LIN WENG | Last modified date：2020.04.30 |

Graduate School

Undergraduate School

Homepage

##### https://kyushu-u.pure.elsevier.com/en/persons/lin-weng

Reseacher Profiling Tool Kyushu University Pure

##### http://www2.math.kyushu-u.ac.jp/~weng/

Field of Specialization

Algebraic and/or Arithmetic and/or Complex Geometry, Number Theory

Outline Activities

1) Between 1989-98, we developed a theory of

relative Bott-Chern secondary characteristic classes,

based on which we established an arithmetic Grothendieck-Riemann-Roch

theorem for l.c.i. morphisms.

2) We also develop an Arakelov theory for surfaces

with respect to singular metrics by establishing an arithmetic

Deligne-Riemann-Roch isometry for them.

Consequently, we study arithmetic aspect of the moduli spaces of punctured

Riemann surfaces by introducing certain natural metrized

line bundles related with Weil-Petersson metrics, Takhtajan-Zograf

metrics. Intrinsic relations among them, some of which

are open problems, are exposed as well. The difficulty here

is that classical approach on determinant metric does not work.

3) We introduce genuine non-abelain L functions for global fields,

based on a new cohomology, stability and Langlands' theory of

Eisenstein series, and expose the relation

between these non-abelian Ls and what we call the Arthur periods.

Basic properties such as meromorphic continuation and

functional equation(s) are established as well.

In particular we show that the rank two non-abelian

zetas for number fields satisfy the Riemann Hypothesis.

4) We develop a Program on what we call Geometric Arithmetic,

in which an approach to non-abelian Class Field Theory using stability

and an approach to the Riemann Hypothesis using intersection,

together with a study on non-abelian L functions, are included.

5) We initiated an Arakelov approach to the study of what we call

Kobayashi-Hitchin correspondence for manifolds aiming at establishing

the equivalence between intersection stability and existence of KE metrics.

I spent several years in discussion with Mabuchi.

These almost weekly discussions prove to be quite crucial to problems

involved. I have no formal publication in it. But

one can trace them from some papers of Mabuchi.

6) Other works such as metrized version of projective flatness

of certain bundles and degenerations of Riemann surfaces

are of some importance to the related fields.

7) We are studying zeta functions and general class field theory

using stability and Galois representations. In particular, together

with Zagier, we establish the Riemann hypothesis for non-abelian zeta

functions of elliptic curves on finite fields.

8) We have just published a book on "Zeta Functions for Reductive Groups and Their Zeros"

with World Scientific in February 2018. In this book, we develop a basic theory for these functions,

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

Recently, I develop a theory of high rank algebraic codes and construct arithmetic characteristic curves.

Most of the works listed above can be found either at xxx.lanl.gov

or at MathSciNet.

relative Bott-Chern secondary characteristic classes,

based on which we established an arithmetic Grothendieck-Riemann-Roch

theorem for l.c.i. morphisms.

2) We also develop an Arakelov theory for surfaces

with respect to singular metrics by establishing an arithmetic

Deligne-Riemann-Roch isometry for them.

Consequently, we study arithmetic aspect of the moduli spaces of punctured

Riemann surfaces by introducing certain natural metrized

line bundles related with Weil-Petersson metrics, Takhtajan-Zograf

metrics. Intrinsic relations among them, some of which

are open problems, are exposed as well. The difficulty here

is that classical approach on determinant metric does not work.

3) We introduce genuine non-abelain L functions for global fields,

based on a new cohomology, stability and Langlands' theory of

Eisenstein series, and expose the relation

between these non-abelian Ls and what we call the Arthur periods.

Basic properties such as meromorphic continuation and

functional equation(s) are established as well.

In particular we show that the rank two non-abelian

zetas for number fields satisfy the Riemann Hypothesis.

4) We develop a Program on what we call Geometric Arithmetic,

in which an approach to non-abelian Class Field Theory using stability

and an approach to the Riemann Hypothesis using intersection,

together with a study on non-abelian L functions, are included.

5) We initiated an Arakelov approach to the study of what we call

Kobayashi-Hitchin correspondence for manifolds aiming at establishing

the equivalence between intersection stability and existence of KE metrics.

I spent several years in discussion with Mabuchi.

These almost weekly discussions prove to be quite crucial to problems

involved. I have no formal publication in it. But

one can trace them from some papers of Mabuchi.

