[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki24\/archives\/292302#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki24\/archives\/292302","headline":"\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570 &#8211; Wikipedia","name":"\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570 &#8211; Wikipedia","description":"\u6570\u5b66\u306b\u304a\u3051\u308b\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570\uff08\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u3060\u3048\u3093\u304b\u3093\u3059\u3046\u3001\u82f1: Weierstrass&#8217;s elliptic functions\uff09\u306f\u3001\u30ab\u30fc\u30eb\u30fb\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306b\u540d\u3092\u56e0\u3080\u3001\u5358\u7d14\u306a\u5f62\u3092\u3057\u305f\u6955\u5186\u51fd\u6570\u306e\u4e00\u7a2e\u3067\u3042\u308b\u3002\u3053\u306e\u30af\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570\u306f\u3001\u30da\u30fc\u51fd\u6570\u3068\u547c\u3070\u308c\u3001\u4e00\u822c\u306b \u2118 \u306a\u308b\u8a18\u53f7\uff08\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u30fb\u30da\u30fc\uff09\u3067\u8868\u3055\u308c\u308b\u3002 \u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u30da\u30fc\u51fd\u6570\u8a18\u53f7 \u8907\u7d20\u6570\u5e73\u9762\u306e\u90e8\u5206\u96c6\u5408\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u30da\u30fc\u51fd\u6570\u3092\u3001\u6a19\u6e96\u7684\u306a\u8996\u899a\u5316\u6cd5\u3068\u3057\u3066\u3001\u6975\u3092\u767d\u304f\u3001\u96f6\u70b9\u3092\u9ed2\u304f\u3001 |f(z)|=1{displaystyle |f(z)|=1} \u3067\u5f69\u5ea6\u304c\u6975\u5927\u306b\u306a\u308b\u3088\u3046\u306b\u8868\u3057\u305f\u3082\u306e\u3002\u6975\u306e\u6210\u3059\u6b63\u5247\u683c\u5b50\u3068\u96f6\u70b9\u306e\u6210\u3059\u4ea4\u4e92\u683c\u5b50\u306b\u6ce8\u610f\u3002 \u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570\u306f\u3001\u8fd1\u3057\u3044\u95a2\u4fc2\u306b\u3042\u308b\u4e09\u7a2e\u985e\u306e\u65b9\u6cd5\u3067\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001\u305d\u308c\u305e\u308c\u4e00\u9577\u4e00\u77ed\u304c\u3042\u308b\u3002\u4e00\u3064\u306f\u3001\u8907\u7d20\u5909\u6570 z \u3068\u8907\u7d20\u6570\u5e73\u9762\u4e0a\u306e\u683c\u5b50 \u039b \u306e\u51fd\u6570\u3068\u3057\u3066\u3001\u3044\u307e\u4e00\u3064\u306f z \u3068\u683c\u5b50\u306e\u4e8c\u3064\u306e\u751f\u6210\u5143\uff08\u5468\u671f\u5bfe\uff09\u3092\u4e0e\u3048\u308b\u8907\u7d20\u6570 \u03c91, \u03c92","datePublished":"2022-04-27","dateModified":"2022-04-27","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki24\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki24\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/5a\/Weierstrass_p.svg\/150px-Weierstrass_p.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/5\/5a\/Weierstrass_p.svg\/150px-Weierstrass_p.svg.png","height":"176","width":"150"},"url":"https:\/\/wiki.edu.vn\/jp\/wiki24\/archives\/292302","about":["Wiki"],"wordCount":17002,"articleBody":"\u6570\u5b66\u306b\u304a\u3051\u308b\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570\uff08\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u3060\u3048\u3093\u304b\u3093\u3059\u3046\u3001\u82f1: Weierstrass&#8217;s elliptic functions\uff09\u306f\u3001\u30ab\u30fc\u30eb\u30fb\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306b\u540d\u3092\u56e0\u3080\u3001\u5358\u7d14\u306a\u5f62\u3092\u3057\u305f\u6955\u5186\u51fd\u6570\u306e\u4e00\u7a2e\u3067\u3042\u308b\u3002\u3053\u306e\u30af\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570\u306f\u3001\u30da\u30fc\u51fd\u6570\u3068\u547c\u3070\u308c\u3001\u4e00\u822c\u306b \u2118 \u306a\u308b\u8a18\u53f7\uff08\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u30fb\u30da\u30fc\uff09\u3067\u8868\u3055\u308c\u308b\u3002  \u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u30da\u30fc\u51fd\u6570\u8a18\u53f7  \u8907\u7d20\u6570\u5e73\u9762\u306e\u90e8\u5206\u96c6\u5408\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u30da\u30fc\u51fd\u6570\u3092\u3001\u6a19\u6e96\u7684\u306a\u8996\u899a\u5316\u6cd5\u3068\u3057\u3066\u3001\u6975\u3092\u767d\u304f\u3001\u96f6\u70b9\u3092\u9ed2\u304f\u3001|f(z)|=1{displaystyle |f(z)|=1} \u3067\u5f69\u5ea6\u304c\u6975\u5927\u306b\u306a\u308b\u3088\u3046\u306b\u8868\u3057\u305f\u3082\u306e\u3002\u6975\u306e\u6210\u3059\u6b63\u5247\u683c\u5b50\u3068\u96f6\u70b9\u306e\u6210\u3059\u4ea4\u4e92\u683c\u5b50\u306b\u6ce8\u610f\u3002\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570\u306f\u3001\u8fd1\u3057\u3044\u95a2\u4fc2\u306b\u3042\u308b\u4e09\u7a2e\u985e\u306e\u65b9\u6cd5\u3067\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001\u305d\u308c\u305e\u308c\u4e00\u9577\u4e00\u77ed\u304c\u3042\u308b\u3002\u4e00\u3064\u306f\u3001\u8907\u7d20\u5909\u6570 z \u3068\u8907\u7d20\u6570\u5e73\u9762\u4e0a\u306e\u683c\u5b50 \u039b \u306e\u51fd\u6570\u3068\u3057\u3066\u3001\u3044\u307e\u4e00\u3064\u306f z \u3068\u683c\u5b50\u306e\u4e8c\u3064\u306e\u751f\u6210\u5143\uff08\u5468\u671f\u5bfe\uff09\u3092\u4e0e\u3048\u308b\u8907\u7d20\u6570 \u03c91, \u03c92 \u3092\u7528\u3044\u3066\u8ff0\u3079\u308b\u3082\u306e\u3001\u6b8b\u308b\u4e00\u3064\u306f z \u3068\u4e0a\u534a\u5e73\u9762\u306b\u304a\u3051\u308b\u6bcd\u6570 (modulus) \u03c4 \u306b\u95a2\u3059\u308b\u3082\u306e\u3067\u3042\u308b\u3002\u6700\u5f8c\u306e\u306f\u305d\u306e\u524d\u306e\u3068\u3001\u4e0a\u534a\u5e73\u9762\u4e0a\u306e\u5468\u671f\u5bfe\u3092\u9078\u3093\u3067 \u03c4 = \u03c92\/\u03c91 \u3068\u3057\u305f\u95a2\u4fc2\u306b\u3042\u308b\u3002\u3053\u306e\u65b9\u6cd5\u3067\u306f\u3001z \u3092\u6b62\u3081\u3066\u3001\u03c4 \u306e\u51fd\u6570\u3068\u898b\u308b\u3068\u3001\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u6955\u5186\u51fd\u6570\u306f \u03c4 \u306e\u30e2\u30b8\u30e5\u30e9\u30fc\u51fd\u6570\u306b\u306a\u308b\u3002\u5468\u671f\u5bfe\u3092\u4e0e\u3048\u308b\u65b9\u6cd5\u3092\u5177\u4f53\u7684\u306b\u66f8\u3051\u3070\u3001\u03c91, \u03c92 \u3092\u4e8c\u3064\u306e\u5468\u671f\u306b\u6301\u3064\u30da\u30fc\u51fd\u6570\u306f\u3001\u2118(z;\u03c91,\u03c92)=1z2+\u2211n2+m2\u22600{1(z+m\u03c91+n\u03c92)2\u22121(m\u03c91+n\u03c92)2}{displaystyle wp (z;omega _{1},omega _{2})={frac {1}{z^{2}}}+sum _{n^{2}+m^{2}neq 0}left{{frac {1}{(z+momega _{1}+nomega _{2})^{2}}}-{frac {1}{left(momega _{1}+nomega _{2}right)^{2}}}right}}\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u5468\u671f\u683c\u5b50\u039b={m\u03c91+n\u03c92:m,n\u2208Z}{displaystyle Lambda ={momega _{1}+nomega _{2}:m,nin mathbb {Z} }}\u3092\u8003\u3048\u308c\u3070\u3001\u683c\u5b50\u306e\u4efb\u610f\u306e\u751f\u6210\u5bfe\u306b\u5bfe\u3057\u3066\u2118(z;\u039b)=\u2118(z;\u03c91,\u03c92){displaystyle wp (z;Lambda )=wp (z;omega _{1},omega _{2})}\u306f\u8907\u7d20\u5909\u6570\u3068\u683c\u5b50\u306e\u51fd\u6570\u3068\u3057\u3066\u306e\u30da\u30fc\u51fd\u6570\u3092\u5b9a\u3081\u308b\u3002\u4e0a\u534a\u5e73\u9762\u306b\u5c5e\u3059\u308b\u8907\u7d20\u6570 \u03c4 \u306b\u5bfe\u3057\u3066\u3001\u2118(z;\u03c4)=\u2118(z;1,\u03c4)=1z2+\u2211n2+m2\u22600{1(z+m+n\u03c4)2\u22121(m+n\u03c4)2}{displaystyle wp (z;tau )=wp (z;1,tau )={frac {1}{z^{2}}}+sum _{n^{2}+m^{2}neq 0}left{{1 over (z+m+ntau )^{2}}-{1 over (m+ntau )^{2}}right}}\u3068\u7f6e\u304f\u3002\u4e0a\u8a18\u306e\u548c\u306f \u22122-\u6b21\u306e\u6589\u6b21\u548c\u3067\u3042\u308b\u3002\u3053\u306e\u30da\u30fc\u51fd\u6570\u3092\u7528\u3044\u308b\u3068\u3001\u5148\u306b\u8ff0\u3079\u305f\u5468\u671f\u5bfe\u306b\u5bfe\u3059\u308b\u30da\u30fc\u51fd\u6570\u306f\u2118(z;\u03c91,\u03c92)=\u2118(z\u03c91;\u03c92\u03c91)\u03c912{displaystyle wp (z;omega _{1},omega _{2})={frac {wp ({frac {z}{omega _{1}}};{frac {omega _{2}}{omega _{1}}})}{omega _{1}^{2}}}}\u3068\u66f8\u3051\u308b\u3002\u30da\u30fc\u51fd\u6570\u306f\u53ce\u6582\u306e\u65e9\u3044\u30c6\u30fc\u30bf\u51fd\u6570\u3092\u7528\u3044\u3066\u8868\u305b\u3070\u3001\u4e0a\u8a18\u306e\u5b9a\u7fa9\u306b\u7528\u3044\u305f\u7d1a\u6570\u3092\u7528\u3044\u308b\u3088\u308a\u3082\u3001\u624b\u65e9\u304f\u8a08\u7b97\u3067\u304d\u308b\u3002\u30c6\u30fc\u30bf\u51fd\u6570\u306b\u3088\u308b\u8868\u793a\u306f\u2118(z;\u03c4)=\u03c02\u03d12(0;\u03c4)\u03d1102(0;\u03c4)\u03d1012(z;\u03c4)\u03d1112(z;\u03c4)\u2212\u03c023[\u03d14(0;\u03c4)+\u03d1104(0;\u03c4)]{displaystyle wp (z;tau )=pi ^{2}vartheta ^{2}(0;tau )vartheta _{10}^{2}(0;tau ){vartheta _{01}^{2}(z;tau ) over vartheta _{11}^{2}(z;tau )}-{pi ^{2} over {3}}left[vartheta ^{4}(0;tau )+vartheta _{10}^{4}(0;tau )right]}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u30da\u30fc\u51fd\u6570\u306f\uff08\u539f\u70b9\u3092\u542b\u3080\uff09\u5468\u671f\u683c\u5b50\u306e\u5404\u9802\u70b9\u306b\u304a\u3044\u3066\u4e8c\u4f4d\u306e\u6975\u3092\u6709\u3059\u308b\u3002\u3053\u308c\u3089\u306e\u5b9a\u7fa9\u306e\u3082\u3068\u3001\u30da\u30fc\u51fd\u6570 \u2118(z) \u306f\u5076\u51fd\u6570\u3001\u305d\u306e z \u306b\u95a2\u3059\u308b\u5c0e\u51fd\u6570 \u2118\u2032 \u306f\u5947\u51fd\u6570\u306b\u306a\u308b\u3002\u3055\u3089\u306b\u6955\u5186\u51fd\u6570\u8ad6\u3092\u63a8\u3057\u9032\u3081\u308c\u3070\u3001\u4e0e\u3048\u3089\u308c\u305f\u5468\u671f\u683c\u5b50\u3092\u6301\u3064\u4efb\u610f\u306e\u6709\u7406\u578b\u51fd\u6570\u306e\u4e2d\u3067\u3001\u30da\u30fc\u51fd\u6570\u306b\u95a2\u3059\u308b\u6761\u4ef6\u306f\u3001\u5b9a\u6570\u3092\u52a0\u3048\u305f\u308a\u975e\u96f6\u5b9a\u6570\u500d\u3057\u305f\u308a\u3059\u308b\u3053\u3068\u3092\u9664\u304d\u3001\u6975\u306b\u95a2\u3059\u308b\u6761\u4ef6\u306e\u307f\u3067\u6c7a\u307e\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u308b\u3002  \u5358\u4f4d\u5186\u677f\u4e0a\u306e\u30ce\u30fc\u30e0 q \u306e\u51fd\u6570\u3068\u3057\u3066\u306e\u3001\u4e0d\u5909\u91cf g3 \u306e\u5b9f\u90e8\u3002  \u5358\u4f4d\u5186\u677f\u4e0a\u306e\u30ce\u30fc\u30e0 q \u306e\u51fd\u6570\u3068\u3057\u3066\u306e\u3001\u4e0d\u5909\u91cf g3 \u306e\u865a\u90e8\u3002\u539f\u70b9\u306e\u8fd1\u508d\u3092\u9664\u304d\u3001\u2118 \u306e\u30ed\u30fc\u30e9\u30f3\u7d1a\u6570\u5c55\u958b\u306f\u2118(z;\u03c91,\u03c92)=z\u22122+120g2z2+128g3z4+O(z6){displaystyle wp (z;omega _{1},omega _{2})=z^{-2}+{frac {1}{20}}g_{2}z^{2}+{frac {1}{28}}g_{3}z^{4}+O(z^{6})}\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u305f\u3060\u3057\u3001g2=60\u2211(m,n)\u2260(0,0)(m\u03c91+n\u03c92)\u22124,{displaystyle g_{2}=60sum _{(m,n)neq (0,0)}(momega _{1}+nomega _{2})^{-4},}g3=140\u2211(m,n)\u2260(0,0)(m\u03c91+n\u03c92)\u22126{displaystyle g_{3}=140sum _{(m,n)neq (0,0)}(momega _{1}+nomega _{2})^{-6}}\u3067\u3042\u308b\u3002\u3053\u308c\u3089\u306e\u6570\u5024 g2, g3 \u306f\u30da\u30fc\u51fd\u6570\u306e\u4e0d\u5909\u91cf (invariant) \u3068\u547c\u3070\u308c\u308b\u3002\u4fc2\u6570 60 \u304a\u3088\u3073 140 \u306e\u5f8c\u308d\u306b\u3042\u308b\u548c\u306f\u30a2\u30a4\u30bc\u30f3\u30b7\u30e5\u30bf\u30a4\u30f3\u7d1a\u6570\u306e\u6700\u521d\u306e\u4e8c\u3064\u3067\u3001\u3053\u308c\u3089\u306f Im(\u03c4)>0 \u306a\u308b \u03c4 = \u03c92\/\u03c91 \u306e\u51fd\u6570 G4(\u03c4) \u304a\u3088\u3073 G6(\u03c4) \u3068\u3057\u3066\u305d\u308c\u305e\u308c\u3092\u898b\u505a\u305b\u3070\u30e2\u30b8\u30e5\u30e9\u30fc\u5f62\u5f0f\u3092\u6210\u3059\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3053\u3053\u3067\u3001g2 \u304a\u3088\u3073 g3 \u306f\u305d\u308c\u305e\u308c\u6b21\u6570 \u22124 \u304a\u3088\u3073 \u22126 \u306e\u6589\u6b21\u51fd\u6570\u3067\u3042\u308b\u3002\u3064\u307e\u308ag2(\u03bb\u03c91,\u03bb\u03c92)=\u03bb\u22124g2(\u03c91,\u03c92){displaystyle g_{2}(lambda omega _{1},lambda omega _{2})=lambda ^{-4}g_{2}(omega _{1},omega _{2})}\u304a\u3088\u3073g3(\u03bb\u03c91,\u03bb\u03c92)=\u03bb\u22126g3(\u03c91,\u03c92).{displaystyle g_{3}(lambda omega _{1},lambda omega _{2})=lambda ^{-6}g_{3}(omega _{1},omega _{2}).}\u3092\u6e80\u305f\u3059\u3002\u5f93\u3063\u3066\u3001\u6163\u7fd2\u7684\u306b\u3001g2 \u304a\u3088\u3073 g3 \u3092\u3001\u4e0a\u534a\u5e73\u9762\u306b\u5c5e\u3059\u308b\u5468\u671f\u6bd4 \u03c4 = \u03c92\/\u03c91 \u3092\u7528\u3044\u3066\u3001g2(\u03c4)=g2(1,\u03c92\/\u03c91)=\u03c914g2(\u03c91,\u03c92),g3(\u03c4)=g3(1,\u03c92\/\u03c91)=\u03c916g3(\u03c91,\u03c92){displaystyle g_{2}(tau )=g_{2}(1,omega _{2}\/omega _{1})=omega _{1}^{4}g_{2}(omega _{1},omega _{2}),quad g_{3}(tau )=g_{3}(1,omega _{2}\/omega _{1})=omega _{1}^{6}g_{3}(omega _{1},omega _{2})}\u3068\u8868\u3059\u3053\u3068\u3082\u3088\u304f\u884c\u308f\u308c\u308b\u3002g2 \u304a\u3088\u3073 g3 \u306f Im(\u03c4)>0 \u306b\u304a\u3044\u3066\u6b63\u5247\u3067[1]\u3001\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\u3001\u30ce\u30fc\u30e0 q = exp(i\u03c0\u03c4) \u306e\u5e73\u65b9\u3092\u7528\u3044\u3066\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u3066\u3001g2(\u03c4)=4\u03c043[1+240\u2211k=1\u221e\u03c33(k)q2k]{displaystyle g_{2}(tau )={frac {4pi ^{4}}{3}}left[1+240sum _{k=1}^{infty }sigma _{3}(k)q^{2k}right]}\u304a\u3088\u3073g3(\u03c4)=8\u03c0627[1\u2212504\u2211k=1\u221e\u03c35(k)q2k]{displaystyle g_{3}(tau )={frac {8pi ^{6}}{27}}left[1-504sum _{k=1}^{infty }sigma _{5}(k)q^{2k}right]}\u3068\u306a\u308b[2]\u3002\u305f\u3060\u3057\u3001\u03c3a(k) \u306f\u7d04\u6570\u51fd\u6570\u3067\u3042\u308b\u3002\u3053\u308c\u3089\u306e\u5f0f\u306f\u30e9\u30f3\u30d9\u30eb\u30c8\u7d1a\u6570\u3092\u7528\u3044\u3066\u66f8\u304d\u76f4\u3059\u3053\u3068\u3082\u3067\u304d\u308b\u3002\u4e0d\u5909\u91cf\u3092\u30e4\u30b3\u30d3\u306e\u30c6\u30fc\u30bf\u51fd\u6570\u3092\u7528\u3044\u3066\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u308b\u304c\u3001\u30c6\u30fc\u30bf\u51fd\u6570\u306e\u53ce\u6582\u306f\u975e\u5e38\u306b\u901f\u304f\u3001\u3053\u308c\u306f\u6570\u5024\u8a08\u7b97\u306b\u975e\u5e38\u306b\u6709\u52b9\u306a\u65b9\u6cd5\u3067\u3042\u308b\u3002Abramowitz &#038; Stegun (1965) \u306e\u8a18\u6cd5\u3067\u3001\u305f\u3060\u3057\u539f\u59cb\u534a\u5468\u671f\u306f \u03c91, \u03c92 \u3068\u66f8\u304f\u3082\u306e\u3068\u3059\u308b\u3068\u3001\u4e0d\u5909\u91cf\u306b\u95a2\u3057\u3066g2(\u03c91,\u03c92)=\u03c0412\u03c914(\u03b82(0,q)8\u2212\u03b83(0,q)4\u03b82(0,q)4+\u03b83(0,q)8){displaystyle g_{2}(omega _{1},omega _{2})={frac {pi ^{4}}{12omega _{1}^{4}}}left(theta _{2}(0,q)^{8}-theta _{3}(0,q)^{4}theta _{2}(0,q)^{4}+theta _{3}(0,q)^{8}right)}\u304a\u3088\u3073g3(\u03c91,\u03c92)=\u03c06(2\u03c91)6[827(\u03b82(0,q)12+\u03b83(0,q)12)\u221249(\u03b82(0,q)4+\u03b83(0,q)4)\u22c5\u03b82(0,q)4\u03b83(0,q)4]{displaystyle g_{3}(omega _{1},omega _{2})={frac {pi ^{6}}{(2omega _{1})^{6}}}left[{frac {8}{27}}left(theta _{2}(0,q)^{12}+theta _{3}(0,q)^{12}right)-{frac {4}{9}}left(theta _{2}(0,q)^{4}+theta _{3}(0,q)^{4}right)cdot theta _{2}(0,q)^{4}theta _{3}(0,q)^{4}right]}\u304c\u6210\u308a\u7acb\u3064\u3002\u305f\u3060\u3057\u3001\u03c4 = \u03c92\/\u03c91 \u306f\u5468\u671f\u6bd4\u3067 q = exp(i\u03c0\u03c4) \u306f\u30ce\u30fc\u30e0\u3067\u3042\u308b\u3002Table of Contents\u7279\u5225\u306e\u5834\u5408[\u7de8\u96c6]\u5fae\u5206\u65b9\u7a0b\u5f0f[\u7de8\u96c6]\u7a4d\u5206\u65b9\u7a0b\u5f0f[\u7de8\u96c6]\u30e2\u30b8\u30e5\u30e9\u30fc\u5224\u5225\u5f0f[\u7de8\u96c6]j-\u4e0d\u5909\u91cf[\u7de8\u96c6]\u5b9a\u6570 e1, e2, e3[\u7de8\u96c6]\u3044\u304f\u3064\u304b\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066[\u7de8\u96c6]\u57fa\u672c\u534a\u5468\u671f 1 \u306e\u5834\u5408[\u7de8\u96c6]\u30e4\u30b3\u30d3\u306e\u6955\u5186\u51fd\u6570\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]\u7279\u5225\u306e\u5834\u5408[\u7de8\u96c6]\u4e0d\u5909\u91cf\u304c g2 = 0, g3 = 1 \u306e\u3068\u304d\u3001\u7b49\u975e\u8abf\u548c\uff08\u82f1\u8a9e\u7248\uff09\u3067\u3042\u308b\u3068\u3044\u3044\u3001g2 = 1, g3 = 0 \u306e\u3068\u304d\u30ec\u30e0\u30cb\u30b9\u30b1\u30fc\u30c8\u6955\u5186\u51fd\u6570\uff08\u82f1\u8a9e\u7248\uff09\u3067\u3042\u308b\u3068\u3044\u3046\u3002\u5fae\u5206\u65b9\u7a0b\u5f0f[\u7de8\u96c6]  \u4e0d\u5909\u91cf\u3092\u7528\u3044\u3066\u3001\u30da\u30fc\u51fd\u6570\u306f\u4ee5\u4e0b\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f[\u2118\u2032(z)]2=4[\u2118(z)]3\u2212g2\u2118(z)\u2212g3{displaystyle [wp &#8216;(z)]^{2}=4[wp (z)]^{3}-g_{2}wp (z)-g_{3}}\u3092\u6e80\u8db3\u3059\u308b\u3002\u3053\u308c\u306f\u5468\u671f\u5bfe \u03c91, \u03c92 \u306e\u53d6\u308a\u65b9\u306b\u4f9d\u5b58\u3057\u3066\u7d71\u5236\u3055\u308c\u308b\u3002\u3053\u306e\u95a2\u4fc2\u5f0f\u306f\u4e21\u8fba\u306e\u6975\u3092\u6bd4\u3079\u308c\u3070\u76f4\u3061\u306b\u78ba\u304b\u3081\u3089\u308c\u308b\u3002\u4f8b\u3048\u3070\u3001\u5de6\u8fba\u306e z = 0 \u306b\u304a\u3051\u308b\u6975\u306f  [\u2118\u2032(z)]2|z=0\u223c4z6\u221224z2\u22111(m\u03c91+n\u03c92)4\u221280\u22111(m\u03c91+n\u03c92)6{displaystyle [wp &#8216;(z)]^{2}|_{z=0}sim {frac {4}{z^{6}}}-{frac {24}{z^{2}}}sum {frac {1}{(momega _{1}+nomega _{2})^{4}}}-80sum {frac {1}{(momega _{1}+nomega _{2})^{6}}}}\u3067\u3042\u308a\u3001\u53f3\u8fba\u7b2c\u4e00\u9805\u306e z = 0 \u306b\u304a\u3051\u308b\u6975\u306f[\u2118(z)]3|z=0\u223c1z6+9z2\u22111(m\u03c91+n\u03c92)4+15\u22111(m\u03c91+n\u03c92)6{displaystyle [wp (z)]^{3}|_{z=0}sim {frac {1}{z^{6}}}+{frac {9}{z^{2}}}sum {frac {1}{(momega _{1}+nomega _{2})^{4}}}+15sum {frac {1}{(momega _{1}+nomega _{2})^{6}}}}\u3067\u3001\u3053\u308c\u3089\u3092\u6bd4\u8f03\u3057\u3066\u4e0a\u8a18\u306e\u95a2\u4fc2\u5f0f\u3092\u5f97\u308b\u3002  \u7a4d\u5206\u65b9\u7a0b\u5f0f[\u7de8\u96c6]\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u30fb\u30da\u30fc\u51fd\u6570\u306f\u6955\u5186\u7a4d\u5206\u306e\u9006\u51fd\u6570\u3068\u3057\u3066\u4e0e\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u3067\u306f g2 \u304a\u3088\u3073 g3 \u306f\u5b9a\u6570\u3067\u3042\u308b\u3082\u306e\u3068\u3057\u3066\u3001u=\u222by\u221eds4s3\u2212g2s\u2212g3{displaystyle u=int _{y}^{infty }{frac {ds}{sqrt {4s^{3}-g_{2}s-g_{3}}}}}\u3068\u304a\u304f\u3068\u3001y=\u2118(u){displaystyle y=wp (u)}\u3068\u306a\u308b\u306e\u3067\u3042\u308b\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u4e0a\u8a18\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u7a4d\u5206\u3057\u3066\u76f4\u622a\u306b\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u30e2\u30b8\u30e5\u30e9\u30fc\u5224\u5225\u5f0f[\u7de8\u96c6]  \u5358\u4f4d\u5186\u677f\u4e0a\u306e\u30ce\u30fc\u30e0 q \u306e\u51fd\u6570\u3068\u3057\u3066\u306e\u5224\u5225\u5f0f\u306e\u5b9f\u90e8\u30e2\u30b8\u30e5\u30e9\u30fc\u5224\u5225\u5f0f (modular discriminant) \u0394 \u306f\u0394=g23\u221227g32{displaystyle Delta =g_{2}^{3}-27g_{3}^{2}}\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3053\u306e\u5224\u5225\u5f0f\u306f\u3001\u305d\u308c\u81ea\u4f53\u304c\uff08\u5468\u671f\u683c\u5b50\u306e\u51fd\u6570\u3068\u3057\u3066\u306e\uff09\u5c16\u70b9\u5f62\u5f0f\u3068\u898b\u3066\u3001\u30e2\u30b8\u30e5\u30e9\u30fc\u5f62\u5f0f\u8ad6\u306b\u304a\u3051\u308b\u7814\u7a76\u306e\u5bfe\u8c61\u306b\u306a\u308b\u3002\u30c7\u30c6\u30ad\u30f3\u30c8\u306e\u30a4\u30fc\u30bf\u95a2\u6570 \u03b7 \u3092\u7528\u3044\u308c\u3070\u3001\u0394=(2\u03c0)12\u03b724{displaystyle Delta =(2pi )^{12}eta ^{24}}\u3068\u66f8\u3051\u308b\u3053\u3068\u306b\u6ce8\u610f[3]\u3002\u3053\u306e 24 \u3068\u3044\u3046\u6570\u306f\u3001\u30a4\u30fc\u30bf\u51fd\u6570\u3068\u30ea\u30fc\u30c1\u683c\u5b50\u306b\u3042\u308b\u3088\u3046\u306a\u3001\u4f55\u304b\u5225\u306e\u73fe\u8c61\u3068\u306e\u95a2\u9023\u306b\u3088\u3063\u3066\u7406\u89e3\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002g2 \u304a\u3088\u3073 g3 \u306f Im(\u03c4)>0 \u306b\u304a\u3044\u3066\u6b63\u5247\u3060\u304b\u3089 \u0394 \u3082 Im(\u03c4)>0 \u306b\u304a\u3044\u3066\u6b63\u5247\u3067\u3042\u308b\u3002\u3055\u3089\u306b Im(\u03c4)>0 \u306b\u304a\u3044\u3066 \u0394(\u03c4) \u2260 0 \u304c\u6210\u308a\u7acb\u3064[4]\u3002\u3055\u3066\u4e0a\u8a18\u5224\u5225\u5f0f\u306f\u91cd\u307f 12 \u306e\u30e2\u30b8\u30e5\u30e9\u30fc\u5f62\u5f0f\u3067\u3042\u308b\u3002\u3059\u306a\u308f\u3061 a,b,c,d \u304c ad \u2212 bc = 1 \u3092\u6e80\u305f\u3059\u6574\u6570\uff08\u3064\u307e\u308a (abcd){displaystyle {begin{pmatrix}a&#038;b\\c&#038;dend{pmatrix}}}\u304c\u30e2\u30b8\u30e5\u30e9\u30fc\u7fa4 SL(2, Z) \u306b\u5c5e\u3059\u308b\uff09\u306e\u3068\u304d Im(\u03c4)>0 \u306b\u304a\u3044\u3066\u0394(a\u03c4+bc\u03c4+d)=(c\u03c4+d)12\u0394(\u03c4){displaystyle Delta left({frac {atau +b}{ctau +d}}right)=left(ctau +dright)^{12}Delta (tau )}\u304c\u6210\u308a\u7acb\u3064[5]\u3002\u307e\u305f\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\u3001\u30ce\u30fc\u30e0 q = exp(i\u03c0\u03c4) \u306e\u5e73\u65b9\u3092\u7528\u3044\u3066\u3001\u0394(\u03c4)=(2\u03c0)12\u2211k=1\u221e\u03c4(k)q2k,\u03c4(k)\u2208Z{displaystyle Delta (tau )=(2pi )^{12}sum _{k=1}^{infty }tau (k)q^{2k},tau (k)in mathbf {Z} }\u3068\u306a\u308b[6]\u3002\u3053\u3053\u3067 \u03c4(1)=1, \u03c4(2)=-24, \u03c4(3)=252, &#8230; \u306f\u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u306e\u30bf\u30a6\u51fd\u6570\u3067\u3042\u308b\uff08\u30aa\u30f3\u30e9\u30a4\u30f3\u6574\u6570\u5217\u5927\u8f9e\u5178\u306e\u6570\u5217 A000594\uff09\u3002\u3055\u3089\u306b\u30c7\u30c7\u30ad\u30f3\u30c8\u306e\u30a4\u30fc\u30bf\u95a2\u6570\u3068\u306e\u95a2\u4fc2\u304b\u3089\u0394(\u03c4)=(2\u03c0)12q2\u220fn=1\u221e(1\u2212q2n){displaystyle Delta (tau )=(2pi )^{12}q^{2}prod _{n=1}^{infty }(1-q^{2n})}\u304c\u6210\u308a\u7acb\u3064\u3002j-\u4e0d\u5909\u91cf[\u7de8\u96c6]  \u8907\u7d20\u5e73\u9762\u5185\u306e\u30af\u30e9\u30a4\u30f3\u306e j-\u4e0d\u5909\u91cf\u4e0a\u8a18\u306e\u4e0d\u5909\u91cf\u3092\u7528\u3044\u3066j(\u03c91,\u03c92)=1728g23\u0394{displaystyle j(omega _{1},omega _{2})={frac {1728g_{2}^{3}}{Delta }}}\u3068\u5b9a\u3081\u308b\u3068\u3001 \u0394 \u304a\u3088\u3073 g23\u306f\u3068\u3082\u306b\u6b21\u6570 \u221212 \u306e\u6589\u6b21\u51fd\u6570\u3067\u3042\u308b\u304b\u3089 j \u306f\u6b21\u6570 0 \u306e\u6589\u6b21\u95a2\u6570\u3067\u3042\u308b\u3002\u3064\u307e\u308a \u03c4 = \u03c92\/\u03c91 \u306a\u3089\u3070\u3064\u306d\u306bj(\u03c91,\u03c92)=j(1,\u03c4){displaystyle j(omega _{1},omega _{2})=j(1,tau )}\u304c\u6210\u308a\u7acb\u3064\u3002\u3057\u305f\u304c\u3063\u3066\u3053\u308c\u306f\u5468\u671f\u6bd4 tau=\u03c92\/\u03c91 \u306b\u3088\u3063\u3066\u306e\u307f\u5b9a\u307e\u308b\u306e\u30671\u5909\u6570\u95a2\u6570j(\u03c4)=j(1,\u03c4)=1728g23(1,\u03c4)\u0394(1,\u03c4){displaystyle j(tau )=j(1,tau )={frac {1728g_{2}^{3}(1,tau )}{Delta (1,tau )}}}\u304c\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3053\u308c\u3092\u30d5\u30a7\u30ea\u30c3\u30af\u30b9\u30fb\u30af\u30e9\u30a4\u30f3\u306e j-\u4e0d\u5909\u91cf\u3001j-\u51fd\u6570\u3001\u3042\u308b\u3044\u306f\u5358\u306b j-\u4e0d\u5909\u91cf (j-invariant) \u3068\u3044\u3046[7]\u3002Im(\u03c4)>0 \u306b\u304a\u3044\u3066 g2 \u304a\u3088\u3073 g3 \u306f\u6b63\u5247\u3067 \u0394(\u03c4) \u2260 0 \u304c\u6210\u308a\u7acb\u3064\u304b\u3089\u3001 j-\u4e0d\u5909\u91cf\u3082 Im(\u03c4)>0 \u306b\u304a\u3044\u3066\u6b63\u5247\u3067\u3042\u308b\u3002\u307e\u305f\u4e0d\u5909\u91cf\u306f\u5468\u671f\u683c\u5b50\u306b\u306e\u307f\u4f9d\u5b58\u3059\u308b\u3053\u3068\u304b\u3089\u30e2\u30b8\u30e5\u30e9\u30fc\u5909\u63db\u306b\u3088\u308a\u4e0d\u5909\u3067\u3042\u308b\u3002\u3064\u307e\u308a a,b,c,d \u304c ad \u2212 bc = 1 \u3092\u6e80\u305f\u3059\u6574\u6570\uff08\u3064\u307e\u308a (abcd){displaystyle {begin{pmatrix}a&#038;b\\c&#038;dend{pmatrix}}}\u304c\u30e2\u30b8\u30e5\u30e9\u30fc\u7fa4 SL(2, Z) \u306b\u5c5e\u3059\u308b\uff09\u306e\u3068\u304d Im(\u03c4)>0 \u306b\u304a\u3044\u3066j(a\u03c4+bc\u03c4+d)=j(\u03c4){displaystyle jleft({frac {atau +b}{ctau +d}}right)=j(tau )}\u304c\u6210\u308a\u7acb\u3064[8]\u3002\u305d\u3057\u3066 j-\u4e0d\u5909\u91cf\u306b\u3064\u3044\u3066\u306f\u30d5\u30fc\u30ea\u30a8\u7d1a\u6570\u306f\u3001\u30ce\u30fc\u30e0 q = exp(i\u03c0\u03c4) \u306e\u5e73\u65b9\u3092\u7528\u3044\u3066\u3001j(\u03c4)=q\u22122+744+196884q2+\u22ef{displaystyle j(tau )=q^{-2}+744+196884q^{2}+cdots }\u3068\u306a\u308b\uff08\u4fc2\u6570\u306f\uff08\u30aa\u30f3\u30e9\u30a4\u30f3\u6574\u6570\u5217\u5927\u8f9e\u5178\u306e\u6570\u5217 A000521\uff09\u306b\u3088\u308a\u4e0e\u3048\u3089\u308c\u308b\uff09[9]\u3002\u5b9a\u6570 e1, e2, e3[\u7de8\u96c6]\u4e09\u6b21\u306e\u591a\u9805\u5f0f\u65b9\u7a0b\u5f0f 4t3 \u2212 g2t \u2212 g3 = 0 \u3068\u305d\u306e\u4e09\u6839 e1, e2, e3 \u3092\u8003\u3048\u308b\u3002\u5224\u5225\u5f0f \u0394 = g23 \u2212 27g32 \u304c\u96f6\u3067\u306a\u3051\u308c\u3070\u3001\u3053\u308c\u3089\u306e\u6839\u306f\u3069\u306e\u4e8c\u3064\u3082\u76f8\u7570\u306a\u308b\u3002\u3053\u306e\u591a\u9805\u5f0f\u306b\u306f\u4e8c\u6b21\u306e\u9805\u304c\u306a\u3044\u304b\u3089\u3001\u6839\u306fe1+e2+e3=0{displaystyle e_{1}+e_{2}+e_{3}=0}\u3092\u6e80\u305f\u3059\u3002\u4e00\u6b21\u306e\u9805\u3068\u5b9a\u6570\u9805\u306e\u4fc2\u6570\uff08\u305d\u308c\u305e\u308c g2 \u3068 g3) \u306f\u6839\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u306b\u3088\u308ag2=\u22124(e1e2+e1e3+e2e3)=2(e12+e22+e32){displaystyle g_{2}=-4(e_{1}e_{2}+e_{1}e_{3}+e_{2}e_{3})=2(e_{1}^{2}+e_{2}^{2}+e_{3}^{2})}\u304a\u3088\u3073g3=4e1e2e3{displaystyle g_{3}=4e_{1}e_{2}e_{3}}\u3092\u6e80\u305f\u3059[10]\u3002\u4e0d\u5909\u91cf\u304c\u5b9f\u6570\u306e\u5834\u5408\u306b\u306f\u3001\u0394 \u306e\u7b26\u53f7\u306f\u6839\u306e\u7279\u6027\u3092\u6c7a\u5b9a\u3059\u308b\u3002\u0394 > 0 \u306a\u3089\u3070\u3001\u4e09\u6839\u306f\u5168\u3066\u5b9f\u6570\u3067\u3001\u6163\u7fd2\u7684\u306b e1 > e2 > e3 \u3067\u3042\u308b\u3082\u306e\u3068\u3059\u308b\u3002\u0394 < 0 \u306a\u3089\u3070\u3001\u6163\u7fd2\u7684\u306b \u03b1 > 0, \u03b2 > 0 \u3092\u7528\u3044\u3066 e1 = \u2212\u03b1 + \u03b2i, e3 \u306f e1 \u306e\u8907\u7d20\u5171\u8edb\u3001e \u306f\u975e\u8ca0\u5b9f\u6570\u3068\u306a\u308b\u3088\u3046\u306b\u3059\u308b\u3002\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u30da\u30fc\u51fd\u6570\u306e\u534a\u5468\u671f \u03c91\/2, \u03c92\/2 \u306f\u3001\u3053\u308c\u3089\u306e\u6839\u3068\u306e\u9593\u306b\u2118(\u03c91\/2)=e1,\u2118(\u03c92\/2)=e2,\u2118(\u03c93\/2)=e3,(\u03c93:=\u2212(\u03c91+\u03c92)){displaystyle wp (omega _{1}\/2)=e_{1},quad wp (omega _{2}\/2)=e_{2},quad wp (omega _{3}\/2)=e_{3},qquad (omega _{3}:=-(omega _{1}+omega _{2}))}\u306a\u308b\u95a2\u4fc2\u3092\u6301\u3064\u3002\u30da\u30fc\u51fd\u6570\u306e\u5c0e\u51fd\u6570\u306e\u5e73\u65b9\u306f\u3001\u4e0a\u3067\u8ff0\u3079\u305f\u51fd\u6570\u5024\u306e\u4e09\u6b21\u591a\u9805\u5f0f\u306b\u7b49\u3057\u3044\u304b\u3089\u3001\u2118\u2032(\u03c9i\/2)2=\u2118\u2032(\u03c9i\/2)=0{displaystyle wp &#8216;(omega _{i}\/2)^{2}=wp &#8216;(omega _{i}\/2)=0}\u304c i = 1, 2, 3 \u306b\u5bfe\u3057\u3066\u6210\u308a\u7acb\u3064\u3002\u9006\u306b\u3001\u51fd\u6570\u5024\u304c\u3053\u306e\u591a\u9805\u5f0f\u306e\u6839\u306b\u7b49\u3057\u3044\u306a\u3089\u3070\u3001\u5c0e\u51fd\u6570\u306f\u96f6\u306b\u306a\u308b\u3002g2, g3 \u304c\u3068\u3082\u306b\u5b9f\u6570\u3067 \u0394 > 0 \u306a\u3089\u3070\u3001ei \u306f\u5168\u3066\u5b9f\u6570\u3067\u3042\u308a\u3001\u30da\u30fc\u51fd\u6570 \u2118 \u306f 0, \u03c93, \u03c91 + \u03c93, and \u03c91 \u3092\u56db\u9802\u70b9\u3068\u3059\u308b\u77e9\u5f62\u306e\u5468\u4e0a\u3067\u5b9f\u6570\u5024\u3092\u3068\u308b\u3002\u4e0a\u3067\u8ff0\u3079\u305f\u3088\u3046\u306b\u6839\u3092 e1 > e2 > e3 \u3068\u9806\u5e8f\u4ed8\u3051\u308b\u306a\u3089\u3070\u3001\u7b2c\u4e00\u534a\u5468\u671f\u306f\u5b9f\u6570\u03c91\/2=\u222be1\u221edz4z3\u2212g2z\u2212g3{displaystyle omega _{1}\/2=int _{e_{1}}^{infty }{frac {dz}{sqrt {4z^{3}-g_{2}z-g_{3}}}}}\u306b\u306a\u308a\u3001\u4e00\u65b9\u7b2c\u4e09\u534a\u5468\u671f\u306f\u7d14\u865a\u6570\u03c93\/2=i\u222b\u2212e3\u221edz4z3\u2212g2z\u2212g3{displaystyle omega _{3}\/2=iint _{-e_{3}}^{infty }{frac {dz}{sqrt {4z^{3}-g_{2}z-g_{3}}}}}\u306b\u306a\u308b\u3002\u3044\u304f\u3064\u304b\u306e\u5b9a\u7406\u306b\u3064\u3044\u3066[\u7de8\u96c6]\u30da\u30fc\u51fd\u6570\u306e\u6e80\u305f\u3059\u3044\u304f\u3064\u304b\u306e\u6027\u8cea\u3092\u4ee5\u4e0b\u306b\u793a\u3059\u3002|\u2118(z)\u2118\u2032(z)1\u2118(y)\u2118\u2032(y)1\u2118(z+y)\u2212\u2118\u2032(z+y)1|=0.{displaystyle {begin{vmatrix}wp (z)&#038;wp &#8216;(z)&#038;1\\wp (y)&#038;wp &#8216;(y)&#038;1\\wp (z+y)&#038;-wp &#8216;(z+y)&#038;1end{vmatrix}}=0.}\u3053\u308c\u306e\u5bfe\u79f0\u7248\u306f u + v + w = 0 \u3068\u3057\u3066|\u2118(u)\u2118\u2032(u)1\u2118(v)\u2118\u2032(v)1\u2118(w)\u2118\u2032(w)1|=0{displaystyle {begin{vmatrix}wp (u)&#038;wp &#8216;(u)&#038;1\\wp (v)&#038;wp &#8216;(v)&#038;1\\wp (w)&#038;wp &#8216;(w)&#038;1end{vmatrix}}=0}\u3068\u66f8\u3051\u308b\u3002\u307e\u305f\u3001\u52a0\u6cd5\u516c\u5f0f\u2118(z+y)=14{\u2118\u2032(z)\u2212\u2118\u2032(y)\u2118(z)\u2212\u2118(y)}2\u2212\u2118(z)\u2212\u2118(y){displaystyle wp (z+y)={frac {1}{4}}left{{frac {wp &#8216;(z)-wp &#8216;(y)}{wp (z)-wp (y)}}right}^{2}-wp (z)-wp (y)}\u304a\u3088\u3073\u30012z \u304c\u5468\u671f\u3067\u306a\u3044\u9650\u308a\u306b\u304a\u3044\u3066\u500d\u6570\u516c\u5f0f\u2118(2z)=14{\u2118\u2033(z)\u2118\u2032(z)}2\u22122\u2118(z){displaystyle wp (2z)={frac {1}{4}}left{{frac {wp &#8221;(z)}{wp &#8216;(z)}}right}^{2}-2wp (z)}\u304c\u6210\u308a\u7acb\u3064\u3002\u57fa\u672c\u534a\u5468\u671f 1 \u306e\u5834\u5408[\u7de8\u96c6]\u03c91 = 1 \u306e\u3068\u304d\u306b\u306f\u3001\u03c92 \u3092\u6163\u7fd2\u7684\u306b \u03c4 \u3068\u66f8\u304d\u3001\u307e\u305f\u3001\u4e0a\u3067\u8ff0\u3079\u305f\u7406\u8ad6\u306e\u591a\u304f\u306f\u3088\u308a\u7c21\u5358\u306a\u5f62\u306b\u306a\u308b\u3002\u4e0a\u534a\u5e73\u9762\u306e\u5143 \u03c4 \u3092\u4e00\u3064\u56fa\u5b9a\u3059\u308b\u3068\u3001\u03c4 \u306e\u865a\u90e8\u306f\u6b63\u3067\u3042\u308a\u3001\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e \u2118-\u51fd\u6570\u306f\u2118(z;\u03c4)=1z2+\u2211(m,n)\u2260(0,0)1(z+m+n\u03c4)2\u22121(m+n\u03c4)2{displaystyle wp (z;tau )={frac {1}{z^{2}}}+sum _{(m,n)neq (0,0)}{1 over (z+m+ntau )^{2}}-{1 over (m+ntau )^{2}}}\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u548c\u306f\u539f\u70b9\u3092\u9664\u304f\u683c\u5b50 {m + n\u03c4\u00a0: m, n \u2208 Z} \u306e\u5168\u3066\u306e\u70b9\u306b\u4e99\u3063\u3066\u53d6\u308b\u3002\u3053\u3053\u3067\u306f\u3001\u03c4 \u3092\u56fa\u5b9a\u3057\u3066\u3001\u2118 \u3092 z \u306e\u51fd\u6570\u3068\u898b\u3066\u3044\u308b\u304c\u3001z \u3092\u56fa\u5b9a\u3057\u3066 \u03c4 \u3092\u52d5\u304b\u305b\u3070\u3001\u6955\u5186\u30e2\u30b8\u30e5\u30e9\u30fc\u51fd\u6570\u306e\u9762\u7a4d\u304c\u5c0e\u304b\u308c\u308b\u3002\u30da\u30fc\u51fd\u6570 \u2118 \u306f\u8907\u7d20\u5e73\u9762\u4e0a\u306e\u6709\u7406\u578b\u51fd\u6570\u3067\u3001\u5404\u683c\u5b50\u70b9\u306b\u304a\u3044\u3066\u4e8c\u4f4d\u306e\u6975\u3092\u6709\u3059\u308b\u3002\u307e\u305f\u30011 \u3068 \u03c4 \u3092\u5468\u671f\u306b\u6301\u3064\u4e8c\u91cd\u5468\u671f\u51fd\u6570\u3001\u3059\u306a\u308f\u3061 \u2118 \u306f\u2118(z+1)=\u2118(z+\u03c4)=\u2118(z){displaystyle wp (z+1)=wp (z+tau )=wp (z)}\u3092\u6e80\u305f\u3059\u3002\u4e0a\u8a18\u306e\u548c\u306f\u6b21\u6570 \u22122 \u306e\u6589\u6b21\u51fd\u6570\u3067\u3001c \u3092\u96f6\u3067\u306a\u3044\u8907\u7d20\u6570\u3068\u3057\u3066\u2118(cz;c\u03c4)=\u2118(z;\u03c4)\/c2{displaystyle wp (cz;ctau )=wp (z;tau )\/c^{2}}\u304c\u6210\u7acb\u3057\u3001\u3053\u308c\u3092\u7528\u3044\u3066\u3001\u4efb\u610f\u306e\u5468\u671f\u5bfe\u306b\u5bfe\u3059\u308b \u2118-\u51fd\u6570\u3092\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002z \u306b\u95a2\u3059\u308b\u5c0e\u51fd\u6570\u3082\u8a08\u7b97\u3067\u304d\u3066\u3001\u2118 \u306b\u95a2\u3057\u3066\u4ee3\u6570\u7684\u306a\u95a2\u4fc2\u5f0f\u2118\u20322=4\u21183\u2212g2\u2118\u2212g3{displaystyle wp &#8216;^{2}=4wp ^{3}-g_{2}wp -g_{3}}\u304c\u5f97\u3089\u308c\u308b\u3002\u3053\u3053\u3067 g2, g3 \u306f \u03c4 \u306e\u307f\u306b\u4f9d\u5b58\u3057\u3066\u6c7a\u307e\u308a\u3001\u307e\u305f \u03c4 \u306e\u30e2\u30b8\u30e5\u30e9\u30fc\u5f62\u5f0f\u306b\u306a\u308b\u3002\u4ee3\u6570\u65b9\u7a0b\u5f0fY2=4X3\u2212g2X\u2212g3{displaystyle Y^{2}=4X^{3}-g_{2}X-g_{3}}\u306f\u6955\u5186\u66f2\u7dda\u3092\u5b9a\u3081\u3001(\u2118, \u2118\u2032) \u304c\u3053\u306e\u66f2\u7dda\u306e\u5f84\u6570\u4ed8\u3051\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u308b\u3002\u4e0e\u3048\u3089\u308c\u305f\u5468\u671f\u3092\u6301\u3064\u4e8c\u91cd\u5468\u671f\u6709\u7406\u578b\u51fd\u6570\u306e\u5168\u57df\u6027\u306f\u3001\u6955\u5186\u66f2\u7dda\u306b\u4ed8\u968f\u3059\u308b\u4ee3\u6570\u51fd\u6570\u4f53\u3092\u5b9a\u3081\u308b\u304c\u3001\u3053\u306e\u4f53\u304cC(\u2118,\u2118\u2032){displaystyle mathbb {C} (wp ,wp &#8216;)}\u3067\u3042\u308b\u3053\u3068\u304c\u793a\u305b\u308b\u306e\u3067\u3001\u305d\u306e\u3088\u3046\u306a\u51fd\u6570\u306f\u30da\u30fc\u51fd\u6570\u3068\u305d\u306e\u5c0e\u51fd\u6570\u306b\u95a2\u3059\u308b\u6709\u7406\u51fd\u6570\u306b\u306a\u308b\u3002\u5358\u72ec\u306e\u5468\u671f\u5e73\u884c\u56db\u8fba\u5f62\u3092\u30c8\u30fc\u30e9\u30b9\uff08\u3064\u307e\u308a\u30c9\u30fc\u30ca\u30c4\u578b\u3092\u3057\u305f\u30ea\u30fc\u30de\u30f3\u9762\uff09\u306b\u5dfb\u304d\u3064\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3089\u3001\u4e0e\u3048\u3089\u308c\u305f\u5468\u671f\u5bfe\u306b\u4ed8\u968f\u3059\u308b\u6955\u5186\u51fd\u6570\u3092\u3001\u3053\u306e\u30ea\u30fc\u30de\u30f3\u9762\u4e0a\u306e\u51fd\u6570\u3068\u898b\u505a\u3059\u3053\u3068\u3082\u3067\u304d\u308b\u3002\u4e09\u6b21\u591a\u9805\u5f0f 4X3 \u2212 g2X \u2212 g3 \u306e\u6839 e2, e3 \u306f \u03c4 \u306b\u4f9d\u5b58\u3057\u3066\u6c7a\u307e\u308a\u3001\u30c6\u30fc\u30bf\u51fd\u6570\u3092\u7528\u3044\u3066e1(\u03c4)=13\u03c02(\u03d14(0;\u03c4)+\u03d1014(0;\u03c4)),{displaystyle e_{1}(tau )={tfrac {1}{3}}pi ^{2}(vartheta ^{4}(0;tau )+vartheta _{01}^{4}(0;tau )),}e2(\u03c4)=\u221213\u03c02(\u03d14(0;\u03c4)+\u03d1104(0;\u03c4)),{displaystyle e_{2}(tau )=-{tfrac {1}{3}}pi ^{2}(vartheta ^{4}(0;tau )+vartheta _{10}^{4}(0;tau )),}e3(\u03c4)=13\u03c02(\u03d1104(0;\u03c4)\u2212\u03d1014(0;\u03c4)){displaystyle e_{3}(tau )={tfrac {1}{3}}pi ^{2}(vartheta _{10}^{4}(0;tau )-vartheta _{01}^{4}(0;tau ))}\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002g2=\u22124(e1e2+e2e3+e3e1),g3=4e1e2e3{displaystyle g_{2}=-4(e_{1}e_{2}+e_{2}e_{3}+e_{3}e_{1}),quad g_{3}=4e_{1}e_{2}e_{3}}\u3060\u304b\u3089\u3053\u308c\u3089\u3082\u30c6\u30fc\u30bf\u51fd\u6570\u3092\u7528\u3044\u3066\u66f8\u3051\u308b\u3002\u30da\u30fc\u51fd\u6570\u3082\u30c6\u30fc\u30bf\u51fd\u6570\u3092\u7528\u3044\u3066\u2118(z;\u03c4)=\u03c02\u03d12(0;\u03c4)\u03d1102(0;\u03c4)\u03d1012(z;\u03c4)\u03d1112(z;\u03c4)+e2(\u03c4){displaystyle wp (z;tau )=pi ^{2}vartheta ^{2}(0;tau )vartheta _{10}^{2}(0;tau ){vartheta _{01}^{2}(z;tau ) over vartheta _{11}^{2}(z;tau )}+e_{2}(tau )}\u3068\u66f8\u3051\u308b\u3002\u30da\u30fc\u51fd\u6570 \u2118 \u306f\uff08\u5468\u671f\u3092\u9664\u3044\u3066\uff09\u4e8c\u3064\u306e\u96f6\u70b9\u3092\u6301\u3061\u3001\u305d\u306e\u5c0e\u51fd\u6570 \u2118\u2032 \u306f\u4e09\u3064\u306e\u96f6\u70b9\u3092\u6301\u3064\u3002\u5c0e\u51fd\u6570 \u2118\u2032 \u306e\u96f6\u70b9\u306e\u65b9\u306f\u7c21\u5358\u306b\u6c42\u3081\u3089\u308c\u308b\u3001\u3068\u3044\u3046\u306e\u3082 \u2118\u2032 \u306f\u5947\u51fd\u6570\u3086\u3048\u96f6\u70b9\u306f\u534a\u5468\u671f\u70b9\u306b\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u304b\u3089\u3067\u3042\u308b\u3002\u4ed6\u65b9\u3001\u30da\u30fc\u51fd\u6570 \u2118 \u81ea\u4f53\u306e\u96f6\u70b9\u306f,\u3001\u6bcd\u6570 \u03c4 \u304c\u7279\u5225\u306a\u5024\u3067\u3042\u308b\u5834\u5408\uff08\u4f8b\u3048\u3070\u3001\u5468\u671f\u683c\u5b50\u304c\u30ac\u30a6\u30b9\u6574\u6570\u5168\u4f53\u306e\u6210\u3059\u96c6\u5408\u306b\u306a\u308b\u3068\u304d\uff09\u3092\u9664\u3051\u3070\u3001\u9589\u3058\u305f\u5f0f\u306b\u8868\u3059\u306e\u306f\u975e\u5e38\u306b\u56f0\u96e3\u3067\u3042\u308b\u3002\u4e00\u3064\u306e\u5f0f\u304c\u3001\u30b6\u30ae\u30a8\u3068\u30a2\u30a4\u30d2\u30e9\u30fc\u306b\u3088\u3063\u3066\u6c42\u3081\u3089\u308c\u3066\u3044\u308b[11]\u3002\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u7406\u8ad6\u306b\u306f\u3001\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u30fb\u30bc\u30fc\u30bf\u51fd\u6570\u3068\u3044\u3046\u3082\u306e\u3082\u3042\u308a\u3001\u3053\u308c\u306f\u30da\u30fc\u51fd\u6570 \u2118 \u306e\u4e0d\u5b9a\u7a4d\u5206\u3067\u3001\u4e8c\u91cd\u5468\u671f\u51fd\u6570\u306b\u306f\u306a\u3089\u306a\u3044\u3002\u307e\u305f\u3001\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u30fb\u30bc\u30fc\u30bf\u3092\u5bfe\u6570\u5c0e\u51fd\u6570\u3068\u3059\u308b\u3088\u3046\u306a\u3001\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u30fb\u30b7\u30b0\u30de\u51fd\u6570\u3068\u547c\u3070\u308c\u308b\u30c6\u30fc\u30bf\u51fd\u6570\u3082\u6301\u3064\u3002\u3053\u306e\u30b7\u30b0\u30de\u51fd\u6570\u306f\u4efb\u610f\u306e\u5468\u671f\u70b9\u306b\u96f6\u70b9\u3092\u6301\u3061\uff08\u304b\u3064\u305d\u308c\u4ee5\u5916\u306b\u96f6\u70b9\u3092\u6301\u305f\u306a\u3044\uff09\u3001\u30e4\u30b3\u30d3\u306e\u6955\u5186\u51fd\u6570\u3092\u7528\u3044\u3066\u8868\u3059\u3053\u3068\u3082\u3067\u304d\u308b\u3002\u3053\u308c\u306b\u3088\u3063\u3066\u3001\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570\u3068\u30e4\u30b3\u30d3\u306e\u6955\u5186\u51fd\u6570\u306e\u9593\u306e\u76f8\u4e92\u5909\u63db\u306e\u4e00\u3064\u306e\u65b9\u6cd5\u304c\u4e0e\u3048\u3089\u308c\u308b\u3002\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u30fb\u30b7\u30b0\u30de\u306f\u6574\u51fd\u6570\u3067\u3042\u308a\u3001J.E.\u30ea\u30c8\u30eb\u30a6\u30c3\u30c9\u306e\u30e9\u30f3\u30c0\u30e0\u6574\u51fd\u6570\u8ad6\u306b\u304a\u3044\u3066\u300c\u5178\u578b\u7684\u300d\u306a\u51fd\u6570\u3068\u3057\u3066\u306e\u5f79\u5272\u3092\u6301\u3064\u3002\u30e4\u30b3\u30d3\u306e\u6955\u5186\u51fd\u6570\u3068\u306e\u95a2\u4fc2[\u7de8\u96c6]\u6570\u5024\u89e3\u6790\u7684\u306a\u5834\u9762\u306b\u304a\u3044\u3066\u3001\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570\u306e\u8a08\u7b97\u306b\u306f\u30e4\u30b3\u30d3\u306e\u6955\u5186\u51fd\u6570\u3092\u7528\u3044\u308b\u3068\u4fbf\u5229\u306a\u3053\u3068\u3082\u591a\u3044\u3002\u57fa\u672c\u95a2\u4fc2\u5f0f\u306f\u2118(z)=e3+e1\u2212e3sn2w=e2+(e1\u2212e3)dn2wsn2w=e1+(e1\u2212e3)cn2wsn2w{displaystyle wp (z)=e_{3}+{frac {e_{1}-e_{3}}{mathrm {sn} ^{2},w}}=e_{2}+left(e_{1}-e_{3}right){frac {mathrm {dn} ^{2},w}{mathrm {sn} ^{2},w}}=e_{1}+left(e_{1}-e_{3}right){frac {mathrm {cn} ^{2},w}{mathrm {sn} ^{2},w}}}\u3067\u4e0e\u3048\u3089\u308c\u308b[12]\u3002\u305f\u3060\u3057\u3001ei (i = 1, 2, 3) \u306f\u4e0a\u3067\u8ff0\u3079\u305f\u4e09\u3064\u306e\u6839\u3001\u30e4\u30b3\u30d3\u306e\u6955\u5186\u51fd\u6570\u306e\u6bcd\u6570 k \u306fk\u2261e2\u2212e3e1\u2212e3{displaystyle kequiv {sqrt {frac {e_{2}-e_{3}}{e_{1}-e_{3}}}}}\u3092\u6e80\u305f\u3057\u3001\u5404\u30e4\u30b3\u30d3\u306e\u6955\u5186\u51fd\u6570\u306e\u5f15\u6570 w \u306fw\u2261ze1\u2212e3{displaystyle wequiv z{sqrt {e_{1}-e_{3}}}}\u3067\u3042\u308b\u3002^ Apostol, Theorem 1.15, p.15^ Apostol, Theorem 1.18, p.20^ Apostol, Theorem 3.3, p.51^ Apostol, Theorem 1.15, p.15^ Apostol, Theorem 3.2, p.50^ Apostol, Theorem 1.19, p.20^ Apostol, Chapter 1.12, p. 15 \u3067\u306f\u4fc2\u65701728\u3092\u4e57\u305c\u305a\u306b\u5b9a\u7fa9\u3057\u3066\u3044\u308b\u3002^ Apostol, Theorem 1.16, p.17^ Apostol, Theorem 1.20, p.21^ Abramowitz and Stegun, p. 629^ Eichler, M.; Zagier, D. (1982). \u201cOn the zeros of the Weierstrass \u2118-Function\u201d. Mathematische Annalen 258 (4):  399\u2013407. doi:10.1007\/BF01453974.\u00a0^ Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp.\u00a0p. 721. LCCN\u00a059-14456\u00a0\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Abramowitz, Milton; Stegun, Irene A., eds. (1965), &#8220;Chapter 18&#8221;,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 627, ISBN 978-0486612720, MR 0167642N. I. Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second Edition (1990), Springer, New York ISBN 0-387-97127-0 (See chapter 1.)K. Chandrasekharan, Elliptic functions (1980), Springer-Verlag ISBN 0-387-15295-4Konrad Knopp, Funktionentheorie II (1947), Dover; Republished in English translation as Theory of Functions (1996), Dover ISBN 0-486-69219-1Serge Lang, Elliptic Functions (1973), Addison-Wesley, ISBN 0-201-04162-6Reinhardt, William P.; Walker, Peter L. (2010), \u201cWeierstrass Elliptic and Modular Functions\u201d, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN\u00a0978-0521192255, http:\/\/dlmf.nist.gov\/23\u00a0E. T. Whittaker and G. N. Watson,  A course of modern analysis, Cambridge University Press, 1952, chapters 20 and 21\u7af9\u5185\u7aef\u4e09\u300e\u6955\u5713\u51fd\u6578\u8ad6\u300f\u5ca9\u6ce2\u66f8\u5e97\u3008\u5ca9\u6ce2\u5168\u66f8\u3009\u30011936\u5e74\u3002\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki24\/archives\/292302#breadcrumbitem","name":"\u30f4\u30a1\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u6955\u5186\u51fd\u6570 &#8211; Wikipedia"}}]}]