10 problems

Let $S^1_{\mathbb Q}$ be the universal solenoid and let $\mathbb C P^1_{\mathbb Q}$ be the adelic projective line. Adelic Fargues–Fontaine object conjecture. There exists a suitabl…

Rank-2 maximal-subbundle conjecture. The set $S(E)$ is nonempty and bounded above, its supremum is attained at $q_1=\max S(E)$, the quotient $E/\mathcal L_{q_1}$ is a Wiener-holomo…

Let $S$ be a variety over a number field $K$, and let $L$ be the biextension line bundle on $S$ whose adelic fiberwise structures have degrees equal to the Beilinson–Bloch heights…

Beilinson--Bloch adelic parametrization conjecture. The Beilinson--Bloch height in the number-field case should be parametrized by an adelic line bundle.

Continuity of envelopes conjecture. Continuity of envelopes holds for every semiample $\mathbb{Q}$-line bundle $L$ on $X$: for every $\varphi\in C^0(L)$, $P(\varphi)=Q(\varphi)$ is…

Let $X$ be the adelic variety considered in the paper, and let $L$ be a line bundle on $X$. Assume that $L$ is nef and big. Equidistribution conjecture. Theorem should also hold wh…

Let $X$ be the underlying projective scheme and let $L$ be an invertible $\mathcal O_X$-module. Assume that $L$ is big. Slope-boundedness conjecture. Any such $L$ is slope-bounded.…

Adelic-sector conjecture. The zeta eigenvalues are adelic in nature, corresponding to prime numbers and their adelic sectors, while the trivial zeros correspond to the infinite pri…

Let $K$ be the underlying base field of an adelic curve, assumed to be perfect, and let $E$ and $F$ be semistable Hermitian adelic vector bundles over that adelic curve. Generalize…

Let $K$ be the function field and $\mathbf A_K$ its adele ring. Let $X\subseteq {\mathbb G}_a^g$ be a subvariety and let $\Gamma\subseteq K^g$ be a finitely generated $\Phi(R)$-mod…