arXiv subjects
Publications and source records attributed to Zijian Yao.
We prove that every de Rham $p$-adic local system on a smooth projective curve over a $p$-adic field is potentially semistable; that is, it becomes semistable after pulling back along a finite cover of the curve. This establishes a relative version of the classical $p$-adic monodromy theorem of Berger and André--Kedlaya--Mebkhout. Along the way, we also establish a $p$-adic monodromy theorem for de Rham $p$-adic local systems on disks and annuli.
Let $X$ be an affinoid rigid analytic space over a $p$-adic field, equipped with a suitable étale map to a unit polydisk. We provide a formalism of \emph{imperfect relative period rings} over $X$, which lie inside the corresponding perfect relative period rings constructed by Kedlaya--Liu. Then we establish a relative version of the Fontaine--Cherbonnier--Colmez equivalence between $p$-adic local systems on $X$ and étale $(φ, Γ)$-modules over such imperfect relative period rings, generalizing previous works of Andreatta--Brinon and others. Using this equivalence, we construct $p$-adic differential equations attached to de Rham local systems, carrying both geometric and arithmetic differential operators. This generalizes the work of Berger to the relative geometric setting. Along the way, we study a relative Fontaine--Sen theory on the decompletion of $Γ$-modules over the relative $\mathbf{B}_{\mathrm{dR}}^+$-period rings.
We develop a theory of logarithmic $A_{\mathrm{inf}}$-cohomology with coefficients for a class of $p$-adic log formal schemes that are ``sufficiently log smooth'', where the coefficients are given by relative log BKF modules. Then we establish comparison isomorphisms with étale, de Rham, and crystalline cohomology, and also extend these results to the derived setting. As an application, we give a new proof of the $C_{\mathrm{st}}$ conjecture for semistable local systems. The proof also uses the prismatic interpretation of semistable local systems established by Du--Liu--Moon--Shimizu.
We prove that Shimura varieties and geometric period images satisfy a $p$-adic extension property for large enough primes $p$. More precisely, let $\mathsf{D}^{\times}\subset \mathsf{D}$ denote the inclusion of the closed punctured unit disc in the closed unit disc. Let $X$ be either a Shimura variety or a geometric period image with torsion-free level structure. Let $F$ be a discretely valued $p$-adic field containing the number field of definition of $X$, where $p$ is a large enough prime. Then, any rigid-analytic map $f: (\mathsf{D}^{\times})^a \times \mathsf{D}^b \rightarrow X_F^{\textrm{an}}$ defined over $F$ whose image intersects the good reduction locus of $X_F^{\textrm{an}}$ (with respect to an integral canonical model) extends to a map $\mathsf{D}^{a+b}\rightarrow X_F^{\textrm{an}}$. We note that this hypothesis is vacuous if $X$ is proper. We also deduce an application to algebraicity of rigid-analytic maps. Our methods also apply to the more general situation of the rigid generic fiber of formal schemes admitting Fontaine-Laffaile modules which satisfy certain positivity conditions.
We study several rigidity properties of $p$-adic local systems on a smooth rigid analytic space $X$ over a $p$-adic field. We prove that the monodromy of the log isocrystal attached to a $p$-adic local system is ''rigid'' along irreducible components of the special fiber. Then we give several applications. First, suppose that $X$ has good reduction. We show that if a family of semistable representations is crystalline at one classical point on $X$, then it is crystalline everywhere. Second, combining with the $p$-adic monodromy theorem recently studied by the authors and their collaborators, we prove the following surprising rigidity result conjectured by Shankar: for any $p$-adic local system on a smooth projective variety with good reduction, if it is potentially crystalline at one classical point, then it is potentially crystalline everywhere. Finally, we show that if a $p$-adic local system on the complement of a reduced normal crossing divisor on a smooth rigid analytic space is crystalline at all classical points, then it extends uniquely to a $p$-adic local system on the entire space. In other words, such a local system cannot have geometric monodromy if it has no arithmetic monodromy everywhere on the complement of a reduced normal crossing divisor.
We extend the construction of A$_{\rm inf}$-cohomology by Bhatt-Morrow-Scholze to the context of log $p$-adic formal schemes over a log perfectoid base. In particular, using coordinates, we prove comparison theorems between log A$_{\rm inf}$-cohomology with other $p$-adic cohomology theories, including log de Rham, log (q-)crystalline, log prismatic, and Kummer étale cohomology, as well as the derived A$_{\rm inf}$-cohomology of certain infinite root stacks. Along the way, we define and give a combinatorial characterization of a new class of maps between saturated log schemes, called pseudo-saturated maps, which is of independent interest. They are related to (and slightly weaker than) the notion of quasi-saturated maps and maps of Cartier type studied by Tsuji.
