テータ関数
Z ( u ) := E ( u ) − E ( k ) u K ( k ) , E ( u ) := ∫ 0 u d t d n 2 t = ∫ 0 s n u d t 1 − k 2 t 2 1 − t 2 = ∫ 0 a m u d θ 1 − k 2 sin 2  θ , {\displaystyle {\begin{aligned}Z(u)&:={\mathcal {E}}(u)-{\frac {E(k)u}{K(k)}},\\{\mathcal {E}}(u)&:=\int _{0}^{u}\mathrm {d} t\,\mathrm {dn} ^{2}t=\int _{0}^{\mathrm {sn} u}\mathrm {d} t{\sqrt {\frac {1-k^{2}t^{2}}{1-t^{2}}}}=\int _{0}^{\mathrm {am} u}\mathrm {d} \theta {\sqrt {1-k^{2}\sin ^{2}\theta }},\end{aligned}}}
E ( u ) {\displaystyle {\mathcal {E}}(u)} はヤコビのイプシロン関数、 E ( k ) {\displaystyle E(k)} は第2種完全楕円積分、 s n u = s n ( u , k ) {\displaystyle \mathrm {sn} \,u=\mathrm {sn} (u,k)} , d n u = d n ( u , k ) {\displaystyle dn\,u=dn(u,k)} は ヤコビの楕円関数、 a m u = a m ( u , k ) {\displaystyle \mathrm {am} \,u=\mathrm {am} (u,k)} は振幅関数である。
θ 1 ( 2 z , τ ) θ 2 ( 0 , τ ) θ 3 ( 0 , τ ) θ 4 ( 0 , τ ) = 2 θ 1 ( z , τ ) θ 2 ( z , τ ) θ 3 ( z , τ ) θ 4 ( z , τ ) θ 2 ( 2 z , τ ) θ 2 3 ( 0 , τ ) = θ 2 4 ( z , τ ) − θ 1 4 ( z , τ ) = θ 3 4 ( z , τ ) − θ 4 4 ( z , τ ) θ 3 ( 2 z , τ ) θ 3 3 ( 0 , τ ) = θ 3 4 ( z , τ ) + θ 1 4 ( z , τ ) = θ 2 4 ( z , τ ) + θ 4 4 ( z , τ ) θ 4 ( 2 z , τ ) θ 4 3 ( 0 , τ ) = θ 4 4 ( z , τ ) − θ 1 4 ( z , τ ) = θ 3 4 ( z , τ ) − θ 2 4 ( z , τ ) {\displaystyle {\begin{aligned}&\vartheta _{1}\left(2z,\tau \right)\vartheta _{2}(0,\tau )\vartheta _{3}(0,\tau )\vartheta _{4}(0,\tau )=2\vartheta _{1}\left(z,\tau \right)\vartheta _{2}(z,\tau )\vartheta _{3}(z,\tau )\vartheta _{4}(z,\tau )\\&\vartheta _{2}\left(2z,\tau \right)\vartheta _{2}^{\;3}(0,\tau )=\vartheta _{2}^{\;4}\left(z,\tau \right)-\vartheta _{1}^{\;4}\left(z,\tau \right)=\vartheta _{3}^{\;4}\left(z,\tau \right)-\vartheta _{4}^{\;4}\left(z,\tau \right)\\&\vartheta _{3}\left(2z,\tau \right)\vartheta _{3}^{\;3}(0,\tau )=\vartheta _{3}^{\;4}\left(z,\tau \right)+\vartheta _{1}^{\;4}\left(z,\tau \right)=\vartheta _{2}^{\;4}\left(z,\tau \right)+\vartheta _{4}^{\;4}\left(z,\tau \right)\\&\vartheta _{4}\left(2z,\tau \right)\vartheta _{4}^{\;3}(0,\tau )=\vartheta _{4}^{\;4}\left(z,\tau \right)-\vartheta _{1}^{\;4}\left(z,\tau \right)=\vartheta _{3}^{\;4}\left(z,\tau \right)-\vartheta _{2}^{\;4}\left(z,\tau \right)\\\end{aligned}}}
