2.1 Notation

Let $K={\mathbb Q}(\sqrt {-D})$ be an imaginary quadratic number field with class number $1$ and let ${\mathcal O}_K$ represent its ring of integers with associated norm $N(\alpha )=\alpha \overline {\alpha }.$ In later definitions, we will also refer to a Euclidean imaginary quadratic number field K, where D is $1,2,3,7,$ or 11. We let

$$ \begin{align*}\mathcal H_3 = \{(z,t)\in {\mathbb C}\times {\mathbb R} \mid t> 0\}\end{align*} $$

designate hyperbolic 3-space, the unique simply-connected Riemannian manifold of dimension 3 and constant sectional curvature $-1$ . Additionally, we define the standard action of ${\text {GL}_2(\mathbb {C})}$ on $\mathcal H_3$ . Explicitly, for a matrix in ${\text {GL}_2(\mathbb {C})}$ of determinant $\Delta $ , we have

$$\begin{align*}\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot (z,t) = \left( \frac{(az+b)\overline{(cz+d)}+a\bar{c}t^2}{|cz+d|^2+|c|^2t^2},\frac{|\Delta| t}{|cz+d|^2+|c|^2t^2} \right ). \end{align*}$$

Alternatively, we may view elements of $\mathcal H_3$ as elements of the Hamiltonians $\mathbb {H}$ via the map $(z,t)\mapsto z+tj$ . In this case, for $u\in \mathbb {H,}$ the action of ${\text {GL}_2(\mathbb {C})}$ is given by

$$\begin{align*}\begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot u = \frac{au+b}{cu+d}. \end{align*}$$

2.2 Bianchi modular forms

Let k be a non-negative integer. Then denote $V_{2k+2}({\mathbb C})$ as the complex vector space of homogeneous polynomials of degree $2k+2$ in variables $X,Y$ . Define the multiplier system

$$ \begin{align*}J(\gamma; (z,t)) = \begin{pmatrix} cz+d & -ct \\ \bar{c}t & \overline{cz+d}\end{pmatrix},\qquad \gamma=\begin{pmatrix} a & b \\ c&d \end{pmatrix} \in \textrm{SL}_2(\mathbb{C}),\ (z,t) \in \mathcal H_3. \end{align*} $$

Given a function $F:\mathcal H_3\to V_{2k+2}({\mathbb C})$ , fixing $(z,t)$ we may represent it as a polynomial as $F(z,t)=P(X,Y)$ in $V_{2k+2}({\mathbb C})$ , equipped with the action

$$\begin{align*}F\left(z,t\right)\left(\gamma\begin{pmatrix} X\\ Y\end{pmatrix}\right) = P\left(\gamma\begin{pmatrix} X\\ Y\end{pmatrix}\right)= P(aX+bY,cX+dY). \end{align*}$$

Following [Reference PalaciosPal23], for example, given $\gamma \in \mathrm {SL}_2({\mathbb C})$ and a function $F:\mathcal H_3 \rightarrow V_{2k+2}({\mathbb C})$ , we define the slash operator as

(2.1) $$ \begin{align} F|_{\gamma}(z,t)\begin{pmatrix} X\\ Y\end{pmatrix}= |\text{det}(\gamma)|^{-k}F\left(\gamma \cdot (z,t)\right)\left( J\left(\frac{\gamma}{\sqrt{\text{det}(\gamma)}}; (z,t) \right)^{-1}\begin{pmatrix} X \\ Y\end{pmatrix}\right), \end{align} $$

where we note that our J is transposed from that of Palacios. Although we may define Bianchi cusp forms for any congruence subgroup of ${\text {SL}_2(\mathcal {O}_K)}$ , we will only be concerned with those of full level ${\text {SL}_2(\mathcal {O}_K)}$ . For the remainder of our exposition, we set $\Gamma = {\text {SL}_2(\mathcal {O}_K)}.$ A more general definition is given by [Reference WilliamsWil17, Definition 1.2].

We say that a function $F:\mathcal H_3\rightarrow V_{2k+2}({\mathbb C})$ is a cuspidal Bianchi modular form of weight $(k,k)$ and level $\Gamma $ if it satisfies the following properties:

1. (i) $F|_\gamma = F$ for every $\gamma \in \Gamma $ .

2. (ii) $\Psi F = 0$ and $\Psi ' F = 0$ .

3. (iii) F has at worst polynomial growth at each cusp of $\Gamma $ .

4. (iv) $\int _{{\mathbb C}/{\mathcal O}_K} F|_\gamma (z,t) dz = 0$ for every $\gamma \in \Gamma $ .

Here $\Psi , \Psi '$ represent the Casimir operators, which are two elements generating the center of the universal enveloping algebra of the Lie algebra associated with the real Lie group $\textrm {PSL}_2(\mathbb {C})$ . If we do not impose (iv) in the above definition, then we say F is a Bianchi modular form of weight $(k,k)$ and level $\Gamma $ , although we will not be making use of Bianchi modular forms which are not cuspidal. Finally, we denote by $S_{k,k}(\Gamma )$ to be the space of such Bianchi cusp forms.

2.3 Fourier–Bessel expansion

Let $K_n(x)$ be the modified Bessel function that solves the differential equation

$$\begin{align*}\frac{d^2K_n}{dx^2}+\frac1x\frac{dK_n}{dx}-\left(1+\frac{n^2}{x^2}\right)K_n=0. \end{align*}$$

It is well established (see [Reference GhateGha99, Reference PalaciosPal23]) that a cuspidal Bianchi form F has a Fourier–Bessel expansion given by

$$\begin{align*}F(z,t)\begin{pmatrix} X\\ Y\end{pmatrix} = \sum_{n=0}^{2k+2}F_n(z,t)X^{2k+2-n}Y^n, \end{align*}$$

where

$$\begin{align*}F_n(z,t) := t\binom{2k+2}{n}\sum_{\alpha\in K^\times} c(\alpha \delta) \left(\frac{\alpha}{i|\alpha|}\right)^{k+1-n}K_{n-k-1}(4\pi |\alpha| t)e^{2\pi i (\alpha z + \overline{\alpha z})}, \end{align*}$$

$\delta = \sqrt {-D}$ , and the Fourier coefficient $c(I)$ is a function on the fractional ideals of K with $c(I)=0$ for I non-integral.

