![\"\"](\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2022\/01\/3-1.jpg\")
<\/p>\n\u4f60\u6c92\u6709\u770b\u932f\uff0c\u8981\u8ddf\u5927\u5bb6\u8ac7\u8ad6\u7684\u984c\u76ee\u5c31\u662f\u300c\u6708\u5149\u3001\u602a\u7269\u300d\u3002\u4e5f\u8a31\u4f60\u6703\u611f\u5230\u7591\u60d1\uff0c\u540d\u5b57\u6703\u90a3\u9ebc\u5947\u602a\uff0c\u300c\u6708\u5149\u3001\u602a\u7269\u300d\u807d\u8d77\u4f86\u6bd4\u8f03\u50cf\u624b\u6a5f\u904a\u6232\u6216\u662f\u79d1\u5e7b\u5c0f\u8aaa\u7684\u540d\u5b57\u3002\u4f46\u6211\u5011\u4e0d\u662f\u8981\u8ac7\u8ad6\u6708\u5149\u602a\u7378\u6216\u662f\u5982\u4f55\u628a\u4f60\u8b8a\u6210\u602a\u7269\u3002\u8ac7\u8ad6\u7684\u662f\u4e00\u500b\u5be6\u5be6\u5728\u5728\u7684\u6578\u5b78\u554f\u984c\u3002\u9996\u5148\uff0c\u6211\u5e0c\u671b\u89e3\u91cb\u4e00\u4e0b\u9019\u4e9b\u602a\u540d\u5b57\u7684\u500b\u4e2d\u7de3\u7531\uff0c\u63a5\u8457\u6703\u8aaa\u660e\u5b83\u6240\u5f15\u4f38\u51fa\u7684\u6578\u5b78\u554f\u984c\u53ca\u7406\u8ad6\u3002<\/p>\n
\u4ec0\u9ebc\u662f\u602a\u7269\uff08Monster\uff09\u548c\u6708\u5149\uff08Moonshine\uff09\uff1f \u6839\u64da\u82f1\u8a9e\u5b57\u5178\u7684\u5b9a\u7fa9\uff0c\u6240\u8b02\u7684\u602a\u7269\uff0c\u662f\u60f3\u50cf\u7684\u3001\u65e2\u5927\u53c8\u919c\u53c8\u53ef\u6015\u3002\u91cd\u9ede\u662f\u5b83\u53c8\u5927\u53c8\u53ef\u6015\u3002\u4e0d\u904e\u5982\u679c\u4f60\u662f30\u6b72\u4ee5\u4e0b\u7684\u5e74\u9752\u4eba\uff0c\u5c0dMonster\u7684\u5370\u8c61\u53ef\u80fd\u662fPok\u00e9Mon\uff0c\u4ed6\u5011\u662f\u5c0f\u5de7\u53ef\u611b\u7684\u3002<\/p>\n
\u90a3Moonshine\u662f\u4ec0\u9ebc\uff1f\u76f4\u63a5\u7ffb\u8b6f\u5c31\u662f\u6708\u5149\uff0c\u662f\u4e0d\u592a\u660e\u4eae\u3001\u6709\u9ede\u6a21\u7cca\u4e0d\u6e05\u3002\u53e6\u4e00\u7a2e\u6bd4\u8f03\u9760\u8fd1\u6211\u5011\u4e3b\u984c\u7684\u89e3\u91cb\u662f\u4e0d\u5be6\u5728\u3001\u7a7a\u5984\u7684\uff0c\u7279\u5225\u662f\u6307\u4e00\u4e9b\u865b\u7121\u98c4\u6e3a\u7684\u60f3\u6cd5\u3002\u53e6\u4e00\u7a2e\u89e3\u91cb\u662f\u82f1\u570b\u7684\u4fda\u8a9e\uff0cMoonshine\u6307\u7684\u662f\u8d70\u79c1\u6216\u79c1\u91c0\u7684\u70c8\u9152\uff0c\u4e00\u822c\u662f\u6307\u5a01\u58eb\u5fcc\u3002\u7b49\u4e00\u4e0b\u5927\u5bb6\u5c31\u6703\u660e\u767d\u70ba\u4ec0\u9ebc\u8981\u547d\u540d\u70baMoonshine\uff0c\u56e0\u70ba\u5b83\u5c31\u50cf\u79c1\u91c0\u9152\u4e00\u6a23\uff0c\u6c92\u6709\u5408\u6cd5\u7684\u5730\u4f4d\u3002<\/p>\n
\u5be6\u969b\u4e0a\u6211\u5011\u6240\u8aaa\u7684Monster\u662f\u4e00\u500b\u6709\u9650\u7fa4\uff08finite group\uff09\u3002\u6709\u9650\u7fa4\u7684\u7406\u8ad6\u662f\u5f9e19\u4e16\u7d00\u672b\u671f\uff0c\u4f3d\u7f85\u74e6\uff08Evariste Galois\uff09\u6709\u95dc\u591a\u9805\u5f0f\u89e3\u7684\u5de5\u4f5c\u958b\u59cb\u3002\u4ed6\u7684\u5de5\u4f5c\u5f8c\uff0c\u7fa4\u7684\u6982\u5ff5\u5c31\u6162\u6162\u6ef2\u5165\u6578\u5b78\u7684\u5404\u9818\u57df\uff0c\u662f\u73fe\u4ee3\u6578\u5b78\u4e2d\u4e00\u500b\u4e0d\u53ef\u7f3a\u7684\u5de5\u5177\u3002\u5176\u4e2d\u6709\u9650\u55ae\u7fa4\u7684\u5206\u985e\u66f4\u88ab\u55bb\u70ba20\u4e16\u7d00\u6578\u5b78\u4e0a\u7684\u4e00\u500b\u91cd\u5927\u6210\u5c31\uff0c\u73fe\u5728\u6211\u5011\u77e5\u9053\u4efb\u610f\u4e00\u500b\u975e\u4ea4\u63db\u7684\u6709\u9650\u55ae\u7fa4G\u6703\u540c\u69cb\u65bc\u4e0b\u5217\u7684\u4e00\u500b\u7fa4\uff1a<\/p>\n
