, in which case this result simplifies as follows.

Theorem 5.4. Suppose that $m,n$ are chosen in Proposition 3.6 with $n_j\ne \overline {{n}}_j$ for some j and in such a way that $D_{b,\eta }$ is Fredholm and has trivial $L^2$ -kernel for each $\eta \in \mathfrak {P}_{\Gamma }$ . If $\dim X$ is odd and $r_1+r_2-1>1$ , then

$$ \begin{align*}T(X,E_{m,n},g_{\operatorname{fc}},h)= \tau(\overline{X},E_{m,n}), \end{align*} $$

where $T(X,E_{m,n}, g_{\operatorname {fc}},h)$ is the analytic torsion of $(X,E_{m,n},g_{\operatorname {fc}},h)$ and $\tau (\overline {X},E_{m,n})$ is the Reidemeister torsion of $(\overline {X},E_{m,n})$ . If instead $\dim X$ is even with $r_1+r_2-1>1$ or $(r_1,r_2)\in \left ( {(2,0),(0,2)} \right )$ , then

$$ \begin{align*}T(X,E_{m,n},g_{\operatorname{fc}},h)= 1. \end{align*} $$

Proof. Since we assume that $n_j\ne \overline {{n}}_j$ for some j, we know by Theorem 4.3 that $H^*_{(2)}(X;E_{m,n})=\{0\}$ . Since $\ker _{L^2_b}D_{b,\eta }=\{0\}$ for each $\eta \in \mathfrak {P}_{\Gamma }$ , this implies by Lemma 4.5 and (4.6) that $H^*(M;\widehat {E}_{m,n})=\{0\}$ and $H^*(\partial \overline {X};E_{m,n})=\{0\}$ . We deduce from the long exact sequence (4.17) that $H^*(\overline {X};E_{m,n})=\{0\}$ as well, hence that this long exact sequence is trivial.

Thus, if $\dim X$ is odd, Theorem 5.4 is just Theorem 5.3 when the long exact sequence (4.17) is trivial. If $\dim X$ is even, the result follows instead from Theorem 4.7 due to the fact that $\widehat {E}_{m,n}$ is self-dual as a flat vector bundle and as a Hermitian vector bundle, so by [Reference MüllerMül93, Proposition 2.9],

$$ \begin{align*}T(M,\widehat{E}_{m,n},g_{\operatorname{fc},{\varepsilon}},h_{{\varepsilon}})= 1 \quad \forall {\varepsilon}\ge 0. \end{align*} $$

6 Construction of acyclic bundles

Following [Reference Bergeron and VenkateshBV13, sect. 8.1], we explain in some detail how acyclic $\Gamma $ -modules are constructed in our case. To construct the corresponding representations we proceed as follows. Let $G_0=\operatorname {SL}(2)/F$ and let $T_0\subset G_0$ be the standard maximal torus. Let $T:=\operatorname {Res}_{F/\mathbb {Q}}(T_0)$ be the corresponding maximal torus of G. Let $\mathbb {G}_m$ be the multiplicative group. We select an isomorphism $\mathbb {G}_m/F\cong T_0/F$ , given by

$$\begin{align*}a\mapsto\begin{pmatrix}a&0\\0& a^{-1}\end{pmatrix} \end{align*}$$

for $a\in \mathbb {G}_m(F)$ . This gives rise to an identification

$$\begin{align*}\operatorname{Res}_{F/\mathbb{Q}}(\mathbb{G}_m)=\operatorname{Res}_{F/\mathbb{Q}}(T_0)=T. \end{align*}$$

Let $E/\mathbb {Q}$ be a Galois extension splitting T such that $\operatorname {Hom}(F,E)\neq 0$ . Such a Galois extension always exists [Reference Borel and TitsBT65, Prop. 1.5]. We denote by $\{\sigma \colon F\to E\}$ the set of embeddings of F into E on which the Galois group $\operatorname {Gal}(E/\mathbb {Q})$ acts transitively. Note that

$$\begin{align*}\#\{\sigma\colon F\to E\}=[F\colon\mathbb{Q}]. \end{align*}$$

Let $G\times _{\mathbb {Q}} E$ and $T\times _{\mathbb {Q}} E$ be the groups obtained from G and T, respectively, by extension of scalars [Reference MilneMil11, I, 4c]. We have

