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=11ks\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:primeG(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}}
limtTamt(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)=limp1ζG/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

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

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

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)}.
ap    pordp(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,
0p    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
aν=0a=0\displaystyle {\left| a \right|}_{\nu} = 0 \Longleftrightarrow a = 0
\displaystyle {\left| ab \right|}_{\nu} = {\left| a \right|}_{\nu} {\left| b \right|}_{\nu}
abν=aνbν\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+bνaν+bν\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+bpmax(ap, bp)\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}

そこで,

\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)

\displaystyle d_{\nu} (a, b) = {\left| a - b \right|}_{\nu}
dν(a,b)=abν\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=νlimnan     (anQ)\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}
aν=limnanν\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
Qpp\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={aQp  ap1}\displaystyle \mathbb{Z}_{p} = \{a \in \mathbb{Q}_{p} ~ \mid ~ {\left| a \right|}_{p} \leqq 1\}

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
νP^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}
apZp\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,), aQ\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}

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=11ks\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:prime111ps.\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:prime111p2=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)πs2Γ(s2)=ζ(1s)π1s2Γ(1s2).\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)=2sπs1sinπs2Γ(1s)ζ(1s),\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(1s)=21sπscosπs2Γ(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}
111ps  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.
πs2Γ(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^(1s).\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}
QQνA\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

\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
tiQ,1i4\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}

\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}
dt1dt2dt3t1\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}

\displaystyle \omega
ω\displaystyle \omega

左不変な微分形式になる.

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

\displaystyle \omega
ω\displaystyle \omega

\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}}

\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:primeG(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 conThe ~ 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:primeG(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:primeX(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:primeG(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={(xyyx) 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:primeE(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  tE(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}}
limtTamt(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})}}
limtTamt(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

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

上の楕円曲線とする.

\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})

\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+1E(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  NE11a(p, E)1ps+p12s)\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}}}
×pNE11a(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  EN11a(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}}
=pEN1 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)

\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(1ρ(Frobp)ps\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(1ρ(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)
ptdet(1ρ(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}}
limtTamt(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)

\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!

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