Researcher Information

ATOBE Hiraku

Associate Professor

Approach Number Theory from the theory of Automorphic Representations

Department of Mathematics, Mathematics


Liftings of automorphic forms and classification of automorphic representations

FieldThe theory of automoprhic represetations
KeywordAutomorhic represetations, Liftings of automorphic forms, Langlands correspondence

Introduction of Research

Automorphic forms, which are functions with rich symmetry, give us several number theoretic information, such as analytic continuation and functional equations of L-functions.
I study automorphic representations, which are representations consists of automorphic forms.
Arthur's classification which is a classification of automorphic representations of classical groups, is difficult to understand.
My main research theme is to study Arthur's classification using liftings of automorphic forms.

Representative Achievements

H. Atobe and W. T. Gan,
Local theta correspondence of tempered representations and Langlands parameters.
Invent. Math. 210 (2017), no. 2, 341–415.
H. Atobe,
On the uniqueness of generic representations in an L-packet.
Int. Math. Res. Not. IMRN 2017, no. 23, 7051–7068.
H. Atobe,
The local theta correspondence and the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case.
Math. Ann. 371 (2018), no. 1-2, 225–295.
H. Atobe, Pullbacks of Hermitian Maass lifts.
J. Number Theory 153 (2015) 158–229.
H. Atobe and W. T. Gan,
On the local Langlands correspondence and Arthur conjecture for even orthogonal groups.
Represent. Theory 21 (2017), 354–415.
Academic degreePh.D.
Self Introduction

I am from Osaka.

Affiliated academic societyThe Mathematical Society of Japan
Room addressFaculty of Science, Building No. 4 4-512

Department of Mathematics, Mathematics

ATOBE Hiraku

Associate Professor

What is the research theme that you are currently focusing on?

My research topic is “number theory”. In particular, I am interested in the question of whether prime numbers can be expressed in a given quadratic form. A prime number is a natural number that cannot be divided by anything other than 1 and itself, and a quadratic form is an expression such as ax2 + bxy + cy2. For example, Fermat’s theorem on sums of two squares states that an odd prime p can be expressed as p = x2 + y2 with x and y integers, if and only if p ≡ 1 mod 4. Many similar assertions have been established, but when the quadratic form is complicated, the question becomes much difficult. The situation is still far from a complete solution to this problem.

Quadratic forms and congruences
Please briefly introduce us to the big project you have been tackling.

When we consider the problem of expressing prime numbers in quadratic form, in the simplest case, an answer can be obtained by classifying prime numbers by congruences. On the other hand, to describing the prime numbers p which can be written in the form p = x2 + xy + 6y2 with x and y integers, we need the notion of modular forms. A modular form is a function satisfying a certain kind of functional equations. Modular forms were also used to show Fermat’s Last Theorem. Currently, by using modular forms in a different way, I am considering a new approach to the problem of expressing prime numbers in quadratic forms. For this purpose, I am also carrying out numerical experiments using a computer.

Numerical calculation of a modular form