{"id":31240,"date":"2023-05-18T00:12:09","date_gmt":"2023-05-17T16:12:09","guid":{"rendered":"https:\/newsletter.sinica.edu.tw\/?p=31240"},"modified":"2023-07-13T15:54:35","modified_gmt":"2023-07-13T07:54:35","slug":"%e3%80%90%e5%b0%88%e6%ac%84%e3%80%91%e6%95%b8%e5%ad%97%e8%88%87%e6%96%b9%e7%a8%8b%e5%bc%8f%e7%9a%84%e5%b0%8d%e7%a8%b1%e6%80%a7-langlands-%e7%b6%b1%e9%a0%98","status":"publish","type":"post","link":"https:\/newsletter.sinica.edu.tw\/31240\/","title":{"rendered":"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818"},"content":{"rendered":"<div class=\"auth-info\">\n<div class='one_fourth'>\n\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" src=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_9.jpg\" alt=\"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818\" width=\"150\" height=\"150\" class=\"aligncenter size-full wp-image-31963\" \/>\n\t\t\t\t<\/div><br \/>\n<div class='three_fourth et_column_last'>\n\t\t\t\t\t\u73fe\u8077\u4e2d\u7814\u9662\u6578\u5b78\u6240\u52a9\u7814\u7a76\u54e1\u3002\u4ee5\u524d\u5728\u7f8e\u570b\u6771\u897f\u5cb8\u5ff5\u904e\u548c\u6559\u904e\u66f8\uff0c\u89ba\u5f97\u56de\u4f86\u81fa\u5317\u6bd4\u8f03\u5feb\u6a02\u3002\u7167\u7247\u662f\u5728\u6cd5\u570b\u7814\u8a0e\u6703\u7684\u9031\u672b\u62cd\u7684\uff0c\u8868\u60c5\u986f\u793a\u89ba\u5f97\u6578\u5b78\u5f88\u96e3\u3002\n\t\t\t\t<\/div><div class='clear'><\/div>\n<\/div>\n<div id=\"attachment_21652\" style=\"width: 792px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-21652\" src=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_2.jpg\" alt=\"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818\" width=\"782\" height=\"403\" class=\"aligncenter size-full wp-image-31254\" srcset=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_2.jpg 782w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_2-300x155.jpg 300w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_2-768x396.jpg 768w\" sizes=\"(max-width: 782px) 100vw, 782px\" \/><p id=\"caption-attachment-21652\" class=\"wp-caption-text\">\u25b2\u4f5c\u8005\u65bc 2022 \u5e74\u6691\u5047\u53c3\u52a0\u6cd5\u570b\u9ad8\u7b49\u79d1\u5b78\u7814\u7a76\u6240 Langlands \u7db1\u9818\u7814\u8a0e\u6703\u4e4b\u6d77\u5831\u622a\u5716\u3002<\/p><\/div>\n<p>\u5728\u8907\u96dc\u7684\u73fe\u4ee3\u793e\u6703\u88e1\uff0c\u6211\u5011\u6703\u5728\u5404\u7a2e\u5c0f\u5730\u65b9\u8207\u6578\u5b57\u6709\u95dc\u7684\u5c0f\u554f\u984c\u4e0d\u671f\u800c\u9047\u3002\u6bd4\u5982\u8aaa\uff0c\u6211\u60f3\u8981\u70ba\u4e00\u6b3e\u904a\u6232\u8a2d\u8a08\u4e00\u500b\u65b9\u683c\u72c0\u7684\u68cb\u76e4\uff0c\u88e1\u9762\u6709\u4e94\u7a2e\u984f\u8272\u7684\u65b9\u683c\uff1a<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_3.jpg\" alt=\"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818\" width=\"705\" height=\"705\" class=\"aligncenter size-full wp-image-31242\" srcset=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_3.jpg 705w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_3-300x300.jpg 300w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_3-150x150.jpg 150w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_3-440x440.jpg 440w\" sizes=\"(max-width: 705px) 100vw, 705px\" \/><\/p>\n<p>\u4e0a\u5716\u7684\u8a2d\u8a08\u6709\u4e00\u4e9b\u5c0d\u7a31\u6027\uff0c\u50cf\u662f\uff1a<br \/>\n1. \u8b93\u76e4\u9762\u7121\u9650\u5ef6\u4f38\u7684\u8a71\uff0c\u6bcf\u7a2e\u984f\u8272\u51fa\u73fe\u7684\u983b\u7387\u548c\u65b9\u5f0f\u90fd\u4e00\u6a23\u3002<br \/>\n2. \u628a\u540c\u8272\u7684\u65b9\u683c\u9023\u8d77\u4f86\uff0c\u4e5f\u6703\u5f62\u6210\u4e00\u500b\u6b63\u65b9\u683c\u72c0\u7684\u7d50\u69cb\u3002<br \/>\n\u8b80\u8005\u53ef\u4ee5\u767c\u73fe\uff0c\u5927\u7684\u6b63\u65b9\u683c\u9762\u7a4d\u662f\u5c0f\u65b9\u683c\u7684\u4e94\u500d\u3002\u9019\u662f\u56e0\u70ba\u6211\u5011\u6709\u7562\u6c0f\u5b9a\u7406\uff1a2<sup>2<\/sup>+1<sup>2<\/sup>=5<br \/>\n\u5982\u679c\u5c0f\u65b9\u683c\u7684\u908a\u9577\u8207\u9762\u7a4d\u662f 1\uff0c\u90a3\u9ebc\u5927\u65b9\u683c\u7684\u908a\u9577\u662f \u221a5\uff0c\u9762\u7a4d\u662f 5\u3002\u9019\u4e5f\u8aaa\u660e\u4e86\u6bcf\u7a2e\u984f\u8272\u4f54\u64da\u7684\u6bd4\u4f8b\u662f\u4e94\u5206\u4e4b\u4e00\uff0c\u6240\u4ee5\u4e94\u7a2e\u984f\u8272\u6b63\u597d\u586b\u6eff\u68cb\u76e4\u3002\u9019\u7d66\u4e86\u6211\u5011\u4e00\u500b\u6709\u8da3\u7684\u5c0f\u89c0\u5bdf\uff1a<\/p>\n<p><strong>\u300c\u540c\u6a23\u7684\u8a2d\u8a08\uff0c\u5c0d\u4e09\u7a2e\u984f\u8272\u6216\u662f\u516d\u7a2e\u984f\u8272\u884c\u4e0d\u901a\uff0c\u56e0\u70ba 3 \u548c 6 \u4e0d\u80fd\u5beb\u6210\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u548c\u3002\u300d<\/strong><\/p>\n<p>\u9019\u5c31\u7d66\u4e86\u6211\u5011\u4e00\u500b\u4fe1\u624b\u62c8\u4f86\u7684\u95dc\u65bc\u6578\u5b57\u7684\u5c0f\u554f\u984c\uff1a\u6709\u54ea\u4e9b\u6578\u5b57\uff08\u6574\u6578\uff09\u53ef\u4ee5\u5beb\u6210\u53e6\u5916\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u4e4b\u548c\u5462\uff1f\u6bd4\u5982\u8aaa 