3.1 The Eichler–Shimura–Harder isomorphism

Let $V_{k,k}=V_{k,k}({\mathbb C})$ denote the complex vector space of polynomials in $X,Y,\overline {X},\overline {Y}$ which are homogeneous of degree k in $X,Y$ and homogeneous of degree k in $\overline {X},\overline {Y}$ . Denote by $\mathcal V_{k,k}$ the locally constant sheaf on $Y_{\Gamma} = \Gamma \backslash \mathbb H$ associated with $V_{k,k}$ . Let $H^1_{\text {cusp}}(Y_{\Gamma} , \mathcal V_{k,k})$ denote the cuspidal cohomology of $Y_{\Gamma} $ . The Eichler–Shimura–Harder isomorphism [Reference HarderHar87] shows that

(3.1) $$ \begin{align} S_{k,k}(\Gamma) \cong H^1_{\text{cusp}}(Y_{\Gamma}, \mathcal V_{k,k}), \end{align} $$

while work by Ghate [Reference GhateGha99, Section 5] gives an explicit form of this isomorphism, which we shall describe.

When K is Euclidean, Karabulut [Reference KarabulutKar22, Section 5] defines the parabolic group cohomology $H^1_{\text {par}}(\Gamma , V_{k,k})\simeq H^1_{\text {cusp}}(Y_{\Gamma} , \mathcal V_{k,k})$ which is the quotient of parabolic cocycles on $\Gamma $ by parabolic coboundaries on $\Gamma $ , which we present in the following section. Karabulut then gives an isomorphism from $H^1_{\text {par}}(\Gamma , V_{k,k})$ to a quotient of an explicit subspace $W_{k,k}\subset V_{k,k}$ , which we denote $\tilde W_{k,k}$ . We will combine these results to obtain a map

(3.2) $$ \begin{align} S_{k,k}(\Gamma) \rightarrow \tilde W_{k,k}, \end{align} $$

generalizing the Eichler integral for classical modular forms into the space of period polynomials.

Williams [Reference WilliamsWil17, Section 2] defines elements of the compactly supported cohomology group $H^1_c(Y_{\Gamma} , \mathcal V_{k,k})$ by identifying it with the space Symb $_{\Gamma} (V_{k,k})$ of Bianchi modular symbols, sending F to $\psi _F$ , where $\psi _F$ is the map from pairs of cusps in $\mathbb {P}^1(K)$ to $V_{k,k}$ defined by

(3.3) $$ \begin{align} \psi_F(r,s) = \int_r^s\omega_F, \end{align} $$

and $\omega _F$ is a $V_{k,k}$ -valued differential form on $\mathcal H_3$ which is defined explicitly. Note that the latter integral gives a perfect pairing between the relevant first homology and cohomology groups. (Williams actually considers spaces $V^*_{k,k}$ , the ${\mathbb C}$ -dual of $V_{k,k}$ , and $V^r_{k,k}$ , and the space $V_{k,k}$ with $\Gamma $ acting from the right; we will use the latter.) The natural image of $H^1_c(Y_{\Gamma} , \mathcal V_{k,k})$ in $H^1(Y_{\Gamma} ,\mathcal V_{k,k})\simeq H^1(\Gamma ,V_{k,k})$ is often denoted $H_!^1(Y_{\Gamma} ,\mathcal V_{k,k})$ [Reference Ash and StevensAS86, Section 1.4] and contains $H^1_{\text {cusp}}(Y_{\Gamma} ,\mathcal V_{k,k})$ [Reference HidaHid94, Section 5] (cf. [Reference Rahm and ŞengünRŞ13, Section 1] for a variation). Putting these together, we have

(3.4) $$ \begin{align} S_{k,k}(\Gamma) \simeq H^1_{\text{cusp}}(Y_{\Gamma}, \mathcal V_{k,k}) \simeq H^1_{\text{par}}(\Gamma, V_{k,k}) \simeq \tilde W_{k,k} \end{align} $$

which we shall make explicit. That is, given a Bianchi modular form F, we can associate to it a cuspidal cohomology class such that its image in Symb $_{\Gamma} (V_{k,k})$ is the modular symbol $\psi _F$ above valued in $V_{k,k}$ , and its image in $V_{k,k}$ can be identified with the quotient $\tilde W_{k,k}$ .

We first give an expression for $\psi _F$ in a form useful to us. Set the binomial coefficient $(\begin {smallmatrix}n\\ m \end {smallmatrix}) = 0$ whenever $m<0$ and $m>n$ .

