Stability and Arithmetic Gormetry

2014 Fiscal Year Final Research Report

• Project/Area Number: 23340009
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: WENG Lin 九州大学, 数理(科)学研究科(研究院), 教授 (60304002)
• Project Period (FY): 2011-04-01 – 2015-03-31
• Keywords: ゼータ函数 / 零点の分布 / Dirac 分布 / 対相関関数 / モチーフなEuler積 / ゼータの特殊統一性 / リーマン予想 / アデリックコホモロジー理論

Outline of Final Research Achievements

Mainly concentrated on non-abelian zeta functions. Historic achievements are (1) introduce pure non-abelian zeta functions for function fields motivated by Driffield's work on counting super-cuspidal Galois representations, (2) prove jointly with Zagier the Riemann hypothesis for pure zetas of elliptic curves, (3) discover two levels of structures for distributions of zeta zeros. We explain (3) in details only. Unlike for distributions of Riemann zeros, the classical delta distributions are of Dirac type. In concrete terms, this means that the delta sequences of pair correlations have accumulating points. This is very different from that of Gaussian unitary ensembles (GUE). However, motivated by the work (2), we blow-up the infinitesimal neighborhood of these accumulating points to introduce totally new big Delta sequences and hence to discover the second level structure of the distributions of our zeta zeros. We conjecture that the structure of big Delta distributions are that of GUE.

Free Research Field

数論幾何