玉河数と Riemann zeta関数

2017.07.01.

@ロマンティック数学ナイト

@Riemann_Zeta_F

\displaystyle \zeta (s) = \sum_{k = 1}^{\infty} \frac{1}{k^s}
ζ(s)=∑k=1∞1ks\displaystyle \zeta (s) = \sum_{k = 1}^{\infty} \frac{1}{k^s}
\displaystyle \mathrm{Tam} (\mathbb{G}) = \prod_{p: \mathrm{prime}} \frac{|\mathbb{G}(\mathbb{F}_{p})|}{p^{\mathrm{dim} \mathbb{G}}}
Tam(G)=∏p:prime∣G(Fp)∣pdimG\displaystyle \mathrm{Tam} (\mathbb{G}) = \prod_{p: \mathrm{prime}} \frac{|\mathbb{G}(\mathbb{F}_{p})|}{p^{\mathrm{dim} \mathbb{G}}}
\displaystyle \tau (\mathbb{G}) = \int_{\mathbb{G} (\mathbb{Q})\backslash\mathbb{G}(\mathbb{A})} \omega_{\mathbb{A}}
τ(G)=∫G(Q)\G(A)ωA\displaystyle \tau (\mathbb{G}) = \int_{\mathbb{G} (\mathbb{Q})\backslash\mathbb{G}(\mathbb{A})} \omega_{\mathbb{A}}
\displaystyle \lim_{t \to \infty} \frac{\mathrm{Tam}_{t} (\mathrm{E})}{(\mathrm{log}~t)^{r}}
limt→∞Tamt(E)(log t)r\displaystyle \lim_{t \to \infty} \frac{\mathrm{Tam}_{t} (\mathrm{E})}{(\mathrm{log}~t)^{r}}
\displaystyle
\displaystyle
\displaystyle \zeta_{\mathbb{G}/\mathbb{F}_{1}} (s) = \lim_{p \to 1} \zeta_{\mathbb{G}/{\mathbb{F}_p}} (s)
ζG/F1(s)=limp→1ζG/Fp(s)\displaystyle \zeta_{\mathbb{G}/\mathbb{F}_{1}} (s) = \lim_{p \to 1} \zeta_{\mathbb{G}/{\mathbb{F}_p}} (s)
\displaystyle \mathrm{ord}_{s = 1} L(s, E/\mathbb{Q}) = \mathrm{rank} E(\mathbb{Q})
ords=1L(s,E/Q)=rankE(Q)\displaystyle \mathrm{ord}_{s = 1} L(s, E/\mathbb{Q}) = \mathrm{rank} E(\mathbb{Q})
\displaystyle \rho : \mathrm{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q}) \longrightarrow GL_{n} (K)
ρ:Gal(Q‾/Q)⟶GLn(K)\displaystyle \rho : \mathrm{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q}) \longrightarrow GL_{n} (K)

自己紹介.

・

\displaystyle 14
14\displaystyle 14

歳，中学

\displaystyle 2
2\displaystyle 2

年生.

・昨年の

\displaystyle 7
7\displaystyle 7

月にある

\displaystyle 2
2\displaystyle 2

人の日曜数学者のブログを

見て，数学に興味を持つ.

・好きな四字熟語は虚数乗法，好きな素数は

と

\displaystyle 37
37\displaystyle 37
\displaystyle 691
691\displaystyle 691

と

\displaystyle 1229
1229\displaystyle 1229
\displaystyle 7758337633
7758337633\displaystyle 7758337633

と

他多数.

・グレブナー基底にはポン酢派.

素数は無数に存在する．

\displaystyle \mathrm{Theorem. ~~~ (Euclid, 3 B.C.)}
Theorem.   (Euclid,3B.C.)\displaystyle \mathrm{Theorem. ~~~ (Euclid, 3 B.C.)}
\displaystyle
\displaystyle

Adele

\displaystyle
\displaystyle

Adele

素数を定義通りに列挙する：

\displaystyle 2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, \cdots
2,3,5,7,11,13,17,23,29,31,37,⋯\displaystyle 2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37, \cdots

というように無限に続く.

\displaystyle
\displaystyle

Adele

ここで記号

\displaystyle \infty
∞\displaystyle \infty

を導入すれば，素数は

\displaystyle 2, 3, 5, 7, 11, 13, 17, 23, 29, 31, \cdots , \infty
2,3,5,7,11,13,17,23,29,31,⋯,∞\displaystyle 2, 3, 5, 7, 11, 13, 17, 23, 29, 31, \cdots , \infty

とストップする.

\displaystyle \infty
∞\displaystyle \infty

を無限素点とよび，通常の

\displaystyle p
p\displaystyle p

を有限素点とよぶ.

素点を表す文字として

\displaystyle \nu
ν\displaystyle \nu

を用いる.

\displaystyle \hat{}
^\displaystyle \hat{}
\displaystyle \mathbb{P}
P\displaystyle \mathbb{P}
\displaystyle = \mathbb{P}
=P\displaystyle = \mathbb{P}
\displaystyle \cup
∪\displaystyle \cup
\displaystyle \infty
∞\displaystyle \infty

素点全体の集合を

\displaystyle \hat{}
^\displaystyle \hat{}
\displaystyle \mathbb{P}
P\displaystyle \mathbb{P}

とすると，

と書ける.