1.\u4ea4\u66ff\u7fa4Altn, n\u22655\uff1b
\n2.\u4e00\u500bLie\u985e\u578b\u7684\u6709\u9650\u55ae\u7fa4\uff08\u670916\u500b\u7121\u9650\u5bb6\u65cf\uff09\uff1b\u6216
\n3.26\u500b\u96f6\u661f\uff08sporadic\uff09\u55ae\u7fa4\u4e4b\u4e00\u3002<\/p>\n
Monster\uff08\u602a\u7269\u7fa4\uff09\u662f\u901926\u500b\u96f6\u661f\u55ae\u7fa4\u4e2d\u4f4d\u6578\uff08order\uff09\u6700\u5927\u7684\u4e00\u500b\u3002\u82f1\u6587\u4e2dsporadic\u6307\u7684\u662f\u96f6\u661f\u3001\u5206\u6563\u7684\uff0c\u6240\u4ee5\u7576\u9019\u500b\u7fa4\u662fsporadic\u6642\uff0c\u5c31\u4e0d\u80fd\u7528\u4e00\u822c\u7684\u65b9\u6cd5\u4f86\u8aaa\u660e\u5b83\uff0c\u4e0d\u592a\u6e05\u695a\u5b83\u5f9e\u4f55\u4ee5\u4f86\uff0c\u5fc5\u9808\u8cbb\u5fc3\u601d\u53bb\u4e86\u89e3\u5b83\u3002 Monster\u662fsporadic\u7fa4\u4e2d\u6700\u5927\u7684\uff0c\u4f46\u5b83\u6709\u591a\u5927\u5462\uff0c\u6703\u5f97\u5230\u4e00\u500b\u8c61\u5fb5\u6050\u6016\u7684\u540d\u5b57\uff1fMonster\u7fa4\u7684\u5927\u5c0f\u70ba
\n|M|= 2^46 3^20 5^9 7^6 11^2 13^3\u202717\u202719\u202723\u202729\u202731\u202741\u202747\u202759\u202771 =808017424794512875886459904961710757005754368000000000
\n\u662f\u4e00\u500b54\u4f4d\u6578\u5b57\u300254\u4f4d\u6578\u807d\u8d77\u4f86\u4f3c\u4e4e\u9084\u597d\uff0c\u4f46\u662f\u5982\u679c\u8a8d\u771f\u7b97\u4e00\u4e0b\uff0c\u5b83\u662f\u975e\u5e38\u53ef\u6015\u7684\uff0c\u5c24\u5176\u7576\u6642\u7684\u96fb\u8166\u9084\u4e0d\u662f\u5f88\u767c\u9054\u3002\u6240\u4ee5\u7576\u767c\u73fe\u6709\u9019\u500b\u53c8\u5927\u53c8\u53ef\u6015\u7684\u5c0d\u8c61\u6642\uff0c\u5c31\u88ab\u547d\u540d\u70baMonster\u3002<\/p>\n
\u90a3Moonshine\u53c8\u662f\u4ec0\u9ebc\u5462\uff1f\u6c92\u8fa6\u6cd5\u7528\u4e09\u8a00\u5169\u8a9e\u8a73\u7d30\u7684\u8aaa\u660e\u3002\u5b83\u4ee3\u8868\u4e00\u500b\u95dc\u65bcMonster\u7fa4\u7684\u6307\u793a\u6a19\uff08character\uff09\u548c\u6a21\u51fd\u6578\uff08Modular Function\uff09\u7684\u795e\u79d8\u95dc\u4fc2\u3002\u6a21\u51fd\u6578\u662f\u6578\u8ad6\u4e2d\u4e00\u985e\u975e\u5e38\u91cd\u8981\u7684\u51fd\u6578\uff0c\u4f46\u5b83\u597d\u50cf\u8ddf\u6709\u9650\u7fa4\u6c92\u6709\u4efb\u4f55\u95dc\u4fc2\uff1f\u7576\u6578\u5b78\u5bb6\u5011\u767c\u73fe\u9019\u6a23\u4e00\u500b\u5947\u602a\u95dc\u4fc2\u6642\uff0c\u7576\u7136\u60f3\u77e5\u9053\u5b83\u662f\u5426\u6b63\u78ba\uff1f\u7b2c\u4e8c\u500b\u554f\u984c\u81ea\u7136\u800c\u7136\u5c31\u662f\u70ba\u4ec0\u9ebc\u662f\u6b63\u78ba\u7684\u3002\u5982\u679c\u662f\u6b63\u78ba\u7684\uff0c\u90a3\u70ba\u4ec0\u9ebc\u662f\u6b63\u78ba\u7684\uff0c\u6709\u6c92\u6709\u4e00\u500b\u597d\u7684\u89e3\u91cb\u3002\u91cd\u9ede\u662f\u9019\u985e\u795e\u79d8\u95dc\u4fc2\u628aMonster\u7684\u7814\u7a76\u63a8\u5ee3\u5230\u6709\u9650\u7fa4\u4e4b\u5916\uff0c\u5c07Monster\u7684\u7814\u7a76\u5e36\u5230\u6578\u8ad6\u3001\u674e\u4ee3\u6578\u3001\u751a\u81f3\u662f\u7269\u7406\u7684\u9818\u57df\u3002<\/p>\n
