arXiv subjects
Publications and source records attributed to Xinwen Zhu.
Let F be a totally real field, and let p be prime. Under standard Taylor--Wiles hypotheses, we show that an irreducible, 2-dimensional, totally odd mod p representation of the absolute Galois group of F admits lifts lying on arbitrary prescribed components of local deformation rings, allowing potentially semistable conditions with arbitrary regular Hodge--Tate weights.
We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.
For a quasi-split tamely connected reductive group G over a p-adic field, we prove that its (monodromic) affine Hecke category is canonically equivalent to its equal characteristic counterpart as monoidal categories.
This article reviews the work of the same authors on the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives, with some new observations and speculations toward the categorical local Langlands developed by X. Zhu.
This article contributes to the study of the generic part of the cohomology of Shimura varieties. Under a mild restriction of the characteristic of the coefficient field, we prove a torsion vanishing result for Shimura varieties of abelian type, confirming a conjecture by Hamann--Lee. Our proofs utilize the unipotent categorical local Langlands correspondence and, in contrast to previous works, do not rely on the endoscopic classification of representations or on other results established through trace formula techniques.
For a possibly twisted loop group $LG$, and any character sheaf of its Iwahori subgroup, we identify the associated affine Hecke category with a combinatorial category of Soergel bimodules. In fact, we prove such results for affine Hecke categories arising from central extensions of the loop group $LG$. Our results work for mod $\ell$ or integral $\ell$-adic coefficients. As applications, we obtain endoscopic equivalences between affine Hecke categories, including the derived Satake equivalence for metaplectic groups, and a series of conjectures by Gaitsgory in quantum geometric Langlands.
In one of our previous articles, we outlined the formulation of a version of the categorical arithmetic local Langlands conjecture. The aims of this article are threefold. First, we provide a detailed account of one component of this conjecture: the local Langlands category. Second, we aim to prove this conjecture in the tame case for quasi-split unramified reductive groups. Finally, we will explore the first applications of such categorical equivalence.
We explain how the geometric Langlands program inspires some recent new prospectives of classical arithmetic Langlands program and leads to the solutions of some problems in arithmetic geometry.
The autonomous driving industry is rapidly advancing, with Vehicle-to-Vehicle (V2V) communication systems highlighting as a key component of enhanced road safety and traffic efficiency. This paper introduces a novel Real-time Vehicle-to-Vehicle Communication Based Network Cooperative Control System (VVCCS), designed to revolutionize macro-scope traffic planning and collision avoidance in autonomous driving. Implemented on Quanser Car (Qcar) hardware platform, our system integrates the distributed databases into individual autonomous vehicles and an optional central server. We also developed a comprehensive multi-modal perception system with multi-objective tracking and radar sensing. Through a demonstration within a physical crossroad environment, our system showcases its potential to be applied in congested and complex urban environments.
We prove that Shimura varieties of abelian type satisfy a $p$-adic Borel-extension property over discretely valued fields. More precisely, let $\mathsf{D}$ denote the rigid-analytic closed unit disc and $\mathsf{D}^{\times} = \mathsf{D} \setminus \{0\}$, let $X$ be a smooth rigid-analytic variety, and let $S(G,\mathcal{H})_{\mathsf{K}}$ denote a Shimura variety of abelian type with torsion-free level structure. We prove every rigid-analytic map defined over a discretely valued $p$-adic field $\mathsf{D}^{\times} \times X \rightarrow S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{an}}$ extends to an analytic map $\mathsf{D} \times X \rightarrow (S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{BB}})^{\textrm{an}}$, where $S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{BB}}$ is the Baily-Borel compactification of $S(G,\mathcal{H})_{\mathsf{K}}$. We also deduce various applications to algebraicity of analytic maps, degenerations of families of abeloids, and to $p$-adic notions of hyperbolicity. Along the way, we also prove an extension result for Rapoport-Zink spaces.
In a celebrated unpublished manuscript Beilinson and Drinfeld quantize the Hitchin integrable system by showing that the global sections of critically twisted differential operators on the moduli stack of G-bundles on an algebraic curve is identified with the ring of regular functions on the space of G-opers; they deduce existence of an automorphic D-module corresponding to a local system carrying a structure of an oper. In this note we show for G=GL(n) that those results admit a short proof by reduction to positive characteristic, where they are deduced from generic Langlands duality established earlier by the first author and A. Braverman. The appendix contains a proof of some properties of the p-curvature map restricted to the space of opers.
On any smooth algebraic variety over a $p$-adic local field, we construct a tensor functor from the category of de Rham $p$-adic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a $p$-adic analogue of Deligne's classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this $p$-adic Riemann--Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.
We develop a theory of log adic spaces by combining the theories of adic spaces and log schemes, and study the Kummer étale and pro-Kummer étale topology for such spaces. We also establish the primitive comparison theorem in this context, and deduce from it some related cohomological finiteness or vanishing results.
Over any smooth algebraic variety over a $p$-adic local field $k$, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham $\mathbb Z_p$-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over $k$. In particular, we prove finiteness of étale cohomology with partial compact support of any $\mathbb Z_p$-local systems, and establish the Poincaré duality for such cohomology after inverting $p$.
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin-Selberg $L$-function, then the Bloch-Kato Selmer group is of rank one.
In this article, we study deformations of conjugate self-dual Galois representations. The study has two folds. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation.
We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet--Rallis relative trace formula, we complete the proof of the global Gan--Gross--Prasad conjecture and its refinement Ichino--Ikeda conjecture for $\mathrm{U}(n)\times\mathrm{U}(n+1)$ in the stable case.
Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $\ell$-adic tautological systems for the case where the ground field is of characteristic $p$.