13 problems

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…

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 $M_{Kah,reg}$ 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…

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…

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

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

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.…

Katz's Lie-algebraic $p$-curvature conjecture. With this notation, $\mathfrak g$ is the smallest algebraic Lie subalgebra of $\operatorname{End}_{\mathcal O_X}V\otimes K(X)$ such t…