22 problems

Grothendieck–Serre conjecture. Every étale $G$-torsor over $A$ that becomes trivial after base change to $K$ is already trivial over $A$.

A generalized affine Springer fiber is the variety associated with a reductive group, a finite-dimensional representation, and an element of the corresponding loop-space representa…

Purity conjecture. The Weil–Deligne representation

Let $R$ be a regular local ring, let $G$ be a reductive group scheme over $R$, and let a $G$-homogeneous space over $R$ be rationally trivial if it becomes trivial over the field o…

Let $Y$ be the additive character variety associated to a reductive group $G$, a genus $g$, and a tuple of $G$-orbits in the Lie algebra, under the additive genericity assumptions.…

Let $X$ be a regular scheme of finite type over $S$, with $S$ pure of characteristic $p>0$. Let $Sh_t(X/S)$ denote tame sheaves, let $i:Z\hookrightarrow X$ be the inclusion of a re…

Let $A_\lambda$ be the operator whose spectral contribution to the purity is given by Eq. (Ifp), and let the kernel contribution arising from the localized left and right generaliz…

Let $R$ be a regular local ring, let $K=\operatorname{Frac}(R)$ be its fraction field, and let $G$ be a reductive $R$-group scheme. Colliot-Thélène--Sansuc purity conjecture. In th…

Let $R$ be a regular local ring, let $K=\operatorname{Frac}(R)$ be its fraction field, and let $G$ be a reductive $R$-group scheme. A $G$-torsor over $R$ is generically trivial if…

Let $k$ be a field, let $G$ be a connected reductive group over $k$, and let $X$ be an irreducible smooth $k$-scheme with function field $K$. Purity for $G$-torsors over $X$ means…

Let $R$ be a regular local ring with fraction field $K$, and let ${\mathbf{G}}$ be a reductive connected group $R$-scheme. Let $R^{(1)}$ denote the set of height-$1$ primes of $R$.…

Let $\mathcal{M}_{\rm B}^{{\bm\mu},{\bf r}}$ be a generic wild character variety and let $\mathcal{M}_{{\bm\mu},{\bf r}}^*$ be the corresponding open de Rham space. Let…

Purity conjecture for flat cohomology. The fppf cohomology with supports vanishes in low degrees:

Let $X$ be a regular scheme and let $Z\subset X$ be a closed subscheme of codimension at least $2$. Write $U=X\setminus Z$. Grothendieck–Auslander–Goldman purity conjecture. The re…

Fundamental-domain purity conjecture. The cohomology of $F_{\gamma}$ is pure in the sense of Deligne.

Let $k$ be the ground field, let $G$ be a non-special semisimple $k$-group scheme, and let $X$ be a smooth affine $k$-variety. Purity failure conjecture. There exists a smooth affi…

Let $X$ be the spectrum of a regular noetherian integral semi-local ring, let $G$ be a smooth reductive algebraic $X$-group scheme, and let $K$ be the function field of $X$. Purity…

Let $C$ be a projective geometrically integral algebraic curve over $k$ whose singularities are all planar, and let $\overline{\operatorname{Jac}}_{C}$ be its compactified Jacobian…

Let $F_{\gamma}$ be the fundamental domain for the action of the free abelian group generated by $\chi(\epsilon)$, with $\chi\in X_{*}(T)$, on the affine Springer fiber…

Let $k=\mathbf{F}_{q}$, let $G$ be a connected reductive group split over $k$, and let $\gamma\in \mathfrak{t}(\mathcal{O})$ be regular, with affine Springer fiber … Here…

Let $\tilde w\in RB^{\aff}$, let $\ell(\tilde w)$ denote its length, and let $j$ be the relevant inclusion defining the sheaf…

Let $M$ be a standard-2 abelian T-motive of type $(\varphi,k)$, where $\varphi$ is written as $(\alpha_0+1,\ldots,\alpha_1)\cdots(\alpha_{i-1}+1,\ldots,\alpha_i)$, and let…