θ 2 ′ ( v , τ ) θ 2 ( v , τ ) = − π tan  π v + 4 π ∑ n = 1 ∞ ( − 1 ) n e 2 n π i τ 1 − e 2 n π i τ sin  2 π n v θ 3 ′ ( v , τ ) θ 3 ( v , τ ) = 4 π ∑ n = 1 ∞ ( − 1 ) n e n π i τ 1 − e 2 n π i τ sin  2 π n v θ 4 ′ ( v , τ ) θ 4 ( v , τ ) = 4 π ∑ n = 1 ∞ e n π i τ 1 − e 2 n π i τ sin  2 π n v {\displaystyle {\begin{aligned}{\frac {\vartheta _{2}'(v,\tau )}{\vartheta _{2}(v,\tau )}}&=-\pi \tan \pi {v}+4\pi \sum _{n=1}^{\infty }{\frac {{(-1)^{n}}e^{2n\pi {i\tau }}}{1-e^{2n\pi {i\tau }}}}\sin 2\pi {nv}\\{\frac {\vartheta _{3}'(v,\tau )}{\vartheta _{3}(v,\tau )}}&=4\pi \sum _{n=1}^{\infty }{\frac {{(-1)^{n}}e^{n\pi {i\tau }}}{1-e^{2n\pi {i\tau }}}}\sin 2\pi {nv}\\{\frac {\vartheta _{4}'(v,\tau )}{\vartheta _{4}(v,\tau )}}&=4\pi \sum _{n=1}^{\infty }{\frac {e^{n\pi {i\tau }}}{1-e^{2n\pi {i\tau }}}}\sin 2\pi {nv}\\\end{aligned}}}
である。
例えば
θ 3 ( x + y , τ ) θ 3 ( x − y , τ ) = ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ e π i τ n 2 + 2 π i n ( x + y ) + π i τ m 2 + 2 π i m ( x − y ) = ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ e 2 π i τ ( n + m 2 ) 2 + 4 π i ( n + m 2 ) x + 2 π i τ ( n − m 2 ) 2 + 4 π i ( n − m 2 ) y {\displaystyle {\begin{aligned}\vartheta _{3}\left(x+y,\tau \right)\vartheta _{3}\left(x-y,\tau \right)&=\sum _{n=-\infty }^{\infty }\sum _{m=-\infty }^{\infty }e^{{\pi }i{\tau }n^{2}+2{\pi }in(x+y)+{\pi }i{\tau }m^{2}+2{\pi }im(x-y)}\\&=\sum _{n=-\infty }^{\infty }\sum _{m=-\infty }^{\infty }e^{2{\pi }i{\tau }\left({\tfrac {n+m}{2}}\right)^{2}+4{\pi }i\left({\tfrac {n+m}{2}}\right)x+2{\pi }i{\tau }\left({\tfrac {n-m}{2}}\right)^{2}+4{\pi }i\left({\tfrac {n-m}{2}}\right)y}\\\end{aligned}}}
θ 1 ( x + y , τ ) θ 2 ( x − y , τ ) θ 3 ( 0 , τ ) θ 4 ( 0 , τ ) = θ 1 ( x , τ ) θ 2 ( x , τ ) θ 3 ( y , τ ) θ 4 ( y , τ ) + θ 3 ( x , τ ) θ 4 ( x , τ ) θ 1 ( y , τ ) θ 2 ( y , τ ) θ 1 ( x + y , τ ) θ 3 ( x − y , τ ) θ 2 ( 0 , τ ) θ 4 ( 0 , τ ) = θ 1 ( x , τ ) θ 3 ( x , τ ) θ 2 ( y , τ ) θ 4 ( y , τ ) + θ 2 ( x , τ ) θ 4 ( x , τ ) θ 1 ( y , τ ) θ 3 ( y , τ ) θ 1 ( x + y , τ ) θ 4 ( x − y , τ ) θ 3 ( 0 , τ ) θ 2 ( 0 , τ ) = θ 1 ( x , τ ) θ 4 ( x , τ ) θ 3 ( y , τ ) θ 2 ( y , τ ) + θ 3 ( x , τ ) θ 2 ( x , τ ) θ 1 ( y , τ ) θ 4 ( y , τ ) {\displaystyle {\begin{aligned}&\vartheta _{1}\left(x+y,\tau \right)\vartheta _{2}\left(x-y,\tau \right)\vartheta _{3}(0,\tau )\vartheta _{4}(0,\tau )=\vartheta _{1}\left(x,\tau \right)\vartheta _{2}\left(x,\tau \right)\vartheta _{3}(y,\tau )\vartheta _{4}(y,\tau )+\vartheta _{3}\left(x,\tau \right)\vartheta _{4}\left(x,\tau \right)\vartheta _{1}(y,\tau )\vartheta _{2}(y,\tau )\\&\vartheta _{1}\left(x+y,\tau \right)\vartheta _{3}\left(x-y,\tau \right)\vartheta _{2}(0,\tau )\vartheta _{4}(0,\tau )=\vartheta _{1}\left(x,\tau \right)\vartheta _{3}\left(x,\tau \right)\vartheta _{2}(y,\tau )\vartheta _{4}(y,\tau )+\vartheta _{2}\left(x,\tau \right)\vartheta _{4}\left(x,\tau \right)\vartheta _{1}(y,\tau )\vartheta _{3}(y,\tau )\\&\vartheta _{1}\left(x+y,\tau \right)\vartheta _{4}\left(x-y,\tau \right)\vartheta _{3}(0,\tau )\vartheta _{2}(0,\tau )=\vartheta _{1}\left(x,\tau \right)\vartheta _{4}\left(x,\tau \right)\vartheta _{3}(y,\tau )\vartheta _{2}(y,\tau )+\vartheta _{3}\left(x,\tau \right)\vartheta _{2}\left(x,\tau \right)\vartheta _{1}(y,\tau )\vartheta _{4}(y,\tau )\\\end{aligned}}}
θ 4 ( v ; τ ) = ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 − e ( 2 m − 1 ) π i τ e 2 π i v ) ( 1 − e ( 2 m − 1 ) π i τ e − 2 π i v ) = ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 − 2 e ( 2 m − 1 ) π i τ cos  2 π v + e 2 ( 2 m − 1 ) π i τ ) {\displaystyle {\begin{aligned}\vartheta _{4}(v;\tau )&=\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1-e^{(2m-1){\pi }i{\tau }}e^{2{\pi }iv}\right)\left(1-e^{(2m-1){\pi }i{\tau }}e^{-2{\pi }iv}\right)}\\&=\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1-2e^{(2m-1){\pi }i{\tau }}\cos {2{\pi }v}+e^{2(2m-1){\pi }i{\tau }}\right)}\\\end{aligned}}}
が得られ、同様にして数十もの恒等式が得られる。
θ 2 ( v ; τ ) = 2 e π i τ / 4 cos  π v ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 + e 2 m π i τ e 2 π i v ) ( 1 + e 2 m π i τ e − 2 π i v ) = 2 e π i τ / 4 cos  π v ∏ m = 1 ∞ ( 1 − e 2 m π i τ ) ( 1 + 2 e 2 m π i τ cos  2 π v + e 4 m π i τ ) {\displaystyle {\begin{aligned}\vartheta _{2}(v;\tau )&=2e^{{\pi }i{\tau }/4}\cos {\pi }v\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1+e^{2m{\pi }i{\tau }}e^{2{\pi }iv}\right)\left(1+e^{2m{\pi }i{\tau }}e^{-2{\pi }iv}\right)}\\&=2e^{{\pi }i{\tau }/4}\cos {\pi }v\prod _{m=1}^{\infty }{\left(1-e^{2m{\pi }i{\tau }}\right)\left(1+2e^{2m{\pi }i{\tau }}\cos {2{\pi }v}+e^{4m{\pi }i{\tau }}\right)}\\\end{aligned}}}
また、ヤコビのエータ関数
