10 problems

Extension conjecture. Every $\rho\in{\sf Fib}_k(U)$ extends to a functor in ${\sf Fib}_k(V)$.

Let $V/k$ be a smooth, geometrically connected, hyperbolic curve over a number field $k$, with smooth completion $V\subset X$ and boundary $Y=X\setminus V$. For each $k$-rational c…

Toric $p$-adic section conjecture. This map is bijective. The statement is supported by the paper's proof of the corresponding result for torsors under abelian varieties over finit…

Toric section conjecture. This map is bijective. The conjecture is proposed as an intermediate step toward Grothendieck's section conjecture; the paper proves injectivity of the na…

Existence conjecture. There is a Kodaira fibration with no topological section. The source motivates this by the universal family of curves, which has no topological sections, but…

Weak topological section conjecture. The fibration $f$ admits an algebraic section if and only if its short exact sequence of fundamental groups splits. This is introduced as a con…

Let $k$ be finitely generated over $\mathbb{Q}$. Let $T/k$ be a smooth variety, and let $X/k$ be a smooth, proper, geometrically connected hyperbolic curve. Grothendieck's Hom conj…

Let $k$ be a number field or a finite extension of $\mathbb Q_p$, and let $X/k$ be a smooth, projective and geometrically connected curve of genus at least $2$, with function field…

Let $k/\mathbb Q_p$ be a finite extension, and let $X/k$ be a smooth, projective and geometrically connected curve of genus at least $2$. A rational point $a\in X(k)$ determines a…

Let $k$ be a number field, let $X$ be a smooth projective curve over $k$ of genus at least $2$, and fix a geometric point $\bar x$ of $X$. Consider the natural exact sequence … A c…