# Graduate School of Science and Engineering Science of Environment and Mathematical Modeling

- Course Outline
- Earth System Science / Environmental Magnetism Laboratory
- Geo-environmental Science Laboratory
- Wild Life Preservation Laboratory
- Advanced Materials Science and Process Systems Laboratory
- New Energy System Laboratory
- Environmental Systems Engineering Laboratory
- Regional Environment Laboratory
- Geometry Laboratory
- Functional Equations Laboratory
- Statistical Finance Laboratory
- Computational Mathematics Laboratory
- Laboratory of Mathematics for Information
- Discrete Mathematics Laboratory
- Algebra Laboratory
- Analysis Laboratory

## Algebra Laboratory

### Staff

KAWAGUCHI Shu

[Professor]

Acceptable course | |
---|---|

Master's degree course | ✓ |

Doctoral degree course |

Telephone : +81-774-65-6971

Office : HS-309

Database of Researchers

### Research Contents

My research lies mainly in algebraic geometry. My research interests include Arakelov geometry, theory of heights,
and analytic torsion.

Algebraic varieties, principal objects in algebraic geometry, are defined locally as zero loci of polynomials. When
the coefficients of polynomials are rational numbers, algebraic varieties are said to be defined over the field of
rational numbers. For a rational point on such a variety, there is a notion called a height, which measures an
arithmetic "size" or "complexity" of the point. Supposing further that algebraic varieties have some "nice"
self-maps, I have constructed heights that behave well under he self-maps, and have studied arithmetical properties
of the varieties. Further, Professor J. H. Silverman and I have studied relationship between the height growth of
rational points and the degree growth of the self-map.

When an algebraic variety is defined over the field of rational numbers, clearing the denominators of defining
polynomials gives a model of the variety over the ring of integers. Specializing the model simultaneously to all
primes and to the field of complex numbers sometimes gives useful information on the original variety. This point of
view is typical in Arakelov geometry, a discipline that has been attracting me. I have also been attracted to
analytic torsion, a quantity that appears naturally in Arakelov geometry but is originated in complex analysis or
differential geometry.

### Keywords

- Algebraic geometry
- Arakelov geometry
- Theory of heights
- Analytic torsion