\displaystyle
\displaystyle

Adele

有理数体

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

の局所化として各素点

\displaystyle \nu \in
ν∈\displaystyle \nu \in
\displaystyle \mathbb{P}
P\displaystyle \mathbb{P}
\displaystyle \hat{}
^\displaystyle \hat{}

に対して局所体

\displaystyle \mathbb{Q}_{\nu}
Qν\displaystyle \mathbb{Q}_{\nu}

が，

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

の絶対値

\displaystyle {\left|~~\right|}_{\nu}
∣  ∣ν\displaystyle {\left|~~\right|}_{\nu}
\displaystyle \nu = \infty
ν=∞\displaystyle \nu = \infty

の定める距離の完備化として得ら

れるわけである.

i.e.

のときは

\displaystyle {\left| a \right|}_{\infty} ~~~~ {\left | a \right|}.
∣a∣∞    ∣a∣.\displaystyle {\left| a \right|}_{\infty} ~~~~ {\left | a \right|}.
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle =
=\displaystyle =
\displaystyle \nu = p
ν=p\displaystyle \nu = p

のときは

\displaystyle {\left| a \right|}_{p} ~~~~ p^{-\mathrm{ord}_{p} (a)}.
∣a∣p    p−ordp(a).\displaystyle {\left| a \right|}_{p} ~~~~ p^{-\mathrm{ord}_{p} (a)}.
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle =
=\displaystyle =
\displaystyle {\left| 0 \right|}_{p} ~~~~ 0,
∣0∣p    0,\displaystyle {\left| 0 \right|}_{p} ~~~~ 0,
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle =
=\displaystyle =
\displaystyle
\displaystyle

Adele

\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle {\left| a \right|}_{\nu} = 0 \Longleftrightarrow a = 0
∣a∣ν=0⟺a=0\displaystyle {\left| a \right|}_{\nu} = 0 \Longleftrightarrow a = 0
\displaystyle {\left| ab \right|}_{\nu} = {\left| a \right|}_{\nu} {\left| b \right|}_{\nu}
∣ab∣ν=∣a∣ν∣b∣ν\displaystyle {\left| ab \right|}_{\nu} = {\left| a \right|}_{\nu} {\left| b \right|}_{\nu}
\displaystyle {\left| a + b \right|}_{\nu} \leqq {\left| a \right|}_{\nu} + {\left| b \right|}_{\nu}
∣a+b∣ν≦∣a∣ν+∣b∣ν\displaystyle {\left| a + b \right|}_{\nu} \leqq {\left| a \right|}_{\nu} + {\left| b \right|}_{\nu}
\displaystyle \mathrm{Theorem.} ~~~~ (\mathrm{Strong ~ Triangle ~ Inequality})
Theorem.    (Strong Triangle Inequality)\displaystyle \mathrm{Theorem.} ~~~~ (\mathrm{Strong ~ Triangle ~ Inequality})
\displaystyle {\left| a + b \right|}_{p} \leqq \mathrm{max} ({\left| a \right|}_{p}, ~ {\left| b \right|}_{p})
∣a+b∣p≦max(∣a∣p, ∣b∣p)\displaystyle {\left| a + b \right|}_{p} \leqq \mathrm{max} ({\left| a \right|}_{p}, ~ {\left| b \right|}_{p})
\displaystyle
\displaystyle

Adele

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

の

そこで，

点

\displaystyle 2
2\displaystyle 2
\displaystyle a, b
a,b\displaystyle a, b

の距離

\displaystyle d_{\nu} (a, b)
dν(a,b)\displaystyle d_{\nu} (a, b)

を

\displaystyle d_{\nu} (a, b) = {\left| a - b \right|}_{\nu}
dν(a,b)=∣a−b∣ν\displaystyle d_{\nu} (a, b) = {\left| a - b \right|}_{\nu}

と定めると

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

は距離空間となり，

\displaystyle d_{\nu}
dν\displaystyle d_{\nu}

による

の完備化と

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

して完備な距離空間

を得る.

\displaystyle \mathbb{Q}_{\nu}
Qν\displaystyle \mathbb{Q}_{\nu}
\displaystyle \mathbb{Q}_{\nu}
Qν\displaystyle \mathbb{Q}_{\nu}

は体となり，

\displaystyle a = \nu - \lim_{n \to \infty} a_{n} ~~~~~ (a_{n} \in \mathbb{Q})
a=ν−limn→∞an     (an∈Q)\displaystyle a = \nu - \lim_{n \to \infty} a_{n} ~~~~~ (a_{n} \in \mathbb{Q})

のとき

\displaystyle {\left| a \right|}_{\nu} = \lim_{n \to \infty} {\left| a_{n} \right|}_{\nu}
∣a∣ν=limn→∞∣an∣ν\displaystyle {\left| a \right|}_{\nu} = \lim_{n \to \infty} {\left| a_{n} \right|}_{\nu}

により

\displaystyle \mathbb{Q}_{\nu}
Qν\displaystyle \mathbb{Q}_{\nu}

に絶対値

\displaystyle {\left| ~~ \right|}_{\nu}
∣  ∣ν\displaystyle {\left| ~~ \right|}_{\nu}

が拡張される.

\displaystyle
\displaystyle

Adele

\displaystyle \mathbb{Q}_{\nu}
Qν\displaystyle \mathbb{Q}_{\nu}

このようにして，各

\displaystyle \nu \in
ν∈\displaystyle \nu \in
\displaystyle \hat{}
^\displaystyle \hat{}
\displaystyle \mathbb{P}
P\displaystyle \mathbb{P}

に対し局所体

が得られた.

