22 problems

Let $X$ be a smooth projective variety over $\bC$, let $A=\operatorname{Spec}R$ where $R$ is a finitely generated $\bZ$-algebra containing $\bZ$, and let $X_A\to A$ be a spreading…

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 $(R,\mathfrak m,k)$ 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…

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 $A$ 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 $k$ 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…

Let $(R,\mathfrak{m},k)$ be a local or graded ring of prime characteristic $p>0$, and let $I\subseteq R$ be an ideal, homogeneous in the graded case. Write $I^{[p^e]}$ for the idea…

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 $k$ be an algebraically closed field of characteristic zero and let $W$ be a finite dimensional $k$-representation for a reductive group $G$. Put … A Frobenius summand means an…

Let $k$ be an algebraically closed field of characteristic zero and let $W$ be a finite dimensional $k$-representation for a reductive group $G$. Put … Here FFRT means that the num…

Let $X/F$ be a variety, let $i$ be a cohomological degree, and let H^i_\u0**eell(X) and $H^i_{c,\u0**eell}(X)$ denote the corresponding Weil--Deligne representations, with the la…

Let $k$ be a finite field, let $f:X\to Y$ be a proper map of varieties over $k$, and let $y$ be a closed point of $Y$. Write $\overline{f^{-1}(y)}$ for the geometric fiber and…

Let $A$ be an abelian variety over a number field $K$, let $L$ be the coefficient field used for its de Rham cohomology, and let $s\in \operatorname{End}(H^1_{\mathrm{dR}}(A,L))$.…

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 $\mathbb{Z}$-…

Assume that $X$ is an ordinary smooth projective curve and that $V'$ is a general stable bundle of rank $r$. Let … be the morphism sending $V$ to the inclusion…

Finite-fibre conjecture. The quotient scheme $\mathscr{Q}^{r,0}_{V'}$ is finite and reduced. This conjecture is motivated by analogous characteristic-zero results; it is presented…

Let $X$ be a smooth, geometrically connected, $n$-dimensional projective variety over ${\mathbf Q}$. For all sufficiently large primes $p$, let $X_p$ be a smooth reduction of $X$ o…

Let $V_l^i$ be the $l$-adic representation attached to the $i$th cohomology of the generic fibre, and let $R\Gamma_f(\mathbb Q_p,V_l^i)$ be the corresponding local finite-cohomolog…

Let ${\cal P}$ be a CY(4)-operator with normalized holomorphic solution … and let ${\cal Q}$ be the associated CY(5)-operator with normalized holomorphic solution … The Dwork congr…

Let ${\cal P}={\cal P}(\theta,z)$ be a CY(4)-operator, and let ${\cal Q}$ be the fifth-order differential operator constructed from ${\cal P}$ by requiring that, for any two linear…