Publications and preprints

We show that one may always glue a separated formal algebraic space \(\mathfrak{X}\) over \(\mathbb{Z}_p\) to a separated algebraic space \(X\) over \(\mathbb{Q}_p\) via an open embedding \(j\colon \mathfrak{X}_\eta\to X^\mathrm{an}\), resulting in an algebraic space. In fact, we show that this gluing procedure gives rise to an equivalence between such gluing triples \((X,\mathfrak{X},j)\) and the category of separated algebraic spaces over \(\mathbb{Z}_p\). Moreover, one may replace \(\mathbb{Z}_p\) by an essentially arbitrary base. This is a sort of Beauville--Laszlo theorem but for (algebraic) spaces, opposed to sheaves. We apply this to better understand some well-documented phenomena in arithmetic geometry.

We study some Tannakian aspects of the theory of prismatic \(F\)-crystals, a recent notion in integral \(p\)-adic Hodge theory developed by Bhatt--Scholze and others. Additionally, we explain how to connect this Tannakian theory to the previous Tannakian theory of shtukas studied by Scholze et al.

We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when \(p > 3\) by showing that the Kisin–Pappas–Zhou integral models of Shimura varieties of abelian type are canonical. In particular, this shows that these models are independent of the choices made during their construction, and that they satisfy functoriality with respect to morphisms of Shimura data.

In this paper we construct an object on the integral canonical models of Shimura varieties of abelian type (and hyperspecial level) which should be thought of as the prismatic \(F\)-gauge realization of the universal \(\mathcal{G}\)-motive on such an object. We use this to obtain new \(p\)-adic Hodge-theoretic information about such Shimura varieties, notably an analogue of the Serre--Tate theorem, as well as characterizing them (even at finite levels). Along the way we develop an integral analogue of the functor \(D_\mathrm{crys}\), and relate it to Fontaine--Lafaille and Dieudonné theory.

In this paper we show that the \(B_\mathrm{dR}^+\)-affine Grassmannian can be computed as the sheafification of the presheaf quotient \(LG/L^+G\) for the analytic open topology (as opposed to the étale topology) for a reductive group \(G\).

In this paper we define a moduli space of \(L\)-parameters over \(\mathbb{Q}\), show it has good geometric properties (i.e., is smooth with explicitly parameterized geometric connected components), and show that there is a morphism from this moduli space to the moduli space of Weil--Deligne parameters which is an isomorphism over a dense open subset of the target.

For a formal scheme \(\mathfrak{X}\) over a height \(1\) valuation ring \(\mathcal{O}\), we construct a continuous specialization homomorphism \(\pi_1^\mathrm{dJ}(\mathfrak{X}_\eta)\to \pi_1^\mathrm{proet}(\mathfrak{X}_s)\) from the de Jong fundamental group of the generic fiber to the Bhatt--Scholze pro-étale fundamental group of the special fiber. We show that it is surjective in good cases. We also show that the notion of a de Jong covering space is étale local on the target for a smooth quasi-compact rigid space over an equicharacteristic \(0\) non-archimedean field \(K\).

In this paper we develop a new notion of covering spaces, called geometric coverings, in rigid geometry. Our definition is modeled after the notion of geometric coverings developed by Bhatt--Scholze but must be modified to account for the more subtle topology of rigid spaces. We show that geometric coverings are closed under composition, disjoint union, and are étale local on the target. We also show that the category of geometric coverings is a tame infinite Galois category, and so supports a notion of fundamental group.

This paper, a companion to Geometric Arcs and Fundamental Groups of Rigid Spaces, uses the theory of geometric coverings to study some previously introduced objects. For a (Grothendieck) topology \(\tau\) on a rigid space \(X\), write \(\mathbf{Cov}^\tau_X\) for the category of morphisms \(Y\to X\) which split into a disjoint union of finite étale pieces \(\tau\)-locally. We answer two questions of de Jong by showing that his notion of covering space is not admissible (and thus certainly not étale) local on the target in general, but that the resulting enlarged categories \(\mathbf{Cov}^\mathrm{adm}_X\) and \(\mathbf{Cov}^\mathrm{et}_X\) are still (essentially) tame infinite Galois categories. We also show that the category \(\mathbf{Loc}(X_\mathrm{proet})\) of pro-étale local systems on \(X\) is equivalent to the category \(\mathbf{Cov}^\mathrm{et}_X\), making the former (essentially) a tame infinite Galois category.

We show that, under suitable hypotheses, the local Langlands conjecture may be characterized by the so-called Scholze--Shin equations.

In previous work of Berg--Jones--Vazirani, a bijection was introduced between two sets of combinatorial objects of representation-theoretic significance: \(n\)-cores with first part equal to \(k\) and \((n-1)\)-cores with first part less than or equal to \(k\). In this article, we develop techniques using the associated affine hyperplane arrangement to interpret this bijection geometrically as a projection of alcoves onto the hyperplane containing their coroot lattice points.