16 problems

Let $H$ be a reductive group, let $G=H\times H$, and let $X=H\backslash(H\times H)$. Let $c:H\to H$ be the Cartan involution and let $fs_H$ be a local system on the dual group. Set…

Let $C$ be a smooth projective curve of genus $g$, let $I$ be a finite index set with $|I|=r$, and let $\underline{\e}\in\{\pm1\}^r_0$. Let $fs_n$ and $fs_{n-1}$ be irreducible Wei…

Let $C$ be a smooth projective curve, let $I$ be a finite index set with $|I|=r$, and let $\underline{\e}\in\{\pm1\}^r_0$. For $d\in\ZZ$, the intersection pairing on the degree-$d$…

Let $C$ be the underlying smooth projective curve, let $I$ be a finite index set with $|I|=r$, and let $fs_n$ be a geometrically irreducible Weil local system of rank $n$ on $C$. W…

Let $\mathcal{B}$ be a hereditary $\mathcal{O}_X$-order that is Iwahori at each $y\in Y$, and let $I_\bullet=(I)$. Write $\widetilde{\mathbf{w}}_{\operatorname{bsc}}$ for the basic…

Assume that $G$ is split over ${\mathbb F}_q$, fix an open compact level $K\subset G({\mathbb O}_F)$, and let $Q$ be the set of places where $K_v\neq G({\mathcal O}_v)$. For a fini…

Let $C$ be the completed algebraic closure of a $p$-adic field, let $\mathcal{Y}_{\operatorname{FF}}=\operatorname{Spa}(A_{\inf},A_{\inf})\setminus V([p^{\flat}])$, and let $A$ be…

Let $r\geq 1$, let $E/K$ be a nonisotrivial elliptic curve with associated Galois representation, and let $w$ be the nonzero invariant cohomology class in the relevant degree-$2r$…

Let ${\mathscr S}_{\sf K}$ be the integral model of a Shimura variety and let ${\mathscr P}_{\sf K}$ be its universal ${\mathcal G}$-shtuka. Prismatic refinement conjecture. The in…

Let $X$ be a smooth, projective, geometrically irreducible curve with function field $K$, and let $E$ be an elliptic curve over $K$ with conductor $N$ and irreducible associated Ga…

Let $q$ be a prime power, let $F:=\mathbb{F}_q(P,Q)$, and let $S_q:=\overline{\operatorname{Sht}}^{2,\operatorname{tr}}(\Gamma_0(N))$, where $N=(0)+(1)+(-1)+(\infty)$ if $q$ is odd…

Lafforgue's conjectures. The cuspidal cohomology $H_{G,I,W}^{j,\; \on{cusp}}$ is finite-dimensional, and

Frobenius semisimplicity conjecture. For the eigenvalue $q$, the generalized eigenspace of the $\operatorname{Fr}$-action on $W'_\pi$ coincides with the eigenspace, and…

Let $C$ be the ambient complete algebraically closed field, and let $G=(V,\mathcal{L})$ be a shtuka with one paw: $V$ is a finite-dimensional $\mathbb{Q}_p$-vector space and…

Let $A$ be the base ring and let $G\in\mathbb{F}_{q}\text{-}\operatorname{Gr}^{\boxplus}_{A}$. Suppose that there is a closed embedding … for some $N\in\mathbb{N}$. Let…

Let $A$ be the base ring, let $q$ be a power of the characteristic prime $p$, and let $G\in\mathbb{F}_{q}\text{-}\operatorname{Gr}^{+}_{A}$. Suppose that there is a closed embeddin…