\displaystyle \nu = \infty
ν=∞\displaystyle \nu = \infty

のとき

\displaystyle \mathbb{Q}_{\nu} = \mathbb{R}.
Qν=R.\displaystyle \mathbb{Q}_{\nu} = \mathbb{R}.
\displaystyle \nu = p
ν=p\displaystyle \nu = p

のとき

\displaystyle \mathbb{Q}_{p} \cdots p
Qp⋯p\displaystyle \mathbb{Q}_{p} \cdots p

- 進数体

\displaystyle .
.\displaystyle .
\displaystyle \mathbb{Q}_{p}
Qp\displaystyle \mathbb{Q}_{p}

の部分集合

\displaystyle \mathbb{Z}_{p} = \{a \in \mathbb{Q}_{p} ~ \mid ~ {\left| a \right|}_{p} \leqq 1\}
Zp={a∈Qp ∣ ∣a∣p≦1}\displaystyle \mathbb{Z}_{p} = \{a \in \mathbb{Q}_{p} ~ \mid ~ {\left| a \right|}_{p} \leqq 1\}

は

の部分環

\displaystyle \mathbb{Q}_{p}
Qp\displaystyle \mathbb{Q}_{p}

(

\displaystyle \mathbb{Q}_{p}
Qp\displaystyle \mathbb{Q}_{p}

の加法について閉じている

)

.

\displaystyle
\displaystyle

Adele

直積

\displaystyle \prod_{\nu \in \hat{\mathbb{P}} } \mathbb{Q}_\nu
∏ν∈P^Qν\displaystyle \prod_{\nu \in \hat{\mathbb{P}} } \mathbb{Q}_\nu

の要素

\displaystyle a = (a_\nu)
a=(aν)\displaystyle a = (a_\nu)

に対して，

\displaystyle a = (a_{\nu})
a=(aν)\displaystyle a = (a_{\nu})
\displaystyle \Longleftrightarrow
⟺\displaystyle \Longleftrightarrow
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}

有限個の

\displaystyle p
p\displaystyle p

を除いて

\displaystyle a_{p} \in \mathbb{Z}_{p}
ap∈Zp\displaystyle a_{p} \in \mathbb{Z}_{p}

がAdele

とし，Adeleの全体を

\displaystyle \mathbb{A}
A\displaystyle \mathbb{A}

と書く.

有理数体

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

は対角写像

\displaystyle a \longmapsto (a, ~ a, ~ \cdots, ~a, \cdots), ~ a \in \mathbb{Q}
a⟼(a, a, ⋯, a,⋯), a∈Q\displaystyle a \longmapsto (a, ~ a, ~ \cdots, ~a, \cdots), ~ a \in \mathbb{Q}

により

\displaystyle \mathbb{A}
A\displaystyle \mathbb{A}

の中の離散的な部分加群とみなされる.

商群

\displaystyle \mathbb{A} \backslash \mathbb{Q}
A\Q\displaystyle \mathbb{A} \backslash \mathbb{Q}

は

Compact.

\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle
\displaystyle

Adele

\displaystyle \mathbb{A}
A\displaystyle \mathbb{A}
\displaystyle \mathbb{Q}_{2} ~~~ \mathbb{Q}_{3} ~~~ \mathbb{Q}_{5} ~~~ \mathbb{Q}_{7} ~~~ \mathbb{Q}_{11} ~ \cdots \cdots ~ \mathbb{Q}_{\infty}
Q2   Q3   Q5   Q7   Q11 ⋯⋯ Q∞\displaystyle \mathbb{Q}_{2} ~~~ \mathbb{Q}_{3} ~~~ \mathbb{Q}_{5} ~~~ \mathbb{Q}_{7} ~~~ \mathbb{Q}_{11} ~ \cdots \cdots ~ \mathbb{Q}_{\infty}
\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}
\displaystyle \mathbb{Z}
Z\displaystyle \mathbb{Z}

Adele ring

Local field

Rational field

Rational integer ring

\displaystyle
\displaystyle

Riemann zeta関数

\displaystyle
\displaystyle

Riemann zeta関数

\displaystyle \zeta (s)
ζ(s)\displaystyle \zeta (s)
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \sum_{k = 1}^{\infty} \frac{1}{k^s}
∑k=1∞1ks\displaystyle \sum_{k = 1}^{\infty} \frac{1}{k^s}
\displaystyle \mathrm{Re} (s) > 1.
Re(s)>1.\displaystyle \mathrm{Re} (s) > 1.
\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle \mathrm{Theorem.} ~~~ (\mathrm{Basel ~ problem)}
Theorem.   (Basel problem)\displaystyle \mathrm{Theorem.} ~~~ (\mathrm{Basel ~ problem)}
\displaystyle \zeta (2) = \frac{\pi^{2}}{6}.
ζ(2)=π26.\displaystyle \zeta (2) = \frac{\pi^{2}}{6}.
\displaystyle
\displaystyle

Riemann zeta関数

\displaystyle \zeta (s) = \prod_{p \mathrm{: prime}} \frac{1}{1 - \frac{1}{p^s}}.
ζ(s)=∏p:prime11−1ps.\displaystyle \zeta (s) = \prod_{p \mathrm{: prime}} \frac{1}{1 - \frac{1}{p^s}}.
\displaystyle \mathrm{Theorem.} ~~~ (\mathrm{Euler ~ product)}
Theorem.   (Euler product)\displaystyle \mathrm{Theorem.} ~~~ (\mathrm{Euler ~ product)}
\displaystyle \mathrm{Example.}
Example.\displaystyle \mathrm{Example.}
\displaystyle \zeta (2) = \prod_{p \mathrm{: prime}} \frac{1}{1 - \frac{1}{p^2}} = \frac{\pi^{2}}{6}.
ζ(2)=∏p:prime11−1p2=π26.\displaystyle \zeta (2) = \prod_{p \mathrm{: prime}} \frac{1}{1 - \frac{1}{p^2}} = \frac{\pi^{2}}{6}.
\displaystyle
\displaystyle

