Explicit Eichler-Shimura for GL2 over imaginary quadratic fields

Let {\mathbf{H}} denote the Hamilton quaternions. Set

\displaystyle \begin{array}{rcl} \mathfrak{H}^{3} & = & \left\{ z=x+jy\mid x\in\mathbf{C}\,\mathrm{and}\, y\in\mathbf{R}_{>0}\right\} \end{array}

Let {\mathbf{i}} be the point with {x=0}, {y=1}. There is a transitive action of {\mathrm{SL}_{2}(\mathbf{C})} given by {\gamma z=(az+b)(cz+d)^{-1}} with multiplication taking place in {\mathbf{H}}. This extends uniquely to an action of {\mathrm{GL}_{2}(\mathbf{C})} by requiring that the center {Z(\mathbf{C})} act trivially. The stabilizer of {\mathbf{i}} is exactly {Z(\mathbf{C})\mathrm{SU}_{2}}, so we get an identification

\displaystyle \begin{array}{rcl} \mathrm{GL}_{2}(\mathbf{C})/Z(\mathbf{C})\mathrm{SU}_{2} & \cong & \mathfrak{H}^{3}\\ gZ(\mathbf{C})\mathrm{SU}_{2} & \mapsto & g\cdot\mathbf{i}.\end{array}

Fix a triple {\lambda=(k,w,w_{c})} with {k\in\mathbf{Z}_{\geq0}} and {w,w_{c}\in k+2\mathbf{\mathbf{Z}}}. Set {\omega_{\lambda}(z)=z^{w}\overline{z}^{w_{c}}}. Let {\mathcal{L}_{k}=\mathbf{C}[X]^{\deg\leq k}\otimes\mathbf{C}[X_{c}]^{\deg\leq k}} with {\mathrm{GL}_{2}(\mathbf{C})}-action

\displaystyle (g\cdot_{\mathcal{L}}p)(X,X_{c})=(a+cX)^{k}(\overline{a}+\overline{c}X_{c})^{k}p\left(\frac{b+dX}{a+cX},\frac{\overline{b}+\overline{d}X_{c}}{\overline{a}+\overline{c}X_{c}}\right).

Note that {\mathcal{L}_{k}} has central character {\mathrm{diag}(z,z)\mapsto(z\overline{z})^{k}}. Set {\mathcal{L}_{\lambda}=\mathcal{L}_{k}\otimes\det^{\frac{w-k}{2}}\overline{\det}^{\frac{w_{c}-k}{2}}}, so {\mathcal{L}_{\lambda}} has central character {\omega_{\lambda}}.

Let {W_{m}} be the right representation of {\mathrm{SL}_{2}(\mathbf{C})} on {\mathbf{C}[U,V]^{\mathrm{homog.\, of\, deg.}\, m}} given by {(p\cdot g)(U,V)=p(aU+bV,cU+dV)}. There is a unique {\mathbf{C}}-bilinear pairing {\left\langle ,\right\rangle _{m}:W_{m}\times W_{m}\rightarrow\mathbf{C}} such that {\left\langle U^{m},V^{m}\right\rangle _{m}=1} and {\left\langle p_{1}\cdot g,p_{2}\cdot g\right\rangle _{m}=\left\langle p_{1},p_{2}\right\rangle _{m}} for all {g\in\mathrm{SL}_{2}(\mathbf{C})}.

Definition. Let {F\subset\mathbf{C}} be an imaginary quadratic field. A cuspidal automorphic form of weight {\lambda} and level {K\subset\mathrm{GL}_{2}(\mathbf{A}_{F,f})} is a function {\mathbf{f}:\mathrm{GL}_{2}(\mathbf{A}_{F})\rightarrow W_{2k+2}} such that

\displaystyle \mathbf{f}(\gamma gz_{\infty}k_{\infty}k_{f})=\omega_{\lambda}(z_{\infty})^{-1}\mathbf{f}(g)\cdot_{W}k_{\infty}

for all {\gamma\in\mathrm{GL}_{2}(F)}, {g\in\mathrm{GL}_{2}(\mathbf{A}_{F})}, {z_{\infty}\in Z(\mathbf{C})}, {k_{\infty}\in\mathrm{SU}_{2}}, and {k_{f}\in K}, and such that {\mathbf{f}} satisfies the usual smoothness and cuspidality conditions, and has Casimir eigenvalues {k+\frac{k^{2}}{2}}. Let {S_{\lambda}(K)\subset\mathcal{C}^{\infty}(\mathrm{GL}_{2}(\mathbf{A}_{F}))\otimes_{\mathbf{C}}W_{2k+2}} denote the space of such functions.