(6.1) $$ \begin{align} G\times_{\mathbb{Q}} E=\prod_{\sigma\colon F\to E}G_0\times_\sigma E= \prod_{\sigma\colon F\to E}\operatorname{SL}(2)/F\times_\sigma E. \end{align} $$

Hence an irreducible representation $\rho $ of $G\times _{\mathbb {Q}} E$ is a tensor product

$$\begin{align*}\rho=\bigotimes_{\sigma\colon F\to E}\rho_\sigma, \end{align*}$$

where $\rho _\sigma $ is the irreducible representation of $\operatorname {SL}(2)/E$ on the E-vector space $W_\sigma :=\operatorname {Sym}^{d_\sigma }(E^2)$ of homogeneous polynomials of degree $d_\sigma $ in two variables. Thus we have an E-rational representation

(6.2) $$ \begin{align} \rho\colon G\times_{\mathbb{Q}} E\to\operatorname{GL}(W), \end{align} $$

where

(6.3) $$ \begin{align} W=\bigotimes_{\sigma\colon F\to E}W_\sigma=\bigotimes_{\sigma\colon F\to E} \operatorname{Sym}^{d_\sigma}(E^2). \end{align} $$

The base change $T\times _{\mathbb {Q}} E$ is a split torus, i.e., we have

(6.4) $$ \begin{align} T\times_{\mathbb{Q}} E=\prod_{\sigma\colon F\to E} T_0\times_{F,\sigma}E= \prod_{\sigma\colon F\to E} T_0. \end{align} $$

Let $X^\ast (T\times _{\mathbb {Q}} E)=\operatorname {Hom}(T\times _{\mathbb {Q}} E,\mathbb {G}_m)$ be the group of characters of $T\times _{\mathbb {Q}} E$ and let $X^\ast _+(T\times _{\mathbb {Q}} E)$ be the dominant characters. By (6.4) we have

(6.5) $$ \begin{align} X^\ast(T\times_{\mathbb{Q}} E)=\bigoplus_{\sigma\colon F\to E} X^\ast(T_0\times_{F,\sigma} E) =\bigoplus_{\sigma\colon F\to E} X^\ast(T_0). \end{align} $$

By (6.5) the highest weight $\lambda _\rho \in X^\ast _+(T\times _{\mathbb {Q}} E)$ of $\rho $ is given by

(6.6) $$ \begin{align} \lambda_\rho=(d_{\sigma_1},...,d_{\sigma_n}). \end{align} $$

and to every highest weight as above there is associated a unique irreducible finite dimensional rational representation of $G\times _{\mathbb {Q}} E$ defined over E.

Now observe that the functor “restriction of scalars” is right adjoint to the functor “extension of scalars” [Reference MilneMil11, I, 4d]. Thus we have

(6.7) $$ \begin{align} \operatorname{Hom}_{\mathbb{Q}}(G,\operatorname{Res}_{E/\mathbb{Q}}(\operatorname{GL}(W)))\cong\operatorname{Hom}_E(G\times_{\mathbb{Q}} E,\operatorname{GL}(W)). \end{align} $$

Hence the representation (6.2) corresponds to a representation

(6.8) $$ \begin{align} \tilde\rho\colon G\to \operatorname{Res}_{E/\mathbb{Q}}(\operatorname{GL}(W)) \end{align} $$

which is defined over $\mathbb {Q}$ . Since $\operatorname {Res}_{E/\mathbb {Q}}$ is a functor, there is a canonical homomorphism

(6.9) $$ \begin{align} \operatorname{Res}_{E/\mathbb{Q}}(\operatorname{GL}(W))\to\operatorname{GL}(\operatorname{Res}_{E/\mathbb{Q}}(W)). \end{align} $$

Let $V=\operatorname {Res}_{E/\mathbb {Q}}(W)$ which is just W considered as $\mathbb {Q}$ -vector space. Combining (6.8) and (6.9), we obtain a representation

(6.10) $$ \begin{align} \varrho\colon G\to \operatorname{GL}(V), \end{align} $$

which is defined over $\mathbb {Q}$ . Now recall that by (6.3), W is the tensor product of the E-vector spaces $\operatorname {Sym}^{d_\sigma }(E^2)$ , $\sigma \colon F\to E$ , and $\operatorname {Sym}^{d_\sigma }(E^2)$ is the space of homogeneous polynomials of degree $d_\sigma $ in two variables with coefficients in E. Choose an integral basis $e_1,...,e_q$ of E over $\mathbb {Q}$ , where $q=[E\colon \mathbb {Q}]$ . Expressing the coefficients of the polynomials in this basis, we obtain a vector space over $\mathbb {Q}$ . So

