Rational points on some PEL-stacks (Q1296401)

From the author's introduction: ``This paper has three aims. In the first place we will associate a certain positive rational number to polarized abelian varieties with additional structure of a very particular sort. When regarded as an element in \(\mathbb Q^\times/(\mathbb Q^\times)^2\), this number even turns out to be an invariant of the isogeny class of the abelian variety with that additional structure. Secondly, we show how to use this invariant to derive a theorem about rational points on corresponding PEL-stacks. Finally if \(s\) is any prime number, we show that the image of that invariant in the group \(\mathbb Q_s^\times/(\mathbb Q_s^\times)^2\) can be extracted from the cohomology of the abelian variety, where by cohomology we mean \(s\)-adic cohomology if \(s\) is prime to the characteristic of the ground field, and crystalline cohomology otherwise.''.