Lemma 3.1 For $\kappa \in K$ , we have

$$ \begin{align*} &{ \psi_F(\kappa,\infty)=} \sum_{p,q=0}^k \left(\binom{k}{p}\binom{k}{q}\sum_{i=0}^p\sum_{j=0}^q c_{i,j}(\kappa,F)(-1)^{p+q-i-j}\binom{k-i}{p-i} \binom{k-j}{q-j} \kappa^{p-i}\bar{\kappa}^{q-j} \right) \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot Y^{p}X^{k-p}\overline{Y}^{q}\overline{X}^{k-q} \end{align*} $$

where

(3.5) $$ \begin{align} c_{i,j}(\kappa,F) = 2\binom{2k+2}{k+i-j+1}^{-1}(-1)^{k+j+1}\int_0^\infty t^{i+j}F_{k+i-j+1}(\kappa,t) dt. \end{align} $$

Proof This is essentially [Reference WilliamsWil17, Proposition 2.9], with the difference being that we have $V_{k,k}$ in place of $V^*_{k,k}$ . We use the following definition of a slash operator on $V_{k,k}$ . For a polynomial $P\in V_{k,k}$ and a matrix $\gamma = \begin {pmatrix} a & b \\ c & d \end {pmatrix}\in {\text {GL}_2(\mathbb {C})}$ , we define

(3.6) $$ \begin{align} P(X,Y,\overline{X},\overline{Y})|_\gamma = P\left(aX+bY,cX+dY,\overline{a}\overline{X}+\overline{b}\overline{Y}, \overline{c}\overline{X}+\overline{d}\overline{Y}\right). \end{align} $$

By [Reference WilliamsWil17, Proposition 2.7], we have $\psi _F(r,s)=\psi _F(\gamma r,\gamma s)|_\gamma $ , and in particular

(3.7) $$ \begin{align} \int_{\gamma(0)}^{\gamma(\infty)} \omega_F = \int_0^\infty \omega_F \Big|_{\gamma^{-1}}. \end{align} $$

Then, by [Reference WilliamsWil17, Proposition 2.9] we have

$$\begin{align*}\phi_F(\kappa,\infty) = \sum_{p,q=0}^k c_{p,q}(\kappa,F)(\mathcal{Y}-\kappa\mathcal{X})^{k-p}\mathcal{X}^p(\overline{\mathcal{Y}}- \bar{\kappa}\overline{\mathcal{X}})^{k-q}\overline{\mathcal{X}}^q, \end{align*}$$

where $\phi _F=\eta \circ \psi _F$ and $\eta $ is the natural isomorphism from $V^r_{k,k}$ to $V_{k,k}^*$ as given in [Reference WilliamsWil17, Proposition 2.6], with $\mathcal {X},\mathcal {Y},\overline {\mathcal {X}},\overline {\mathcal {Y}}\in V^*_{k,k}$ . We note that the above equality is corrected slightly as explained in [Reference PalaciosPal23, Theorem 3.2].

Rewriting the latter summand as

$$\begin{align*}\left(\sum_{i=0}^p\sum_{j=0}^q c_{i,j}(\kappa,F)(-1)^{p+q-i-j}\binom{k-i}{p-i}\binom{k-j}{q-j} \kappa^{p-i}\bar{\kappa}^{q-j} \right) \mathcal{X}^{p}\mathcal{Y}^{k-p}\overline{\mathcal{X}}^{q}\overline{\mathcal{Y}}^{k-q}, \end{align*}$$

and undoing the isomorphism $\eta $ via the inverse map

$$\begin{align*}\mathcal{X}^{p}\mathcal{Y}^{k-p}\overline{\mathcal{X}}^{q}\overline{\mathcal{Y}}^{k-q} \longmapsto \binom{k}{p}\binom{k}{q}X^{k-p}Y^p\overline{X}^{k-q}\overline{Y}^q \end{align*}$$

then gives the desired expression.

3.2 Bianchi period polynomials

For the rest of this section, we assume $K={\mathbb Q}(\sqrt {-D})$ is Euclidean so that ${D=1,2,3,7,11.}$ For each value of D, define $\omega = i, i\sqrt {2},\frac {-1+i\sqrt {3}}{2},\frac {1+i\sqrt {7}}{2},$ and $\frac {1+i\sqrt {11}}{2}$ , respectively. Furthermore, let

$$\begin{align*}S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},\quad T= \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix},\quad T_\omega= \begin{pmatrix} 1 & \omega \\ 0 & 1 \end{pmatrix},\quad U=TS. \end{align*}$$

When $D=1$ , let $L=\begin {pmatrix} i & 0 \\ 0 & -i \end {pmatrix},$ when $D=3$ , let $L=\begin {pmatrix} \omega ^2 & 0 \\ 0 & \omega \end {pmatrix}$ , and when $D=11$ , let $E=T_\omega ^{-1}ST_\omega ST.$

Let $C(\Gamma , V_{k,k})$ be the space of 1-cocycles on $\Gamma $ as in [Reference KarabulutKar22, Section 5]. Then, we define the subspace $C_p(\Gamma , V_{k,k})$ of parabolic cocycles by

$$\begin{align*}C_p(\Gamma, V_{k,k})=\{f\in C(\Gamma, V_{k,k})| f(T)=f(T_\omega) = f(L)=0\}\end{align*}$$

when $D=1,3$ and

$$\begin{align*}C_p(\Gamma, V_{k,k})=\{f\in C(\Gamma, V_{k,k})| f(T)=f(T_\omega) =0\}\end{align*}$$

when $D=2,7,11.$

Karabulut [Reference KarabulutKar22, Section 5] shows that the evaluation of a cocycle at S is an isomorphism from the parabolic cocycles $C_p(\Gamma )$ to a subspace $W_{k,k}$ of $V_{k,k}$ defined for each D as the subspace of polynomials $P\in V_{k,k}$ satisfying the given relations:

$$ \begin{align*} &D=1, 3: && P|_{(I+S)}=P|_{(I-L)}= P|_{(I+U+U^2)} =P|_{(I+T_\omega SL+(T_\omega SL)^2)}= 0,\\ &D=2: && P|_{(I+S)} = P|_{(I+U+U^2)}=P|_{(I+ST_\omega+T_\omega S+T_\omega^{-1}ST_\omega S)} = 0,\\ &D=7: && P|_{(I+S)} = P|_{(I+U+U^2)}=P|_{(T+ST_\omega+T_\omega ST+ST_\omega^{-1}ST_\omega)} = 0,\\ &D=11: && P|_{(I+S)} = P|_{(I+U+U^2)} =P|_{(T+ST_\omega+TE+ST_\omega E^{-1}+T_\omega ST+ST_\omega^{-1}ST_\omega)} = 0. \end{align*} $$

More generally, we may define $W_{k,k}$ as the subspace of $V_{k,k}$ defined by the relations amongst the generators of $\Gamma $ .

