44 problems

Vojta's stronger conjecture. For every $\epsilon>0$, there is a proper Zariski-closed subset $Z$ of $X$ such that, for all $c\in\mathbb{R}$,

Let $\alpha$ be a non-zero algebraic integer of degree $n$ that is not a root of unity, and define its house by … where the $\alpha_k$ are the conjugates of $\alpha$. An algebraic…

Let $\mathfrak{X}$ be a “nice” algebraic stack defined over a number field $K$, and let $\mathcal{E}$ be a “nice” vector bundle on $\mathfrak{X}$. Let…

Let $E$ be a rational elliptic curve, let $Q\in E(\mathbb{Q})$ be a rational point with $x(Q)=a/b$, and let $\psi_n$ denote the $n$-th division polynomial. A compact $\mathbb{Z}^2$…

Let $m$ be fixed, let $f(x)=f_0+\cdots+f_mx^m$, and let $g(x)=g_0+\cdots+g_nx^n$ be a degree $n$ polynomial. Here $H$ denotes the height of an integer polynomial or integer, and…

Let $R_1,R_2,\dots$ be finite, simply-connected, domino-tileable regions that grow without bound, with suitably rescaled copies converging to a compact subset $R^*$ of the plane. S…

Let $\mathbb{K}$ be a number field, and let $\tilde\Gamma\subseteq\overline{\mathbb{Q}}^\times$ be a subgroup of finite rank. Write $\tilde\Gamma^{\mathrm{div}}$ for its divisible…

Function-field Vojta conjecture. Let $\mathcal{A}$ be an ample line bundle on $\mathcal{X}$ and let $r$ be a positive integer. For every $\epsilon>0$, there is a proper Zariski-clo…

Gaussian free field scaling-limit conjecture. For a broad class of height functions, the scaling limit should be a continuum two-dimensional Gaussian free field.

Let $\vec b=(b_1,b_2,\ldots,b_T)$ be a realization of the barcode process with $b_j\in\{0,1\}$, and define its height function by $h(t)=\sum_{j=K}^t b_j$, where $K$ is a fixed offs…

Uniform bounded-height conjecture. There are constants $c_g,d_g>0$ such that for any $f\in\mathcal{F}(X)$ and any $P=(\alpha,\beta)\in C_f^0(\mathbb{Q})$, the inequality above hold…

Direct convergence conjecture. This convergence should also hold without a regularization step, in a direct sense analogous to the doubly periodic height-function convergence theor…

Let $\mathbb{Q}^{(d)}$ be the compositum of all number fields of degree at most $d$, and let the Weil height be the absolute multiplicative Weil height on algebraic numbers. The No…

Let $f:X\to X$ be an endomorphism of a smooth projective variety, let $H$ be a Weil height, and let $\delta(f)>1$ be the dynamical degree of $f$. For a point $P$ whose orbit is Zar…

Let $X/K$ be a smooth projective variety over a number field, let $H$ be a Weil height associated to an ample divisor, and define the arithmetic order $\mathsf{A}(X)$ using the ite…

Let $X$ be a projective variety over a number field $K$, let $f:X\dashrightarrow X$ be a dominant rational self-map, and let $V$ be an irreducible subvariety of dimension $k$. Supp…

A height function $h$ on a planar lattice is delocalised when it does not admit a translation-invariant Gibbs measure; the discrete Gaussian free field (GFF) is the Gaussian field…

Let $\operatorname{Hilb}^{2}(\mathbb{P}^{2})(\mathbb{Q})$ denote the rational points of the Hilbert scheme of two points on the projective plane, and let $N_{s,t}(B)$ count points…

For a stacky curve $\mathfrak{X}=\mathfrak{X}(\mathbb{P}^1:({\mathbf a},{\mathbf m}))$, let $\gamma(\mathfrak{X})$ be the infimum of the real numbers $\delta$ for which the modifie…

Let $V$ be the convex nearest-neighbour potential satisfying the quantitative requirement used in the paper, namely … The paper's main delocalisation result is proved for shift-inv…

A height function model is a statistical-mechanical model whose configurations assign heights to vertices, with fluctuations governed by a convex nearest-neighbour potential. A mod…

Let $\mathcal{F}(n)$ be the set of height functions under consideration, let $R(f)$ denote the size of the range of $f$, and choose $f$ uniformly from $\mathcal{F}(n)$. Benjamini,…

Let $X$ be a quasi-projective variety, let $\Phi\colon X\dashrightarrow X$ be a rational self-map, and let $f\colon X\dashrightarrow \mathbb{P}^1$ be a non-constant rational functi…

Let $X$ be a quasi-projective variety, let $\Phi\colon X\dashrightarrow X$ be a rational self-map, and let $f\colon X\dashrightarrow \mathbb{P}^1$ be a non-constant rational functi…

Let $d\geq 3$, and consider uniformly sampled homomorphism height functions on a domain in $\mathbb{Z}^d$ with zero boundary conditions. Bounded-fluctuation conjecture. The fluctua…