Papers

Proof of the CM BSD conjecture and the congruent number problem←NEW

For an elliptic curve E over K, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group E(K) is equal to the order of the zero of L(E/ K,s) at s=1. In this paper, we shall give a proof for elliptic curves with complex multiplications. The key method of the proof is to reduce the Galois action of infinite order on the Tate module of an elliptic curve to that of finite order by using the p-adic Hodge theory. As a corollary, we can determine whether a given natural number is a congruent number (congruent number problem). This problem is one of the oldest unsolved problems in mathematics.

SU(3) Grand Unified Theory←NEW

The current mainstream model of the grand unified theory is based on the SU(5) gauge group and assumes that SU(3)×SU(2)×U(1)_Y gauge symmetry remains after the symmetry breaking. According to this theory, there exist 24 gauge fields with equal forces at the beginning of the universe. Indeed, one coupling constant unifies all the interactions, but the number of parameters is as many as 24, which is somewhat unsatisfactory for a unified theory. In this paper, we assume that the universe begins with U(1) gauge symmetry and construct the grand unified theory as a SU(3) gauge symmetry model.

現在の大統一理論のモデルとして主流であるのはゲージ群としてSU(5)を用いたものであり, 対称性の破れにより, SU(3)× SU(2)× U(1)_Yゲージ対称性が残ったとするものである. この理論によれば, 宇宙が始まったときに24個の等しい力を持ったゲージ場が存在することになる. 確かに, １つの結合定数ですべての相互作用が統一されることにはなるが, 24個ものパラメーターを持つことになり, 統一理論としてはいささか不満が残るところではある. この小論では宇宙がU(1)ゲージ対称性を持つことから始まると仮定して, 大統一理論をSU(3)ゲージ対称性モデルとして構成したい.

Notes on the particle spin (スピンについての考察)←NEW

The nature of the spin, which is believed to link matter, force and space, is still largely unexplained, even for fundamental things. In this essay, I would like to give a consideration of the spin from a simple point of view.

(物質・力・空間を結びつけているとされているスピンの性質は基本的なことに対しても未だに解明されてないことが多い. この小論では, 素朴な観点に立ち, スピンに対する考察を与えたい.)

Discrete property of the differential calculus (微分の離散性について)←NEW

For example, when we drive from Kyoto to Tokyo (in Japan), it takes an infinite number of measurements to work out the precise path. In this paper, however, if everything is written in polynomial form, we shall illustrate that these measurements can be done a finite number of times by using the discrete property of the differential calculus. This also means that a finite number of Orbis can be used to enforce all speed violations.

(例えば, 京都から東京まで車で行く場合, 正確な経路を求めるには無限の測定が必要である. しかし, この論説ではすべてのことが多項式の形式で書かれていれば, 微分がもつ離散性を利用して, これらの測定が有限回で済むことを論じる. また, これを使えば, 有限個のオービスによって, すべての速度違反も取り締まれることになる.)

Elliptic Curves and Birch and Swinnerton-Dyer conjecture (BSD conjecture)

Observation note on arithmetic elliptic curves

In this note, we shall give a self-righteous observation on rational points on elliptic curves and on Birch and Swinnerton-Dyer conjecture from the classical number theoretic view point.

Moving the rational points on arithmetic elliptic curve

In this short note, we shall construct a certain topological family which contains all elliptic curves over Q and, as an application, show that this family provides some geometric interpretations of the Hasse-Weil L-function of an elliptic curve over Q whose Mordell-Weil group is of rank ≤ 1.

Algebraic Cycles and Mixed Motives

Construction and Definition of the theory of mixed motives

In this paper, we shall give a candidate for the t-structure on the triangulated category of mixed motives due to Voevodsky. Through the regulator map, this gives rise to the generalization of the category of mixed Hodge structures which is constructed in [M].

Generalization of the mixed Hodge structure

In this paper, we shall generalize the theory of mixed Hodge structures due to Deligne and obtain a subcategory GMHS in the category of mixed Hodge structures such that we have Ext^2_{GMHS}(Q,−) ̸= 0 in general.

Motivation of my research

Physics from the standpoint of Arithmetic Geometry

Construction and Definition of the theory of quantum gravity

In this paper, by using mathematical interpretations on space-time obtained in [Mo3], we shall give a candidate for a particle picture of continuous gravitational fields.

Search for the smallest objects in the universe

In this paper, from the standpoint of quantum field theory, we shall study an analogous theory with the deformation theory of elliptic curves over Q obtained in [Mo2].

Relation between the number theory and physics

It is well-known that the value of Casimir energy coincides with the special value of the zeta function. In this paper, we shall give some observations on this relationship from the standpoint of number theory and research on its
generalizations by putting the new ideas obtained in [Mo] and the idea of the
discretization of physical quantities in string theory together.

Relation between the special relativity and quantum mechanics

In this paper, we shall give mathematical interpretations of the particle picture by using the momentum expansion of the quantized wave function of a Schr¨odinger equation and the theory of special relativity. As a result, we will introduce a new notion of time.

p-adic Hodge theory

1. Generalization of  (Phi, Gamma)-module. We construct  some differential operator which is closely related to the monodromy operator N and satisfies the Griffiths transeversality.

2. Comparison of p-adic representations in the perfect and imperfect residue field case by using  (phi, Gamma)-modules. As an application, we deduce the p-adic monodromy theorem of Fontaine in the imperfect residue field case.

3. Computation of p-adic Galois cohomology by using (phi, Gamma)-modules in the imperfect residue field case.

MathSciNet

書いたもの

楕円曲線上の有理点の求め方 (BSD周辺)

Cremona [C] に従って, Q上の楕円曲線E のMordell-Weil 群E(Q) についての様々な不変量を具体的に求めるアルゴリズムを紹介する. 特に, Mordell-Weil群E(Q) の生成元やランクなどが明示的に計算できることを解説するのが目的である.

楕円曲線の方程式の求め方

Cremona [C] に従って, 楕円曲線上の不変量の具体的な計算の仕方を紹介する. 特に, modular form f に付随する楕円曲線E_fの方程式を具体的に求めたい.

グレブナー基底の解説 (菅崎賢人くんによる)

1). 15分で理解するグレブナー基底

2). グレブナー基底について以下の文献をまとめたもの

『グレブナ基底と代数多様体入門　上・下』　丸善出版
著者: デビッド・コックス，ドナル・オシー，ジョン・リトル
翻訳: 落合啓之，西山享，山本敦子，示野信一，室政和

日本語訳