18 problems

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…

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 $\mathrm{U}(n+1,1)$ 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…

Let $X_\Gamma^\Sigma$ be a toroidal compactification of an orthogonal type Shimura variety, or of a unitary type Shimura variety of signature $(n+1,1)$. The special cycles of codim…

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…

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 $E/F$ 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 $X$ be an orthogonal Shimura variety arising from an even unimodular lattice of signature $(b,2)$, where $b>2$. The cone of special cycles of codimension $2$ on $X$ is the cone…

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 $(V,Q)$ be a quadratic space over $\mathbb{Q}$ of signature $(n,2)$, let $D^+$ be a connected component of the associated orthogonal hermitian symmetric domain, let…

Kudla's conjecture. The formal generating series

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…

Let $L$ be a lattice with discriminant group $L'/L$, let $\bar\rho_L$ be its associated representation, and let $\mathcal{X}_K$ be a regular projective flat integral model over…