11 problems

Conjectured invariants of $S_q$. If $q$ is even, then

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, 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…

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 $m\le n$ be fixed. Let $X'/X$ be the given double cover, let $\operatorname{U}_{2m}$ be the associated quasi-split unitary group, and let…

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 $X$ 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…

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, 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…