We introduce a notion of volume for an ℓ-adic local system over an algebraic curve and, under some conditions, give a symplectic form on the rigid analytic deformation space of the corresponding geometric local system. These constructions can be viewed as arithmetic analogues of the volume and the Chern-Simons invariants of a representation of the fundamental group of a 3-manifold which fibers over the circle and of the symplectic form on the character varieties of a Riemann surface. We show that the absolute Galois group acts on the deformation space by conformal symplectomorphisms which extend to an ℓ-adic analytic flow. We also prove that the locus of local systems which are arithmetic over a cyclotomic extension is the critical set of a collection of rigid analytic functions. The vanishing cycles of these functions give additional invariants.

我们在代数曲线上为ℓ -adic 局部系统引入体积概念，并在某些条件下给出相应几何局部系统的刚性解析变形空间的辛形式。这些构造可以被看作是体积和陈-西蒙斯不变量的算术模拟，它代表了在圆上纤维化的 3 流形的基本群和黎曼曲面的字符变体上的辛形式。我们证明了绝对伽罗瓦群通过共形辛同胚作用于变形空间，这些共形辛同胚扩展到一个ℓ-adic 分析流。我们还证明了在分圆扩展上进行算术的局部系统的轨迹是刚性解析函数集合的临界集。这些函数的消失循环给出了额外的不变量。