\u63a5\u4e0b\u4f86\u8b93\u6211\u5011\u8a0e\u8ad6\u4e00\u4e0b\u4ec0\u9ebc\u662f\u6709\u9650\u7fa4\u3002\u6709\u9650\u7fa4\u8ad6\u662f\u4f3d\u7f85\u74e6\u6709\u95dc\u591a\u9805\u5f0f\u65b9\u7a0b\u7684\u5de5\u4f5c\u958b\u59cb\u3002\u4e2d\u5b78\u7684\u6642\u5019\uff0c\u5927\u5bb6\u53ef\u80fd\u5b78\u904e\u5982\u4f55\u6c42\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u7684\u89e3\u3002\u4e8c\u6b21\u65b9\u7a0b\u3016ax\u3017^2+bx+c=0 \u53ef\u4ee5\u5229\u7528\u958b\u65b9\u6839\u5f97\u5230\u5169\u500b\u89e3
\nx=(-b+\u221a(b^2-4ac))\/2a or x=(-b-\u221a(b^2-4ac))\/2a
\n\u91cd\u9ede\u662f\u4f60\u53ea\u8981\u5229\u7528\u958b\u65b9\u9019\u500b\u7a0b\u5e8f\u53ca\u4e00\u822c\u904b\u7b97\u5c31\u53ef\u4ee5\u5f97\u5230\u65b9\u7a0b\u5f0f\u7684\u89e3\u3002\u90a3\u540c\u6a23\u7684\u65b9\u6cd5\u53ef\u4ee5\u5957\u7528\u5728\u66f4\u9ad8\u6b21\u65b9\u7684\u65b9\u7a0b\u5f0f\u55ce\uff1f\u5982\u679c\u662f\u4e09\u6b21\u65b9\u3001\u56db\u6b21\u65b9\u7684\u65b9\u7a0b\uff0c\u662f\u6709\u985e\u4f3c\u7684\u65b9\u6cd5\uff0c\u89e3\u4e0d\u662f\u9019\u6a23\u7c21\u55ae\uff0c\u4f46\u6709\u500b\u6c42\u89e3\u7684\u6f14\u7b97\u6cd5\uff0c\u53ea\u9700\u8981\u5229\u7528\u958b\u65b9\u9019\u500b\u7a0b\u5e8f\uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210\u3002\u5927\u7d04\u572819\u4e16\u7d00\u6642\uff0cAbel\u7d66\u51fa\u4f8b\u5b50\uff0c\u8a3c\u660e5\u6b21\u65b9\u4ee5\u4e0a\u7684\u65b9\u7a0b\uff0c\u4e0d\u4e00\u5b9a\u6709\u958b\u65b9\u89e3\u3002\u4f3d\u7f85\u74e6\u5247\u662f\u80fd\u4f7f\u7528\u958b\u65b9\u8655\u7406\u8207\u4e0d\u53ef\u80fd\u7684\u689d\u4ef6\u6e05\u695a\u5beb\u51fa\u3002\u4ed6\u7684\u60f3\u6cd5\u662f\u9769\u547d\u6027\u7684\uff0c\u628a\u91cd\u9ede\u653e\u5728\u7814\u7a76\u89e3\u672c\u8eab\u7684\u5c0d\u7a31\u6027\u3002\u4f55\u8b02\u89e3\u7684\u5c0d\u7a31\u6027\uff1f\u4e00\u500b\u65b9\u7a0b\u901a\u5e38\u6709\u591a\u500b\u4e0d\u540c\u7684\u89e3\uff0c\u6709\u4e9b\u89e3\u64c1\u6709\u76f8\u540c\u7684\u7279\u6027\uff0c\u6709\u4e9b\u537b\u4e0d\u4e00\u6a23\u3002\u4f8b\u5982\uff0c x^2-2=0 \u9019\u500b\u65b9\u7a0b\u6709\u5169\u500b\u89e3\uff0c\u4e00\u500b\u662f \u221a2\uff0c\u53e6\u4e00\u500b\u662f-\u221a2\u3002\u5f9e\u4ee3\u6578\u7684\u89d2\u5ea6\u4f86\u770b\uff0c\u5169\u500b\u89e3\u662f\u5b8c\u5168\u4e00\u6a23\uff0c\u53ef\u4ee5\u96a8\u610f\u8abf\u63db\u3002\u4f46 x^4-1=0 \u76844\u500b\u89e31,-1,\u221a(-1),-\u221a(-1)\u5247\u9700\u5206\u6210\u5169\u985e\uff0c1,-1\u548c \u221a(-1),-\u221a(-1)\u672c\u8cea\u4e0a\u6709\u6240\u4e0d\u540c\uff0c\u89e3\u5c31\u7121\u6cd5\u96a8\u610f\u8abf\u63db\u3002\u4f3d\u7f85\u74e6\u63d0\u51fa\u7814\u7a76\u89e3\u7684\u6392\u5217\u7d50\u69cb\u53ca\u7f6e\u63db\u7684\u60f3\u6cd5\uff0c\u4ed6\u7a31\u89e3\u7684\u7f6e\u63db\u70ba\u4e00\u500b\u300c\u7f6e\u63db\u7684\u7fa4\u7d44\u300d\uff0c\u9019\u662f\u7fa4\u7684\u7531\u4f86\u3002\u4f3d\u7f85\u74e6\u5f9e\u7fa4\u7684\u7d50\u69cb\u6392\u5217\u4f86\u78ba\u5b9a\u65b9\u7a0b\u6709\u6c92\u6709\u958b\u65b9\u89e3\u3002\u73fe\u5728\u7fa4\u4e00\u822c\u8996\u70ba\u7814\u7a76\u7279\u5b9a\u5c0d\u8c61\u5c0d\u7a31\u6027\u7684\u5de5\u5177\u3002<\/p>\n
<\/p>\n
![\"\"](\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2022\/01\/3-2.jpg\")
<\/p>\n
