翁 林（うぇん りん） | データ更新日：2023.11.27 |

教授 ／
数理学研究院
代数幾何部門
数学部門

**著書**

1. | Lin Weng, Zeta functions of reductive groups and their zeros, World Scientific Publishing Co. Pte Ltd, 10.1142/9789813230651, 2018.02, [URL], This book provides a systematic account of several breakthroughs in the modern theory of zeta functions. It contains two different approaches to introduce and study genuine zeta functions for reductive groups (and their maximal parabolic subgroups) defined over number fields. Namely, the geometric one, built up from stability of principal lattices and an arithmetic cohomology theory, and the analytic one, from Langlands' theory of Eisenstein systems and some techniques used in trace formula, respectively. Apparently different, they are unified via a Lafforgue type relation between Arthur's analytic truncations and parabolic reductions of Harder-Narasimhan and Atiyah-Bott. Dominated by the stability condition and/or the Lie structures embedded in, these zeta functions have a standard form of the functional equation, admit much more refined symmetric structures, and most surprisingly, satisfy a weak Riemann hypothesis. In addition, two levels of the distributions for their zeros are exposed, i.e. a classical one giving the Dirac symbol, and a secondary one conjecturally related to GUE. This book is written not only for experts, but for graduate students as well. For example, it offers a summary of basic theories on Eisenstein series and stability of lattices and arithmetic principal torsors. The second part on rank two zeta functions can be used as an introduction course, containing a Siegel type treatment of cusps and fundamental domains, and an elementary approach to the trace formula involved. Being in the junctions of several branches and advanced topics of mathematics, these works are very complicated, the results are fundamental, and the theory exposes a fertile area for further research.. |

2. | Lin WENG, Zeta Functions for Reductive Groups and Their Zeros, World Scientific, pp 528+xxvii, 2018.02, [URL], In this book, we develop a basic theory for new yet genuine zeta functions functions for number fields, establish the spacial uniformity of zeta functions on the equivalence of rank n non-abelian zeta functions and SL(n)-zeta functions, based on Siegel-Langland' theory of Eisenstein series. In particular, we confirm a central conjecture on "Parabolic Reduction, Stability and the Volumes". The key to this is an analytic version of the Mumford'S GIT correspondence between un-stable principle bundles and the parabolic subgroups of the associated reductive groups. This itself is based on an equivalence between Arthur's analytic truncation on the adelic spaces and the geo-arithmetic truncation of stability on principal bundles. Finally, we prove the Riemann hypothesis for our zeta functions. The book consists of 7 parts: Part 1 Non-Abelian Zeta Function Part 2 Rank 2 Zeta Functions Part 3 Eisensetin Periods and Multiple Zeta Functions Part 4 Zeta Functions for Reductive Groups Part 5 Algebraic and Analytic Structures and Riemann Hypothesis Part 6 Geometric Structures and Riemann Hypothesis Appendices (with K. Sugahara) Five Essays On Arithmetic Cohomology. |

3. | Iku Nakamura and Lin WENG (eds), Algebraic and Arithmetic Structures of Moduli Spaces, Japan Math Soc, 58, 479, 2010.06. |

4. | Weng, Lin; Kaneko, Masanobu (eds), Conference on L-Functions, World Scientific, 2007.01, [URL]. |

5. | Weng, Lin; Nakamura, Iku (eds), Arithmetic Geometry and Number Theory, World Scientific, 2006.08, [URL]. |

6. | Weng, Lin, Hyperbolic Metrics, Selberg Zeta Functions and Arakelov Theory for Punctured Riemann Surfaces, Math Dept, Osaka University, Lecture Note Series in Math. Vol.6, 1998.05. |

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