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# Professor Atsushi Murase

Area and Subject Taught | Number Theory |
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Research Theme(s) | Theory of Automorphic Forms of Several Variables |

Academic Degrees | Doctor of Science, University of Tokyo |

Keywords for Research Field | Number Theory, Automorphic Forms, L-Functions, Symmetries |

Office Phone Number | 81-75-705-1611 |

## Research Overview

Number theory began by consideration of the properties of the integers 1, 2, 3...,particularly the prime numbers. My field of research expands the focus from the world of ordinary integers to things called "automorphic forms of several variables," and considers objects corresponding to prime numbers in that domain. Automorphic forms are functions of complex variables with strong symmetry. They appear not only in number theory, but in a variety of other fields including group theory,analysis,geometry, and physics, and have extremely beautiful properties. The one variable case has been extensively investigated since the era of Abel and Gauss, and played a major role recently in the proof in Fermat's Last Theorem. Research on automorphic forms of several variables, on the other hand, has a relatively recent and short history, and current knowledge is far less developed than for the one variable case. There are many interesting, unsolved problems in this area. Research is carried out using a variety of tools, such as algebra, analysis and geometry; in my case, I frequently rely on analytical techniques such as the theory of group representation and harmonic analysis. At present, I am focusing on the automorphic forms on orthogonal groups. This is a fertile subject, which is also related to infinite dimensional Lie algebras and algebraic geometry, and I find it interesting because the more I research it, the more the puzzles deepen.

## Notable Publications and Works in the Last Three Years

- Heim, B. and Murase, A.: A Characterization of Holomorphic Borcherds Lifts by Symmetries, International Mathematics Research Notices 2015; (DOI: 10.1093/imrn/rnv021).
- Heim, B. and Murase, A.: Symmetries for Borcherds lifts on Hilbert modular groups and Hirzeburch-Zagier divisors, International J. Math. 24 (2013), 1350065, 23pp. (DOI: 10.1142/S0129167X13500651).
- Heim, B. and Murase, A.: Symmetries for Siegel theta functions, Borcherds lifts and automorphic Green functions, J. Number Theory 133 (2013), no. 10, 3485-3499. (DOI: 10.1016/j.jnt.2013.04.008).