Duality for complexes of tori over a global field of positive characteristic

Duality via cycle complexes

We show that Bloch's complex of relative zero-cycles can be used as a dualizing complex over perfect fields and number rings. This leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields, finite fields, local fields of mixed characteristic, and rings of integers in number rings, generalizing results which so far have only been known for smooth schemes or in low dimensions, and unify the p-adic and l-adic theory. As an application, we generalize Rojtman's theorem to normal, projective schemes.

环面的Mayer–Vietoris序列

Mayer--Vietoris sequences for complexes of tori

本文在拼接框架下，针对有限维向量空间拼接成立的域因子化逆系统，构造了交叉模对应的六项Mayer–Vietoris序列，推广了Harbater–Hartmann–Krashen关于线性代数群的经典结果；当交叉模为环面的二项复形时，该序列扩展为九项正合序列，无需逆系统Galois上同调全局支配假设，并应用于非交换第二Galois上同调，得到弱局部-整体原理。

Invariants of Toric Seiberg Duality

Three-branes at a given toric Calabi-Yau singularity lead to different phases of the conformal field theory related by toric (Seiberg) duality. Using the dimer model/brane tiling description in terms of bipartite graphs on a torus, we find a new invariant under Seiberg duality, namely the Klein j-invariant of the complex structure parameter in the distinguished isoradial embedding of the dimer, determined by the physical R-charges. Additional number theoretic invariants are described in terms of the algebraic number field of the R-charges. We also give a new compact description of the a-maximi…

Poitou--Tate sequence for complex of tori over $p$-adic function fields

We complete the picture of local and global arithmetic duality theorems for short complexes of finite Galois modules and tori over $p$-adic function fields. In view of the duality theorems, we deduce a $12$-term Poitou--Tate exact sequence which relates global Galois cohomology groups to restricted topological products of local Galois cohomology groups.

pro-étale $\mathbf{Q}_p$-局部系的庞加莱对偶

Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems

本文在Banach-Colmez空间框架下证明了真p-adic刚性解析空间上pro-étale Q_p-局部系的有限性、庞加莱对偶及最优上同调消失界，并推广至算术上同调。

Nombre de classes des tores de multiplication complexe et bornes inferieures pour orbites Galoisiennes de points speciaux

We establish a Brauer-Siegel type formula for class numbers of CM tori. We deduce lower bounds for Galois orbits of certain Shimura varieties.