Note that if {\mathbf{f}\in S_{\lambda}(K)} is a Hecke eigenform, say with {T_{v}\mathbf{f}=a_{v}(\mathbf{f})\mathbf{f}} for almost all places {v}, there is a unique cuspidal automorphic representation {\pi(\mathbf{f})=\pi_{\infty}\otimes\bigotimes_{v<\infty}\pi_{v}} such that for almost all {v}, {\pi_{v}^{\mathrm{GL}_{2}(\mathcal{O}_{v})}} is one-dimensional and {T_{v}} acts through the scalar {a_{v}(\mathbf{f})}. Furthermore, {\pi_{\infty}} satisfies {H^{\ast}(\mathfrak{g},K_{\infty};\pi_{\infty}\otimes\mathcal{L}_{\lambda})\neq0} (and this determines {\pi_{\infty}} uniquely in terms of {\lambda}). In fact, we have

\displaystyle S_{\lambda}(K)\simeq\oplus_{\pi\,\mathrm{with}H^{\ast}(\mathfrak{g},K_{\infty};\pi_{\infty}\otimes\mathcal{L}_{\lambda})\ne0}\pi_{f}^{K}\otimes\pi_{\infty}^{\mathrm{min}}

as {\mathbf{T}(\mathrm{GL}_{2}(\mathbf{A}_{F,f}),K)\times\mathrm{SU}_{2}}-modules, where {\pi_{\infty}^{\mathrm{min}}\simeq W_{2k+2}} is the lowest {\mathrm{SU}_{2}}-type of {\pi_{\infty}}. Furthermore, fixing {\iota:\overline{\mathbf{Q}_{p}}\overset{\sim}{\rightarrow}\mathbf{C}}, there should be a Galois representation {\rho_{\mathbf{f}}:G_{F}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{Q}_{p}})} such that {\iota\mathrm{tr}\rho_{\mathbf{f}}(\mathrm{Frob}_{v})=a_{v}(\mathbf{f})}, and my normalization of things implies that {\rho_{\mathbf{f}}} should have Hodge-Tate weights {\{\frac{w-k}{2},\frac{w+k}{2}+1\}} at one embedding and {\left\{ \frac{w_{c}-k}{2},\frac{w_{c}+k}{2}+1\right\} } at the other.

Let {Y(K)=\mathrm{GL}_{2}(F)\backslash\left(\mathfrak{H}^{3}\times\mathrm{GL}_{2}(\mathbf{A}_{F,f})\right)/K}. My goal here is to explicitly define a canonical Hecke-equivariant linear injection

\displaystyle S_{\lambda}(K)\hookrightarrow H_{c}^{1}\left(Y(K),\mathcal{L}_{\lambda}\right).

I hope the actual construction below will illuminate my use of the word “explicit”. Some version of this kind of thing lurks in the literature in many places, e.g. in a paper of Hida referenced below and in Urban’s thesis (Compositio. 99, 1995) (and in many other papers, but these were my primary references). A version of this over general number fields will appear in my book on p-adic L-functions, but the imaginary quadratic case is the key thing to understand, since once you understand how to do one complex place you just tensor madly over all complex places, and real places are easy.

