LIN WENG | Last modified date：2016.09.28 |

Graduate School

Undergraduate School

E-Mail

Homepage

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

Phone

092-802-4490

Academic Degree

Ph. D (University of Science and Technology of China)

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.

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.

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

or at MathSciNet.

Research

**Research Interests**

- 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**

Educational

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