6) Other works such as metrized version of projective flatness

of certain bundles and degenerations of Riemann surfaces

are of some importance to the related fields.

7) We are studying zeta functions and general class field theory

using stability and Galois representations. In particular, together

with Zagier, we establish the Riemann hypothesis for non-abelian zeta

functions of elliptic curves on finite fields.

8) We have just published a book on "Zeta Functions for Reductive Groups and Their Zeros"

with World Scientific in February 2018. In this book, we develop a basic theory for these functions,

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

Recently, I develop a theory of high rank algebraic codes and construct arithmetic characteristic curves.

Most of the works listed above can be found either at xxx.lanl.gov

or at MathSciNet.

Research

**Research Interests**

- arithmetic characteristic curves, arithmetic Higgs bundles

keyword : arithmetic characteristic curves, arithmetic Higgs bundles

2019.01～2020.03. - Zeta Functions for Reductive Groups and Their Zeros

keyword : Zeta Functions, Reductive Groups, Riemann Hypothesis

2000.12～2020.04. - Uniformity of zeta functions

keyword : zeta function, uniformity

2015.04～2020.03. - Stability and Arithmetic Geometry

keyword : stability, zeta function

2010.04～2015.03. - Geometric Arithmetic

keyword : Non-abelian Class Field Theory, Abelian and Non-Abelian Zeta Functions

1999.06Geometric Arithmetic. - Arithmetic Aspects of Moduli Spaces of Punctured Riemann Surfaces

keyword : Weil-Petersson, Takhtajan-Zograf

2000.03Arithmetic Aspects of Moduli Spaces of Punctured Riemann Surfaces. - Relative Bott-Chern Secondary Characteristic Classes and Arithmetic Grothendieck-Riemann-Roch Theorem for L.C.I. Morphisms

keyword : Relative Bott-Chern Secondary Class, Grothendieck-Riemann-Roch Theorem

1990.06Relative Bott-Chern Secondary Characteristic Classes and Arithmetic Grothendieck-Riemann-Roch Theorem.

**Academic Activities**

**Books**

1. | 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. |

**Papers**

1. | ＃Lin WENG（九大の教員),
他＠Don Zagier, Higher-rank zeta functions and $SL_n$-zeta functions for curves, 米国科学アカデミー紀要(PNAS), PNAS March 24, 2020 117 (12), PNAS March 24, 2020 117 (12), 6398-6408, 2020.03, [URL], In earlier papers L.Weng introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finitefield, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve, and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case when G = $SL_n$ and $P$ is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding to n = 1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.. |

2. | L. Weng, D. Zagier, Higher rank zeta functions for elliptic curves, 米国科学アカデミー紀要(PNAS), PNAS March 3, 2020 117 (9), PNAS March 3, 2020 117 (9), 4546-4558, 2020.02, [URL], The rank n non-abelian zeta functions for curves over finite fields was defined by the first author in 2005, using moduli stacks of rank n semi-stable vector bundles on curves. These zeta functions satisfy the standard zeta properties such as the rationality, the functional equation and are conjectured to satisfy the Riemann Hypothesis. In particular, when n=1, these zeta functions coincide with the famous Artin zeta functions. In this paper, we prove the Riemann hypothesis for non-abelian zeta functions of elliptic curves.. |

Educational

**Educational Activities**

For the undergraduate study,

we currently follow the tradition of this institute in choosing textbooks and

in evaluating studends. Particularly, we assign many tutorial problems for our

students in helping them to understand what has been taught in the class.

Normally, we spend lots of time to explain many of these problems as well.

The results are quite encouraging. The first half of 2014 was a tough one.

Simply too heavy.

For graduate study, we bring the students to the up-most frontier of the

current research while explaining the basic materials, and encourage the students

to have their independent thinking towards mathematics.

we currently follow the tradition of this institute in choosing textbooks and

in evaluating studends. Particularly, we assign many tutorial problems for our

students in helping them to understand what has been taught in the class.

Normally, we spend lots of time to explain many of these problems as well.

The results are quite encouraging. The first half of 2014 was a tough one.

Simply too heavy.

For graduate study, we bring the students to the up-most frontier of the

current research while explaining the basic materials, and encourage the students

to have their independent thinking towards mathematics.

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