For the isomorphism with $H^1_{\text {par}}(\Gamma , V_{k,k})$ , it is necessary to identify the image of $B_p(\Gamma )$ of parabolic coboundaries, which Karabulut claims has dimension $1$ . We will prove the explicit form of this image, which is also mentioned without proof in [Reference CombesCom24].

Proof Consider an element $f\in B_p(\Gamma )$ . [Reference KarabulutKar22] shows f is a function from $\Gamma $ to $V_{k,k}$ of the form $\gamma \mapsto P|_\gamma - P,$ where $f(T)=f(T_\omega )=0,$ and additionally $f(L)=0$ for the cases where $d=1,3.$ Let $Q(X,\overline {X})=P(X,1,\overline {X},1).$ We will show that Q is constant.

For this to occur, we must, in particular, have

$$\begin{align*}\!\!\!\!\!\!\!0=Q|_T-Q=Q(X+1,\overline{X}+1)-Q(X,\overline{X}), \end{align*}$$

$$\begin{align*}0=Q|_{T_{\omega}}-Q=Q(X+\omega,\overline{X}+\bar{\omega})-Q(X,\overline{X}). \end{align*}$$

Thus, $Q(0,0)=Q(n,n)=Q((a+b\omega )n,(a+b\overline {\omega })n)$ for all $n\in {\mathbb N}$ and fixed $a,b\in {\mathbb N}$ . Thus, shows that $Q(Z,Z)$ and $Q((a+b\omega ) Z, (a+b\bar {\omega })Z)$ are both constant polynomials in Z since they attain the same value infinitely many times. In particular, they are the same constant as seen when $Z=0$ . This shows that

$$\begin{align*}Q(Z,Z)=Q\left(\frac{a+b\omega}{a+b\bar{\omega}}Z,Z\right), \end{align*}$$

so for all $z\in {\mathbb C}^\times $ , $Q(w,z)$ attains the same value infinitely many times as long $a,b\in {\mathbb N}$ are chosen so that $\frac {a+b\omega }{a+b\bar {\omega }}$ is not a root of unity.

Therefore, $Q(X,\overline {X})$ is constant as a polynomial in X. An analogous argument shows it is also constant as a polynomial in $\overline {X}$ , so Q is constant. This shows P is of the form $cY^k\overline {Y}^k$ , for some $c\in {\mathbb C}$ . Then, $f(\gamma )=P|_\gamma -P$ and its image under evaluation by S is thus

$$\begin{align*}(cY^k\overline{Y}^k)|_S-cY^k\overline{Y}^k = -c(Y^k\overline{Y}^k-X^k\overline{X}^k), \end{align*}$$

and so the result follows.

Hence, the evaluation at S map is an isomorphism from $H^1_{\text {par}}(\Gamma , V_{k,k})$ to $W_{k,k}$ modulo the space generated by $X^k\overline {X}^k-Y^k\overline {Y}^k$ , which we denote by $\tilde W_{k,k}$ . We summarize this as follows.

Theorem 3.3 Let K be Euclidean. The map

$$ \begin{align*} {F\mapsto \psi_F(S) =} -2\sum_{p,q=0}^k\binom{k}{p}\binom{k}{q}c_{p,q}(0,F)X^{k-p}Y^{p}\overline{X}^{k-q}\overline{Y}^{q}, \end{align*} $$

where $c_{p,q}(0,F)$ is given in (3.5), is an isomorphism from $S_{k,k}(\Gamma )$ to $\tilde W_{k,k}$ .

Proof We evaluate the Bianchi modular symbol $\psi _F$ in (3.3) at the pair of cusps $(r,s) = (0,\infty )$ to obtain an element of $V_{k,k}$ , whereby

$$\begin{align*}\psi_F(0,\infty)= \int_{0}^{\infty}\omega_F. \end{align*}$$

Denoting this temporarily by P, we then form the 1-cocycle $P|_\gamma - P$ and evaluate at the element $\gamma = S$ to obtain

$$\begin{align*}P|_S - P = \int_{S^{-1}(0)}^{S^{-1}(\infty)}\omega_F - \int_{0}^{\infty}\omega_F = -2\int_{0}^{\infty}\omega_F \end{align*}$$

which is an element of $\tilde W_{k,k}$ . (This realizes the isomorphism (3.4), and note that the latter integral is referred to as the canonical period polynomial associated with F in [Reference CombesCom24, (5.1)].) Then expanding the integral as in Lemma 3.1 with $\kappa =0$ gives the result.

4.1 An intermediate expression

Let $0\le p,q\le k$ be integers. Then, from Lemma 3.1, we may write

$$\begin{align*}\psi_F(\kappa,\infty) = \sum_{p,q=0}^k\binom{k}{p} \binom{k}{q} r_{p,q}(\kappa,F) Y^pX^{k-p}\overline{Y}^{q}\overline{X}^{k-q},\end{align*}$$

where we denote here

(4.1) $$ \begin{align} r_{p,q}(\kappa, F) = \sum_{i=0}^p\sum_{j=0}^q c_{i,j}(\kappa,F)(-1)^{p+q-i-j}\binom{k-i}{p-i}\binom{k-j}{q-j} \kappa^{p-i}\bar{\kappa}^{q-j} \end{align} $$

and $c_{i,j}(\kappa ,F)$ is given in (3.5). We wish to write the coefficients $r_{p,q}(\kappa ,F)$ as an ${\mathcal O}_K$ -linear combination of $c_{i,j}(0,F).$ Notice that when $\kappa =0$ , $r_{p,q}(0,F)=c_{p,q}(0,F).$ When the context is clear, we will denote $r_{p,q}=r_{p,q}(F)=r_{p,q}(0,F) = c_{p,q}(0,F)$ and $r^{\kappa }_{p,q}=r_{p,q}(\kappa ,F)$ .

