27 problems

Let $R\subset\mathbb{C}$ be a finitely-generated $\mathbb{Z}$-algebra, and let $S$ be a smooth $R$-scheme. Let $(\mathscr{E},\nabla)$ be a flat vector bundle on $S/R$. Vanishing…

Let a foliation on a variety over a finitely generated base be reduced modulo almost all primes, and suppose it is closed under $p$-th powers modulo $p$. Ekedahl–Shepherd-Barron–Ta…

Grothendieck–Katz and Bombieri–Dwork conjectures. The Lie algebra of the differential Galois group of $D$ is minimal with the property that, for almost all primes $\mathfrak p$ of…

Relative Etingof–Lee $p$-curvature conjecture. The connection satisfies

Let $R\subset\mathbb{C}$ be a finitely-generated $\mathbb{Z}$-algebra, and let $M$ be a smooth $R$-scheme. A foliation $\mathscr{F}\subset T_{M/R}$ is a sub-bundle of Lie algebras.…

Let … psiA:mathbb{F}p(x)^ntomathbb{F}p(x)^n, qquad Y(x)mapsto(partial-Ap)^pY(x). … admits a basis of algebraic solutions if and only if its $p$-curvature vanishes for almost all, t…

Let $M_{Kah,reg}$ be the specialized quantum $D$-module, let $x\in H^2_{T\times\mathbb{G}_m}(X)$, and let $s$ be the Artin--Schreier map defining $p$-curvature. Quantum-side p-curv…

Let $M_{eq,reg}$ be the $D$-module of twisted traces, let $\mathcal{A}_0^2$ be the degree-$2$ part of the relevant quantization algebra, and let $s$ be the Artin--Schreier map defi…

Let $g\in \mathbb{Q}(z,y_0,\ldots,y_{n-1})$ be a rational function, let $t_0,\ldots,t_{n-1}\in\mathbb{Q}$ be initial values for which $g(0,t_0,\ldots,t_{n-1})$ is defined, and let…

Let a first-order system of linear differential equations on $\mathbb P^1$ be defined over $\overline{\mathbb Q}$. Call it globally nilpotent if its reductions modulo almost all pr…

Let $X$ be a conical symplectic resolution, and consider its mod $p$ equivariant quantum cohomology, quantum Steenrod operations on divisor classes, and mod $p$ equivariant quantum…

Grothendieck's $p$-curvature conjecture. If $L_py=0$ has a basis of $\mathbb{F}_p[[x^p]]$-linearly independent solutions in $\mathbb{F}_p[[x]]$ for almost all prime numbers $p$, th…

Let $\mathscr{L}\in\mathbb{Q}(x)\langle\partial_x\rangle$ be the differential operator attached to the paper's linear differential equation, and let $\mathscr{L}_p$ denote its redu…

Let $\mathbb L$ be a geometric local system on a smooth variety $X$, with underlying vector bundle with connection $(E,\nabla)$. For all sufficiently large primes $p$, require that…

Let $X$ be a smooth quasi-projective variety over $\mathbb C$, and let $(E,\nabla)$ be an integrable algebraic connection. Choose a finite-type scheme $S$ over $\mathbb Z$ over whi…

Let $X=\operatorname{Spec}\mathcal O(X)$ with $\mathcal O(X)=\mathbb C[t,t^{-1},(t-1)^{-1}]$, and let $b=(b_{ij})$ and $c=(c_{ij})$ be matrices in $M_r(\overline{\mathbb Q})$. Cons…

Let $\overline{\mathbb Q}(z)[\frac{d}{dz}]$ be the ring of linear differential operators over $\overline{\mathbb Q}(z)$. A differential operator is globally nilpotent if its $p$-cu…

André's nilpotency conjecture. Then $(V,\nabla)$ is motivic.

Let $K$ be a finitely generated field of characteristic zero, let $C$ be a curve over $K$, and let $(\mathscr{E},\nabla)$ be a vector bundle with connection on $C$. Choose an embed…

Semisimple monodromy conjecture. If almost all $p$-curvatures of $(V,\nabla)$ vanish, then $(V,\nabla)$ has semisimple monodromy.

Grothendieck's p-curvature conjecture. If $M$ satisfies $\frak{P}$, then there exists a finite étale cover $h:Y\to X$ such that $h^*M$ is trivial as a connection; equivalently, $M$…

Consider Lamé equations with parameter $n=-\frac{1}{2}$ defined over $\overline{\mathbb{Q}}$, and the four classes numbered $1,2,3,4$ above. Chudnovsky and Chudnovsky's conjecture.…

Let a differential equation have $p$-curvature at almost all primes, and call it globally nilpotent when this $p$-curvature is nilpotent for almost all primes. A differential equat…

Let a differential equation be defined over a number field, and let its $p$-curvature be the reduction modulo $p$ of its connection-theoretic $p$-curvature for almost all primes…

Let $\mathcal{L}(S)=0$ be a differential equation over $\mathbb{Q}(x)$, and let $M_p$ denote its $p$-curvature matrix for a prime $p$. Grothendieck's conjecture. The equation posse…