26 problems

Let $L$ be a lattice of signature $(n,2)$, let $X$ be the associated orthogonal Shimura variety, and for $1\leq g\leq n$ let $\mathcal Z(M)$ be the codimension-$g$ special cycle as…

Let $X$ be a Shimura variety of Type I or Type IV, and let $X\hookrightarrow X^\Sigma$ be a smooth toroidal compactification. Let $\overline{\mathcal Z}(m)$ denote the Zariski clos…

Let $\mathbb{V}$ be the relevant hermitian space. For a basis $\{x_1,\dots,x_{2n-m},y_1,\dots,y_m\}$ of $\mathbb{V}$, let $\mathcal{Z}(x_i)$ and $\mathcal{Y}(y_j)$ be the associate…

Let $E/F$ be the unramified quadratic extension, let ${\mathbb V}$ be the space of special homomorphisms with hermitian form $h$, and let $\lbrace x_1,\dots,x_{2n}\rbrace$ be a bas…

Let $M_K$ be the orthogonal Shimura variety, let $e>1$, let $r$ be a positive integer, and let $\phi_f\in S(V^r(\mathbb{A}_f))^K$. For…

Derived-intersection conjecture.

Let $X$ be a Shimura variety for $\operatorname{SU}(2,2)\sim\operatorname{SO}(4,2)$ that is simultaneously of unitary and orthogonal type. Let…

Let $X$ be a Shimura variety with special cycles $\mathcal Z(m)$ whose generating series is modular in cohomology, and let $X^*$ be its Satake–Baily–Borel compactification. Looijen…

Let $\mathcal M$ be a PEL-type Shimura variety with special cycles $\mathcal Z(m)$ and closures $\overline{\mathcal Z}(m)$ in a compactification. Generalized Kudla compactification…

Let $1<r\leq n/2$, let $\mathcal Z_r(m)$ be the special cycles on $\mathcal A_n$ associated with principally polarized abelian varieties of dimension $r$ as isogeny factors, and de…

Greer–Lian's modularity conjecture. The series $\Theta_{\mathcal A_n}(q)$ is a holomorphic modular form of weight $2n$ for $\operatorname{SL}_2(\mathbb Z)$, valued in…

Let $C$ be a smooth projective curve of genus $g$, let $I$ be a finite index set with $|I|=r$, and let $\underline{\e}\in\{\pm1\}^r_0$. Let $fs_n$ and $fs_{n-1}$ be irreducible Wei…

Let $\mathrm{U}(n+1,1)$ be the unitary group considered in the paper, and let $g$ denote the codimension of the special cycles. The paper's main completion and explicit-form statem…

Kudla's modularity conjecture. For any toroidal compactification $X^{\operatorname{tor}}$ of $X$, there exists a natural class…

Let $\{x_1,\dots,x_n\}$ be a basis of $\mathbb{V}$, let $\mathcal{Z}(x_1),\dots,\mathcal{Z}(x_n)$ be the associated special cycles in $\mathcal{N}^{[h]}_n$, and let…

Let $\{x_1,\dots,x_{n+h},y_1,\dots,y_{n-h}\}$ be a basis of $\mathbb{V}$ with moment matrix … where $T$ is an $n\times n$ matrix, and let…

Let $\mathcal{N}^{[h]}_n$ be the Rapoport–Zink space and let $L^{\flat}\subset \mathbb{V}$ be an $O_F$-lattice of rank $n-1$. Let $\mathcal{Z}(x)$ and $\mathcal{Y}(x)$ denote the c…

Feng-Yun-Zhang's modularity conjecture. There is an automorphic form

Let $\rho^{?}\in\{\rho,\rho^*(1)\}$, let $K^p\subset \mathrm{H}(\mathbf{A}^{p\infty})$ be open compact, and let $K_p^\circ$ be the chosen hyperspecial subgroup. Let…

Let $x_1,\dots,x_n,y_1,\dots,y_n$ be a basis of the hermitian space ${\mathbb V}$, and let $\mathbf{x}=(x_1,\dots,x_n)$ and $\mathbf{y}=(y_1,\dots,y_n)$. Define … Let…

Let $0\leq h\leq 2n$, and let $x_1,\dots,x_{2n-h},y_1,\dots,y_h\in {\mathbb V}$ be the special homomorphisms defining cycles ${\mathcal Z}(x_i)$ and ${\mathcal Y}(y_j)$. Let $B$ be…

Let $x_1,\dots,x_n\in {\mathbb V}$, where ${\mathbb V}$ is the hermitian space of special homomorphisms, and let $h$ be its hermitian form. Put … The special cycles…

Let $Y_K$ be the Shimura variety in the paper. Let $\widetilde{{\rm SC}}^\ast(Y_K)$ be the subring of the Chow ring generated by the weighted special cycles, and let…

Let $Y$ be the arithmetic manifold, let $E(\lambda)$ be the coefficient system, and let $H^{n\times q}_{\mathrm{cusp}}(Y,E(\lambda))$ denote the indicated cuspidal cohomology subsp…

Let $(V,Q)=(B^{\operatorname{Tr}=0},D_0\cdot\operatorname{nr})$, let $\varphi$ be a Schwartz function, and let $\{Z(t,\varphi;H)\}$ be the refined cohomological class attached to t…