2015 Fiscal Year Final Research Report
Generalization of quantum ergodicity
| Project/Area Number |
23540031
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Toyo University |
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Principal Investigator |
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| Project Period (FY) |
2011-04-28 – 2016-03-31
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| Keywords | ゼータ関数 / 量子エルゴード性 / 量子カオス / 合同部分群 / 関数体 |
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| Outline of Final Research Achievements |
Quantum ergodicity in the level aspect for Eisenstein series of congruence subgroups of the modular group was known only over the rational integer ring in case of the charactristic zero. We gneralized this phenomena to the positive characteristic case over the ring of polynomials over finite fields.
A key in the proof is a nontrivial estimate of automorphic L-functions. We solved this problem by combining the results of Lafforgue and Conrey-Ghosh. Lafforgue's theorem is an analogue of the Riemann hypothesis over function fields, which he proved in 2002. On the other hand, the theorem of Conrey and Ghosh was shown in 2006, when they proved that any zeta function satisfying the analogue of the Riemann hypothesis also satisfies the Lindelof hypothesis, as long as it belongs to the Selberg class. Since the automorphic L-functions satisfy functional equations, we see that it belongs to the Selberg class, even if it is over function fields. Hence we could successfully reached the result.
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| Free Research Field |
数学・整数論・ゼータ関数論
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