$2$$\tilde{X}$$X$$\overline{X}$$4$$3$$V$$(p,q)$$SL_2(R) \times Orth(V_R)$$\mathcal{S}(V_R)$$V_R$$G = \SO_0(V_R)$$K$$\mathfrak{g}$$\mathfrak{k}$$\mathfrak{g} = \mathfrak{p} \oplus \mathfrak{k}$$G$$\mathcal{S}(V)$$\varphi$$r$$\tilde{\varphi}$$D= G/K$$pq$$\mathcal{S}(V)$$\mathcal{L}$$V$$\Theta=\Theta_{\mathcal{L}}$$\Theta = \sum_{\ell \in \mathcal{L}} \delta_{\ell}$$\ell$$\Theta$$\G = \Stab(\mathcal{L}) \subset G$$\Gamma'$$SL(2,\Z)$$\Theta$$\Gamma'$$\tilde{\varphi}$$r$$\theta_{\varphi}$$X = \Gamma \backslash D$$\varphi$$k$$\SO(2) \subset SL_2(R)$$\theta_{\varphi}$$\h$$k$$\G'$$\theta_{\varphi}$$f$$(pq-r)$$\eta$$r$$C$$X$$f$$\eta$$C$$\varphi$$\varphi^V_{q}$$(\mathcal{S}(V) \otimes \wedge^q \mathfrak{p}^{\ast})^K$$(p+q)/2$$SL_2$$X$$\SO(3,2)$$\SO(p,q)$$p+q>6$$p \geq q$$\varphi_q^V$$H^q(X)$$\varphi^V_{q}$$V$$(2,2)$$\Q$$1$$D \simeq \h \times \h$$X$$\overline{X}$$X$$P$$e'(P)$$3$$\overline{X}$$4$$X$$\partial \overline{X} = e'(P)$$k: \partial \overline{X} \hookrightarrow \overline{X}$$C_n$$C_n$$X$$T_n$$\tilde{X}$$n \in \N$$H_2(X, \partial X,\Q)$$\varphi^V_{2}$$C$$X$$2$$C_n$$q = e^{2\pi i \tau}$$\tau \in \h$$\theta_{\varphi^V_{2}}$$2$${X}$$H^2_c(X)$$\theta_{\varphi^V_{2}}$$X$$\overline{X}$$k^{\ast}$$\theta_{\varphi^V_{2}}$$\partial \overline{X}$$\partial\overline{X}$$\theta_{\phi_1^W}$$W$$(1,1)$$2$$1$$\partial \overline{X}$$\theta_{\phi_1^W}$$k^{\ast} \theta_{\varphi^V_{2}}$$k: \partial \overline{X} \hookrightarrow \overline{X}$$[\theta_{\varphi^V_{2}}, \theta_{\phi_1^W}]$$H^2_c(X)$$C$$\overline{X}$$H_2({X},\partial {X},\Z)$$[\theta_{\varphi^V_{2}}, \theta_{\phi_1^W}]$$C$$C_n$$\partial \overline{X}$$C_n$$\partial \overline{X}$$2$$A_n$$\partial \overline{X}$$C_n$$\partial \overline{X}$$C_n$$A_n$$C_n^c$$\overline{X}$$H_2({X},\Q)$$C$$\overline{X}$$2$$C^c_n$$2$$\overline{X}$$H^2(X)$$H_2({X}, \partial {X})$$H^2(X)$$Orth(p,q)$$\theta_{\varphi^V_{q}}$$3$$\theta_{\phi_1^W}$$1$$a$$b$$3$$M$$a$$b$$M$$A$$2$$M$$a$$c$$1$$\partial \overline{X}$$\partial C_n$$2$$\int_c \theta_{\phi_1^W}$$\sum_{n>0}\Lk(\partial C_n,c) q^n$$3$$M$$\sum_{n>0}\Lk(\partial C_n,c) q^n$$2$$3/2$$j:X \hookrightarrow \tilde{X}$$T_n$$X$$C_n$$\tilde{X}$$T_n^c$$T_n$$H_2(\tilde{X},\Q)$$\tilde{X}$$\sum_{n \geq 0} [T_n^c] q^n$$2$$H_2(\tilde{X},\Q)$$j_{\ast} C_n^c = T_n^c$$\tilde{X}$$T^c_n$$T_m$$2$$F$$T_n^c \cdot T_m$$T_n^c \cdot T_m = (T_n \cdot T_m)_X + ({T}_n \cdot {T}_m)_{\infty}$$T_n$$T_m$$X$$({T}_n \cdot {T}_m)_{\infty}$$\sum_{n=0}^{\infty} (T_n \cdot T_m)_X q^n$$\sum_{n=0}^{\infty} (T_n \cdot T_m)_{\infty} q^n$$F_X$$F_{\infty}$$F(\tau)$$C=C_m$$(T_n \cdot T_m)_X$$F_X$$F_{\infty}$$\partial \overline{X}$$\theta_{\phi_1^W}$$\theta_{\phi_1^W}$$\varphi^V_2$$n$$\theta_{\varphi^V_{2}}$$C_n$$\varphi^V_2$$C_n$$\Xi(n)$$C_n$$\Xi(n)$$T_n^c$$\tilde{X}$$\Xi(n)$$V$$4$$(\,,\,)$$(2,2)$$\underline{G} = \SO(V)$$\Q$$G=\underline{G}_0(R) \simeq \SO_0(2,2)$$\underline{G}$$D= D_V$$2$$V(R)$$(\,,\,)$$\dim z =2$$(\,,\,)|_z < 0$$\{e_1,e_2,e_3,e_4\}$$V_{R}$$(e_1,e_1)=(e_2,e_2)=1$$(e_3,e_3)=(e_4,e_4)=-1$$x$$x_i$$D$$z_0=[e_3,e_4]$$e_3$$e_4$$K \simeq \SO(2)\times \SO(2)$$G$$z_0$$D \simeq G/K$$D \simeq \H \times \h$$\underline{P}$$\ell$$P= \underline{P}_0(R)$$\underline{N}$$N = \underline{N}(R)$$u =(e_1+e_4)/\sqrt{2}$$u' =(e_1-e_4)/\sqrt{2}$$(u,u')=1$$u,u'$$\Q$$\ell = \Q u$$\ell'=\Q u'$$W = \ell^{\perp} \cap {\ell'}^{\perp}$$W_{R} = \Span_{R}(e_2,e_3)$$u'$$\underline{P}$$u,e_2,e_3,u'$$N \simeq W_{R}$$D$$z=z(t,s,w)$$z$$V_{R}$$z=[n(w)a(t)m(s)e_3,n(w)a(t)m(s)e_4]$$M$$u_2,u_2'$$W_{R}$$m(s) = m'(e^s)$$M \simeq \SO_0(W_{R})$$\frak{n},\frak{a},\frak{m}$$P$$\omega_{\alpha\mu}$$\sigma: \frak{n}\frak{a}\frak{m} \to \frak{g} \to \frak{g}/\frak{k} \simeq \frak{p}$$W_{R}$$w= wu_2+w'u_2'$$D \simeq \h \times \h$$V_{R} \simeq M_2(R)$$u = \kzxz{1}{0}{0}{0}$$u' = \kzxz{0}{0}{0}{1}$$q(x) = (x,x)/2$$q(x) = \det(x)$$e_2= \tfrac1{\sqrt{2}}\kzxz{0}{1}{-1}{0}$$e_3= \tfrac1{\sqrt{2}}\kzxz{0}{1}{1}{0}$$SL_2(R) \times SL_2(R)$$M_2(R)$$(g_1,g_2)x = g_1x\, {^{t}g_2}$$\Spin(2,2) \simeq SL_2{R} \times SL_2(R)$$D \simeq \h \times \h$$(z_1,z_2)= (x_1+iy_1,x_2+iy_2) \in \h \times \h$$L$$V$$N$$L \subseteq L^{\#}$$(x,x) \in 2 \Z$$x \in L$$q(L^{\#}) \Z = \tfrac1{N}\Z$$h \in L^{\#}$$\Gamma \subseteq \Stab{L}$$\mathcal{L}:=L+h$$G$$\ell =\Q u$$u$$L$$V$$\Q$$\underline{G}$$1$$V$$\Q$$d>0$$K = \Q(\sqrt{d})$$\Q$$\mathcal{O}_K$$x \mapsto x'$$K$$V \subset M_2(K)$$M_2(K)$$^tx' =-x$$M_2(K)$$V$$(2,2)$$\Q$$1$$L$$d$$SL_2(K)$$SL_2{R} \times SL_2(R)$$g \mapsto (g,g')$$SL_2(\mathcal{O}_K)$$L$$\g.x = \g x{^t\g'}$$d \equiv 1 \pmod{4}$$X$$X$$\Q$$X$$Y$$X$$Y$$Y$$\Phi$$X$$\Phi$$Y$$X$$\Phi$$Y$$P$$\G_P = \G \cap P$$\G_N = \G_P \cap N$$\G_P/\G_N$$\underline{P}/\underline{N}$$\underline{P}/\underline{N}$$\G_P/\G_N$$1$$\ell^{\perp}/\ell$$(1,1)$$\G_P/\G_N \simeq \Z$$g \in \G_P$$\bar{g}$$\G_P/\G_N$$g$$M$$g$$\G_M :=\G_P \cap M$$P = NAM$$\overline{D}$$D$$\underline{P}$$P=NAM$$D_W \simeq M \simeq R$$W$$\overline{D}$$\underline{P}$$\G$$D$$\overline{D}$$X = \G \back D$$\overline{X}$$[\underline{P}]$$\G$$X_W := \G_M \back D_W$$e'(P)$$T^2$$\G_N \back N$$\kappa: e'(P) \to X_W$$e(P)$$\overline{D}$$e'(P)$$\overline{X}$$[(T,\infty] \times e'(P)]$$T$$z(t,s,w)$$t>T$$i: X \hookrightarrow \overline{X}$$\overline{X}$$X$$X'$$X$$\overline{X}$$e'(P)$$X'$$\tilde{X}$$\pi:\tilde{X} \to X'$$j:X \hookrightarrow \tilde{X}$$\tilde{X}$$X^{in}$$X$$X^{out}$$e'(P)$$X^{in}$$X^{out}$$X^{in} \cap X^{out}$$\Gamma_N =\pi_1(T^2)$$\Gamma_P$$H_1(T^2,\Q)$$H_1(e'(P),\Q)$$a_P \in H_1(e'(P),\Z)$$\kappa:e'(P) \to X_W$$b_P \in H_2(e'(P),\Z)$$\kappa$$a_P$$b_P$$1$$a_P$$b_P$$a_P$$H_1(e'(P),\Q)$$b_P$$e'(P)$$a_P$$e'(P)$$b_P$$\Z$$\Omega_P$$P$$2$$e'(P)$$b_P$$T^2$$H_2(e'(P),\Z)$$\Omega_P$$T^2$$W_{R} \simeq N$$T^2=\G_N \back