Riemann zeta関数

\displaystyle \mathrm{Theorem.} ~~~ (\mathrm{Functional ~ equation)}
Theorem.   (Functional equation)\displaystyle \mathrm{Theorem.} ~~~ (\mathrm{Functional ~ equation)}
\displaystyle \zeta (s) \pi^{- \frac{s}{2}} \Gamma {\left( \frac{s}{2} \right)} = \zeta (1 - s) \pi^{- \frac{1 - s}{2}} \Gamma {\left( \frac{1 - s}{2} \right)}.
ζ(s)π−s2Γ(s2)=ζ(1−s)π−1−s2Γ(1−s2).\displaystyle \zeta (s) \pi^{- \frac{s}{2}} \Gamma {\left( \frac{s}{2} \right)} = \zeta (1 - s) \pi^{- \frac{1 - s}{2}} \Gamma {\left( \frac{1 - s}{2} \right)}.
\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle \prod_{n = 1}^{\infty} ~ n = \sqrt{2\pi}.
∏n=1∞ n=2π.\displaystyle \prod_{n = 1}^{\infty} ~ n = \sqrt{2\pi}.
\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle \zeta (-1) = \sum_{k = 1}^{\infty} ~ k = -\frac{1}{12}.
ζ(−1)=∑k=1∞ k=−112.\displaystyle \zeta (-1) = \sum_{k = 1}^{\infty} ~ k = -\frac{1}{12}.
\displaystyle
\displaystyle

Riemann zeta関数

\displaystyle \zeta (s) = 2^{s} \pi^{s-1} \sin \frac{\pi s}{2} \Gamma (1 - s) \zeta (1 - s),
ζ(s)=2sπs−1sinπs2Γ(1−s)ζ(1−s),\displaystyle \zeta (s) = 2^{s} \pi^{s-1} \sin \frac{\pi s}{2} \Gamma (1 - s) \zeta (1 - s),
\displaystyle \zeta (1 - s) = 2^{1 - s} \pi^{-s} \cos \frac{\pi s}{2} \Gamma (s) \zeta (s).
ζ(1−s)=21−sπ−scosπs2Γ(s)ζ(s).\displaystyle \zeta (1 - s) = 2^{1 - s} \pi^{-s} \cos \frac{\pi s}{2} \Gamma (s) \zeta (s).
\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle \mathrm{Conjecture.}
Conjecture.\displaystyle \mathrm{Conjecture.}
\displaystyle The ~ real ~ part ~ of ~ every ~ non ~~~ trivial ~ zero ~ of ~ the
The real part of every non   trivial zero of the\displaystyle The ~ real ~ part ~ of ~ every ~ non ~~~ trivial ~ zero ~ of ~ the
\displaystyle Riemann ~ zeta ~ function ~ is ~ \frac{1}{2}.
Riemann zeta function is 12.\displaystyle Riemann ~ zeta ~ function ~ is ~ \frac{1}{2}.
\displaystyle -
−\displaystyle -
\displaystyle
\displaystyle

Riemann zeta関数

素点に関する zeta関数 place zeta関数

\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle \zeta_{\nu} (s)
ζν(s)\displaystyle \zeta_{\nu} (s)

を定義する.

\displaystyle \zeta_\nu(s)
ζν(s)\displaystyle \zeta_\nu(s)
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \Biggl\{
{\displaystyle \Biggl\{
\displaystyle \frac{1}{1 - \frac{1}{p^s}} ~ \cdots \cdots ~ \mathrm{if} ~ p ~ \mathrm{is ~ prime ~ number}
11−1ps ⋯⋯ if p is prime number\displaystyle \frac{1}{1 - \frac{1}{p^s}} ~ \cdots \cdots ~ \mathrm{if} ~ p ~ \mathrm{is ~ prime ~ number}
\displaystyle \pi^{-\frac{s}{2}} \Gamma \left(\frac{s}{2}\right) ~ \cdots ~ \mathrm{if} ~ p = \infty.
π−s2Γ(s2) ⋯ if p=∞.\displaystyle \pi^{-\frac{s}{2}} \Gamma \left(\frac{s}{2}\right) ~ \cdots ~ \mathrm{if} ~ p = \infty.
\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle \hat{\zeta} (s)
ζ^(s)\displaystyle \hat{\zeta} (s)
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \prod_{\nu}~ \zeta_{\nu}(s).
∏ν ζν(s).\displaystyle \prod_{\nu}~ \zeta_{\nu}(s).
\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle \hat{\zeta}(s) = \hat{\zeta}(1 - s).
ζ^(s)=ζ^(1−s).\displaystyle \hat{\zeta}(s) = \hat{\zeta}(1 - s).
\displaystyle
\displaystyle

玉河数

\displaystyle
\displaystyle

玉河数

\displaystyle \mathbb{Q} \longrightarrow \mathbb{Q}_{\nu} \longrightarrow \mathbb{A}
Q⟶Qν⟶A\displaystyle \mathbb{Q} \longrightarrow \mathbb{Q}_{\nu} \longrightarrow \mathbb{A}

このプロセスは

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

で定義された代数多様体

\displaystyle X
X\displaystyle X

に適用できる:

零点をとるような体や環を変化させることで，

\displaystyle X(\mathbb{Q}) \longrightarrow X(\mathbb{Q}_{\nu}) \longrightarrow X(\mathbb{A})
X(Q)⟶X(Qν)⟶X(A)\displaystyle X(\mathbb{Q}) \longrightarrow X(\mathbb{Q}_{\nu}) \longrightarrow X(\mathbb{A})

というプロセスも考えることができる.