For {g\in\mathrm{SL}_{2}(\mathbf{C})} and {z\in\mathfrak{H}^{3}}, set {j(g,z)=\left(\begin{array}{cc} cx+d & -cy\\ \overline{c}y & \overline{cx+d}\end{array}\right)}. A straightforward calculation gives {j(gh,z)=j(g,hz)j(h,z)}. It’s not hard to see that {j} extends to a well-defined one-cocycle {j:\mathrm{GL}_{2}(\mathbf{C})\times\mathfrak{H}^{3}\rightarrow\mathrm{GL}_{2}(\mathbf{C})/\{\pm I\}} by setting {j(g,z)=\pm I} for {g\in Z(\mathbf{C})}. If {u=\left(\begin{array}{cc} \alpha & \beta\\ -\overline{\beta} & \overline{\alpha}\end{array}\right)\in\mathrm{SU}_{2}}, then {j(u,\mathbf{i})=\left(\begin{array}{cc} \overline{\alpha} & \overline{\beta}\\ -\beta & \alpha\end{array}\right)=u^{-t}}.

Set {dz=\left(\begin{array}{c} -d\overline{x}\\ dt\\ dx\end{array}\right)} and {\eta(z)=(dz)^{t}}. Following Hida’s 1993 Duke paper “p-ordinary cohomology groups for {\mathrm{SL}(2)} over number fields”, define {\rho_{2}:\mathrm{GL}_{2}\rightarrow\mathrm{GL}_{3}} by

\displaystyle \rho_{2}\left(\begin{array}{cc} a & b\\ c & d\end{array}\right)\left(\begin{array}{c} S^{2}\\ ST\\ T^{2}\end{array}\right)=\left(\begin{array}{c} (aS+bT)^{2}\\ (aS+bT)(cS+dT)\\ (cS+dT)^{2}\end{array}\right).

Set {g^{\iota}=g^{-1}\det g}. According to a calculation in Hida’s paper, we have {dz=\rho_{2}(j(g,z)^{\iota})d(gz)} for any {g\in\mathrm{SL}_{2}(\mathbf{C})}, so trivially {\eta(z)=\eta(gz)\rho_{2}(j(g,z)^{\iota})^{t}}. The cocycle {J(g,z)=\rho_{2}(j(g,z)^{\iota})^{t}} extends to {g\in\mathrm{GL}_{2}(\mathbf{C})} by requiring the center to act trivially ({\rho_{2}} kills the ambiguity caused by {\mathrm{SL}_{2}(\mathbf{C})\cap Z(\mathbf{C})=\{\pm I\}}), and the formula {\eta(z)=\eta(gz)J(g,z)} then remains true for all {g\in\mathrm{GL}_{2}(\mathbf{C})}.
Define functions {\Phi_{\varepsilon}(g)\in\mathcal{C}^{\infty}(\mathrm{GL}_{2}(\mathbf{C}))\otimes\mathcal{L}_{k}\otimes W_{2k+2}} for {\varepsilon\in\{1,0,-1\}} by setting

\displaystyle \left((\begin{array}{cc} 1 & X\end{array})g\left(\begin{array}{c} U\\ V\end{array}\right)\right)^{k}\left((\begin{array}{cc} 1 & X_{c}\end{array})\overline{g}\left(\begin{array}{c} V\\ -U\end{array}\right)\right)^{k}\left((\begin{array}{cc} A & B\end{array})j(g,\mathbf{i})^{\iota t}\left(\begin{array}{c} U\\ V\end{array}\right)\right)^{2}=\left(\begin{array}{c} A^{2}\\ AB\\ B^{2}\end{array}\right)^{t}\left(\begin{array}{c} \Phi_{1}(g)\\ \Phi_{0}(g)\\ \Phi_{-1}(g)\end{array}\right).

Proposition. The functions {\Phi_{\varepsilon}(g)} satisfy the equivariance formula {\Phi_{\varepsilon}(gu)=\Phi_{\varepsilon}(g)\cdot_{W}u} for any {u\in\mathrm{SU}_{2}}.