N$$1$$X$$\tilde{X}$$S_{P}$$P$$H_3(\tilde{X}) =0$$P$$a_P$$H_1(X^{out})$$b_P$$b_P$$\tilde{X}$$\oplus_P H_2(e'(P)) \to H_2(X) \to j_{\ast} H_2(X)$$H_2(\tilde{X})$$j_{\#}$$\oplus_P H_2(e'(P)) \to H_2(X) \to j_{\ast} H_2(X)$$H_2(\partial \overline{X})$$j_{\ast}$$X$$A_c^{\bullet}(X)$$X$$H_c^{\bullet}(X)$$X$$C^{\bullet}$$i^*$$i: X \hookrightarrow \overline{X}$$C^{\bullet}$$d(a,b) = (da, i^*a - db)$$(a,b)$$C^{\bullet}$$[[a,b]]$$A_c^{\bullet}(X) \to C^{\bullet}$$c \mapsto (c,0)$$C^{\bullet}$$V$$\partial \overline{X}$$\pi:V \to \partial \overline{X}$$b$$\partial \overline{X}$$\pi^{\ast} b$$f$$t$$1$$t=\infty$$t \leq T$$T$$f$$V$$\partial \overline{X}$$f$$V$$(a,b)$$C^i$$\mu$$\partial \overline{X}$$[a,b]$$H^i_c(X)$$\alpha$$[[a,b]] \mapsto [a,b]$$[a,b]$$\eta$$\overline{X}$$C$$\overline{X}$$$\langle[a, b], [\eta]\rangle = \int_{\overline{X}}a\wedge \eta - \int_{\partial \overline{X}} b \wedge i^*\eta, \ \text{and} \ \ \langle [a,b],C \rangle = \int_{C}a - \int_{\partial C} b.$$$x \in V$$(x,x)>0$$D_x$$D$$\G_x \subset \G$$x$$C_x$$X$$n \in \Q$$\calL_n = \{ x \in \mathcal{L}; \, \tfrac12(x,x)= n\}$$C_n$$H_2(X,\partial X,\Z)$$C_x^c$$C_x$$\partial \overline{X}$$\overline{C}_x$$\partial C_x$$e'(P)$$(x,u) \neq 0$$U_{\infty}$$e(P)$$$D_x \cap U_{\infty} = \emptyset.$$$(x,u) = 0$$\overline{D}_x \cap e(P)$$p$$s(x)$$s(x)$$R$$s(x)$$\overline{D}_x \cap e(P)$$W$$c_x \subset \partial C_x$$s(x)$$\overline{D}_x \cap e(P)$$e(P) \to e'(P)$$1$$\partial C_x$$P$$\kappa: e'(P) \to X_W$$c_x$$c_y$$\overline{C}_n$$\partial C_n$$e'(P)$$\calL_V=\calL = L +h$$L_{W,k} \subset W$$h_{W,k} \in L^{\#}_{W,k}$$W \simeq N$$\G_N = N \cap \G$$\Lambda_W$$W$$u$$n(w) x= x + (w,x)u$$x \in u^{\perp}$${\calL}_W$$\Lambda_W$$\partial C_n \cap e'(P)$$\min'$$\partial C_{n,P} := \partial C_n \cap e'(P)$$\calL_{n,u} = \{ x \in \calL \cap u^{\perp};\, (x,x)=2n\}$$\calL_{n,u}$$\partial C_{n,P}$$\Gamma$$V$$\sim_{\Gamma}$$\G_p \back \calL_{n,u} \subset V$$[x_i]= [x_i]_P, 1 \leq i \leq k$$R$$R = \coprod _{i=1}^k \coprod_{ y \in [x_i]} c_y.$$(\partial C_{x_i})_P = \coprod_{ y \in [x_i]} c_y$$1$$e'(P)$$\partial X$$y \in [x_i]$$D_y$$C_{x_i}$$D$$e(P)$$c_y$$R= \coprod_{ \sim_{\Gamma} \back \calL_{n,u}} \partial C_{x_i}.$$\partial C_{n,P}$$\coprod_{ \substack{x\in \G_M \back \mathcal{L}_W \\ (x,x)=2n}} \coprod_{0 \leq k < \min'_{\la \in \Lambda_W} |(\la,x)|} x+ku$$\G_P$$\calL_{n,u}$$c_{x+ku}$$x \in \mathcal{L}_{n,u}$$n>0$$2$$a_x$$e'(P)$$\partial a_x = c_x$$\int_{a_x} \Omega_P = 0$$\Omega_P$$a_x$$\int_{a_x} \Omega_P \in \Q$$(A_x)_P$$(A_x)_P = \sum_{y \in [x]} a_x$$e'(P)$$A_x$$2$$\partial X$$(\partial C_x)_P = \sum_{y \in [x]} c_y$$2$$\overline{X}$$2$$A_x$$\partial \overline{X}$$C_x^c$$H_2(\overline{X}) = H_2(X)$$C_n^c$$\tilde{X}$$T_n^c$$T_n$$\tilde{X}$$C_n$$\tilde{X}$$T_n$$H_2(\tilde{X})$$\tilde{X}$$T_n^c$$T_n$$j_{\ast} H_2(X)$$X$$3$$e'(P)$$T_n$$T_n = T_n \cap X^{in} + T_n \cap X^{out}$$2$$\tilde{X}$$j_{\ast} \overline{C}_n = T_n \cap X^{in}$$2$$B_n = T_n \cap X^{out}$$\partial C_n = - \partial B_n$$T_n = j_{\ast} C_n^c + B_n^c$$2$$\tilde{X}$$B_n^c$$B_n$$e'(P)$$A_n$$C_n^c$$j_*C_n^c$$S_P$$X^{in}$$X^{out}$$T_n = j_*C_n^c + B_n^c$$T_n$$H_2(\tilde{X}) = j_*H_2(X) \oplus S_P$$T_n^c = j_*C_n^c$$\alpha$$e(P)$$\partial C_x$$2$$A$$A$$P+ T +\mathcal{M}(\gamma_0)$$2$$M$$P$$T$$\Omega$$\mathcal{M}(\gamma_0)$$1$$C_1(T^2)$$T^2$$k$$S$$Y$$k$$|S|$$Y$$k$$C_k(Y)$$3$$M$$e'(P)$$f \in SL(2,\Z)$$3$$M$$2$$T^2 = W/ \Z^2$$\pi: R\times T^2 \to M$$T^2$$c$$c$$T^2$$W$$[\alpha]$$\alpha$$T^2$$x$$y$$W$$\overline{xy}$$x$$y$$\overrightarrow{xy}$$\overline{xy}$$\alpha$$\alpha$$z(x)$$s=0$$\alpha_0$$\alpha$$P$$\widetilde{P}$$W \to T^2$$Z_1(T^2,\Q)$$1$$\widetilde{P}$$\int_{P} \Omega = \int_{\widetilde{P}} \Omega \in \Q$$A$$2$$\gamma_0 \subset T^2$$0$$2$$\mathcal{M}(\gamma_0)$$\gamma_0 \times [0,1] \subset T^2 \times R$$M$$Z_1(T^2,\Q)$$f$$f^{-1}(\gamma_0)$$f^{-1}$$|\tr(f^{-1})| >2$$\det(f^{-1} -I)= det( I - f) = \tr(f) -2 \neq 0$$N= \det(f^{-1} -I)$$[\gamma_0] \in H_1(T^2,\Z)$$[\gamma_0] = N \{(f^{-1} - I)^{-1} ([\alpha_0]) \}$$1$$Z_1(T^2,\Q)$$\gamma_0 \in [\gamma_0]$$2$$\mathcal{M}(\gamma_0)$$Z_1(T^2,\Q)$$Z_1(T^2,\Z)$$h_1$$h_2$$\pi$$\alpha_0$$\gamma_0$$T^2$$c_1$$c_2$$W$$c_1 = Nh_1(0)$$c_2=h_2(0)$$d \in W$$d =f^{-1}(c_2)$$W$$\widetilde{T}$$0,c_2,d$$\widetilde{T}$$2$$$\partial \widetilde{T} = \overline{0c_2} + \overline{c_2d} - \overline{0d}.$$$T$$\widetilde{T}$$\pi$$\pi$$\overline{c_2d}$$h_2$$\overline{0c_1}$$N\alpha_0$$\overline{c_2d}$$N\alpha_0$$Z_1(T^2,\Z)$$$\partial (\mathcal{M}(\gamma_0) + T ) = f^{-1}(\gamma_0) -\gamma_0 +\gamma_0 + \alpha_0 - f^{-1}(\gamma_0)= N\alpha_0.$$$A_0 = \mathcal{M}(\gamma_0) +T$$Z_1(M,\Z)$$A$$A = \frac{1}{N} (NP + A_0) = P + \frac{1}{N}T + \frac{1}{N} \mathcal{M}(\gamma_0)$$M$$Z_1(M,\Q)$$$\partial A = \alpha.$$$\Omega$$A$$P$$\widetilde{T}$$\widetilde{T}$$\Omega$$T$$\Omega$$\mathcal{M}(c)$$1$$a$$b$$3$$M$$\Lk(a,b) = \langle A,b \rangle$$A$$2$$M$$a$$b$$M$$A$$M$$a$$b$$a$$b$$b$$s$$a,b \in H_1(T^2,\Z)$$a$$b$$\Z^2$$T^2$$a$$b$$R \times T^2$$M$$a=a(0)=0 \times a$$b=b(\eps)= \eps \times b$$Lk(a, b(\epsilon))$$A$$c$$(f^{-1} - I) (c) =a$$M(c)$$2$$c$$\partial M(c) = (f^{-1} - I) (c) =a$$\cdot$$M$$\cdot$$1$$\epsilon \times T^2$$\cdot$$1$$0 \times T^2$$\langle \cdot, \cdot \rangle$$H_1(T^2,\Q)$$f \in SL(2,\Z)$$H_1(T^2, \Z)$$1$$T^2$$R^3$$Lk(\partial C_n, \partial C_m)$$Jx$$\Lambda_W$$(Jx,x)=0$$u= \kzxz{\sqrt{p}}{0}{0}{0}$$W = \{ \kzxz{0}{\la}{-\la'}{0};\; \la \in K \} \simeq K$$K$$\langle \la, \mu \rangle = \frac{1}{\sqrt{p}} (\la \mu' - \la'\mu)$$N= \left\{ n(\la)= \kzxz{1}{\la}{0}{1} \right\}$$\mu \in K$$n(\la) \mu = \mu + \langle \la, \mu \rangle u$$\partial C_{\mu}$$R \mu = \{\la \in K_R; \; \langle \la, \mu \rangle =0 \}$$\mathcal{O}_K$$\eps$$U_+$$\mathcal{O}_K$$f$$\eps'$$d \equiv 1 \pmod{4}$$m=1$$C_1$$x =1 \in K$$C_1 \simeq SL_2(\Z) \back \h$$\min'$$\langle\,,\, \rangle$$U$$(p,q)$$m$$U=V$$U=W$$G = \SO_0(U_{R})$$K$$D=G/K$$\calS(U_{R})$$U_{R}$$SL_2(R)$$\omega$$\tau \in \h$$z\in D$$\varphi \in \calS(U_{R})$$\SO(2)$$SL_2(R)$$r$$g'_{\tau} \in SL_2(R)$$\varphi^0(x) = \varphi(x) e^{\pi (x,x)}$$E$$G$$g_z \in G$$z_0$$D$$z \in D$$\varphi \in [\calS(U_{R}) \otimes E]^K$$E$$K$$U_{R}$$\varphi(x,z)$$\varphi(x,\tau,z)$$x \in U, z \in D, \tau \in \mathbb{H}$$V$$G$$\mathfrak{g}= \mathfrak{k} \oplus \mathfrak{p}$$\mathfrak{g}$$K$$\mathfrak{g} \simeq \wwedge{2} V_{R}$$X_{ij} = e_i \wedge e_j \in \mathfrak{g}$$\mathfrak{p}$$X_{ij}$$1 \leq i \leq 2$$3 \leq j \leq 4$$\omega_{ij}$$D$$\omega_{13} \wedge \omega_{14} \wedge \omega_{23} \wedge \omega_{24}$$G$$D$$V$$\varphi_2$$\calA^2(D)$$2$$D$$G$$\varphi_2$$\varphi_0(x) := e^{-\pi(x,x)_{0}}$$(x,x)_0= \sum_{i=1}^4 x_i^2$$D$$\varphi_2$$2$$\psi_1$$0$$\omega(L)$$SL_2$$\calS(V_{R})$$d$$D$$\h$$L$$L$$f$$\h$$\tilde{\psi}_1$$\tilde{\psi}_1$$x\ne 0$$\tilde{\psi}^0_{2,0}(x) = \tilde{\psi}_1(x) e^{\pi (x,x)}$$\tilde{\psi}_1(x,z)$$\tilde{\psi}_1$$x \notin \Span[e_3,e_4]^{\perp}$$\tilde{\psi}_1(x,z)$$x$$z \notin D_x$$\tilde{\psi}_1$$2$$\tilde{\psi}_1(x,z)$$1$$D_x$$D_{x}$$d$$D$$(x,x)\leq 0$$\varphi_2(x)$$L\tilde{\psi}_1(x,\tau) = \psi_1(x,\tau)$$\tilde{\psi}$$Orth(p,q)$$\psi$$\varphi$$r-2$$r$$\tilde{\psi}$$d \tilde{\psi} = \varphi$$\varphi_{q}$$\psi_{q-1}$$\tilde{\psi}$$C_x$$W$$W\subset V$$(1,1)$$V$$W$$\mathfrak{m} \simeq R$$M = \SO_0(W_{R})$$X_{23} = e_2 \wedge e_3$$\omega_{23}$$D_W$$M$$W_{R}$$(\,,\,)$$D_W$${\bf s}_0$$e_3$$D_W \simeq R$${\bf s} = \Span x(s)$$s$${\bf s}$$x \in W$$D_{W,x}$$D$$D_{W,x} = \{ {\bf s} \in D; \; {\bf s} \perp x \}$${\bf s} = D_{W,x}$$(x,x({\bf s})) =0$${\bf s}(x)=D_{W,x}$$W$$\varphi_{1,1}$$W_{R}$$\calA^1(D_W) \otimes W_{\C}$$M$$\varphi_{1,1}$$2$$\varphi_{1,1}(x,s)$$\varphi_{1,1}^0$$\psi_{0,1}$$0$$\psi_{0,1}(x,s)$$\psi_{0,1}^0$$\psi_{0,1}$$-\psi_{1,1} - \tfrac12 \Lambda_{1,1}$$\varphi_{1,1}$$\psi_{0,1}$$\tilde{\psi}_{0,1}$$V$$x \in W$$x=0$$\tilde{\psi}_{0,1}^0(x)$$\tilde{\psi}_{0,1}^0(x,s)$$\tilde{\psi}_{0,1}(x,s)$$D_{w,x}$$A$$B$$D_W$$\G(\tfrac12,a) = \int_a^{\infty} e^{-u} u^{-1/2} du$$\G$$s=1/2$$B$$D_W$$A$$A$$B$$D_W$$W$$A$$B$$C^2$$W$$A(x)- (1/2) x_2 \frac{x_3}{|x_3|} e^{-\pi (x,x)}$$C^1$$W$$x_3=0$$|x|x^n$$C^n$$n>0$$\tilde{\psi}_{0,1}$$D_{W,x}$$\tilde{\psi}_{0,1}'$$A'(x)$$B'(x)$$W$$B'(x) + \tfrac12|x_3|e^{- \pi (x,x)}$$C^2$$W$$C^2$$M$$A'(x) + \tfrac12 x_2 \frac{x_3}{|x_3|}e^{- \pi (x,x)}$$C^1$$W$$C^1$$M$$\tilde{\psi}'_{0,1}$$\tilde{\psi}_{0,1}'(x,\tau,s) = v^{-1/2} m(s) \tilde{\psi}_{0,1}'(m^{-1}(s)\sqrt{v}x) e^{\pi i (x,x)\tau}$$\tilde{\psi}_{0,1}'(x)$$D_W$$D_{W,x}$$\tau$$D_{W,x}$$\tilde{\psi}_{0,1}(x)$$\tilde{\psi}_{0,1}'(x)$$D_W$$(x,x)>0$$D_{W,x} \otimes