Unramified correspondences

Unramified two-dimensional Langlands correspondence

In this paper we describe the unramified Langlands correspondence for two-dimensional local fields, we construct a categorical analogue of the unramified principal series representations and study its properties. The main tool for this description is the construction of a central extension. For this (and other) central extension we prove noncommutative reciprocity laws (i.e. the splitting of the central extensions over some subgroups) for arithmetic surfaces and projective surfaces over a finite field. These reciprocity laws connect central extensions which are constructed locally and globally…

Moduli of unramified irregular singular parabolic connections on a smooth projective curve

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the modu…

特殊正交群和酉群的法尔格斯 - 肖尔茨对应与内窥镜分类

Fargues-Scholze correspondence and endoscopic classification for special orthogonal and unitary groups

研究在特定条件下特殊正交群和酉群的法尔格斯 - 肖尔茨对应，证明其与经典对应半简化一致，以此为基础构造对应、推导猜想并建立志村簇新挠消失结果。

曲面的torsion birational motives与unramified cohomology

Torsion birational motives of surfaces and unramified cohomology

该研究探讨了曲面的torsion birational motives与unramified cohomology的关系，并扩展了Kahn的结果，同时给出了违反积分Hodge猜想的例子。

Local Geometric Langlands Correspondence: the Spherical Case

A module over an affine Kac--Moody algebra g^ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical g^-modules of critical level. In this paper we prove that this category is equivalent to the category of quasi-coherent sheaves on the ind-scheme of opers on the punctured disc which are unramified as local systems. This result is a categorical version of the well-known description of spherical vectors in representations of groups over local non-archimedian fields. It may be vie…

A Geometric Proof of Mordell's Conjecture for Function Fields

Let $\Cal C,\Cal C'$ be curves over a base scheme $S$ with $g(\Cal C)\ge 2$. Then the functor $T\mapsto\{$generically smooth $T$-morphisms $T\times_S\Cal C'\to T\times_S\Cal C\}$ from $((S$-schemes)) to ((sets)) is represented by a quasi-finite unramified $S$-scheme. From this one can deduce that for any two integers $g\ge 2$ and $g'$, there is an integer $M(g,g')$ such that for any two curves $C,C'$ over any field $k$ with $g(C)=g$, $g(C')=g'$, there are at most $M(g,g')$ separable $k$-morphisms $C'\to C$. It is conjectured that the arithmetic function $M(g,g')$ is bounded by a linear functio…