4=2<sup>2<\/sup>+0<sup>2<\/sup>\u30018=2<sup>2<\/sup>+2<sup>2<\/sup>\u30019=3<sup>2<\/sup>+0<sup>2<\/sup>\u300110=3<sup>2<\/sup>+1<sup>2<\/sup>\u300113=3<sup>2<\/sup>+2<sup>2<\/sup>\u3002\u4e2d\u9593\u88ab\u6211\u5011\u8df3\u904e\u7684 3, 6, 7, 11, 12 \u5247\u4e0d\u80fd\u5beb\u6210\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u548c\u3002\u6cd5\u570b\u6578\u5b78\u5bb6\u5409\u54c8\uff08Girard\uff09\u548c\u8cbb\u99ac\uff08Fermat\uff09\u90fd\u66fe\u63d0\u51fa\u904e\u4ee5\u4e0b\u7684\u89c0\u5bdf\uff1a<\/p>\n<p><strong><u>\u5b9a\u7406 A<\/u>\uff1a\u4e00\u500b\u6b63\u6574\u6578 n \u53ef\u4ee5\u5beb\u6210\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u548c\u7684\u689d\u4ef6\u5982\u4e0b\uff1an \u53ef\u4ee5\u5beb\u6210\u4e00\u500b\u5e73\u65b9\u6578\u548c\u4e00\u4e9b\u8cea\u6578\u7684\u4e58\u7a4d\uff0c\u4f7f\u5f97\u9019\u4e9b\u8cea\u6578\u90fd\u53ef\u4ee5\u5beb\u6210\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u548c\u3002<\/strong><\/p>\n<p>\u6bd4\u5982\u8aaa\uff0c45=3<sup>2<\/sup>\u00d75\uff0c\u800c\u6211\u5011\u4e5f\u53ef\u4ee5\u5f9e 5=2<sup>2<\/sup>+1<sup>2<\/sup> \u63a8\u5c0e\u51fa 45=6<sup>2<\/sup>+3<sup>2<\/sup>\u3002\u53cd\u904e\u4f86\u8aaa 12=2<sup>2<\/sup>\u00d73\uff0c\u800c 3 \u4e0d\u80fd\u5beb\u6210\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u548c\uff0c12 \u4e5f\u4e0d\u884c\u3002\u53c8\u6bd4\u5982 15=3\u00d75\uff0c \u5176\u4e2d 3 \u4e0d\u80fd\u5beb\u6210\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u548c\uff0c\u65bc\u662f 15 \u4e5f\u4e0d\u884c\uff08\u5118\u7ba1 5 \u53ef\u4ee5\uff09\u3002\u4e00\u822c\u4f86\u8aaa\uff0c\u54ea\u4e9b\u8cea\u6578\u53ef\u4ee5\u5beb\u6210\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u548c\u5462\uff1f\u6211\u5011\u6709\uff1a<\/p>\n<p><strong><u>\u5b9a\u7406 B<\/u>\uff1a\u4e00\u500b\u8cea\u6578 p \u53ef\u4ee5\u5beb\u6210\u5169\u500b\u6574\u6578\u7684\u5e73\u65b9\u548c\u7684\u689d\u4ef6\u662f\uff1ap \u9664\u4ee5 4 \u7684\u9918\u6578\u4e0d\u7b49\u65bc 3\u3002<\/strong><\/p>\n<p>\u597d\u6bd4\u50cf 2=1<sup>2<\/sup>+1<sup>2<\/sup>\u30015=2<sup>2<\/sup>+1<sup>2<\/sup>\u300113=3<sup>2<\/sup>+2<sup>2<\/sup>\u300117=4<sup>2<\/sup>+1<sup>2<\/sup>\u300129=5<sup>2<\/sup>+2<sup>2<\/sup> \u9019\u4e9b\u9664\u4ee5 4 \u7684\u9918\u6578\u90fd\u4e0d\u662f 3\u3002\u81f3\u65bc 3, 7, 11, 19, 23 \u9019\u4e9b\u9664\u4ee5 4 \u9918 3 \u7684\u8cea\u6578\uff0c\u5c31\u53ea\u80fd\u4e7e\u77aa\u773c\u4e86\u3002<\/p>\n<p>\u5728\u6211\u5011\u770b\u66f4\u591a\u4f8b\u5b50\u4e4b\u524d\uff0c\u8b93\u6211\u5011\u8ac7\u8ac7\u4e0a\u9762\u9019\u5169\u500b\u5b9a\u7406\u7684\u539f\u7406\u3002\u9996\u5148\uff0c\u8b93\u6211\u5011\u63d0\u7149\u4e00\u500b\u5728\u672c\u6587\u6703\u4e00\u76f4\u82e5\u96b1\u82e5\u73fe\u7684\u54f2\u5b78\uff1a<\/p>\n<p><strong>\u300c\u95dc\u65bc\u6574\u6578\u7684\u554f\u984c\uff0c\u7d93\u5e38\u53ef\u4ee5\u5206\u89e3\u6210\u95dc\u65bc\u6bcf\u500b\u53ef\u80fd\u7684\u8cea\uff08\u56e0\uff09\u6578\u7684\u554f\u984c\u300d<\/strong><\/p>\n<p>\u6709\u8208\u8da3\u95b1\u8b80\u5b8c\u6574\u5b9a\u7406 A, B \u8b49\u660e\u7684\u8b80\u8005\u53ef\u4ee5\u53c3\u8003\u7dad\u57fa\u767e\u79d1\u95dc\u65bc\u300c\u8cbb\u99ac\u5e73\u65b9\u548c\u5b9a\u7406\u300d\u7684\u689d\u76ee\uff08\u82f1\u6587\u689d\u76ee\u300cFermat&#8217;s theorem on sums of two squares\u300d\u6709\u6bd4\u8f03\u591a\u7684\u5167\u5bb9\uff09\u3002\u901a\u5e38\u5728\u5927\u5b78\u4ee3\u6578\u5982\u679c\u8b1b\u5230\u9019\u500b\u5b9a\u7406\u7684\u8a71\uff0c\u6703\u5f9e\u5982\u4e0b\u7684\u5f15\u7406\u51fa\u767c\uff08\u8a73\u898b\u82f1\u6587\u689d\u76ee\u88e1 Dedekind \u7684\u7b2c\u4e8c\u500b\u8b49\u660e\uff09\uff1a<\/p>\n<p><strong><u>\u5f15\u7406 C<\/u>\uff1a\u5c0d\u65bc\u4e00\u500b\u8cea\u6578 p\uff0c\u5b58\u5728\u6574\u6578 n \u4f7f\u5f97 n<sup>2<\/sup>+1 \u662f p \u7684\u500d\u6578\u7684\u689d\u4ef6\u7b49\u540c\u65bc\u5b9a\u7406 B \u7684\u689d\u4ef6\uff0c\u4e5f\u5c31\u662f p \u9664\u4ee5 4 \u7684\u9918\u6578\u4e0d\u7b49\u65bc 3\u3002<\/strong><\/p>\n<p>\u524d\u9762\u5b9a\u7406 A\u3001 B \u548c\u5f15\u7406 C \u90fd\u662f\u6578\u5b78\u5bb6\u6240\u7814\u7a76\u7684\u5c0d\u7a31\u6027\uff1a<\/p>\n<p><strong>\u300c\u7528\u4e00\u7a2e\u65b9\u5f0f\u8868\u8ff0\u7684\u7d50\u69cb\uff08\u597d\u6bd4\u4e00\u500b\u2606\u7b26\u865f\u3001\u6216\u662f\u80fd\u5beb\u6210\u5169\u5e73\u65b9\u548c\u7684\u8cea\u6578\uff09\uff0c\u5728\u53e6\u4e00\u7a2e\u64cd\u4f5c\u5e95\u4e0b\uff08\u65cb\u8f49 72 \u5ea6\u3001\u6216\u662f\u9664\u4ee5 4 \u53d6\u9918\u6578\uff09\u4fdd\u6301\u4e0d\u8b8a\u3002\u300d<\/strong><\/p>\n<p>\u5728\u6578\u8ad6\uff08\u7814\u7a76\u6574\u6578\u7684\u6578\u5b78\uff09\u4e2d\uff0c\u6709\u4e00\u7cfb\u5217\u50cf\u524d\u8ff0\u9019\u6a23\u5b50\uff0c\u76f8\u7576\u5947\u7279\u7684\u5c0d\u7a31\u6027\u3002\u8b93\u6211\u5011\u770b\u770b\u66f4\u591a\u4f8b\u5b50\u3002\u6bd4\u5982\u8aaa\uff0c\u5f15\u7406 