12 problems

Let $\mathcal{S}\to\operatorname{Spec}(\mathcal{O}_k)$ be a proper, regular model of a smooth projective curve over a number field $k$. For a finite set of finite Galois extensions…

Sarnak's conjecture. There should be infinitely many such isolated points.

Let $S=\operatorname{Spec}\mathcal O_F$ for a number field $F$, and let $X$ be an arithmetic surface over $S$, with special zeta value $\zeta^*(X,1)$ and generalized Euler characte…

Let $\mathcal X$ be the regular proper arithmetic scheme under consideration, of dimension $d$, and work on its étale site $\mathcal X_{et}$. For any integer $n$, let…

Fix a prime number $p$ and let $\mathcal X$ be regular, proper and flat over $\operatorname{Spec}(\mathbb{Z})$. Write $R\Gamma_{dR}(\mathcal{X}_{\mathbb{Q}_p}/\mathbb{Q}_p)/F^n$ fo…

Let $\pi: X\rightarrow B=\operatorname{Spec}(\mathcal{O}_{K})$ be an integral, flat, projective scheme of dimension $2$, and let $L$ be a metrized line bundle on $X$. Suppose there…

Let $K$ be a complete discretely valued field with algebraically closed residue field of characteristic $p$. Let $L/K$ be a $\mathbb{Z}/p$-extension, let $s$ be the jump in its ram…

Unramified zeta integral conjecture. The zeta integral meromorphically extends to the complex plane and satisfies

Let $K$ be a number field and let $g\geq 2$ be an integer. Let $X$ be a curve over $K$ of genus $g$. For $x\in X(\bar{K})$, let $h_{NT}(x)$ be the Néron–Tate height of the divisor…

Let $K$ be a number field and let $g\geq 2$ be an integer. Let $X$ be a curve over $K$ of genus $g$, with set of bad reduction places $S$, and suppose that $X$ has semi-stable redu…

Let $X$ be a semistable regular arithmetic surface of genus $g>1$ over $O_K$. Let $\overline\omega_X$ be its relative dualizing sheaf with its Arakelov metric, let $\delta_X$ denot…

Let $\mathcal X=\{\mathcal X_{N,M}\}=\overline{\operatorname{Spec}\mathbb Z}\prod_{\mathbb F[\pm1]}\overline{\operatorname{Spec}\mathbb Z}$, let…