x$$0$$D_{W,x}$$x \in W$$\phi_{0,1}$$W$$\tilde{\psi}_{0,1}$$\tilde{\psi}_{0,1}'$$W$$\phi_{0,1}(x,s)$$B(x) + B'(x)$$C^2$$W$$C^2$$M$$A(x) + A'(x)$$C^1$$W$$C^1$$M$$X_{23}(B + B') = -(A + A')$$W$$x$$\phi_{0,1}(x,s)$$C^1$$D_W$$W_{\C}$$\varphi_{1,1}$$D_W$$\phi_{0,1}$$K'=\SO(2)$$2$$\chi$$\SO(2) \simeq U(1)$$\phi_{0,1}$$B(x)+B'(x)$$B(x)+B'(x)$$\omega(k')$$B+B'$$L^1$$\omega(k')(B+B')$$[\omega(k')(B+B')] = \chi^2(k')[B+B']$$K'$$x_2^2-x_3^2=0$$B$$\tilde{B}(x)$$\frac{-i}{4\pi} \square + \pi i r^2$$\tilde{B}(x)$$x_3=0$$\frac{\partial}{\partial x_3} \Gamma(\tfrac12,2 \pi x_3^2) = - 2 \sqrt{2\pi} \sgn(x_3) e^{-2 \pi x_3^2}$$H[B]$$H[B']$$B$$B'$$C^2$$|x_3|e^{-\pi(x_2^2-x_3^2)}$$f$$W$$H[B+B'] = [H(B+B')]= 2i[B+B']$$\iota_P$$\mathfrak{n} \simeq W \wedge R u \in \bigwedge^{2} V_R \simeq \mathfrak{g}$$W$$(\,,\,)$$\mathfrak{n}^{\ast} \simeq W \wedge R u'$$\iota_P$$\iota_P$$N$$\calS(W_{R})$$e_2$$e_3$$W$$\iota_P$$1$$w_2,w_3$$W$$w=w_2e_2+w_3e_3$$\iota_P$$W$$\varphi_{1,1}^P$$\phi_{0,1}^P$$\psi_{0,1}^P$${\psi'}_{0,1}^P$$W$$\calL_W$$\G_P$$W$$\G_N$$W$$\varphi_{1,1}$$\psi_{0,1}$$\phi_{0,1}$$\theta_{\varphi_{1,1}}(\tau,{\calL_W})$$\theta_{\phi_{0,1}}(\calL_W)$$2$$SL_2(\Z)$$\theta_{\phi_{0,1}}$$\phi_{0,1}$$B+B'$$\phi_{0,1}$$C^2$$W$$X_{23}$$A+A'$$\phi_{0,1}$$W$$\calL_W$$\theta_{\phi_{0,1}}$$\Q$$V$$2$$\iota_P$$W$$e'(P)$$\theta^P_{\psi_{0,1}}$$\theta^P_{\phi_{0,1}}$$\iota_P$$c$$1$$e'(P)$$\partial C_n$$c=\partial C_y$$C_y$$C_n$$\int_{c} \theta^P_{\phi}(\tau,\calL_{W_P})$$c$$e'(P)$$\phi_{0,1} = \tilde{\psi}_{0,1} + \tilde{\psi'}_{0,1}$$\beta(s) = \tfrac1{16\pi} \int_1^{\infty} e^{-st}t^{-3/2} dt$$\mathcal{W}(\tau)$$R^3$$S^3$$H^3$$c$$1$$3$$U$$c$$\beta$$M-U$$U$$1$$c$$1$$a$$M-U$$M$$c$$c$$U$$c$$\eta$$c$$U$$\eta$$1$$c$$\eta_M$$\eta$$M$$\eta_M$$2$$M$$c$$c$$1$$\beta$$M$$d \beta = \eta_M$$\beta$$c$$a$$1$$M -U$$M$$A$$\partial A = a$$\eta_M$$V$$a$$U$$\eta_{M-V}$$\eta_M$$M-V$$c$$(M-V, \partial (M-V))$$\beta$$M-U$$\beta$$c$$1$$e^{2 \pi n} \tilde{\psi'}_{0,1}(n)$$(\partial C_n)_P$$P$$e(P)$$F_n$$\partial C_n$$F_x$$c_x$$c_x$$D_x \cap e(P)$$e'(P)$$n>0$$1$$e^{2 \pi n} \tilde{\psi'}_{0,1}(n)$$\partial C_n$$e'(P)$$U$$F_n$$c$$1$$e'(P)$$F_n$$c=c_y$$F_x$$\partial C_n$$c$$F_n$$c=c_y$$n>0$$\eta$$2$$e'(P)$$F_n$$\eta =\Omega_P$$A_n$$\Omega \wedge \tilde{\psi'}_{0,1}(n) =0$$\Omega$$(0,2)$$\tilde{\psi'}_{0,1}(n)$$(0,1)$$e'(P)$$\eta$$\eta = d \omega$$1$$\omega$$\eta$$F_n$$c_{x+ku}$$c_x$$F_x$$\int_{a_{x+ku}} \eta = \int_{c_{x+ku}} \omega = \int_{c_x} \omega = \int_{a_x} \eta$$\eta$$x \in \calL_W$$x = \mu e_2$$\mu = \pm \sqrt{2n}$$\sum_{ \g \in \G_M} \g^{\ast} \tilde{\psi'}_{0,1}(x)$$e'(P)$$s=0$$U_\eps= (-\eps,\eps) \times T^2$$e'(P)$$F_x$$\eta \wedge \tilde{\psi'}_{0,1}(x) = d(\omega \wedge \tilde{\psi'}_{0,1}(x))$$U_{\eps}$$\g \ne 1$$\omega(s,w) \wedge \tilde{\psi'}_{0,1}(\g^{-1}x,s,w)$$s=0$$\g=1$$T^2/ c_{e_2}$$c_{e_2}$$0 \times S^1$$T^2$$\omega_3$$dw_3$$\omega$$\partial D_{x}$$w_3$$W$$\omega$$c_{e_2}$$\omega$$F_x$$\int_{c_{e_2}} \omega(0,w_2,w_3)$$w_2$$\left( \int_{T^2/ c_{e_2}} dw_2 \right)\left( \int_{c_{e_2}} \omega \right)e^{- \pi \mu^2}$$\int_{c_{e_2}} \omega = \int_{A_{e_2}} \eta$$W \to R$$w \mapsto (w,e_2)$$T^2/ \partial C_{e_2} \simeq R / (\min_{\la \in \Lambda_W}'|(\la,e_2)|)\Z$$c$$e'(P)$$3$$1$$e'(P)$$N(c)$$c$$N(c)$$F_n$$\eta_c$$2$$N(c)$$1$$N(c)$$N(c)$$\int_{e'(P)} \eta_c \wedge \tilde{\psi'}_{0,1}(n)= \int_{e'(P)} \tilde{\psi'}_{0,1}(n) \wedge \eta_c$$\eta = \eta_c$$V_n$$F_n$$e'(P) - V_n$$N(c)$$\tilde{\psi'}_{0,1}(n)$$e'(P) - V_n \supset \supp (\eta_c)$$\eta_c$$\eta_c$$e'(P) - V_n$$PD(c)$$c$$e'(P) -V_n$$c$$F_n$$c=c_y$$F_x$$\partial C_n$$x = \mu e_2$$c$$e_3 \in W$$s(x )=0$$e_3$$(0,R e_3)$$e'(P)$$\widetilde{\psi}'_{0,1}(x)$$s=0$$c$$\sum_{\gamma \in \Gamma_M} \gamma^* \widetilde{\psi}'_{0,1}(x)$$x$$c(\epsilon)$$F_n$$c \subset F_x \subset F_n$$c_1,\cdots,c_k$$\partial C_n$$F_x$$c$$c_i,1 \leq i \leq k$$\partial C_n$$c$$c_i = c$$Lk(c,c)$$c_i$$c$$c$$c_i$$c$$c(\eps)$$c_i$$c \times [0,\eps]$$c_i$$c_i$$e'(P)$$\Lk(c_i, c(\eps)) =\ Lk(c_i, c)$$C_n^c$$H^2(X)$$X$$\varphi_2$$\calL = L+h$$n \in \Q$$\theta_{\varphi_2}(\tau,\calL)$$\varphi_2(n)$$2$$X$$\theta_{\varphi_2}(\tau,\calL)$$\tau$$2$$\G(N)$$\mathcal{L} = L$$\theta_{\varphi_2}(\tau,\calL)$$\G_0(d)$$2$$\eta$$X$$\delta_{h0}$$\omega$$D$$\omega_{13}\wedge \omega_{14}+\omega_{23}\wedge \omega_{24}$$2$$\G(N) \subset SL_2(\Z)$$C$$2$$X$$H_2(X,\Z)$$\Lambda(C,\tau)$$C_0$$-\frac{1}{2\pi}\delta_{h0} [\omega]$$n>0$$\varphi_2(n)$$C_n$$\varphi_2(n)$$n \leq 0$$\theta_{\varphi_2}(\calL_V)$$\theta_{\psi_1}(\calL_V)$$X$$\overline{X}$$i_P^{\ast}$$e'(P)$$\overline{X}$$\theta_{\varphi_2}(\calL_V)$$\theta_{\psi_1}(\calL_V)$$L = \Z u + L_W + \Z u'$$h=0$$t \to \infty$$z=z(t,0,0)$$\varphi_0(x,z) = \exp\left(-\pi[ t^{-2}y_1^2+ 2q(x')+t^2y_1'^2]\right)$$x = y_1u+x'+y_1'u' \in V$$x' \in W$$t \to \infty$$\theta(\tau,\psi_1^V,\calL_V)$$y'=0$$L_W$$t$$y_1$$x' \in W$$t \to \infty$$\theta_{\varphi_2}(\calL_V)$$\overline{X}$${ \tilde{\psi}_{0,1}}$$\tilde{\psi}_{0,1}$$X$$X$$\tilde{\psi}_{{2,0}}(n)$$\varphi_2(n)$$x \in \calL_V$$n \in \Q$$1$$X$$n>0$$C_n$$1$$e'(P)$$\tilde{\psi'}_{0,1}(n)$$\phi_{0,1}^P(n)$$1$$\tilde{\psi}_{{1}}(n)$$e'(P)$$P$$\ell=\Q u$$x = au + x_W + bu'$$z=(w,t,s)$$(\,,\,)_s$$W$$x \in \calL_V$$b \ne 0$$\tilde{\psi}_1(n)$$t \to \infty$$x_W \in \calL_W$$x_W +(a+h)u \in \calL_V$$a \in \Z$$h \in \Q/\Z$$\calL_V \cap u^{\perp}$$\sum_{a \in \Z} \tilde{\psi}_1(x_W +(a+h)u,z)$$t \to \infty$$w=0$$s=0$$a \in \Z$$k \ne 0$$k=0$$\tilde{\psi}_{0,1}(x_W)$$x_W=0$$n=0$$2$$\theta_{\varphi_2}$$\overline{X}$$\theta_{\varphi_2}$$\partial \overline{X}$$\varphi$$\varphi_2$$\phi$$\phi_{0,1}$$C^{\bullet}$$(\theta_{\varphi_2}(\calL_V), \sum_{[P]} \theta^P_{\phi_{0,1}}(\calL_{W_P}))$$2$$C^{\bullet}$$(\theta_{\varphi},\theta_{\phi})$$[[\theta_{\varphi},\theta_{\phi}]]$$H^2(C^{\bullet})$$[\theta_{\varphi},\theta_{\phi}]$$H^2_c(X)$$[\theta_{\varphi}, \theta_{\phi}]$$\Lambda^c$$2$$\overline{X}$$H^2(\overline{X}) = H^2(X)$$[[\theta_{\varphi}, \theta_{\phi}]]$$[\theta_{\varphi}, \theta_{\phi}]$$X$$\Lambda^c$$\tau$$[\theta_{\varphi}, \theta_{\phi}](\tau)$$2$$\eta$$\overline{X}$$2$$\G(N) \subseteq SL_2(\Z)$$C$$2$$X$$H_2(\overline{X},\partial \overline{X},\Z)$$\omega$$\omega$$[\theta_{\varphi}, \theta_{\phi}]$$H^2(\tilde{X})$$j_{\#}: H_c^2(X) \to H^2(\tilde{X})$$j_{\ast} C_n^c = T_n^c$$\omega$$X$$\omega$$C$$[\omega] = \PD(C)$$\omega$$\PD(C)$$\tilde{\omega}$$N(C)$$C$$X$$\tilde{\omega}$$N(C)$$\tilde{\omega}$$C$$M$$N(C)$$M = \tilde{X}$$\Lambda$$H_2(X)$$H_2(X)$$H_2(\partial X)$$\theta_{\varphi_2}$$j_{\ast} H_2(X) \simeq H_2(X)/ H_2(\partial X)$$j_{\ast} H_2(X)$$\partial X$$C_y$$C_y$${\partial C_y}$$(C^c_n \cdot C_y)$$1$$C_n$$C_y$$C_n$$C_y$$\int_{C_y} \theta_{\varphi_2}$$C_y$$C_n$$Orth(p,2)$$\varphi_2$$\tilde{\psi}_1$$\tilde{\psi}_1(n)$$\Lk(C_n,C_y) = \sum_{[P]} \Lk((\partial C_n)_P, (\partial C_y)_P)$$\partial C_n$$\partial C_y$$\int_{(\partial C_y)_P} \theta^P_{\phi}(\tau,\calL_{W_P})$$\partial C_y$$e'(P)$$C_y$$e'(P)$$A_n$$C_n^c \cdot C_y$$T^c_n \cdot T_m$$(T_n \cdot T_m)_X$$(T_n \cdot T_m)_{\infty}$$(T_n \cdot T_m)_X = (C_n \cdot C_m)_X$$(T_n \cdot T_m)_X$$(T_n \cdot T_m)_X$$\int_{C_y} \theta_{\varphi_2}(\tau,\calL_V)$$\Lambda^c$$C_n^c$$n>0$$\varphi_2(n)$$C_n$$\eta$$2$$\eta$$\eta$$2$$\overline{X}$$\varphi_2$$(p,q)$$n>0$$C_n$$\tilde{\psi}_1(n)$$1$$X$$2$$\eta$$\psi$$\tilde{\psi}_1$$\tilde{\psi}$$\tilde{\psi}_1$$\tilde{\psi}_1$$\xi$$Orth(p,2)$$\Xi(n) = \sum_{x\in\calL_n} \xi(x)$$C_n$$dd^c \xi = \varphi_2$$d^c = \tfrac{1}{4\pi i}(\partial - \overline{\partial})$$d^c \xi = \tilde{\psi}_1$$d^c \varphi_0 = -\psi_1$$n \in \Q$$n>0$$\varphi_2^c(n)$$C_n^c$$\pi^{\ast} \phi^P_{0,1}(n)$$V$$\partial \overline{X}$$f$$V$$t$$1$$t=\infty$$\varphi_2^c(n)$$n$$[\theta_{\varphi},\theta_{\phi}]$$X$$\eta$$\overline{X}$$e'(P)$$\eta_P$$e'(P)$$\eta_P$$e(P)$$N$$\overline{X}$$\theta_{\varphi_2}$$n>0$$C^c_n$$\tilde{\psi}_1^c(n)$$1$$X$$2$$\overline{X}$$2$$\eta$$\overline{X}$$n\leq 0$$\varphi^c_{2}(n)$$\tilde{\psi}^c_{2}(n)$$n \leq 0$$C_n^c = \emptyset$$x=0$$\eta$$\Xi(n)$$T_n^c$$\tilde{X}$$X$$X$$P$$\rho_{T}$$\calF$$\G$$D$$t$$1$$t\leq T$$0$$T+1$$\rho_T\eta$$\left(\int_{C_n} \eta\right)e^{-2\pi n}$$T \to \infty$$T$$d(\rho_T \eta) = \rho_T'(t) dt \wedge \eta + \rho_T d\eta$$\rho_T'(t)=0$$[T,T+1]$$T$$f \equiv 1$$\tilde{\psi}_1(n) = \pi^{\ast} \tilde{\psi}_{0,1}(n) + O(e^{-Ct})$$\phi_{0,1}(n) = \tilde{\psi}_{0,1}(n)+\tilde{\psi}'_{0,1}(n)$$\tilde{\psi}_1(n) - f \pi^{\ast}\phi_{0,1}(n)$$-\pi^{\ast} \tilde{\psi'}_{0,1}(n)$$\eta$$t$$\eta = \Omega$$\eta$$\omega$$C_n^c = C_n \coprod (-A_n)$