C \u7684\u9ad8\u6b21\u65b9\u8b8a\u9ad4\u662f\uff1a<\/p>\n<p><strong><u>\u5b9a\u7406 D<\/u>\uff1a\u5c0d\u65bc\u4efb\u610f\u5169\u500b\u76f8\u7570\u8cea\u6578 p \u548c q\u3002\u4ee5\u4e0b\u5169\u500b\u689d\u4ef6\u7b49\u50f9\uff1a<br \/>\n\u2160. \u5b58\u5728\u6574\u6578n\u4f7f\u5f97n<sup>p-2<\/sup>+n<sup>p-3<\/sup>+\u2026+1 \u662f q \u7684\u500d\u6578\u3002<br \/>\n\u2161. q \u9664\u4ee5 p \u7684\u9918\u6578\u662f 1<\/strong>\u3002<\/p>\n<p>\u63a5\u4e0b\u4f86\u9019\u500b\u4f8b\u5b50\u66f4\u795e\u5947\u4e86\u3002\u8003\u616e\u7121\u7aae\u7d1a\u6578<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_5.jpg\" alt=\"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818\" width=\"350\" height=\"100\" class=\"aligncenter size-full wp-image-31248\" srcset=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_5.jpg 350w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_5-300x86.jpg 300w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><br \/>\n=u\uff081\u0096u\uff09\uff081\u0096u<sup>2<\/sup>\uff09\u2026\uff081\u0096u<sup>22<\/sup>\uff09\uff081\u0096u<sup>23<\/sup>\uff09<sup>2<\/sup>\uff081\u0096u<sup>24<\/sup>\uff09\u2026\uff081\u0096u<sup>45<\/sup>\uff09\uff081\u0096u<sup>46<\/sup>\uff09<sup>2<\/sup>\u2026<br \/>\n=u\u0096u<sup>2<\/sup>\u0096u<sup>3<\/sup>+u<sup>6<\/sup>+u<sup>8<\/sup>\u0096u<sup>13<\/sup>\u0096u<sup>16<\/sup>+u<sup>23<\/sup>\u0096u<sup>24<\/sup>+u<sup>25<\/sup>+u<sup>26<\/sup>+u<sup>27<\/sup>\u0096u<sup>29<\/sup>\u0096u<sup>31<\/sup>+u<sup>39<\/sup>\u0096u<sup>41<\/sup>\u0096u<sup>46<\/sup>\u0096u<sup>47<\/sup>+u<sup>48<\/sup>+u<sup>49<\/sup>\u0096u<sup>50<\/sup>\u0096u<sup>54<\/sup>+u<sup>58<\/sup>+2u<sup>59<\/sup>+\u2026<br \/>\n\uff08\u559c\u6b61\u8a08\u7b97\u7684\u8b80\u8005\uff0c\u53ef\u4ee5\u4eab\u53d7\u4e00\u4e0b\u628a\u4e0a\u4e00\u884c\u4e58\u8d77\u4f86\u5f97\u5230\u4e0b\u4e00\u884c\u7684\u904e\u7a0b\uff09<\/p>\n<p><strong><u>\u5b9a\u7406 E<\/u>\uff1a\u5c0d\u65bc\u8cea\u6578 p\uff0c\u4ee5\u4e0b\u5169\u500b\u689d\u4ef6\u7b49\u50f9\uff1b<br \/>\n\u2160. \u5b58\u5728\u6574\u6578 n \u4f7f\u5f97 n<sup>3<\/sup>\uff0dn\uff0d1 \u662f p \u7684\u500d\u6578\u3002<br \/>\n\u2161. \u5728\u7121\u7aae\u7d1a\u6578 h\uff08u\uff09\u88e1\uff0cu<sup>p<\/sup> \u7684\u4fc2\u6578\u975e\u8ca0\u3002\uff08\u9019\u6642\u5b83\u5011\u7684\u4fc2\u6578\u4e00\u5b9a\u662f -1\uff09<\/strong><\/p>\n<p>\u6bd4\u5982\u8aaa\uff0c\u5728\u4e0a\u9762\u7684\u7d1a\u6578\u4e2d\uff0c\u5c0d\u65bc