\u4ee5\u5168\u7b49\u4e09\u89d2\u5f62\u70ba\u4f8b\uff0c\u5229\u7528\u53cd\u5c04\u4f7f\u5b83\u5de6\u53f3\u4ea4\u63db\uff0c\u4ea4\u63db\u5f8c\uff0c\u9019\u500b\u4e09\u89d2\u5f62\u662f\u6c92\u6709\u4efb\u4f55\u6539\u8b8a\uff0c\u9084\u662f\u540c\u6a23\u7684\u4e09\u89d2\u5f62\u3002\u90a3\u5982\u679c\u540c\u6642\u505a\u4e8c\u6b21\u4e0d\u540c\u7684\u53cd\u5c04\uff0c\u53ef\u4ee5\u767c\u73fe\uff0c\u4e09\u89d2\u5f62\u7684\u9802\u9ede\u88ab\u65cb\u8f49\u4e86\uff0c\u4e09\u89d2\u5f62\u6c92\u6709\u6539\u8b8a\u3002\u9019\u6a23\u4e00\u4f86\uff0c\u9019\u500b\u4e09\u89d2\u5f62\u7684\u5c0d\u7a31\u6027\u7fa4\u6709\u516d\u500b\u5143\u7d20\uff0c\u4e09\u500b\u662f\u53cd\u5c04\uff0c\u53e6\u5916\u4e09\u500b\u662f\u65cb\u8f49\u3002\u6211\u5011\u4e5f\u53ef\u4ee5\u8a0e\u8ad6\u66f4\u9ad8\u7dad\u7684\u5e7e\u4f55\u7269\u6216\u4e00\u4e9b\u8907\u96dc\u7684\u4ee3\u6578\u7cfb\u7d71\u3002<\/p>\n
\u63a5\u4e0b\u4f86\u8b93\u6211\u5011\u8aaa\u660e\u751a\u9ebc\u662f\u6307\u793a\u6a19\uff08character\uff09\u3002\u7fa4\u662f\u62bd\u8c61\u7684\uff0c\u6240\u4ee5\u5e0c\u671b\u7528\u6bd4\u8f03\u719f\u6089\u7684\u5de5\u5177\u4f86\u7814\u7a76\u5b83\uff0c\u6700\u5e38\u4f7f\u7528\u7684\u5de5\u5177\u662f\u77e9\u9663\uff0c\u56e0\u70ba\u53ef\u4ee5\u505a\u5f88\u591a\u5be6\u8cea\u7684\u8a08\u7b97\u3002\u9996\u5148\uff0c\u6211\u5011\u77e5\u9053\u5728\u8907\u6578\u57dfC\u4e0a\uff0c\u6240\u6709\u7684n\u968e\u53ef\u9006\u77e9\u9663\u751f\u6210\u4e00\u500b\u7fa4\uff0c\u9019\u500b\u7fa4\u7a31\u70ba\u4e00\u822c\u7dda\u6027\u7fa4\u53ca\u8a18\u6210GL_n (C) \u3002\u4e00\u500b\u5f9e\u7fa4G\u5230GL_n (C)\u7684\u7fa4\u540c\u614b\u03c1:G\u2192GL_n (C) \u7a31\u70ba\u7fa4G\u7684n\u7dad\u8868\u73fe\u3002\u4e00\u500b\u7279\u5225\u7684\u4f8b\u5b50\u662f\u628aG\u5168\u90e8\u5c0d\u5230\u55ae\u4f4d\u77e9\u9663\uff0c\u9019\u662fG\u7684\u8868\u73fe\uff0c\u7a31\u70ba trivial representation\u3002\u6709\u6642\u5019\u8655\u7406\u77e9\u9663\u4ecd\u7136\u56f0\u96e3\uff0c\u6240\u4ee5\u6703\u8003\u616e\u77e9\u9663\u7684\u8de1\uff08trace\uff09\u3002\u8de1\u662f\u6307\u77e9\u9663A\u5c0d\u89d2\u7dda\u4e0a\u5404\u500b\u5143\u7d20\u7684\u7e3d\u548c\uff0c\u4e00\u822c\u8a18\u4f5ctr A\u3002\u7576\u03c1:G\u2192GL_n (C) \u662fG\u7684\u8868\u73fe\uff0c\u03c7_\u03c1 (g)=tr(\u03c1(g)) \u88ab\u7a31\u70ba\u6307\u793a\u6a19\u3002\u8de1\u662f\u6578\u5b57\u6bd4\u8f03\u5bb9\u6613\u8655\u7406\uff0c\u4f46\u5982\u679c\u4f60\u53ea\u8003\u616e\u8907\u6578\u57df\u4e0a\u7684\u8868\u73fe\u6642\uff0c\u6307\u793a\u6a19\u672c\u8eab\u5c31\u8db3\u4ee5\u6c7a\u5b9a\u6240\u6709\u7684\u4e0d\u53ef\u7d04\u8868\u73fe\u3002\u7576Monster\u7fa4\u525b\u88ab\u767c\u73fe\u6642\uff0c\u6578\u5b78\u5bb6\u5c31\u5617\u8a66\u8a08\u7b97\u5b83\u7684\u6307\u793a\u6a19\u53ca\u4e0d\u53ef\u7d04\u8868\u73fe\uff0c\u5176\u4e2d\u4e00\u500b\u767c\u73fe\u662f\uff0c\u5982\u679cMonster\u5b58\u5728\uff0c\u5b83\u7684\u6700\u4f4e\u7dad\u8868\u73fe\u81f3\u5c11\u662f196883\u7dad\u3002<\/p>\n
\u90a3Moonshine\u662f\u600e\u9ebc\u88ab\u767c\u73fe\u7684\uff1f \u6545\u4e8b\u6e90\u81ea\u4ee5\u4e0b\u4e00\u500b\u975e\u5e38\u7c21\u55ae\u7684\u7b49\u5f0f\uff1a
\n1+196883=196884
\n\u9019\u7b49\u5f0f\u770b\u4f86\u5f88\u7c21\u55ae\uff0c\u5c0f\u5b78\u751f\u90fd\u53ef\u4ee5\u7406\u89e3\uff0c\u4f46\u5b83\u537b\u5be6\u662f\u4e00\u500b\u975e\u5e38\u91cd\u8981\u7684\u767c\u73fe\u3002
