Curves, dynamical systems, and weighted point counting
Curves, dynamical systems, and weighted point counting
复制标题
曲线、动力系统和加权点计数
DOI：
--
复制
发表时间：
2012
影响因子：
11.1
通讯作者：
G. Cornelissen
中科院分区：
文献类型：
--
作者：
G. Cornelissen
设X是有限域k上的一条（光滑射影不可约代数）曲线。在k的所有有限域扩张上计算X上的点数不会唯一地确定曲线。实际上，泰特的一个著名定理意味着两条这样的曲线在k上具有相同的zeta函数（即，在k）的所有扩张上的相同数目的点当且仅当它们对应的雅可比行列式是同构的。我们通过证明，如果两条曲线的所有Dirichlet L-级数通过它们的Dirichlet特征群的同构而不是仅仅zeta函数相等，则曲线同构到“Frobenius扭曲”，即，直到基场的自同构。因为L系列以“加权”的方式计算曲线上的点，所以我们看到加权点计数决定了曲线。在某种意义上，这个结果解决了有限域上曲线的等谱性问题（也被称为“算术等价问题”）的类似问题：它指出曲线是由“谱”数据确定的，即k的弗罗贝纽斯算子作用在对应于狄利克雷特征标的所有λ-adic层的上同调群上的特征值。证明的方法是表明，这是等价于各自的类场论的曲线是同构的动力系统，在某种意义上说，我们使精确。
Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobians are isogenous. We remedy this situation by showing that if, instead of just the zeta function, all Dirichlet L-series of the two curves are equal via an isomorphism of their Dirichlet character groups, then the curves are isomorphic up to “Frobenius twists”, i.e., up to automorphisms of the ground field. Because L-series count points on a curve in a “weighted” way, we see that weighted point counting determines a curve. In a sense, the result solves the analogue of the isospectrality problem for curves over finite fields (also know as the “arithmetic equivalence problem”): It states that a curve is determined by “spectral” data, namely, eigenvalues of the Frobenius operator of k acting on the cohomology groups of all ℓ-adic sheaves corresponding to Dirichlet characters. The method of proof is to show that this is equivalent to the respective class field theories of the curves being isomorphic as dynamical systems, in a sense that we make precise.