p=2, 3, 13, 29 \u7b49\u8cea\u6578\uff0cup \u7684\u4fc2\u6578\u662f -1\u3002\u5176\u4ed6\u4fc2\u6578\u4e0d\u662f -1 \u7684\u8cea\u6578\uff08\u6700\u5e38\u898b\u7684\u662f 0\uff0c\u7b2c\u4e00\u500b\u6b63\u4fc2\u6578\u7684\u8cea\u6578\u6b21\u65b9\u662f u<sup>23<\/sup>\uff0c\u518d\u4f86\u662f 2u<sup>59<\/sup>\uff09\u50cf\u662f p=5, 7, 11 \u7b49\uff0c\u5c31\u5b58\u5728\u6574\u6578 n \u4f7f\u5f97 n<sup>3<\/sup>\uff0dn\uff0d1 \u662f p \u7684\u500d\u6578\uff0c\u6bd4\u5982 2<sup>3<\/sup>\uff0d2\uff0d1=5 \u662f  5 \u7684\u500d\u6578\uff0c5<sup>3<\/sup>\uff0d5\uff0d1=119 \u662f 7 \u7684\u500d\u6578\u7b49\u7b49\u3002<\/p>\n<p>\u8b93\u6211\u5011\u518d\u77aa\u8457\u4e0a\u9762\u7684\u7d1a\u6578\u770b\u5e7e\u773c\uff0c\u5b83\u6709\u5e7e\u500b\u4ee4\u4eba\u60f3\u5410\u69fd\u7684\u554f\u984c\u3002\u597d\u6bd4\u8aaa\uff0c\u70ba\u4ec0\u9ebc\u8981\u6709 u \u7684 23 \u6b21\u65b9\u5462\uff1f23 \u9019\u500b\u795e\u5947\u6578\u5b57 x<sup>3<\/sup>\uff0dx\uff0d1 \u9019\u500b\u4e09\u6b21\u591a\u9805\u5f0f\u7684<strong>\u5224\u5225\u5f0f<\/strong>\uff0c\u4f46\u66f4\u4ee4\u4eba\u6293\u72c2\u7684\u662f\u2026\u2026\u597d\u5427\uff01\u5c31\u7b97\u6709 23 \u597d\u4e86\uff0c\u90a3\u70ba\u4ec0\u9ebc\u662f\u4e0a\u9762\u9019\u500b\u795e\u7955\u7684\u7121\u7aae\u7d1a\u6578\u5462\uff1f<\/p>\n<p>\u518d\u770b\u4e00\u6b21\u9019\u500b\u7121\u7aae\u7d1a\u6578\uff1a<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_5.jpg\" alt=\"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818\" width=\"350\" height=\"100\" class=\"aligncenter size-full wp-image-31248\" srcset=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_5.jpg 350w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_5-300x86.jpg 300w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><br \/>\n=u\u0096u<sup>2<\/sup>\u0096u<sup>3<\/sup>+u<sup>6<\/sup>+u<sup>8<\/sup>\u0096u<sup>13<\/sup>\u0096u<sup>16<\/sup>+u<sup>23<\/sup>\u0096u<sup>24<\/sup>+u<sup>25<\/sup>+u<sup>26<\/sup>+u<sup>27<\/sup>\u0096u<sup>29<\/sup>\u2026<br \/>\n\u5b83\u6709\u4e00\u7cfb\u5217\u7f8e\u5999\u7684\u6027\u8cea\uff0c\u6bd4\u5982\uff1a<br \/>\n\u2160. \u6bcf\u7576 a, b \u662f\u5169\u500b\u4e92\u8cea\u7684\u6b63\u6574\u6578\uff0c\u5728\u7121\u7aae\u7d1a\u6578 h\uff08u\uff09\u88e1\u6211\u5011\u90fd\u6703\u6709 u<sup>ab<\/sup> \u7684\u4fc2\u6578 = u<sup>a<\/sup> \u7684\u4fc2\u6578 \u00d7 u<sup>b<\/sup> \u7684\u4fc2\u6578\u3002<br \/>\n\u4f8b\u5982\uff1a<br \/>\n\uff0e u<sup>2<\/sup> \u548c