H ( u ) := − i exp  ( ( 2 u + i K ′ ) π i / ( 4 K ) ) Θ ( u + i K ′ ) , i := − 1 , H 1 ( u ) := H ( u + K ) , {\displaystyle {\begin{aligned}H(u)&:=-i\exp \left((2u+iK')\pi i/(4K)\right)\Theta (u+iK'),\quad i:={\sqrt {-1\,}},\\H_{1}(u)&:=H(u+K),\end{aligned}}}
を含めて、 Θ ( u ) {\displaystyle \Theta (u)} , Θ 1 ( u ) {\displaystyle \Theta _{1}(u)} , H ( u ) {\displaystyle H(u)} , H 1 ( u ) {\displaystyle H_{1}(u)} のことをヤコビのテータ関数と呼ぶこともある。ただし、 K ′ := K ( k ′ ) {\displaystyle K':=K(k')} である。ヤコビのテータ関数は、後述の楕円テータ関数と以下の関係で結ばれている。
θ 3 ( v , τ ) θ 4 ( v , τ ) = ( ∑ n = − ∞ ∞ e π i τ n 2 + 2 π i n v ) ( ∑ m = − ∞ ∞ ( − 1 ) m e π i τ m 2 + 2 π i m v ) = ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ ( − 1 ) m e π i τ ( n 2 + m 2 ) + 2 π i ( n + m ) v {\displaystyle {\begin{aligned}\vartheta _{3}(v,\tau )\vartheta _{4}(v,\tau )&=\left(\sum _{n=-\infty }^{\infty }e^{\pi {i\tau }n^{2}+2\pi {inv}}\right)\left(\sum _{m=-\infty }^{\infty }(-1)^{m}e^{\pi {i\tau }m^{2}+2\pi {imv}}\right)\\&=\sum _{n=-\infty }^{\infty }\sum _{m=-\infty }^{\infty }(-1)^{m}e^{\pi {i\tau }(n^{2}+m^{2})+2\pi {i(n+m)v}}\\\end{aligned}}}
となるが、 n ± m {\displaystyle {n}\pm {m}} が奇数の項は n ≷ m {\displaystyle {n}\gtrless {m}} で打ち消し合うから
以下のように定義された、添え字を 2 つ持つテータ関数のことを指標付きのテータ関数と呼ぶ。
θ a , b ( z , τ ) := ∑ n = − ∞ ∞ e π i ( n + a ) 2 τ + 2 π i ( n + a ) ( z + b ) , a , b ∈ R . {\displaystyle \vartheta _{a,b}(z,\tau ):=\sum _{n=-\infty }^{\infty }e^{\pi i(n+a)^{2}\tau +2\pi i(n+a)(z+b)},\quad a,b\in \mathbb {R} .}
なお、指標付きのテータ関数の定義には 2 つの流儀があって統一的に用いられていないため、文献を読むときには注意しなければならない 。 この記事で使われているのは、Mumford 2006 で使われているのと同じ定義である。
θ 3 4 ( 0 , τ ) = θ 2 4 ( 0 , τ ) + θ 4 4 ( 0 , τ ) {\displaystyle \vartheta _{3}^{\;4}(0,\tau )=\vartheta _{2}^{\;4}\left(0,\tau \right)+\vartheta _{4}^{\;4}\left(0,\tau \right)}
θ 3 ( v , τ ) = ∑ n = − ∞ ∞ e π i τ n 2 + 2 π i n v = ∑ n = − ∞ ∞ e π i τ ( 2 n ) 2 + 2 π i ( 2 n ) v + ∑ n = − ∞ ∞ e π i τ ( 2 n + 1 ) 2 + 2 π i ( 2 n + 1 ) v ( n ↦ 2 n , 2 n + 1 ) = θ 3 ( 2 v , 4 τ ) + θ 2 ( 2 v , 4 τ ) {\displaystyle {\begin{aligned}\vartheta _{3}\left(v,\tau \right)&=\sum _{n=-\infty }^{\infty }e^{\pi {i}\tau {n^{2}}+2\pi {i}nv}\\&=\sum _{n=-\infty }^{\infty }e^{\pi {i}\tau {(2n)^{2}}+2\pi {i}(2n)v}+\sum _{n=-\infty }^{\infty }e^{\pi {i}\tau {(2n+1)^{2}}+2\pi {i}(2n+1)v}\qquad ({n}\mapsto {2n,2n+1})\\&=\vartheta _{3}\left(2v,4\tau \right)+\vartheta _{2}\left(2v,4\tau \right)\end{aligned}}}