We first compute more general period polynomials.

Lemma 4.1 Let $\gamma =\begin {pmatrix} a & b \\ c & d \end {pmatrix}\in \Gamma $ such that $\gamma (\infty ) =\infty $ . Then

$$\begin{align*}\psi_{F}(\gamma(0),\gamma(\infty)) = \sum_{p,q=0}^k\binom{k}{p} \binom{k}{q} s_{p,q}(\gamma,F) Y^pX^{k-p}\overline{Y}^{q}\overline{X}^{k-q},\end{align*}$$

where $s_{p,q}(\gamma , F)$ is equal to

$$\begin{align*}\sum_{i,j=0}^{k}(-1)^{p+q-i-j}e_{i,j}^{p,q}r_{i,j}(0,F), \end{align*}$$

and

$$\begin{align*}e_{i,j}^{p,q} = \sum_{u,v\in{\mathbb Z}}\binom{p}{u}\binom{k-p}{i-u}a^{u}b^{p-u}c^{i-u}d^{k-p-i+u}\binom{q}{v}\binom{k-q}{j-v}\bar{a}^{v}\bar{b}^{q-v}\bar{c}^{j-v}\bar{d}^{k-q-j+v}.\end{align*}$$

In particular, $e_{i,j}^{p,q}\in {\mathcal O}_K$ .

Proof Using the identity (3.7), we obtain

$$\begin{align*}\psi_{F}(\gamma(0),\gamma(\infty)) = \int_{\gamma(0)}^{\gamma(\infty)}\omega_{F} =\left(\int_{0}^{\infty}\omega_{F}\right)\bigg|_{\gamma^{-1}}. \end{align*}$$

Expanding the second integral and applying the $\gamma $ action, we have

$$\begin{align*}\sum_{p,q=0}^k \binom{k}{p} \binom{k}{q}r_{p,q}(0,F) (-cX+aY)^p(dX-bY)^{k-p}(-\overline{c}\overline{X}+\overline{a}\overline{Y})^q(\overline{d}\overline{X}-\overline{b}\overline{Y})^{k-q}. \end{align*}$$

The summand can also be written by evaluating the first integral as

$$\begin{align*}\sum_{p,q=0}^k\binom{k}{p} \binom{k}{q}r_{p,q}(\gamma(0),F) Y^{p}X^{k-p}\overline{Y}^{q}\overline{X}^{k-q}. \end{align*}$$

Comparing coefficients, the identity follows.

4.2 Continued fractions

In order to transform the endpoints of the period integral as desired, we turn to the Hurwitz continued fractions algorithm for Euclidean imaginary quadratic fields, given in [Reference KarabulutKar22]. Let $\kappa \in K$ , and let $\lfloor {\kappa _{n}}\rceil $ denote the element of ${\mathcal O}_K$ nearest to $\kappa _n$ in the complex plane with respect to Euclidean distance; to break ties, we round down in both the real and imaginary components. Then, our continued fraction algorithm is of the form

$$\begin{align*}\kappa_{0}=\kappa, \; \beta_{n}=\lfloor{\kappa_{n}}\rceil, \; \kappa_{n+1}=\frac{1}{\kappa_{n}-\beta_{n}},\end{align*}$$

with $\beta _n\in {\mathcal O}_K$ , and for any $\kappa \in K$ is guaranteed to eventually terminate at some point where $\kappa _m=\beta _m$ for some m. We define the n-th convergent of the continued fraction for $\kappa $ as $\mu _n/\nu _n=\left [\beta _{1},\beta _{2},\ldots ,\beta _{n}\right ]$ , where

(4.2) $$ \begin{align} \begin{aligned} \mu_{-2}=0,\hspace{1em} \mu_{-1}=1,\hspace{1em}\mu_{n}=\beta_{n}\mu_{n-1}+\mu_{n-2} \\ \nu_{-2}=1,\hspace{1em} \nu_{-1}=0,\hspace{1em} \nu_{n}=\beta_{n}\nu_{n-1}+\nu_{n-2} \end{aligned} \end{align} $$

and $\lim _{n\to \infty } \frac {\mu _{n}}{\nu _{n}}=\kappa $ . Hence, for $\kappa =\mu _m/\nu _m$ , also denoted as $\kappa =\mu /\nu $ , we express

$$\begin{align*}\frac{\mu}{\nu} =\beta_{0}+\frac{1}{\beta_{1}+\frac{1}{\beta_{2}+\frac{1}{\ddots+\frac{1}{\beta_{m}}}}} =\left[\beta_{1},\beta_{2},\ldots,\beta_{m}\right]. \end{align*}$$

Analogous to the classical algorithm as in [Reference Cohen and StrömbergCS17], we find it useful to express the recursion relations of (4.2) in matrix form, with

$$ \begin{align*} g_{0}=\begin{pmatrix} 1 & \beta_{0} \\ 0 & 1 \end{pmatrix},\quad g_{n}=\begin{pmatrix} (-1)^{n}\mu_{n-1} & \mu_{n} \\ (-1)^{n}\nu_{n-1} & \nu_{n} \end{pmatrix} ,\quad g_{n}=g_{n-1}\begin{pmatrix} 0 & (-1)^{n-1} \\ (-1)^{n} & \beta_{n} \end{pmatrix}. \end{align*} $$

Note that det $(g_n)=1$ , and hence $g_n\in \Gamma $ , for all n. Because we retain the same recursion matrices as the classical algorithm with $g_n(0)=\mu _n/\nu _n$ and $g_n(\infty )=\mu _{n-1}/\nu _{n-1}=g_{n-1}(0)$ , we can immediately conclude

(4.3) $$ \begin{align} \sum_{n=0}^m \int_{g_{n}(0)}^{g_{n}(\infty)}\omega_F=\int_{\frac{\mu_m}{\nu_m}}^{\frac{\mu_{-1}}{\nu_{-1}}} \omega_F=\int_{\frac{\mu}{\nu}}^{\infty}\omega_F. \end{align} $$

4.3 Integral formulas

We now show that the coefficients $r_{p,q}(\kappa ,F)$ can be given as an ${\mathcal O}_K$ -linear combination of periods $r_{i,j}(0,F)$ . From there, we also show that the $c_{p,q}(\kappa ,F)$ coefficients can be given as a linear combination of the $r_{p,q}(\kappa ,F)$ coefficients.