このプロセスを

\displaystyle X
X\displaystyle X

の

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

(

に関する

)

Adele化とよぶ.

\displaystyle
\displaystyle

玉河数

\displaystyle G(\mathbb{Q}) = \left\{x = \left( \begin{array}{ccc} t_{1} & t_{2} \\ t_{3} & t_{4} \end{array} \right) ~ \middle| ~ \mathrm{det} ~ x = 1 \right\}
G(Q)={x=(t1t2t3t4) undefined det x=1}\displaystyle G(\mathbb{Q}) = \left\{x = \left( \begin{array}{ccc} t_{1} & t_{2} \\ t_{3} & t_{4} \end{array} \right) ~ \middle| ~ \mathrm{det} ~ x = 1 \right\}

ただし

\displaystyle t_{i} \in \mathbb{Q}, 1 \leqq i \leqq 4
ti∈Q,1≦i≦4\displaystyle t_{i} \in \mathbb{Q}, 1 \leqq i \leqq 4
\displaystyle \mathbb{G}\left(\mathbb{Q}\right)
G(Q)\displaystyle \mathbb{G}\left(\mathbb{Q}\right)
\displaystyle \mathbb{G}\left(\mathbb{A}\right)
G(A)\displaystyle \mathbb{G}\left(\mathbb{A}\right)
\displaystyle \mathbb{G}\left(\mathbb{Z}\right)
G(Z)\displaystyle \mathbb{G}\left(\mathbb{Z}\right)
\displaystyle \mathbb{G}\left(\mathbb{Q}_{2}\right)
G(Q2)\displaystyle \mathbb{G}\left(\mathbb{Q}_{2}\right)
\displaystyle \mathbb{G}\left(\mathbb{Q}_{3}\right)
G(Q3)\displaystyle \mathbb{G}\left(\mathbb{Q}_{3}\right)
\displaystyle \mathbb{G}\left(\mathbb{Q}_{5}\right)
G(Q5)\displaystyle \mathbb{G}\left(\mathbb{Q}_{5}\right)
\displaystyle \mathbb{G}\left(\mathbb{Q}_{\infty}\right)
G(Q∞)\displaystyle \mathbb{G}\left(\mathbb{Q}_{\infty}\right)
\displaystyle \cdots \cdots
⋯⋯\displaystyle \cdots \cdots

Modular group

Group of rational points

Local group

Adele group

\displaystyle
\displaystyle

玉河数

\displaystyle \mathbb{G} = \mathbb{SL}_{2}
G=SL2\displaystyle \mathbb{G} = \mathbb{SL}_{2}

は

\displaystyle 3
3\displaystyle 3

次元だから，

\displaystyle \omega
ω\displaystyle \omega
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \frac{dt_{1} \land dt_{2} \land dt_{3}}{t_1}
dt1∧dt2∧dt3t1\displaystyle \frac{dt_{1} \land dt_{2} \land dt_{3}}{t_1}

という微分形式(体積要素)を考えると，

\displaystyle \mathbb{G}
G\displaystyle \mathbb{G}

は

\displaystyle \omega
ω\displaystyle \omega

の

左不変な微分形式になる.

\displaystyle \omega_{\mathbb{A}}
ωA\displaystyle \omega_{\mathbb{A}}

を

\displaystyle \omega
ω\displaystyle \omega

が

\displaystyle \mathbb{G}(\mathbb{A})
G(A)\displaystyle \mathbb{G}(\mathbb{A})

に定める

Haar測度としたとき

\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle \tau(\mathbb{G})
τ(G)\displaystyle \tau(\mathbb{G})
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \int_{\mathbb{G}(\mathbb{Q}) \backslash \mathbb{G}(\mathbb{A})} \omega_{\mathbb{A}}
∫G(Q)\G(A)ωA\displaystyle \int_{\mathbb{G}(\mathbb{Q}) \backslash \mathbb{G}(\mathbb{A})} \omega_{\mathbb{A}}

が

\displaystyle \mathbb{G}
G\displaystyle \mathbb{G}

の玉河数.

