论文标题
Zeta统计和Hadamard功能
Zeta statistics and Hadamard functions
论文作者
论文摘要
我们在理性功能的Witt环上介绍了Hadamard拓扑，并同时完善了体重和点计数拓扑。代数品种在有限领域的Zeta功能是理性witt环的要素，Hadamard拓扑允许在算术和动机统计数据中猜想统一的结果：Hadamard拓扑的Witt环的完成，可以通过我们呼叫任何ZON的功能来确定HadAmard拓扑的空间，并使您可以自然地进行META，并且可以使“ META”和“我们” META，“我们”，我们可以”。在Hadamard拓扑结构中，在重量和点计数拓扑中收敛到Hadamard功能的功能也收敛。对于由Bertini问题，零循环或Batyrev-Manin猜想引起的统计数据，这产生了对现有结果的明确猜想统一，以动机和算术统计数据的现有结果，这些结果以前仅通过类比而以前仅通过类比进行连接。作为我们的猜想的证据，我们表明，Hadamard融合对于零循环产生的许多自然统计数据以及与动机Batyrev-manin问题有关的动机高度Zeta功能而产生的许多自然统计数据。
We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any "natural" sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.
