论文标题
在数字字段上的某些代数托里的明确tamagawa数字
Explicit Tamagawa numbers for certain algebraic tori over number fields
论文作者
论文摘要
给定一个数字字段扩展名$ k/k $，带有一个中间字段$ k^+$由相应的prime订单$ p $的中心元素修复，我们通过$ k^\ times $ of $ k^\ times $ of $ k^\ times $ k $ k^\ times $ y的$ k $ by -k $构建了一个代数圆环，该元素通过nard Map $ n__ {k/k/k/k/k^+。目的是通过Ono的公式明确计算该圆环的tamagawa数，该公式将其表示为同胞不变的比例。当$ k/k $是galois时，给出了相当完整的详细描述。当延伸不是galois时，或更一般的圆环定义时，将给出包括分子在内的部分结果。 我们还为此目的介绍了为此目的开发的工具，使我们能够建立和计算同事，并探索这种代数圆环的局部全球原则。 当$ [k：k^+] = 2 $和$ k $是CM场时，请特别注意这种情况。这种情况对应于$ \ mathrm {gsp} _ {2n} $中的tori，大多数示例将在该设置中。这是由于在有限领域的阿贝尔品种和双线性形式的Hasse原理上的应用而动机。
Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^\times$ sent to $k^\times$ via the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number of that torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when $K/k$ is Galois. Partial results including the numerator are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SAGE for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when $[K:K^+]=2$ and $K$ is a CM-field. This case corresponds to tori in $\mathrm{GSp}_{2n}$, and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.
