Akihiko Yukie , Professor 
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Research field



Automorphic Forms, and Geometric Invariant Theory


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Research of interest 


@@An algebraic group is a group in which the product can be expressed by polynomials. For example, the group of nonsingular n by n matrices is an algebraic group called a general linear group. Thinking of such an algebraic group over the field of rational numbers, various problems associated with number theory can arise. The field of studying this kind of problems is called gNumber Theory of Algebraic Groupsh. In number theory, the theory of quadratic forms is classic. The general linear group acts on the space of quadratic forms and so it is deeply related with the theory of algebraic groups. Whether or not two quadratic forms are equivalent over the field of rational numbers is a typical problem in number theory of algebraic groups. In number theory of algebraic groups, analytical methods are often used. The theory of automorphic forms is part of number theory of algebraic groups. I am mainly studying analytic apect of the subject called gprehomogeneous vector spacesh associated with algebraic groups. I am interested in density theorems of number theoretic objects concerning algebraic groups. 
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Advice on Research 

What is necessary is that you read books and papers in the field precisely, and make good presentations to the audience and have them understand you. Number theory might sound easy to study, but it uses various methods including algebra and analyses as one progresses to higher levels. I would like you to acquire such a level of technique, and simultaneously learn to know that you can experience such beautiful theorems in number theory.

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Titles of master's theses recently advised 

 gOn generalized $p$adic Whittaker functions on GSp (4)h
 gOn Jesmanowicz conjectureh
 gOn relations of Jacobi sumsh
 gOn difference sets for power residuesh
 gOn the structure of higherdimensional local class field of Parshin.h

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Titles of master's theses recently advised 

 gOn the density of unnormalized Tamagawa numbers of orthogonal groupsh