We show that the category of logarithmic prismatic F-crystals on $(\mathcal{O}_K, \varpi^{\mathbb{N}})$ is equivalent to the category of $\mathbb{Z}_p$-lattices in semistable $\text{Gal}_K$-representations. We then apply our method to describe such Galois representations using linear algebraic data via various "logarithmic" versions of Breuil--Kisin modules.
We continue to study the logarithmic prismatic cohomology defined by the first author, and complete the proof of the de Rham comparison and étale comparison generalizing those of Bhatt and Scholze. We prove these comparisons for a derived version of logarithmic prismatic cohomology, and, along the way, we construct a suitable Nygaard filtration and explain a relation between $F$-crystals and $\mathbb{Z}_p$-local systems in the logarithmic setting.
We prove the Halo conjecture on the geometry of the eigencurve over the boundary of the weight space, predicted by Coleman-Mazur and Buzzard-Kilford.
This article generalizes the geometric quadratic Chabauty method, initiated over $\mathbb{Q}$ by Edixhoven and Lido, to curves defined over arbitrary number fields. The main result is a conditional bound on the number of rational points on curves that satisfy an additional Chabauty type condition on the Mordell-Weil rank of the Jacobian. The method gives a more direct approach to the generalization by Dogra of the quadratic Chabauty method to arbitrary number fields.
For an abelian variety $A$ over an algebraically closed non-archimedean field of residue characteristic $p$, we show that there exists a perfectoid space which is the tilde-limit of $\varprojlim_{[p]}A$. Our proof also works for the larger class of abeloid varieties.
We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.
We provide a new formalism of de Rham--Witt complexes in the logarithmic setting. This construction generalizes a result of Bhatt--Lurie--Mathew, and agrees with those of Hyodo--Kato and Matsuue for log-smooth schemes of log-Cartier type. We then apply our formalism to obtain a more direct proof of the log crystalline comparison of A_inf-cohomology in the case of semistable reduction, which is established by Cesnavicius--Koshiwara.
Let $K$ be a local function field of characteristic $l$, $\mathbb{F}$ be a finite field over $\mathbb{F}_p$ where $l \ne p$, and $\overlineρ: G_K \rightarrow \text{GL}_n (\mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod $p$ cycle map $\overline{\text{cyc}}$, from mod $p$ representations of $\text{GL}_n (\mathcal{O}_K)$ to the mod $p$ fibers of the framed universal deformation ring $R_{\overlineρ}^\square$. This allows us to obtain a function field analog of the Breuil--Mézard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-Mézard conjecture for local number fields in the case of $l \ne p$, obtained by Shotton.
For any graded poset $P$, we define a new graded poset, $\mathcal E(P)$, whose elements are the edges in the Hasse diagram of P. For any group, $G$, acting on the boolean algebra, $B_n$, we conjecture that $\mathcal E(B_n/G)$ is Peck. We prove that the conjecture holds for "common cover transitive" actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.
We discuss several partial results towards proving Dennis White's conjecture on the extreme rays of the $(N,2)$-Schur cone. We are interested in which vectors are extreme in the cone generated by all products of Schur functions of partitions with $k$ or fewer parts. For the case where $k =2$, White conjectured that the extreme rays are obtained by excluding a certain family of "bad pairs," and proved a special case of the conjecture using Farkas' Lemma. We present an alternate proof of the special case, in addition to showing more infinite families of extreme rays and reducing White's conjecture to two simpler conjectures.
The pentagram map has been studied in a series of papers by Schwartz and others. Schwartz showed that an axis-aligned polygon collapses to a point under a predictable number of iterations of the pentagram map. Glick gave a different proof using cluster algebras, and conjectured that the point of collapse is always the center of mass of the axis-aligned polygon. In this paper, we answer Glick's conjecture positively, and generalize the statement to higher and lower dimensional pentagram maps. For the latter map, we define a new system -- the mirror pentagram map -- and prove a closely related result. In addition, the mirror pentagram map provides a geometric description for the lower dimensional pentagram map, defined algebraically by Gekhtman, Shapiro, Tabachnikov and Vainshtein.
In this paper, we discuss rotation number on the invariant curve of a one parameter family of outer billiard tables. Given a convex polygon $η$, we can construct an outer billiard table $T$ by cutting out a fixed area from the interior of $η$. $T$ is piece-wise hyperbolic and the polygon $η$ is an invariant curve of $T$ under the billiard map $ϕ$. We will show that, if $β$ is a periodic point under the outer billiard map with rational rotation number $τ= p / q$, then the $n$th iteration of the billiard map is not the local identity at $β$. This proves that the rotation number $τ$ as a function of the area parameter is a devil's staircase function.