Lemma 4.2 For $\kappa \in K$ , we have

$$\begin{align*}\psi_F(\kappa,\infty) = \sum_{p,q=0}^k\binom{k}{p} \binom{k}{q} r_{p,q}(\kappa,F) Y^pX^{k-p}\overline{Y}^{q}\overline{X}^{k-q},\end{align*}$$

where $r_{p,q}(\kappa ,F)$ is an ${\mathcal O}_K$ -linear combination of $r_{i,j}(0,F)$ .

Proof By (4.3), we can write $\psi _{F}(\kappa ,\infty )$ as

$$ \begin{align*} \int_{\frac{\mu}{\nu}}^{\infty}\omega_F=\sum_{n=0}^m\int_{g_n(0)}^{g_n(\infty)}\omega_{F} &=\sum_{n=0}^m\sum_{p,q=0}^k \binom{k}{p} \binom{k}{q} s_{p,q}(g_n,F) Y^{p}X^{k-p}\overline{Y}^{q}\overline{X}^{k-q} \\ &= \sum_{p,q=0}^k \binom{k}{p} \binom{k}{q} r_{p,q}(\kappa,F) Y^pX^{k-p}\overline{Y}^q\overline{X}^{k-q}, \end{align*} $$

equating coefficients and using Lemma 4.1 shows $r_{p,q}(\kappa ,F)$ is equal to

(4.4) $$ \begin{align} \sum_{n=0}^m \sum_{i,j=0}^{k} (-1)^{p+q-i-j} r_{i,j}(0,F) (e_{i,j}^{p,q})_n, \end{align} $$

where $(e_{i,j}^{p,q})_n\in {\mathcal O}_K$ is equal to

$$ \begin{align*} \sum_{u,v \in {\mathbb Z}} (-1)^{n(i+j)}\binom{p}{u}\mu_{n-1}^u&\mu_n^{p-u}\binom{k-p}{i-u}\nu_{n-1}^{i-u} \nu_n^{k-p-i+u}\\ & \binom{q}{v}\overline{\mu_{n-1}}^v\overline{\mu_n}^{q-v}\binom{k-q}{j-v} \overline{\nu_{n-1}}^{j-v}\overline{\nu_n}^{k-q-j+v}. \end{align*} $$

Recalling from (4.2) that $\mu _n,\nu _n\in {\mathcal O}_K$ for all n, we have that $r_{p,q}(\kappa ,F)$ is an ${\mathcal O}_K$ -linear combination of the periods $r_{i,j}(0,F)$ .

We can now give a formula for the coefficients $c_{p,q}(\kappa ,F)$ in terms of periods $r_{i,j}(\kappa ,F)$ as follows.

Lemma 4.3 For $\kappa \in {\mathcal O}_K$ (alternatively K), the coefficients $c_{p,q}(\kappa ,F)$ are an ${\mathcal O}_K$ -linear (K-linear) combination of periods $r_{i,j}(\kappa ,F)$ , given explicitly by the formula

$$\begin{align*}c_{p,q}(\kappa,F)=\sum_{i=0}^p\sum_{j=0}^q \binom{k-i}{p-i}\binom{k-j}{q-j}\kappa^{p-i}\bar{\kappa}^{q-j}r_{i,j}(\kappa,F). \end{align*}$$

Proof By (4.1), we have

$$\begin{align*}r_{p,q}(\kappa,F) = \sum_{i=0}^p\sum_{j=0}^q c_{i,j}(\kappa,F)\binom{k-i}{p-i}\binom{k-j}{q-j} (-\kappa)^{p-i}(-\bar{\kappa})^{q-j}. \end{align*}$$

To invert this linear transformation, consider the polynomial

$$ \begin{align*} &\sum_{p,q=0}^k r_{p,q}(\kappa,F) Y^pX^{k-p}\overline{Y}^q\overline{X}^{k-q} \\ &= \sum_{p,q=0}^k\left(\sum_{i=0}^p\sum_{j=0}^q c_{i,j}(\kappa,F)\binom{k-i}{p-i}\binom{k-j}{q-j} (-\kappa)^{p-i}(-\bar{\kappa})^{q-j}\right)Y^pX^{k-p}\overline{Y}^q\overline{X}^{k-q} \\ &= \sum_{p,q=0}^k c_{p,q}(\kappa,F)(X-\kappa Y)^{k-p}Y^{p}(\overline{X}-\bar{\kappa}\overline{Y})^{k-q}\overline{Y}^{q}. \end{align*} $$

Mapping $X \mapsto X+\kappa Y$ and $\overline {X} \mapsto \overline {X}+\bar {\kappa }\overline {Y}$ yields

$$ \begin{align*} &\sum_{p,q=0}^k c_{p,q}(\kappa,F)X^{k-p}Y^p\overline{X}^{k-q}\overline{Y}^q \\ &= \sum_{p,q=0}^k r_{p,q}(\kappa,F)(X+\kappa Y)^{k-p}Y^{p}(\overline{X}+\bar{\kappa}\overline{Y})^{k-q}\overline{Y}^{q} \\ &= \sum_{p,q=0}^k \left(\sum_{i=0}^p\sum_{j=0}^q \binom{k-i}{p-i}\binom{k-j}{q-j}\kappa^{p-i}\bar{\kappa}^{q-j}r_{i,j}(\kappa,F)\right)Y^pX^{k-p}\overline{Y}^q\overline{X}^{k-q}, \end{align*} $$

from which equating coefficients gives the desired result.