\n\u4e4b\u524d\u63d0\u904e1\u662ftrivial\u8868\u73fe\u7684\u7dad\u6578\uff0c196883\u662fMonster\u7fa4\u6700\u5c0f\u4e0d\u53ef\u7d04\u8868\u73fe\u7684\u7dad\u6578\u3002196884\u5247\u8ddf\u6a62\u5713j\uff0d\u51fd\u6578(elliptic j-function) \u7684q\u5c55\u958b\u6709\u95dc\u3002\u6a62\u5713j-\u51fd\u6578\u662f\u4e00\u500b\u91cd\u8981\u7684\u6a21\u51fd\u6578\uff0c\u5b83\u662f\u4e00\u500b\u5168\u7d14\u9031\u671f\u51fd\u6578\uff0c\u6240\u4ee5\u53ef\u4ee5\u628a\u5b83\u5c55\u958b\u6210\u5085\u7acb\u8449\u7d1a\u6578\u3002\u8a2dq=e^2\u03c0i\u03c4 \uff0cj \u7684q\uff0d\u5c55\u958b\u5f0f\u53ef\u5beb\u6210
\nj(q)=q^(-1)+744+196884q + higher order terms
\nJohn McKay \u767c\u73fe\u5f0f\u5b50\u4e2d\u7684196884\uff0c\u597d\u50cf\u548cMonster\u6709\u9ede\u95dc\u4fc2\uff0c\u4f46\u53c8\u4e0d\u592a\u78ba\u5b9a\uff0c \u6240\u4ee5\u5beb\u4e86\u4e00\u5c01\u4fe1\u7d66John Thompson \uff08\u7576\u6642\u6700\u6709\u540d\u7684\u7fa4\u8ad6\u5b78\u8005\uff09\uff0c\u8a62\u554f\u4ed6\u5c0d\u9019\u4e8b\u7684\u770b\u6cd5\u3002\u7d93\u904e\u5f88\u591a\u7684\u8a08\u7b97\u5f8c\uff0cThompson\u8ddfMcKay \u731c\u6e2c\u602a\u7269\u7fa4\u5b58\u5728\u4e00\u500b\u7121\u7aae\u7dad\u7684\u6574\u6578\u5206\u7d1a\u8868\u73feV=\u2295_(n\u22650) V_n\uff0c\u4f7f\u5f97\u5b83\u7684graded dimension
\nchV=\u2211_(n=0)^\u221e dimV_n q^(n-1)=j(q)-744
\n\u4ed6\u5011\u9084\u731c\u6e2c\uff0c\u5982\u679c\u8003\u616e\u5206\u7d1a\u8de1\uff08graded trace\uff09
\nT_g (q)=\u2211_(n=0)^\u221e trg|_(V_n ) q^(n-1),q=e^2\u03c0iz,g\u2208Monster,
\n\u9019\u4e5f\u662f\u4e00\u500b\u6a21\u51fd\u6578\u3002McKay\uff0dThompson\u731c\u60f3\u5f37\u70c8\u6697\u793a\u602a\u7269\u7fa4\uff08\u4e00\u500b\u6709\u9650\u7fa4\uff09\u6700\u81ea\u7136\u7684\u8868\u73fe\u53ef\u80fd\u662f\u7121\u9650\u7dad\u7684\uff0c \u9019\u8868\u793a\u80cc\u5f8c\u53ef\u80fd\u6709\u66f4\u591a\u6211\u5011\u4e0d\u4e86\u89e3\u7684\u7d50\u69cb\u5b58\u5728\u3002
\n\u9019\u500b\u731c\u60f3\u51fa\u73fe\u5f8c\uff0c\u6578\u5b78\u5bb6\u5011\u5c31\u958b\u59cb\u8a08\u7b97\u9019\u4e9b\u7d1a\u6578\u7684\u4e00\u4e9b\u6027\u8cea\u3002Conway\u548cNorton\u82b1\u4e86\u5f88\u4e45\u7684\u6642\u9593\u53bb\u8a08\u7b97T_g (q)\u5404\u7d1a\u7684\u4fc2\u6578\uff0c\u5f97\u5230\u8a31\u591a\u7684\u6578\u5b57\u4f46\u53c8\u4e0d\u77e5\u9053\u5982\u4f55\u8655\u7406\uff0c\u5c31\u53bb\u5716\u66f8\u9928\uff0c\u7ffb\u958b\u4e00\u4e9b\u6709\u95dc\u6a21\u51fd\u6578\u7684\u66f8\u7c4d\uff0c\u5728\u4e00\u672c19\u4e16\u7d00\u672b\u7684\u66f8\u4e2d\uff0c\u770b\u5230\u8a31\u591a\u6578\u5b57\u8ddf\u4ed6\u5011\u7b97\u51fa\u4f86\u7684\u4e00\u6a21\u4e00\u6a23\u3002\u4ed6\u5c31\u731c\u6e2cMcKay\uff0dThompson \u7d1a\u6578\u61c9\u8a72\u5c31\u662f\u66f8\u4e2d\u6240\u8a0e\u8ad6\u7684\u51fd\u6578\u3002\u9019\u4e4b\u9593\u5145\u6eff\u5de7\u5408\uff0c\u4f46\u662f\u5f9e\u9019\u4e9b\u5de7\u5408\u53ef\u4ee5\u770b\u5230McKay\uff0dThompson\u731c\u60f3\u4e26\u4e0d\u662f\u4e00\u500b\u5f88\u8f15\u7387\u7684\u6771\u897f\uff0c\u800c\u662f\u6709\u5be6\u969b\u8b49\u64da\uff0c\u5b83\u80cc\u5f8c\u61c9\u6709\u975e\u5e38\u6f02\u4eae\u7684\u7406\u8ad6\u53ca\u5f88\u591a\u6211\u5011\u4e0d\u7406\u89e3\u7684\u4e8b\u60c5\u9700\u8981\u6211\u5011\u53bb\u63a2\u7d22\u3002Conway\u548cNorton\u767c\u73feMcKay\uff0dThompson\u7d1a\u6578\u8ddf\u5176\u4e2d\u7684171\u6a21\u51fd\u6578\u76f8\u95dc\uff0c\u9019\u4e9b\u51fd\u6578\u5f88\u7279\u5225\uff0c\u5b83\u5011\u7684modular