We define a formal algebraic analogue of hypertoric Hitchin systems, whose complex-analytic counterparts were defined by Hausel–Proudfoot. These are algebraic completely integrable systems associated to a graph $\Gamma $. We study the variation of the Tamagawa number of the resulting family of abelian varieties and show that it is described by the Kirchhoff polynomial of the graph $\Gamma $. In particular, this allows us to compute their $p$-adic volumes. We conclude the article by remarking that these spaces admit a volume preserving tropicalisation.

中文翻译：

---

Hypertoric Hitchin 系统和 Kirchhoff 多项式

我们定义了过度 Hitchin 系统的形式代数类似物，其复分析对应物由 Hausel-Proudfoot 定义。这些是与图 $\Gamma $ 相关的代数完全可积系统。我们研究了所得到的阿贝尔变体家族的多摩川数的变化，并表明它是由图 $\Gamma $ 的基尔霍夫多项式描述的。特别是，这使我们能够计算它们的 $p$-adic 体积。我们通过评论这些空间允许保留体积的热带化来结束这篇文章。