A short proof of de Shalit's cup product formula

Self cup products and the theta characteristic torsor

We give a general formula relating self cup products in cohomology to connecting maps in nonabelian cohomology, and apply it to obtain a formula for the self cup product associated to the Weil pairing.

非齐次Tate上链的杯积，以及分裂于循环扩张的局部域上环面的Galois上同调

Cup product of inhomogeneous Tate cochains, and Galois cohomology of tori over local fields that split over cyclic extensions

本文给出Tate上同调中杯积的非齐次上链公式，并用于计算分裂于循环扩张的局部域上环面的一阶上同调类的显式上循环。

Modular cocycles and cup product

We extend to positive real weights Haberland's formula giving a cohomological description of the Petersson scalar product of modular cusp forms of positive even weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We also consider the cup product of two Eichler cocycles attached to modular forms. In the classical context of integral weights at least $2$ this cup product is uninteresting. We show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by du…

A Cup Product in the Galois Cohomology of Number Fields

Let K be a number field containing the group of n-th roots of unity and S a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal S-ramified extension of K with coefficients in n-th roots of unity, which yields a pairing on a subgroup of the multiplicative group of K containing the S-units. In this general situation, we determine a formula for the cup product of two elements which pair trivially at all local places. Our primary focus is the case that K is the cyclotomic field of p-th ro…

Cup products in the etale cohomology of number fields

This paper concerns cup product pairings in étale cohomology related to work of M. Kim and of W. McCallum and R. Sharifi. We will show that by considering Ext groups rather than cohomology groups, one arrives at a pairing which combines invariants defined by Kim with a pairing defined by McCallum and Sharifi. We also prove a formula for Kim's invariant in terms of Artin maps in the case of cyclic unramified Kummer extensions. One consequence is that for all $n > 1$, there are infinitely many number fields $F$ over which there are both trivial and non-trivial Kim invariants associated to cyclic…