18 problems

Let $d$ be the dimension parameter used in the paper, let $f\in H_{1-d,\bar\rho_L}$, and let $\hat{\mathcal{Z}}(f)$ and $\mathcal{Z}(W)$ be the associated arithmetic divisor and cy…

Let $\Cal A$ be the universal abelian scheme over $\Cal M$, with Hodge line bundle $\o=\e^*\O^2_{\Cal A/\Cal M}$ and metrized class $\ohat$ in $\CH^1_\R(\Cal M)$. Let $D(B)$ be the…

Kudla's vanishing conjecture. The constant $\bold c$ is zero.

Let $\mathbb{B}^{0}$ be the trace-zero subspace of the quaternion algebra $\mathbb{B}$, and let $\mathcal{N}_0(x_0)$ be the Rapoport–Zink space associated with the modular curve…

Let $(\mathcal E,a)$ be as in the arithmetic Siegel-Weil formula, and let $\chi\colon\mathbb A_{F'}^\times\to\mathbb C^\times$ be an unramified Hecke character whose restriction to…

Assume $n=2m$, let $\chi\colon\mathbb A_{F'}^\times\to\mathbb C^\times$ be an unramified Hecke character trivial on $\mathbb A_F^\times$, and assume the modularity conjecture so th…

Let $\mathcal{X}$ be a suitable regular integral model of the relevant orthogonal or unitary Shimura variety, let $n=\dim\mathcal{X}$, and let $\mathcal{Z}(m_i)$ be integral models…

Let $F$ be the base local field, let $K/K_3$ be unramified, and let … be embeddings such that $(\Phi_1,\Phi_2)$ is a regular semisimple pair and $(\Phi_0,\Phi_3)$ is regular semisi…

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 ${\mathcal I}_n^n(\mathbf{x},\mathbf{y})$ be the arithmetic intersection number associated with $n$ vectors $\mathbf{x}=(x_1,\dots,x_n)$ and $n$ vectors…

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…

Gillet–Soulé's arithmetic Standard Conjecture of Hodge Index type. The height should satisfy

Let $F$ be a real quadratic field, let $K=F(\sqrt{\Delta})$ be a quartic non-biquadratic CM field, and let $\mathcal M$ be the moduli stack of principally polarized abelian surface…

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…

Let $X_v$ be the base change of a curve $X$ to a non-archimedean place $v$, and let $\varphi(X_v)$ be its non-archimedean $\varphi$-invariant. For each…

Let $K$ be a number field, let $mathcal{X}$ be an arithmetic variety over $OK$, and let $overline{mathcal{L}}$ be a hermitian line bundle on $mathcal{X}$ whose generic fibre…

Moriwaki's convergence conjecture. In the arithmetic case, the sequence