45 problems

Let $(R,\mathfrak m,k)$ be a local ring of characteristic $p>0$ and dimension $d$, let $F^n$ denote the $n$th Frobenius functor, and let $M$ be an $R$-module of finite length. Szpi…

Let $A$ be an abelian surface over a number field $k$, and let $\operatorname{ST}_{A}\subset\operatorname{USp}(4)$ be its Sato–Tate group. For each prime $\mathfrak{p}$ of good red…

Let $V$ be an $n$-dimensional projective simple normal crossing variety over an algebraically closed field of characteristic zero, and let $A$ be a finitely generated $\mathbb{Z}$-…

Let $A$ be the coefficient ring, let $\Phi$ be a rank-$2$ Drinfeld $A$-module with characteristic ideal $\chi_{\Phi}$, and let $P$ be the characteristic polynomial datum appearing…

Let $q=2^d$, let $\mathbb{Z}_q$ be the Witt vectors with values in the finite field $\mathbb{F}_q$, and let $\sigma$ be the canonical lift of the absolute $2$-Frobenius. Let $E$ be…

Let $A$ be the coefficient ring, let $[?]$ be the $A$-characteristic of a finite field $L$, and let $\chi _{\Phi }$ denote the characteristic ideal of a rank-two Drinfeld $A$-modul…

Let $R={\mathbb Z}[u,v,w,x,y,z]$, let ${\mathfrak a}=(\Delta_1,\Delta_2,\Delta_3)R$ be generated by the size-two minors of … where $\Delta_1=vz-wy$, $\Delta_2=wx-uz$, and…

Let $X$ be a smooth projective variety over a finite field $K$ of characteristic $p$, let $k(a)$ be an algebraic closure, and let $G(K)$ act on … Here $m$ is a non-negative integer…

Let $(R,\mathfrak m,k)$ be a local ring of characteristic $p>0$, let $n>0$, and regard $R$ as the $R$-module ${}^{\varphi^n}\!R$ via the $n$th Frobenius map. Let $M$ be an $R$-modu…

Reflexivity conjecture. The module ${{}^{\operatorname{F}_n}\!}R$ is reflexive for some $n$.

Let $U$ be a smooth variety over $\mathbb{F}_q$, and let $\mathcal{L}$ be an irreducible local system on $U$. For every closed point $u$ of $U$, let $\operatorname{Frob}_u$ denote…

Let $F$ be a number field, let $v$ be a finite place above a prime $p$, and let $J/F$ be a Jacobian with semistable ordinary reduction at $v$. Let $\Theta$ be a theta divisor and s…

Let $f\colon X\to Y$ be a fibration of complex smooth projective varieties. Assume that the asymptotic multiplier ideal $\mathcal{I}(X_y,0;\lVert -K_X\rVert_y)$ is trivial for gene…

Let $\E$ be the elliptic surface over $\mathbf{F}_q$ considered in the source, let $\overline{\mathbf{F}}_q$ be an algebraic closure, and let $\Fr_q$ denote Frobenius. Frobenius se…

Let $M$ be a hypergeometric motive, let $p$ be a prime, and let $k=\operatorname{ord}_p(t)$. Let $F_p(M,x)$ be the Frobenius polynomial and let $f_p(M,x)$ be the factor obtained by…

For a family of Calabi–Yau threefolds with $h^{2,1}=1$ near a point of maximal unipotent monodromy, let $U(\varphi)$ denote the Frobenius matrix on the four-dimensional cohomology…

The $d=2$ converse conjecture. When $d=2$, this relation implies

Let $X$ be an algebraic variety over a finite field, and let a Frobenius element act on its $\ell$-adic étale cohomology. Semisimplicity conjecture. The Frobenius element always ac…

Let $A$ be as above, put $L:=K_A^{\operatorname{conn}}$, and let $\operatorname{MT}_A^\sharp$ denote the set of conjugacy classes in $\operatorname{MT}_A$. For a nonzero prime idea…

Let $X_S\to S$ be a smooth, projective family over an integral, mixed characteristic base of finite type over $\operatorname{Spec}\mathbb{Z}$, with generic point $\eta$. For a clos…

Let $\mathcal{J}(r)$ be the Frobenius instability locus in the coarse moduli space of semistable vector bundles of rank $r$ and degree $0$. For a vector bundle, its Harder–Narasimh…

Let $\mathcal{J}(r)$ be the Frobenius instability locus in the coarse moduli space of semistable vector bundles of rank $r$ and degree $0$ over the curve of genus $g$. Dimension co…

Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a reductive group, and let $W$ be a finite-dimensional $k$-representation of $G$. The invariant ring…

Existence conjecture for higher Frobenius sandwiches. There should exist a $p^\bullet$-basis of $R$ over $S$.

Let $(R,\mathfrak{m},\mathbb{C})$ be a reduced standard graded $\mathbb{C}$-algebra with an isolated singularity at $\mathfrak{m}$. Let $A$ be a regular subring of $\mathbb{C}$ suc…