6.1 Combinatorial lemmas

We make use of our previous discussion on Hecke operators to prove an analog of Manin’s rationality result for periods of classical modular forms. We will prove the main Theorem 1.1 following the strategy of the proof of the classical case [Reference Cohen and StrömbergCS17, Theorem 11.11.2]. First, for a cusp form F, we define

$$\begin{align*}r(F)=\left(r_{0,0}(F),r_{0,1}(F),r_{0,2}(F), \dots , r_{k,k}(F)\right)^{\mathsf{T}}, \end{align*}$$

where we interpret $r(F)$ as a column vector mapping $S_{k,k}(\Gamma )$ to ${\mathbb C}^{(k+1)^2}$ . For any ${\mathfrak n} = n{\mathcal O}_K$ , note that since $r_{p,q}(T_{{\mathfrak n}}F)$ is an ${\mathcal O}_K$ -linear combination of $r_{i,j}(F)$ , then there exists some matrix $A(n)$ with entries in ${\mathcal O}_K$ such that $r(T_{{\mathfrak n}}F)=A(n)r(F)$ . Additionally, let

similar to the standard sum of divisors function.

First, we note some observations about $r_{0,0}(T_{{\mathfrak n}}F)$ in terms of the following lemma.

Proof By Theorem 5.1, we have

Let $b_\ell /d_\ell $ represent the $\ell $ th convergent of $b/d$ and let m be the index such that $b_m/d_m=b/d.$ When $p=q=0$ , the proof of Lemma 4.2 gives

$$ \begin{align*} r_{0,0}^{b/d} &= \sum_{\ell=0}^m \sum_{i,j=0}^{k} \left[(-1)^{(n-1)(i+j)} \binom{k}{i}d_{\ell-1}^{i}d_\ell^{k-i}\binom{k}{j}\overline{d_{\ell-1}}^{j} \overline{d_\ell}^{k-j}\right]r_{i,j}(F); \end{align*} $$

putting these results together gives that $r_{0,0}(T_{{\mathfrak n}}F)$ is equal to

In particular, the coefficient of $r_{0,0}(F)$ within the linear combination for $r_{0,0}(T_{{\mathfrak n}}F)$ is given by

and the coefficient of $r_{k,k}(F)$ within the linear combination for $r_{0,0}(T_{{\mathfrak n}}F)$ is given by

Finally, note that since $\psi (F)\in W_{k,k}$ , we have $\psi (F)|_{(I+S)}=0$ . This implies $r_{0,0}(F)=r_{k,k}(F)$ and so the difference between the coefficients are given by

At this point, let

Note that since ${\mathcal O}_K$ is a Euclidean domain,

is also equal to $|({\mathcal O}_K/n{\mathcal O}_K)^\times |$ , and hence, is multiplicative by the Chinese Remainder Theorem for number fields. We claim that

Let $\pi $ denote a prime of ${\mathcal O}_K$ , and $e_\pi $ the largest power of $\pi $ dividing n. We have

Since the norm is multiplicative, it suffices to show that

We then compute

by

$$ \begin{align*} &\left|\left\{ m \in {\mathcal O}_K: N(m)<N(\pi^\nu) \right\}\right| - \left|\left\{ m \in {\mathcal O}_K: N(m)<N(\pi^\nu), \pi | m \right\}\right| \\ &= \left|\left\{ m \in {\mathcal O}_K: N(m)<N(\pi^\nu) \right\}\right| - \left|\left\{ m \in {\mathcal O}_K: N(m)<N(\pi^{\nu-1}) \right\}\right| \\ &= N(\pi^\nu) - N(\pi^{\nu-1}), \end{align*} $$

and so

Using this, we get our desired result as

6.2 Proof of Theorem 1.1

We are now ready to prove the rationality theorem for the Bianchi case. Let F be a normalized Hecke-eigenform. It follows that $T_{{\mathfrak n}}(F)=c(n\delta )(F)$ , with $\delta = \sqrt {-D}$ . Thus,

$$\begin{align*}r(F) \in E_{F}= \bigcap_{{ n\in {\mathcal O}_K\backslash\{0\}}} \text{Ker}(A(n)-c(n\delta)\textit{I}), \end{align*}$$

defined over $K(F)$ . We shall identify the algebraic closure of K with a subset of ${\mathbb C}$ , so that r maps $S_{k,k}(\Gamma )$ to a subspace $R_k$ of ${\mathbb C}^{(k+1)^2}$ , where the relations defining this subspace are completely determined by the action of $\Gamma $ on periods. It follows then that we can identify $R_k$ with $W_{k,k}$ , and moreover $r(F) \in E_F \cap W_{k,k}$ , a ${\mathbb C}$ -vector space.

