第 13 期白眉研究者紹介● グローバル型（従来型）／ Global Type

Introduction of 13th-batch Hakubi Researchers★ 部 局 連 携 型（卓越研究員型）／ Tenure ‒Track Type● 高松 哲平 特定助教
Teppei TAKAMATSU (Assistant Professor)
研 究 課 題： 混標数の幾何学による既約シンプレクティック多様体の数論の研究（Study of Aarithmetic of Iirreducible SsymplecticVvarieties via Mmixed Ccharacteristic Ggeometry）
専 門 分 野： 数論幾何学（Arithmetic Geometry）
受 入 先 部 局： 理学研究科（Graduate School of Science）
前職の機関名： 京都大学大学院 理学研究科（Graduate School of Science, Kyoto University）

私の専門は，数論幾何学という，整数の性質を幾何学

0\VSHFLDOW\LVDULWKPHWLFJHRPHWU\D¿HOGWKDWDLPVWRXQFRY-的背景から解き明かす分野です．数論幾何学の大きな

er the properties of integers through a geometric perspective.
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問題意識に，方程式の整数解・有理数解を知るために，

WKHJHRPHWULFSURSHUWLHVRIWKHVKDSHVGH¿QHGE\HTXDWLRQVLQ

その方程式の定める図形の幾何学的性質を研究するこRUGHUWRXQGHUVWDQGWKHLQWHJHUUDWLRQDOVROXWLRQVRIWKRVHHTXD-とが挙げられます．このためには．図形の「正標数還元」

tions.

をとるというテクニックが非常に強力です．正標数還元$QLPSRUWDQWLQJUHGLHQWLQWKLVSXUVXLWLVWKHWHFKQLTXHVRとは，各素数 p ごとに定まる，方程式を「p で割った余FDOOHG³PRGSUHGXFWLRQ´ZKLFKPHDQVUHWKLQNLQJHTXDWLRQV

りの世界」で考え直して得られる，いわば「素数によin “the world of remainders modulo each prime number p”,る影」の様な図形であり，多くの情報を持っています．

yielding shapes that are, in a sense, “shadows cast by primesp" and contain a wealth of information. However, it is known

しかしながら，これらの影は元の図形とは住む世界が違that these shadows live in a completely different world fromい，その幾何学的な様相も大きく異なることが知られて

the original shapes and have significantly different geometricいます．

aspects.

私の白眉プロジェクトの研究では，既約シンプレク,QP\VWXG\LQWKH+DNXELSURMHFW,ZLOOLQYHVWLJDWHWKHEHKDY-ティック多様体という，興味深い対称性を持った図形に

ior of “mod p reduction” for shapes with interesting symmetries

ついて，近年急速に発展した混標数の幾何学というをcalled irreducible symplectic varieties, using revently devel-用いることで「正標数還元」の様子を調べます．更に，

oped theory of geometry in mixed characteristic.
I hope that this study lead us to reveal the nature of “mod p

これらの研究を応用することで，より一般の図形の正標reduction” for more general varieties, and I also want to apply

数還元の考察や，整数解の有限性を始めとした整数論へ

these ideas to number theory.の応用を目指します．

What is arithmetic geometry?

actually obtain a one-to-one correspondence, completely de-'HVSLWHEHLQJWKHPRVWIDPLOLDUPDWKHPDWLFDOREMHFWLQWHJHUV

scribing the set of solutions.

still hold many mysterious open problems. The goal of arithmetic geometry is to solve these problems using geometric1RZZKDWKDSSHQVZKHQZHFRQVLGHURWKHUVKDSHVEHVLGHV

methods.

circles,)RUH[DPSOHOHW VVD\ZHZDQWWRNQRZKRZPDQ\WULDQJOHVIRUH[DPSOH[A\A ":KDWKDSSHQVZKHQZHDGGPRUH

with integer side lengths exist. This problem can be reduced

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WR¿QGLQJUDWLRQDOVROXWLRQVWRWKHHTXDWLRQ[A\A E\WKH

vast and profound theory called arithmetic geometry.

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rational solutions?