Proof. Easy calculation; note that {j(gu,\mathbf{i})^{\iota t}=j(g,\mathbf{i})^{\iota t}u}. {\square}

Proposition. The functions {\Phi_{\varepsilon}(g)} satisfy the equivariance formula

\displaystyle \left(\begin{array}{c} \Phi_{1}(hg)\\ \Phi_{0}(hg)\\ \Phi_{-1}(hg)\end{array}\right)=J(h,g\mathbf{i})\left(\begin{array}{c} h\cdot_{\mathcal{L}_{k}}\Phi_{1}(g)\\ h\cdot_{\mathcal{L}_{k}}\Phi_{0}(g)\\ h\cdot_{\mathcal{L}_{k}}\Phi_{-1}(g)\end{array}\right)

for any {h\in\mathrm{GL}_{2}(\mathbf{C})}.

Proof. Set

\displaystyle P_{k}(g)=\left((\begin{array}{cc} 1 & X\end{array})g\left(\begin{array}{c} U\\ V\end{array}\right)\right)^{k}\left((\begin{array}{cc} 1 & X^{c}\end{array})\overline{g}\left(\begin{array}{c} V\\ -U\end{array}\right)\right)^{k}.

An easy calculation gives

\displaystyle \left((\begin{array}{cc} U & V\end{array})h\left(\begin{array}{c} A\\ B\end{array}\right)\right)^{2}=(\begin{array}{ccc} U^{2} & 2UV & V^{2}\end{array})\rho_{2}(h)\left(\begin{array}{c} A^{2}\\ AB\\ B^{2}\end{array}\right),

so replacing {h} by {j(g,\mathbf{i})^{\iota}} and taking transposes everywhere we obtain

\displaystyle \left((\begin{array}{cc} A & B\end{array})j(g,\mathbf{i})^{\iota t}\left(\begin{array}{c} U\\ V\end{array}\right)\right)^{2}=\left(\begin{array}{c} A^{2}\\ AB\\ B^{2}\end{array}\right)^{t}J(g,\mathbf{i})\left(\begin{array}{c} U^{2}\\ 2UV\\ V^{2}\end{array}\right).

Comparing this to the definition of {\Phi_{\varepsilon}} gives

\displaystyle \left(\begin{array}{c} \Phi_{1}(g)\\ \Phi_{0}(g)\\ \Phi_{-1}(g)\end{array}\right)=J(g,\mathbf{i})\left(\begin{array}{c} U^{2}P_{k}(g)\\ 2UVP_{k}(g)\\ V^{2}P_{k}(g)\end{array}\right),

and now the result is clear, since {J(hg,\mathbf{i})=J(h,g\mathbf{i})J(g,\mathbf{i})} and {P_{k}(hg)=h\cdot_{\mathcal{L}_{k}}P_{k}(g)}. {\square}

Proposition. The function

\displaystyle \begin{array}{rcl} \Phi_{\varepsilon}(\mathbf{f})(z,g_{f}) & = & \left\langle \mathbf{f}(g_{z},g_{f}),\Phi_{\varepsilon}(g_{z})\right\rangle _{2k+2}(\det g_{z})^{\frac{w-k}{2}}(\overline{\det g_{z}})^{\frac{w_{c}-k}{2}}\\ & \in & \mathcal{C}^{\infty}(\mathfrak{H}^{3}\times\mathrm{GL}_{2}(\mathbf{A}_{F,f}))\otimes\mathcal{L}_{\lambda}\end{array}

is well-defined, where {g_{z}\in\mathrm{GL}_{2}(\mathbf{C})} is any element such that {g_{z}\cdot\mathbf{i}=z}.

Proof. The element {g_{z}} is well-defined up to right multiplication by an arbitrary element {z_{\infty}k_{\infty}\in Z(\mathbf{C})\mathrm{SU}_{2}}. Set

\displaystyle \Phi_{\varepsilon}(\mathbf{f})(g_{\infty},g_{f})=\left\langle \mathbf{f}(g_{\infty},g_{f}),\Phi_{\varepsilon}(g_{\infty})\right\rangle _{2k+2}(\det g_{\infty})^{\frac{w-k}{2}}(\overline{\det g_{\infty}})^{\frac{w_{c}-k}{2}}.