curve\u90fd\u662fgenus zero\uff0c\u540c\u6642\u5b83\u5011\u53ef\u4ee5\u5728\u6a21\u66f2\u7dda\u4e0a\u751f\u6210\u6574\u500b\u51fd\u6578\u57df\u3002\u9019\u767c\u73fe\u975e\u5e38\u6709\u610f\u601d\uff0c\u70ba\u4ec0\u9ebcMonster\u6703\u8ddf\u66f2\u7dda\u7684genus\u6709\u95dc\uff1f\u9019\u500bgenus 0\u7684\u7279\u6027\u5230\u76ee\u524d\u90fd\u6c92\u6709\u5f88\u597d\u7684\u89e3\u91cb\u3002<\/p>\n
Modular Curves, Modular group \u548c\u76f8\u5c0d\u7684\u51fd\u6578\u57df\u5728Monster \u9084\u6c92\u51fa\u73fe\u4e4b\u524d\uff0c\u5728\u6578\u8ad6\u4e2d\u5c31\u88ab\u5ee3\u6cdb\u8a0e\u8ad6\u53ca\u7814\u7a76\u3002\u5176\u4e2dOgg\u8b49\u660e\u4ee5\u4e0b\u7684\u5b9a\u7406\uff1a
\n\u5b9a\u7406(Ogg): \u82e5 p \u70ba\u8cea\u6578, \u0393_0^+ (p) \u662f\u4e00\u500b genus 0\u7fa4\u82e5\u4e14\u60df\u82e5 p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71\u3002<\/p>\n
\u5982\u679c\u4ed4\u7d30\u5c0d\u6bd4\uff0c\u4f60\u6703\u767c\u73fe\u901913\u500b\u8cea\u6578\u6b63\u597d\u5c31\u662f\u53ef\u6574\u9664Monster\u968e\u6578\uff08order\uff09\u7684\u8cea\u6578\u3002Ogg\u6709\u4e00\u6b21\u53bb\u807d\u6f14\u8b1b\u6642\uff0c\u807d\u5230\u6709\u95dcMonster\u7684\u63cf\u8ff0\uff0c\u5176\u4e2d\u6574\u9664Monster\u968e\u6578\u7684\u8cea\u6578\u548c\u4ed6\u5b9a\u7406\u4e2d\u7684\u8cea\u6578\u525b\u597d\u4e00\u6a21\u4e00\u6a23\uff0c\u9019\u6a23\u7684\u5de7\u5408\u7576\u7136\u5f15\u8d77\u5927\u5bb6\u7684\u597d\u5947\u3002Ogg\u61f8\u8cde\u4e86\u4e00\u74f6Jack Daniels\uff08\u5a01\u58eb\u5fcc\u7684\u54c1\u724c\uff09\u7d66\u4efb\u4f55\u4e00\u500b\u53ef\u4ee5\u8aaa\u660e\u5169\u8005\u4e4b\u9593\u95dc\u4fc2\u7684\u4eba\u3002<\/p>\n
\u73fe\u5728\u4f60\u53ef\u4ee5\u6bd4\u8f03\u7406\u89e3Moonshine\u9019\u500b\u540d\u7a31\u7684\u7531\u4f86\uff0cConway\uff0dNorton\u8a66\u5716\u53bb\u89e3\u91cb\u602a\u7269\u7fa4\u548c\u6a21\u51fd\u6578\u7684\u67d0\u7a2e\u95dc\u4fc2\uff0c\u4e5f\u5c0dOgg\u7684\u554f\u984c\u7d66\u51fa\u67d0\u4e9b\u8b49\u64da\uff0c\u4f46\u4e26\u6c92\u6709\u7d66\u51fa\u597d\u7684\u89e3\u91cb\uff0c\u4e5f\u6c92\u6709\u8b49\u660e\u3002\u4ed6\u5011\u53ef\u80fd\u61c9\u8a72\u7372\u5f97\u4e00\u74f6\u5a01\u58eb\u5fcc\u4f5c\u70ba\u734e\u54c1\uff0c\u4f46\u6240\u8a0e\u8ad6\u7684\u95dc\u4fc2\u53ca\u73fe\u8c61\uff0c\u56b4\u683c\u4f86\u8aaa\u6c92\u6709\u8a3c\u660e\uff0c\u5728\u6578\u5b78\u4e0a\u4ecd\u672b\u7372\u5f97“\u5408\u6cd5\u201d \u7684\u5730\u4f4d\uff0c\u6709\u9ede\u50cf\u300c\u79c1\u91c0\u7684\u5a01\u58eb\u5fcc\u300d\uff0c\u88ab\u7a31\u70baMoonshine\u4e5f\u5c31\u4e00\u9ede\u90fd\u4e0d\u5947\u602a\u3002<\/p>\n
1983\u5e74\uff0cFrenkel-Lepowsky-Meurman\u4f7f\u7528\u7269\u7406\u5b78\u4e0a\u7684\u9802\u9ede\u7b97\u5b50(vertex operator)\u69cb\u9020\u51faMcKay-Thompson\u731c\u60f3\u4e2d\u90a3\u500bMonster\u7fa4\u7684\u8868\u73fe\uff0c\u4e00\u822c\u7a31\u70baMoonshine\u6a21\u3002\u4ed6\u5011\u548cRichard Borcherds\u5f15\u5165\u4e86\u4e00\u500b\u65b0\u7684\u4ee3\u6578\u7cfb\u7d71\uff0c\u7a31\u70ba\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\uff08vertex operator algebra\uff09\u3002\u5f8c\u4f86\uff0c\u53c8\u8b49\u660eMoonshine\u6a21\u662f\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\uff0c\u4e26\u8a3c\u660eMonster\u7fa4\u662f\u9019\u500b\u4ee3\u6578\u7684\u81ea\u540c\u69cb\u7fa4\uff0c\u90a3\u5c31\u662f\u8aaaMonster\u7fa4\u662f\u63cf\u8ff0\u4e00\u500b\u9f90\u5927\u4ee3\u6578\u7cfb\u7d71\u7684\u5c0d\u7a31\u6027\u3002\u9019\u500b\u6a21\u73fe\u5728\u7a31\u70ba\u6708\u5149\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u3002<\/p>\n
