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## Graduate School of Science and Engineering

Science of Environment and Mathematical Modeling

### Algebra Laboratory

#### Staff

#### 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.

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