\displaystyle
\displaystyle

玉河数

\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle \tau_{0}(\mathbb{G})
τ0(G)\displaystyle \tau_{0}(\mathbb{G})
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \prod_{p:\mathrm{prime}} \int_{\mathbb{G}(\mathbb{Z}_{p})} \omega_{p},
∏p:prime∫G(Zp)ωp,\displaystyle \prod_{p:\mathrm{prime}} \int_{\mathbb{G}(\mathbb{Z}_{p})} \omega_{p},
\displaystyle \tau_{\infty}(\mathbb{G})
τ∞(G)\displaystyle \tau_{\infty}(\mathbb{G})
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \int_{\mathbb{G}(\mathbb{Z}) \backslash \mathbb{G}(\mathbb{R})} \omega_{\infty}.
∫G(Z)\G(R)ω∞.\displaystyle \int_{\mathbb{G}(\mathbb{Z}) \backslash \mathbb{G}(\mathbb{R})} \omega_{\infty}.
\displaystyle \mathrm{Conjecture.}
Conjecture.\displaystyle \mathrm{Conjecture.}
The ~ Tamagawa ~ number ~ ~~~~~~~~ ~ of ~ a ~ simply ~ con-
The Tamagawa number          of a simply con−The ~ Tamagawa ~ number ~ ~~~~~~~~ ~ of ~ a ~ simply ~ con-
\displaystyle \tau(\mathbb{G})
τ(G)\displaystyle \tau(\mathbb{G})
nected ~ simple ~ algebraic ~ group ~ defined ~ over ~ a
nected simple algebraic group defined over anected ~ simple ~ algebraic ~ group ~ defined ~ over ~ a
number ~ field ~ is ~~~.
number field is   .number ~ field ~ is ~~~.
\displaystyle 1
1\displaystyle 1
\displaystyle
\displaystyle

玉河数

\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle \mathrm{Tam}(\mathbb{G})
Tam(G)\displaystyle \mathrm{Tam}(\mathbb{G})
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \prod_{p:\mathrm{prime}} \frac{\left|\mathbb{G}(\mathbb{F}_{p})\right|}{p^{\mathrm{dim} ~ \mathbb{G}}},
∏p:prime∣G(Fp)∣pdim G,\displaystyle \prod_{p:\mathrm{prime}} \frac{\left|\mathbb{G}(\mathbb{F}_{p})\right|}{p^{\mathrm{dim} ~ \mathbb{G}}},
\displaystyle \mathrm{Tam}(X)
Tam(X)\displaystyle \mathrm{Tam}(X)
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \prod_{p:\mathrm{prime}} \frac{\left|X(\mathbb{F}_{p})\right|}{p^{\mathrm{dim} ~ X}}.
∏p:prime∣X(Fp)∣pdim X.\displaystyle \prod_{p:\mathrm{prime}} \frac{\left|X(\mathbb{F}_{p})\right|}{p^{\mathrm{dim} ~ X}}.
\displaystyle
\displaystyle

玉河数

\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle \mathrm{Tam} (\mathbb{SL}_{2}) = \frac{1}{\zeta (2)} = \frac{6}{\pi^{2}}.
Tam(SL2)=1ζ(2)=6π2.\displaystyle \mathrm{Tam} (\mathbb{SL}_{2}) = \frac{1}{\zeta (2)} = \frac{6}{\pi^{2}}.
\displaystyle \tau(\mathbb{G}) = \int_{\mathbb{G}(\mathbb{Z}) \backslash \mathbb{G}(\mathbb{A})} \omega_{\infty} \times \prod_{p:\mathrm{prime}} \int_{\mathbb{G}(\mathbb{Z}_{p})} \omega_{p}
τ(G)=∫G(Z)\G(A)ω∞×∏p:prime∫G(Zp)ωp\displaystyle \tau(\mathbb{G}) = \int_{\mathbb{G}(\mathbb{Z}) \backslash \mathbb{G}(\mathbb{A})} \omega_{\infty} \times \prod_{p:\mathrm{prime}} \int_{\mathbb{G}(\mathbb{Z}_{p})} \omega_{p}
\displaystyle
\displaystyle

玉河数

\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle \mathrm{Tam} (\mathbb{SL}_{n}) = \frac{1}{\zeta (2) \zeta(3) \cdots \zeta(n)}.
Tam(SLn)=1ζ(2)ζ(3)⋯ζ(n).\displaystyle \mathrm{Tam} (\mathbb{SL}_{n}) = \frac{1}{\zeta (2) \zeta(3) \cdots \zeta(n)}.
\displaystyle \mathrm{Example.}
Example.\displaystyle \mathrm{Example.}
\displaystyle \mathrm{Tam} (\mathbb{SL}_{3}) = \frac{1}{\zeta (2) \zeta(3)}
Tam(SL3)=1ζ(2)ζ(3)\displaystyle \mathrm{Tam} (\mathbb{SL}_{3}) = \frac{1}{\zeta (2) \zeta(3)}
\displaystyle = \frac{6}{\pi^{2} \zeta (3)}.
=6π2ζ(3).\displaystyle = \frac{6}{\pi^{2} \zeta (3)}.
\displaystyle
\displaystyle

玉河数

\displaystyle \mathbb{SO}_{2} = \left\{\left( \begin{array}{ccc} x & -y \\ y & x \end{array} \right) ~ \middle| ~ x^{2} + y^{2} = 1\right\}
SO2={(x−yyx) undefined x2+y2=1}\displaystyle \mathbb{SO}_{2} = \left\{\left( \begin{array}{ccc} x & -y \\ y & x \end{array} \right) ~ \middle| ~ x^{2} + y^{2} = 1\right\}
\displaystyle \cong \left\{(x, ~y) ~ \middle| ~ x^{2} + y^{2} = 1\right\}
≅{(x, y) undefined x2+y2=1}\displaystyle \cong \left\{(x, ~y) ~ \middle| ~ x^{2} + y^{2} = 1\right\}
\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle \mathrm{Tam} \left(\mathbb{SO}_{2}\right) = \frac{~4~}{~\pi ~}.
Tam(SO2)= 4  π .\displaystyle \mathrm{Tam} \left(\mathbb{SO}_{2}\right) = \frac{~4~}{~\pi ~}.
\displaystyle
\displaystyle

