Mathematisch Instituut, Universiteit Utrecht, 3508 TA Utrecht, The Netherlands.
Proc Natl Acad Sci U S A. 2013 Jun 11;110(24):9669-73. doi: 10.1073/pnas.1217710110. Epub 2013 May 28.
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.
假设 X 是有限域 k 上的一条（光滑的射影不可约代数）曲线。计算 X 在 k 的所有有限域扩张上的点的数量不会唯一确定曲线。实际上， Tate 的一个著名定理表明，如果两个这样的曲线在 k 上具有相同的zeta 函数（即在 k 的所有扩张上的点数相同），那么它们对应的 Jacobians 是同构的。我们通过证明如果不是仅仅 zeta 函数，而是通过它们的 Dirichlet 特征群的同构，两个曲线的所有 Dirichlet L 级数都相等，那么曲线在“Frobenius 扭曲”下是同构的，即通过基域的自同构。因为 L 级数以“加权”的方式计数曲线上的点，所以我们看到加权点计数确定了一条曲线。从某种意义上说，该结果解决了有限域上曲线的等谱问题（也称为“算术等价问题”）的类似问题：它指出一条曲线由“谱”数据确定，即 Frobenius 算子在所有对应于 Dirichlet 特征的 ℓ-adic 束的上同调群上的特征值。证明方法是证明这等价于曲线的各自的类域理论作为动力系统是同构的，在我们精确说明的意义上。