We have

\displaystyle \begin{array}{rcl} \Phi_{\varepsilon}(\mathbf{f})(g_{\infty}k_{\infty},g_{f}) & = & \left\langle \mathbf{f}(g_{\infty}k_{\infty},g_{f}),\Phi_{\varepsilon}(g_{\infty}k_{\infty})\right\rangle _{2k+2}(\det g_{\infty})^{\frac{w-k}{2}}(\overline{\det g_{\infty}})^{\frac{w_{c}-k}{2}}.\\ & = & \left\langle \mathbf{f}(g_{\infty},g_{f})\cdot k_{\infty},\Phi_{\varepsilon}(g_{\infty})\cdot k_{\infty}\right\rangle _{2k+2}(\det g_{\infty})^{\frac{w-k}{2}}(\overline{\det g_{\infty}})^{\frac{w_{c}-k}{2}}\\ & = & \Phi_{\varepsilon}(\mathbf{f})(g_{\infty}k_{\infty},g_{f})\end{array}

by the invariance of the pairing {\left\langle ,\right\rangle _{2k+2}}. An easier calculation likewise gives {\Phi_{\varepsilon}(\mathbf{f})(g_{\infty}z_{\infty},g_{f})=\Phi_{\varepsilon}(\mathbf{f})(g_{\infty},g_{f})}, so {\Phi_{\varepsilon}(\mathbf{f})(g_{\infty},g_{f})} descends to a function on {\mathfrak{H}^{3}\times\mathrm{GL}_{2}(\mathbf{A}_{F,f})} as claimed. {\square}

Proposition. The functions {\Phi_{\varepsilon}(\mathbf{f})(z,g_{f})} satisfy the equivariance formula

\displaystyle \left(\begin{array}{c} \Phi_{1}(\mathbf{f})(\gamma z,\gamma g_{f})\\ \Phi_{0}(\mathbf{f})(\gamma z,\gamma g_{f})\\ \Phi_{-1}(\mathbf{f})(\gamma z,\gamma g_{f})\end{array}\right)=J(\gamma,z)\cdot\left(\begin{array}{c} \gamma\cdot_{\mathcal{L}_{\lambda}}\Phi_{1}(\mathbf{f})(z,g_{f})\\ \gamma\cdot_{\mathcal{L}_{\lambda}}\Phi_{0}(\mathbf{f})(z,g_{f})\\ \gamma\cdot_{\mathcal{L}_{\lambda}}\Phi_{-1}(\mathbf{f})(z,g_{f})\end{array}\right)

for any {\gamma\in\mathrm{GL}_{2}(F)}.

Corollary. For any {\mathbf{f}\in S_{\lambda}(K)}, the differential form

\displaystyle \omega(\mathbf{f})(z,g_{f})=\eta(z)\cdot\left(\begin{array}{c} \Phi_{1}(\mathbf{f})(z,g_{f})\\ \Phi_{0}(\mathbf{f})(z,g_{f})\\ \Phi_{-1}(\mathbf{f})(z,g_{f})\end{array}\right)\in\Omega^{1}(\mathfrak{H}^{3}\times\mathrm{GL}_{2}(\mathbf{A}_{F,f}))\otimes_{\mathbf{C}}\mathcal{L}_{\lambda}

satisfies the equivariance relation {\omega(\mathbf{f})(\gamma z,\gamma g_{f}k_{f})=\gamma\cdot_{\mathcal{L}_{\lambda}}\omega(\mathbf{f})(z,g_{f})} for any {\gamma\in\mathrm{GL}_{2}(F)} and {k_{f}\in K}. In particular, {\omega(\mathbf{f})} defines a global section of the automorphic vector bundle {\Omega_{Y(K)}^{1}\otimes_{\mathbf{C}}\mathcal{L}_{\lambda}} on {Y(K)}.

Definition. The Eichler-Shimura map is given by

\displaystyle \begin{array}{rcl} S_{\lambda}(K) & \rightarrow & H_{c}^{1}\left(Y(K),\mathcal{L}_{\lambda}\right)\\ \mathbf{f} & \mapsto & [\omega(\mathbf{f})].\end{array}

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