7 problems

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

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

Let $k=\mathbb{F}_q$ be a finite field of characteristic $p$, let $X$ be a geometrically connected, smooth, projective $k$-surface, and let $P_2(X,t)$ be the characteristic polynom…