Mod p reduction of varieties

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How can we extract arithmetic properties from the geometric

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By associating each of these points with the slope viewed7KHSRLQWLVWKDWWKHLUHTXDWLRQVKDYHLQWHJHUUDWLRQDOFRH൶-IURP WKH SRLQW   ZKLFK LV D UDWLRQDO QXPEHU ZH FDQcients!

30

白眉プロジェクト 2023
7KLVDOORZVXVWRUHWKLQNHTXDWLRQVLQWKHZRUOGRIUHPDLQGHUVDespite their interest, there is still not much progress in theafter dividing by each prime number p. The resulting newstudy of irreducible symplectic varieties.\shapes are called the mod p reductions of the original shape,0\UHVHDUFKDLPVWRVWXG\WKHUHGXFWLRQRILUUHGXFLEOHV\P-and they can be thought of as shadows of the original shapeplectic varieties and reveal their arithmetic properties.

cast by the prime p.
Using these shadows for each prime, we can extract arithme-Method:Geometry in mixed characteristic

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I wanna study mod p reduction using the theory of geometryarithmetic geometry.

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*HRPHWU\LQPL[HGFKDUDFWWHULVWLFLVRQHRIWKHXQL¿HGWKHR-¿JXUHULHVLQJHRPHWU\WKDWLQWHJUDWHVWKHVHFRPSOHWHO\GL൵HUHQWJHometries, where “mixed” means that various prime numbers
are mixed.
Among them, the minimal model program in mixed characteristic, which has been rapidly developing in recent years, isvery powerful. The minimal model program is a grand theory
originally developed for non-mixed characteristic ordinary
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Theories like these are highly compatible with my goal of

the curve is smooth for other primes.

classifying mixed characteristic geometries and can be ex-6XFKLQIRUPDWLRQLVYHU\YDOXDEOH)RUH[DPSOHLWLVNQRZQSHFWHGWRSURGXFHVLJQL¿FDQWUHVXOWVWKDWWKHUHDUHRQO\ ¿QLWHQHVVLVLPSRUWDQW FXELFFXUYHV

+RZHYHUPL[HGFKDUDFWHULVWLFL]DWLRQKDVEHHQYHU\GL൶FXOWZLWKWKHVDPHSURSHUWLHV VXFKDUHVXOWLVFDOOHGD¿QLWHQHVVThe reason is that the concept of “singularity” which is essen-

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tial in the development of geometry, had not been formulatedof the main goals of my research.

well. However, in recent years, singularities that behave wellin mixed characteristic world and mod p worlds have been

Irreducible symplectic varieties

GH¿QHGDQGPL[HGFKDUDFWHULVWLFJHRPHWU\KDVHQWHUHGDPD-Higher-dimensional shapes have much more complicatedMRUWXUQLQJSRLQW

structures compared to the one-dimensional case, making,QWKH+DNXELSURMHFW,ZLOOFRQWLQXHWRH[SORUHWKHUHGXFWLRQ

it difficult to study general shapes directly. However, it isof irreducible symplectic varieties while further advancing

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the basic theory of mixed characteristic geometry.

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Among them, the arithmetic of abelian varieties is relatively wellunderstood and leads to a wealth of applications to number theory.

Bibliography:

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LFK &RQMHFWXUHV IRU +\SHUNlKOHU9DULHWLHVPreprint, arX-

DULWKPHWLFJHRPHWU\)LJXUHVKRZVDWZRGLPHQVLRQDOLUUH-iv:2203.10391

GXFLEOHV\PSOHFWLFYDULHW\DOVRNQRZQDVD.VXUIDFH

Takamatsu, T. & Yoshikawa S. Minimal Model Program for
6HPL6WDEOH7KUHHIROGVLQ0L[HG&KDUDFWHULVWLFWRDSSHDULQ
Journal of Algebraic Geometry, arXiv:2012.07324.DZDNDPL7<RVKLNDZD6 7DNDPDWVX7)HGGHU7\SH
&ULWHULDIRU4XDVL)6SOLWWLQJPreprint, arXiv:2204.10076

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