(6.11) $$ \begin{align} W_\sigma=\operatorname*{\mathrm{\operatorname{Res}}}_{E/\mathbb{Q}}\operatorname{Sym}^{d_\sigma}(E^2)=\bigoplus_{i=1}^q W_i, \end{align} $$

where $W_i$ consists of homogeneous polynomials

$$\begin{align*}\sum_{k=1}^{d_\sigma} a_{ki} e_iX^kY^{d_\sigma-k},\quad a_{ki}\in\mathbb{Q}. \end{align*}$$

Correspondingly, the tensor product W becomes a vector space over $\mathbb {Q}$ , which is denoted by V.

Next we determine how $V\otimes _{\mathbb {Q}}\mathbb {C}$ decomposes into irreducible representations of $G(\mathbb {C})$ . Let $S_\infty $ denote the set of Archimedean places of F. For $v\in S_\infty $ let $F_v$ be the completion of F with respect to v. Let $G_v/\mathbb {R}:= \operatorname *{\mathrm {\operatorname {Res}}}_{F_v/\mathbb {R}}(\operatorname {SL}(2)/F_v)$ . We have

(6.12) $$ \begin{align} G\times_{\mathbb{Q}}\mathbb{R}=\prod_{v\in S_\infty} G_v. \end{align} $$

Fix an embedding $\iota \colon E\to \mathbb {C}$ . Then

$$\begin{align*}W\otimes_{E,\iota}\mathbb{C}=\bigotimes_{v\in S_\infty} W_v. \end{align*}$$

If v is a real place, it corresponds to an embedding $\sigma \colon F\to \mathbb {R}$ and if v is complex then it corresponds to a pair of conjugate embeddings

$$\begin{align*}\sigma,\bar\sigma\colon F\to \mathbb{C}, \end{align*}$$

which are viewed as the two continuous isomorphisms $\sigma ,\bar \sigma \colon F_v\cong \mathbb {C}$ . In the first case, we have $W_v=W_\sigma \otimes _E\mathbb {C}$ , and in the second case

$$\begin{align*}W_v\cong (W_\sigma\otimes_E\mathbb{C})\otimes(W_{\bar\sigma}\otimes_E\mathbb{C}). \end{align*}$$

Let $\sigma _1,...,\sigma _{r_1}$ denote the real embeddings and $(\nu _1,\bar \nu _1),...,(\nu _{r_2},\bar \nu _{r_2})$ the complex embeddings of F in $\mathbb {C}$ . Then the highest weight $\lambda _\rho $ of $\rho $ takes the form

(6.13) $$ \begin{align} \lambda_\rho=(d_{\sigma_1},...,d_{\sigma_{r_1}},(d_{\nu_1},d_{\bar\nu_1}),..., (d_{\nu_{r_2}},d_{\bar\nu_{r_2}})). \end{align} $$

An embedding $\tau \colon E\to \mathbb {C}$ induces an isomorphism $X^\ast (T\times E) \cong X^\ast (T\times \mathbb {C})$ . From (6.11) follows that $V\otimes _{\mathbb {Q}}\mathbb {C}$ is the direct sum of irreducible representations whose highest weights are obtained from $\lambda _\rho $ by applying the various embeddings $E\hookrightarrow \mathbb {C}$ . Now assume that $r_2\ge 1$ and the highest weight $\lambda _\rho =(d_{\sigma _1},...,d_{\sigma _n})$ of the representation $\rho $ of $G\times _{\mathbb {Q}} E$ satisfies

(6.14) $$ \begin{align} d_{\sigma_i}\neq d_{\sigma_j},\quad i\neq j. \end{align} $$

Then it follows from the considerations above that $V\otimes _{\mathbb {Q}}\mathbb {C}$ decomposes into the direct sum of irreducible, non-selfconjugate representations with respect to the standard Cartan involution $\vartheta $ of $G_{\infty }$ with respect to $K_{\infty }$ .

We summarize the properties of the representation $\varrho \colon G\to \operatorname {GL}(V)$ which is defined as (6.10).