Related papers: $p$-integrability

Let $X\to S$ be a smooth projective morphism. Katz proved the Grothendieck-Katz $p$-curvature conjecture for the Gauss-Manin connection on the $i$-th cohomology of $X/S$: if its $p$-curvature vanishes mod $p$ for infinitely many $p$, then…

We investigate an analogue of the Grothendieck $p$-curvature conjecture, where the vanishing of the $p$-curvature is replaced by the stronger condition, that the module with connection mod $p$ underlies a $\mathcal{D}_X$-module structure.…

We prove the $p$-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments.

Originally conjectured unpublished by Grothendieck, then formulated precisely by Katz, the $p$-curvature conjecture is a local-global principle for algebraic differential equations. It is at present open, though various cases are known.…

The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes…

The Grothendieck--Katz $p$-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo $p$ has vanishing $p$-curvatures for {\em almost all} $p,$ has finite monodromy. It is known that it suffices to prove…

We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Pad\'e approximation. In conjunction with Honda's proof of the…

Our aim is to formulate and prove a weak form in equal characteristic $p>0$ of the $p$-curvature conjecture. We also show the existence of a counterexample to a strong form of it.

Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $\mathrm{Br}(X_{k^s})^{G_k}$ is equivalent to the Tate…

We prove a version of quantum geometric Langlands conjecture in characteristic $p$. Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline $\mathcal D$-modules on the stack of rank $N$…

In this paper a new conjecture equivalent to Collatz conjecture is presented. In particural, showing that (all) the solution(s) of newly introduced iterative functional equation(s) have a given property is equivalent to prove Collatz…

As already noted by Niels Borne and Michel Emsalem, there is a natural generalization of the section conjecture for proper orbicurves. Combined with the reformulation by Niels Borne and Angelo Vistoli of the conjecture in terms of the…

We propose a generalization of the Witten conjecture, which connects a descendent enumerative theory with a specific reduction of KP integrable hierarchy. Our conjecture is realized by two parts: Part I (Geometry) establishes a…

In this paper we formulate and lay the foundations for the K-theoretic Farrell-Jones Conjecture for the Hecke algebra of totally disconnected groups. The main result of his paper is the proof that it passes to closed subgroups. Moreover, we…

Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental isomorphism functor holds for all additive functors, like K-theory, cyclic homology,…

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-manifold. We prove the K-theoretic Farrell-Jones Conjecture (up to dimension one) and the L-theoretic Farrell-Jones Conjecture for G, where we…

We verify Curtis conjecture on a class of elements of ${_2\pi_*^s}$ that satisfy a certain factorisation property. To be more precise, suppose $f\in{_2\pi_n^s}$ pulls back to $g\in{_2\pi_n^s}P$ through the Kahn-Priddy map $\lambda:QP\to…

We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point.…

The Pappas-Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to the line bundles on spherical Schubert varieties in the…