玉河数

\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

上の楕円曲線

\displaystyle E
E\displaystyle E

に対し

\displaystyle \mathrm{Tam} (E) = \prod_{p:\mathrm{prime}} \frac{\left| E(\mathbb{F}_{p})\right|}{p}
Tam(E)=∏p:prime∣E(Fp)∣p\displaystyle \mathrm{Tam} (E) = \prod_{p:\mathrm{prime}} \frac{\left| E(\mathbb{F}_{p})\right|}{p}

および

\displaystyle \mathrm{Tam}_{t} (E) = \prod_{p:\mathrm{prime} ~ \leqq ~ t} \frac{\left| E(\mathbb{F}_{p})\right|}{p}.
Tamt(E)=∏p:prime ≦ t∣E(Fp)∣p.\displaystyle \mathrm{Tam}_{t} (E) = \prod_{p:\mathrm{prime} ~ \leqq ~ t} \frac{\left| E(\mathbb{F}_{p})\right|}{p}.
\displaystyle
\displaystyle

玉河数

\displaystyle \mathrm{Conjecture.}
Conjecture.\displaystyle \mathrm{Conjecture.}
\displaystyle Let ~ r = \mathrm{rank}~E(\mathbb{Q}). ~ Then
Let r=rank E(Q). Then\displaystyle Let ~ r = \mathrm{rank}~E(\mathbb{Q}). ~ Then
\displaystyle \lim_{t \to \infty} \frac{\mathrm{Tam}_{t} (E)}{(\mathrm{log} ~ t)^{r}}
limt→∞Tamt(E)(log t)r\displaystyle \lim_{t \to \infty} \frac{\mathrm{Tam}_{t} (E)}{(\mathrm{log} ~ t)^{r}}
\displaystyle converges ~ to ~ finite ~ value ~ which ~ is ~ not ~ 0.
converges to finite value which is not 0.\displaystyle converges ~ to ~ finite ~ value ~ which ~ is ~ not ~ 0.
\displaystyle \mathrm{Conjecture.}
Conjecture.\displaystyle \mathrm{Conjecture.}
\displaystyle \mathrm{ord}_{s = 1} L(s, ~ E/\mathbb{Q}) = \mathrm{rank} ~ E(\mathbb{Q}).
ords=1L(s, E/Q)=rank E(Q).\displaystyle \mathrm{ord}_{s = 1} L(s, ~ E/\mathbb{Q}) = \mathrm{rank} ~ E(\mathbb{Q}).
\displaystyle
\displaystyle

玉河数

\displaystyle \mathrm{Theorem.}
Theorem.\displaystyle \mathrm{Theorem.}
\displaystyle \lim_{t \to \infty} \frac{\mathrm{Tam}_{t} (E)}{(\mathrm{log} ~ t)^{\mathrm{rank} ~ E(\mathbb{Q})}}
limt→∞Tamt(E)(log t)rank E(Q)\displaystyle \lim_{t \to \infty} \frac{\mathrm{Tam}_{t} (E)}{(\mathrm{log} ~ t)^{\mathrm{rank} ~ E(\mathbb{Q})}}
\displaystyle \mathrm{ord}_{s = 1} L(s, ~ E/\mathbb{Q}) = \mathrm{rank} ~ E(\mathbb{Q}).
ords=1L(s, E/Q)=rank E(Q).\displaystyle \mathrm{ord}_{s = 1} L(s, ~ E/\mathbb{Q}) = \mathrm{rank} ~ E(\mathbb{Q}).
\displaystyle E
E\displaystyle E

を

\displaystyle \mathbb{Q}
Q\displaystyle \mathbb{Q}

上の楕円曲線とする.

が

\displaystyle 0
0\displaystyle 0

でない有限値に収束すると仮定すると次が成立: 

(1)

(2)

\displaystyle L(s, ~ E/\mathbb{Q})
L(s, E/Q)\displaystyle L(s, ~ E/\mathbb{Q})

は

\displaystyle \mathrm{Re} (s) > 1
Re(s)>1\displaystyle \mathrm{Re} (s) > 1

において零点なし.

\displaystyle
\displaystyle

玉河数

楕円曲線

E
EE

の導手を

\displaystyle N_{E}
NE\displaystyle N_{E}

とする.

，

\displaystyle a(p, ~E)
a(p, E)\displaystyle a(p, ~E)
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle p + 1- |E( \mathbb{F}_{p})|
p+1−∣E(Fp)∣\displaystyle p + 1- |E( \mathbb{F}_{p})|
\displaystyle L(s, ~E/\mathbb{Q}) = \prod_{p~|~N_E} \frac{1}{1 - a(p, ~ E)\frac{1}{p^{s}} + p^{1-2s})}
L(s, E/Q)=∏p ∣ NE11−a(p, E)1ps+p1−2s)\displaystyle L(s, ~E/\mathbb{Q}) = \prod_{p~|~N_E} \frac{1}{1 - a(p, ~ E)\frac{1}{p^{s}} + p^{1-2s})}
\displaystyle \times \prod_{p | N_{E}} \frac{1}{1 - a(p, ~ E) \frac{1}{p^{s}}}
×∏p∣NE11−a(p, E)1ps\displaystyle \times \prod_{p | N_{E}} \frac{1}{1 - a(p, ~ E) \frac{1}{p^{s}}}
\displaystyle
\displaystyle

