We construct a descent-of-scalars criterion for $K$-linear abelian categories. Using advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti–Tate groups on complete curves over a finite field. We conjecture that such Barsotti–Tate groups “come from” a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over ${\mathbb{𝔽}}_q$. Along the way we formulate a conjecture on the field-of-coefficients of certain compatible systems.

Let $K$ be a finite extension of ${\mathbb{Q}}_p$. The field of norms of a strictly APF extension $K_{\infty }∕K$ is a local field of characteristic $p$ equipped with an action of $\mathrm{Gal}(K_{\infty }∕K)$. When can we lift this action to characteristic zero, along with a compatible Frobenius map? In this article, we explain what we mean by lifting the field of norms, explain its relevance to the theory of $(\phi ,\mathrm{\Gamma })$-modules, and show that under a certain assumption on the type of lift, such an extension is generated by the torsion points of a relative Lubin–Tate group and that the power series giving the lift of the action of the Galois group of $K_{\infty }∕K$ are twists of semiconjugates of endomorphisms of the same relative Lubin–Tate group.

Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega }$ of $\omega$ is essentially finitely generated over the valuation ring $V_{\nu }$ of $\nu$ if and only if the initial index $\epsilon (\omega |\nu )$ is equal to the ramification index $e(\omega |\nu )$ of the extension. This gives a positive answer, for characteristic zero algebraic function fields, to a question posed by Hagen Knaf.

We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a nonconstant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa’s inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.

For a regular scheme and a prime number $p$, we define the FW-cotangent bundle as a vector bundle on the closed subscheme defined by $p=0$, under a certain finiteness condition.

For a constructible complex on the étale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the FW-cotangent bundle. At the end of the article, we compute the singular supports in some cases.

T. Dupuy, E. Katz, J. Rabinoff, and D. Zureick-Brown introduced the module of total $p$-differentials for a ring over $\mathbb{Z}∕p^2\mathbb{Z}$. We study the same construction for a ring over ${\mathbb{Z}}_{(p)}$ and prove a regularity criterion. For a local ring, the tensor product with the residue field is constructed in a different way by O. Gabber and L. Ramero.

In another article we use the sheaf of FW-differentials to define the cotangent bundle and the microsupport of an étale sheaf.

Let $k$ be an algebraically closed field of characteristic $p$ and $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite étale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group $S_n$ or alternating group $A_n$ for infinitely many $n$. These covers come from the moduli spaces of elliptic curves with ${\mathrm{PSL}}_2({\mathbb{𝔽}}_{\ell })$-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder about Markoff triples modulo $\ell$.

Let $U$ be a regular connected affine semilocal scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W{\times }_{k}U$ is trivial if it is trivial over the generic fiber of the projection $W{\times }_{k}U\to U$.

We also simplify the proof of the Grothendieck–Serre conjecture: let $U$ be a regular connected affine semilocal scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is trivial over the generic point of $U$.

We generalize some other related results from the simple simply connected case to the case of arbitrary reductive group schemes.

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Knörrer’s periodicity theorem is a powerful tool to study Cohen–Macaulay representation theory since it reduces the number of variables in computing the stable category $\underset{¯}{\mathrm{CM}}(A)$ of maximal Cohen–Macaulay modules over a hypersurface $A$. In this paper, we prove a noncommutative graded version of Knörrer’s periodicity theorem. Moreover, we prove another way to reduce the number of variables in computing the stable category ${\underset{¯}{\mathrm{CM}}}^{\mathrm{\mathbb{Z}}}\left(A\right)$ of graded maximal Cohen–Macaulay modules if $A$ is a noncommutative quadric hypersurface. Under the high rank property defined in this paper, we also show that computing ${\underset{¯}{\mathrm{CM}}}^{\mathrm{\mathbb{Z}}}\left(A\right)$ over a noncommutative smooth quadric hypersurface $A$ in up to six variables can be reduced to one or two variable cases. In addition, we give a complete classification of ${\underset{¯}{\mathrm{CM}}}^{\mathrm{\mathbb{Z}}}\left(A\right)$ over a smooth quadric hypersurface $A$ in a skew ${\mathbb{P}}^{n-1}$, where $n\le 6$, without high rank property using graphical methods.

We prove that, in certain situations, intersection numbers on formal schemes that come in profinite families vary locally constantly in the parameter. To this end, we define the product $S\times M$ of a profinite set $S$ with a locally noetherian formal scheme $M$ and study intersections thereon. Our application is to the arithmetic fundamental lemma of W. Zhang where the result helps to overcome a restriction in its recent proof. Namely, it allows to spread out the validity of the AFL identity from an open to the whole set of regular semisimple elements.