Proof By Theorem 3.3, we have a map from $S_{k,k}(\Gamma )$ to the space of period polynomials $\tilde W_{k,k} = W_{k,k}/U,$ where U is spanned by $X^k\overline {X}^k-Y^k\overline {Y}^k$ . This can be represented by the coefficient vector $(x, 0, \ldots , 0, -x)^{\mathsf {T}}$ generated by the cocycle $\gamma \mapsto (1|_{\gamma }-1)$ . Thus, if $v\in E_F \cap W_{k,k}$ , then there exists a Bianchi cusp form $G\in S_{k,k}(\Gamma )$ such that

$$\begin{align*}v=r(G)+(x, 0, \ldots , 0, -x)^{\mathsf{T}}. \end{align*}$$

Consider the first component of the product

$$\begin{align*}A(n)(x,0,\dots, -x)^{\mathsf{T}} = A(n)(v-r(G)).\end{align*}$$

From Lemma 6.1, this is equal to . However, we also see that

$$\begin{align*}A(n)(v-r(G))=A(n)v-A(n)r(G)=a(n)v-r(T_{{\mathfrak n}}G),\end{align*}$$

where $a(n)$ is the n-th Fourier coefficient and $T_{{\mathfrak n}}$ eigenvalue of G, and has the first component

$$\begin{align*}a(n)(r_{0,0}(G)+x)-r_{0,0}(T_{{\mathfrak n}}G).\end{align*}$$

By the recent proof of the Ramanujan conjecture for cuspidal Bianchi eigenforms [Reference Boxer, Calegari, Gee, Newton and ThorneBCG+25], we have that $|a(n)|=O(N(n)^{(k+1)/2})$ (though any weaker nontrivial bound suffices). Furthermore, it is known [Reference GarrettGar18, Theorem 2.6.2] that

$$\begin{align*}G=\sum_{i=1}^g \lambda_i F_i \end{align*}$$

for $F_i$ normalized eigenforms. Therefore,

$$\begin{align*}r_{0,0}(T_{{\mathfrak n}}G)=\sum_{i=1}^g \lambda_i r_{0,0}(T_{{\mathfrak n}}F_i) = \sum_{i=1}^g \lambda_i a_i(n)r_{0,0}(F_i), \end{align*}$$

where $a_i(n)$ is the Fourier coefficient of $F_i$ , which shows that $|r_{0,0}(T_{{\mathfrak n}}G)| = O(N(n)^{(k+1)/2})$ also. But since , this is a contradiction unless $x=0$ . Therefore, $v=r(G)$ for some $G\in S_{k,k}(\Gamma )$ .

We now complete the proof of the main theorem. First, assume that K is Euclidean. If $v\in E_F \cap W_{k,k}$ , then by Lemma 6.2 there exists a Bianchi cusp form $G\in S_{k,k}(\Gamma )$ such that $v=r(G)$ . Since $v\in E_F$ , we have by definition that $A(n)v = c(n\delta )v$ , and by the definition of $A(n)$ that

$$\begin{align*}r\left(T_{{\mathfrak n}}G-c(n\delta)G\right) = 0.\end{align*}$$

On the other hand, it follows from our realization of the Eichler–Shimura–Harder in Theorem 3.3 and the proof of the previous lemma that $S_{k,k}(\Gamma )$ is isomorphic to a codimension 1-subspace of $R_k$ not containing the vector $(x,0,\dots , -x)^{\mathsf {T}} $ . In particular, r is injective and thus

$$\begin{align*}T_{{\mathfrak n}}G=c(n\delta)G.\end{align*}$$

It follows by multiplicity one (e.g., [Reference HidaHid94, p. 432]) that if two eigenfunctions for all $T_{{\mathfrak n}}$ have the same eigenvalues, then they must be scalar multiples of one another, so $G=\lambda F$ for some $\lambda \in {\mathbb C} $ . Thus, every element of $E_F \cap W_{k,k}$ has the form $r(\lambda F) = \lambda r(F)$ by the definition of r and the properties of $r_{i,j}$ . But from this it follows that $ E_F \cap W_{k,k}$ has dimension one, and hence

$$\begin{align*}\langle r(F)\rangle_{\mathbb C} = E_F \cap W_{k,k}. \end{align*}$$

Note however that $E_F$ , $W_{k,k}$ , and $T_{{\mathfrak n}}$ are all defined over $K(F)$ . Thus, for each $r_{p,q} (F),$ there exists $\Omega \in {\mathbb C}^{\times }$ such that

$$\begin{align*}\frac{1}{\Omega}r_{p,q}(F) \in K(F), \forall \text{ } 0 \leq p,q \leq k. \end{align*}$$

Remark 6.3 Let K be an arbitrary imaginary quadratic field with class number h. We briefly outline the idea of the proof of the main Theorem 1.1. Consider the intersection

$$\begin{align*}E^{\prime}_{F}= \bigcap_{(0)\neq {\mathfrak n} \subset {\mathcal O}_K} \text{Ker}(T_{{\mathfrak n}}-c({\mathfrak n})) \subset \text{Symb}_{\mathrm{GL}_2(\hat{\mathcal{O}}_K)}(V_{k,k}) \end{align*}$$

in the space of modular symbols valued in $V_{k,k}$ , with notation as in [Reference WilliamsWil17, Section 2]. Let $W_{k,k}$ be the subspace of $V_{k,k}$ determined by the Eichler–Shimura–Harder isomorphism, evaluating (3.3) at the pair of cusps $(r,s) = (0,\infty )$ as before. Again by multiplicity one and Eichler–Shimura–Harder, $E^{\prime }_{F}\cap W_{k,k}$ is one-dimensional and spanned by $\psi _F=(\psi _{F^1},\dots ,\psi _{F^h}),$ where $F^1,\dots ,F_h$ are the components of the Bianchi modular form F as in [Reference WilliamsWil17, Proposition 2.7]. As before, the action of $T_{{\mathfrak n}}$ on modular symbols over $K(F)$ is again defined over $K(F)$ , so the intersection

$$\begin{align*}\text{Symb}_{\mathrm{GL}_2(\hat{\mathcal{O}}_K)}(V_{k,k}(K(F)))\cap E^{\prime}_F \end{align*}$$

is again one-dimensional and spanned by $\psi _F/\Omega $ for some fixed $\Omega \in {\mathbb C}^\times $ (see [Reference WilliamsWil17, Proposition 2.12]). We then obtain $\psi _{F^i}(0,\infty )/\Omega \in V_{k,k}(K(F))$ for each $i = 1,\dots ,h$ .