玉河数

すると，形式的には

\displaystyle L(1, ~ E/\mathbb{Q}) \cong \prod_{p~|~E_{N}} \frac{1}{1 - \frac{a(p, ~ E)}{p} + \frac{1}{p}}
L(1, E/Q)≅∏p ∣ EN11−a(p, E)p+1p\displaystyle L(1, ~ E/\mathbb{Q}) \cong \prod_{p~|~E_{N}} \frac{1}{1 - \frac{a(p, ~ E)}{p} + \frac{1}{p}}
\displaystyle = \prod_{p | E_{N}} \frac{1}{\frac{~|E(\mathbb{F}_{p}|~}{p}}
=∏p∣EN1 ∣E(Fp∣ p\displaystyle = \prod_{p | E_{N}} \frac{1}{\frac{~|E(\mathbb{F}_{p}|~}{p}}
\displaystyle \cong \frac{1}{\mathrm{Tam} (E)}.
≅1Tam(E).\displaystyle \cong \frac{1}{\mathrm{Tam} (E)}.

つまり，

\displaystyle \mathrm{Tam} (E)
Tam(E)\displaystyle \mathrm{Tam} (E)

は

\displaystyle L(s, ~ E)
L(s, E)\displaystyle L(s, ~ E)

の中心Euler積の逆数.

\displaystyle
\displaystyle

Deep Riemann Hypothesis

\displaystyle
\displaystyle

Deep Riemann Hypothesis

Galois表現

\displaystyle \rho : \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow GL_{n} (K)
ρ:Gal(Q‾/Q)⟶GLn(K)\displaystyle \rho : \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow GL_{n} (K)

に対して考える.

\displaystyle L(s, ~\rho)
L(s, ρ)\displaystyle L(s, ~\rho)
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \prod_{p} \frac{1}{\mathrm{det} ( 1- \rho (\mathrm{Frob}_{p})p^{-s}}
∏p1det(1−ρ(Frobp)p−s\displaystyle \prod_{p} \frac{1}{\mathrm{det} ( 1- \rho (\mathrm{Frob}_{p})p^{-s}}

簡単のため，Frobenius元

\displaystyle \mathrm{Frob}_{p}
Frobp\displaystyle \mathrm{Frob}_{p}

によって決められる

という

\displaystyle L
L\displaystyle L

関数は極を持たないものとする.

\displaystyle \mathrm{Definition.}
Definition.\displaystyle \mathrm{Definition.}
\displaystyle \mathrm{Tam} (\rho)
Tam(ρ)\displaystyle \mathrm{Tam} (\rho)
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \prod_{p} \mathrm{det} \left(1 - \rho (\mathrm{Frob}_{p}) \frac{1}{p^{\frac{k}{2}}}\right)
∏pdet(1−ρ(Frobp)1pk2)\displaystyle \prod_{p} \mathrm{det} \left(1 - \rho (\mathrm{Frob}_{p}) \frac{1}{p^{\frac{k}{2}}}\right)
\displaystyle \mathrm{Tam}_{t} (\rho)
Tamt(ρ)\displaystyle \mathrm{Tam}_{t} (\rho)
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \prod_{p \leqq t} \mathrm{det} \left(1 - \rho (\mathrm{Frob}_{p}) \frac{1}{p^{\frac{k}{2}}}\right)
∏p≦tdet(1−ρ(Frobp)1pk2)\displaystyle \prod_{p \leqq t} \mathrm{det} \left(1 - \rho (\mathrm{Frob}_{p}) \frac{1}{p^{\frac{k}{2}}}\right)
\displaystyle
\displaystyle

Deep Riemann Hypothesis

\displaystyle \mathrm{Conjecture.} ~~~ (\mathrm{Deep ~ Riemann ~ Hypothesis)}
Conjecture.   (Deep Riemann Hypothesis)\displaystyle \mathrm{Conjecture.} ~~~ (\mathrm{Deep ~ Riemann ~ Hypothesis)}
\displaystyle Let
Let\displaystyle Let
\displaystyle r
r\displaystyle r
\displaystyle =
=\displaystyle =
\displaystyle \mathrm{def}
def\displaystyle \mathrm{def}
\displaystyle \mathrm{ord}_{s = \frac{k}{2}} L(s, ~ \rho).
ords=k2L(s, ρ).\displaystyle \mathrm{ord}_{s = \frac{k}{2}} L(s, ~ \rho).
\displaystyle Then
Then\displaystyle Then
\displaystyle \lim_{t \to \infty} \frac{\mathrm{Tam}_{t} (\rho)}{(\mathrm{log} ~ t)^{r}}
limt→∞Tamt(ρ)(log t)r\displaystyle \lim_{t \to \infty} \frac{\mathrm{Tam}_{t} (\rho)}{(\mathrm{log} ~ t)^{r}}
\displaystyle converges ~ to ~ finite ~ value ~ which ~ is ~ not ~ 0.
converges to finite value which is not 0.\displaystyle converges ~ to ~ finite ~ value ~ which ~ is ~ not ~ 0.

これは，Riemann予想

「

\displaystyle L(s, ~\rho)
L(s, ρ)\displaystyle L(s, ~\rho)

は

\displaystyle \mathrm{Re} (s) > \frac{k}{2}
Re(s)>k2\displaystyle \mathrm{Re} (s) > \frac{k}{2}

において

零点を持たない

」

を導き，より "深い" 予想.

\displaystyle
\displaystyle

Thank you for your attention!

ロマ酢

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