\u300c\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u300d\u7684\u5f15\u5165\u5c0d\u300c\u602a\u7269\u7fa4\u300d\u7684\u7814\u7a76\u6709\u975e\u5e38\u6df1\u9060\u7684\u5f71\u97ff\u3002\u76ee\u524d\uff0c\u6211\u5011\u53ef\u4ee5\u900f\u904e\u7814\u7a76\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u7684\u7d50\u69cb\u548c\u8868\u73fe\u7406\u8ad6\u4f86\u4e86\u89e3\u602a\u7269\u7fa4\uff0c\u5f9e\u524d\u5f88\u591a\u611f\u5230\u795e\u79d8\u7684\u73fe\u8c61\uff0c\u73fe\u5728\u90fd\u53ef\u4ee5\u5229\u7528\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u4f86\u89e3\u91cb\u3002\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u4e5f\u5c0d\u7269\u7406\u4e0a\u7684\u4fdd\u89d2\u5834\u8ad6\uff08conformal field theory\uff09\u63d0\u4f9b\u4e86\u4e00\u500b\u56b4\u683c\u7684\u516c\u7406\u7cfb\u7d71\u548c\u4ee3\u6578\u65b9\u6cd5\uff0c\u800cMonster\u662f\u9019\u7a2e\u4ee3\u6578\u7d50\u69cb\u7684\u5c0d\u7a31\u6027\u3002<\/p>\n
\u300c\u6708\u5149\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u300d\u662f\u4e00\u500b\u9f90\u5927\u4ee3\u6578\u7cfb\u7d71\uff0c\u7814\u7a76\u5b83\u7684\u5c0d\u7a31\u6027\u975e\u5e38\u56f0\u96e3\uff0c\u6240\u4ee5\u5fc5\u9808\u5c0d\u5b83\u505a\u4e00\u4e9b\u7d30\u7dfb\u5206\u89e3\u3002\u90a3\u5c31\u662f\u8aaa\uff0c\u8a66\u5716\u627e\u5c0b\u4e00\u4e9b\u6709\u826f\u597d\u6027\u8cea\u7684\u5b50\u4ee3\u6578\uff0c\u4e26\u5229\u7528\u9019\u4e9b\u5b50\u4ee3\u6578\u548c\u8868\u73fe\u7406\u8ad6\u4f86\u4e86\u89e3\u6574\u9ad4\u3002\u6709\u9ede\u50cf\u5efa\u7d44\u5408\u5c4b\uff0c\u5148\u628a\u6846\u67b6\uff08\u5b50\u4ee3\u6578\uff09\u8a2d\u597d\uff0c\u518d\u52a0\u5165\u6a21\u7d44\uff08\u5b50\u4ee3\u6578\u7684\u4e0d\u53ef\u7d04\u8868\u73fe\uff09\u7d44\u6210\u4e00\u500b\u8907\u96dc\u7684\u7d50\u69cb\u30021990\u4ee3\u521d\u671f\uff0c\u5927\u5bb6\u5c31\u767c\u73fe\u6708\u5149\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u6709\u4e00\u7d44\u5f88\u597d\u7684\u5b50\u4ee3\u6578\uff0c\u5b83\u662f48\u500b\u4e2d\u5fc3\u96fb\u8377c=\u00bd Virasoro \u4ee3\u6578\u4e0d\u53ef\u7d04\u8868\u73fe\u7684\u5f35\u91cf\u7a4d\u3002\u9019c=\u00bd Virasoro\u4ee3\u6578\u7684\u8868\u73fe\u76f8\u5c0d\u7c21\u55ae\uff0c\u50cf\u62fc\u7a4d\u6728\u4e00\u6a23\uff0c\u6162\u6162\u62fc\u6e4a\u53ef\u4ee5\u5f97\u5230\u8907\u96dc\u7684\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\uff0c\u904e\u7a0b\u985e\u4f3c\u62fc\u5716\u904a\u6232\u6216\u6a02\u9ad8\u7a4d\u6728\uff0c\u8981\u4ed4\u7d30\u4e86\u89e3\u6bcf\u500b\u6a21\u584a\u7684\u95dc\u4fc2\uff0c\u5c0f\u5fc3\u62fc\u6e4a\uff0c\u4e00\u4e0d\u5c0f\u5fc3\u5c31\u8981\u5f9e\u982d\u958b\u59cb\u3002\u6709\u8da3\u7684\u662f\u62fc\u6e4a\u7684\u65b9\u6cd5\u53ef\u4ee5\u75282\u9032\u4f4d\u6578\u78bc\u4f86\u63cf\u8ff0\uff0c\u4e0d\u932f\u5c31\u662f\u96fb\u8166\u548c\u96fb\u8166\u6e9d\u901a\u4f7f\u7528\u76840\u82071\u5e8f\u5217\u3002\u5bae\u672c\u96c5\u5f65\u57282000\u5e74\u5de6\u53f3\u5c31\u5229\u7528\u9019\u500b\u65b9\u6cd5\uff0c\u628a\u6708\u5149\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u91cd\u65b0\u69cb\u9020\u51fa\u4f86\uff0c\u4e26\u5f97\u5230\u5f88\u591a\u6709\u95dcMonster\u7684\u7d50\u679c\u3002\u9019\u7a2e\u65b9\u6cd5\u53ef\u4ee5\u63a8\u5ee3\u5230\u5176\u5b83\u7684\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\uff0c\u9019\u985e\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u73fe\u5728\u7a31\u70ba\u6846\u67b6\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u3002 \u6211\u5011\u7684\u5718\u968a\u5c0d\u9019\u985e\u4ee3\u6578\u6709\u5f88\u591a\u7684\u6210\u679c\u3002\u9996\u5148\uff0c\u6211\u5011\u8a3c\u660e\u4e86\u5982\u679c\u8981\u5f97\u5230\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\uff0c\u7528\u4f86\u62fc\u6e4a\u76842\u9032\u78bc\u5fc5\u9808\u8981\u7b26\u5408\u4e00\u4e9b\u689d\u4ef6\uff0c\u540c\u6642\uff0c\u6211\u5011\u4e5f\u5f97\u5230\u6846\u67b6\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u4e0d\u53ef\u7d04\u8868\u73fe\u7684\u5177\u9ad4\u63cf\u8ff0\u3002\u5229\u7528\u9019\u4e9b\u63cf\u8ff0\uff0c\u6211\u5011\u77e5\u9053\u6846\u67b6\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u5728\u90a3\u7a2e\u60c5\u6cc1\u4e0b\u662f\u5168\u7d14\u5f62\uff0c\u5373\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u81ea\u8eab\u53ea\u6709\u4e00\u500b\u4e0d\u53ef\u7d04\u6a21\u3002\u6211\u5011\u6210\u529f\u7684\u5efa\u69cb\u4e86\u5f88\u591a\u548c\u6708\u5149\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u76f8\u4f3c\u7684\u5168\u7d14\u5f62\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\uff0c\u66f4\u5f97\u5230\u4e2d\u5fc3\u96fb\u8377c=24\u5168\u7d14\u5f62\u6846\u67b6\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u7684\u5206\u985e\u3002\u904e\u7a0b\u4e2d\uff0c\u53c8\u767c\u73fe\u4e86\u5f88\u591a\u548cMonster\u76f8\u95dc\u7684\u4ee3\u6578\uff0c\u5305\u62ec\u6240\u8b02\u7684W\u4ee3\u6578\u548cparafermion\u4ee3\u6578\u3002\u6709\u8da3\u7684\u662f\u9019\u4e9b\u4ee3\u6578\u56e0\u5176\u5b83\u539f\u56e0\uff0c\u5728\u7269\u7406\u5b78\u548c\u8868\u73fe\u7406\u8ad6\u4e2d\u88ab\u5ee3\u6cdb\u7814\u7a76\uff0c\u90a3\u70ba\u4ec0\u9ebc\u5b83\u5011\u8ddfMonster\u76f8\u95dc\uff1f<\/p>\n
\u76ee\u524d\u4ecd\u672a\u6709\u5be6\u8cea\u7684\u7b54\u6848\u3002\u9019\u4e9b\u7d50\u679c\u5f15\u9818\u6211\u5011\u53bb\u8655\u7406\u4e00\u822c\u4e2d\u5fc3\u96fb\u8377c=24\u5168\u7d14\u5f62\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u7684\u5206\u985e\u554f\u984c\u3002\u7d93\u904e\u9019\u5e7e\u5e74\u7684\u52aa\u529b\uff0c\u6211\u5011\u5df2\u7d93\u5927\u81f4\u4e86\u89e3\u53ca\u5b8c\u6210\u5206\u985e\uff0c\u4f46\u6700\u91cd\u8981\u7684\u554f\u984c\uff0c\u5373\u6708\u5149\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u7684\u552f\u4e00\u6027\u4ecd\u7136\u672a\u89e3\u6c7a\uff0c\u5e0c\u671b\u5c0d\u5176\u5b83c=24\u5168\u7d14\u5f62\u9802\u9ede\u7b97\u5b50\u4ee3\u6578\u7684\u4e86\u89e3\u80fd\u5e6b\u52a9\u6211\u5011\u89e3\u6c7a\u9019\u500b\u554f\u984c\u3002<\/p>\n
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