u<sup>3<\/sup> \u7684\u4fc2\u6578\u90fd\u662f -1\uff0c\u800c u<sup>6<\/sup> \u7684\u4fc2\u6578\u679c\u7136\u662f -1 \u00d7 -1= 1\u3002<br \/>\n\uff0e u<sup>4<\/sup> \u7684\u4fc2\u6578\u662f 0\uff0c\u5c0d\u6240\u6709\u5947\u6578 k\uff0c u<sup>4k<\/sup> \u7684\u4fc2\u6578\uff08\u4f8b\u5982 u<sup>12<\/sup>, u<sup>20<\/sup>, u<sup>28<\/sup>\uff09\u90fd\u662f 0\u3002<br \/>\n\uff0e u<sup>3<\/sup> \u7684\u4fc2\u6578\u662f -1\uff0cu<sup>8<\/sup> \u7684\u4fc2\u6578\u662f 1\uff0c\u800c u<sup>24<\/sup> \u7684\u4fc2\u6578\u679c\u7136\u662f -1 \u00d7 1= -1\u3002<br \/>\n\u9806\u5e36\u4e00\u63d0\uff0c\u6211\u5011\u7684\u7121\u7aae\u7d1a\u6578\u82e5\u662f\u6c92\u6709\u4e58\u4e0a\uff081\u0096u<sup>23n<\/sup>\uff09\u7684\u9805\uff0c\u90a3\u9ebc u<sup>24<\/sup> \u7684\u4fc2\u6578\u5c31\u6703\u4e0d\u4e00\u6a23\uff0c\u9019\u500b\u7279\u5225\u7684\u6027\u8cea\u4e5f\u5c31\u4e0d\u6703\u6210\u7acb\u4e86\u3002<\/p>\n<p>\u2161. \u4e0a\u9762\u90a3\u4e9b\u4fc2\u6578\u662f -1 \u7684\u8cea\u6578\uff08p=2, 3, 13, 29\uff09\u7b49\u7b49\u90fd\u53ef\u4ee5\u5beb\u6210 2x<sup>2<\/sup>+xy+3y<sup>2<\/sup> \u7684\u6a23\u5b50\uff0c\u5176\u4e2d x, y \u662f\u6574\u6578\u3002<br \/>\n\u4f8b\u5982\uff1a13 = 2 \u00d7 2<sup>2<\/sup>+ 2 \u00d7 -1+ 3 \u00d7\uff08-1\uff09<sup>2<\/sup>\u3002\u9019\u88e1 2x<sup>2<\/sup>+xy+3y<sup>2<\/sup> \u7684\u5224\u5225\u5f0f\u4e5f\u662f 1<sup>2<\/sup>\u0096 4 \u00d7 2 \u00d7 3 = -23\uff08\u6211\u5011\u7684 23 \u53c8\u51fa\u73fe\u5566\uff01\uff09\u3002\u7e3d\u4e4b\uff0c\u9019\u500b\u7121\u7aae\u7d1a\u6578\u9084\u6709\u5e7e\u500b\u9019\u6a23\u7279\u6027\u3002<\/p>\n<p>\u5176\u4e2d\u6700\u53b2\u5bb3\u4e5f\u6700\u795e\u7955\u7684\u7279\u6027\uff0c\u9700\u8981\u8003\u616e\u5c07 u \u5e36\u5165\u8907\u6578\u7684\u503c\u3002\u8b93\u6211\u5011\u5c0d\u7121\u7aae\u7d1a\u6578 h\uff08u\uff09\u5e36\u5165 u = e<sup>2\u03c0iz<\/sup>\uff0c\u5176\u4e2d z \u662f\u4efb\u4f55\u865b\u90e8\u5927\u65bc 0 \u7684\u8907\u6578\u3002\u8b93\u6211\u5011\u5beb\u505a f\uff08z\uff09=h\uff08e<sup>2\u03c0iz<\/sup>\uff09\u3002\u7531\u65bc\u8907\u6578\u6307\u6578\u51fd\u6578 e<sup>2\u03c0iz<\/sup> \u7684\u6027\u8cea\uff0c\u6211\u5011\u6709\uff1a<br \/>\na. \u5728 z \u7684\u865b\u90e8\u5927\u65bc 0 \u6642\uff0c|u|&lt;1\uff0c\u5f9e\u6b64\u53ef\u4ee5\u8b49\u660e\u7121\u7aae\u7d1a\u6578\u6536\u6582\u3002<br \/>\nb. \u7531\u65bc e<sup>2\u03c0i\uff08z+1\uff09<\/sup> = e<sup>2\u03c0iz<\/sup>\uff0c\u6211\u5011\u6709 f\uff08z+1\uff09=f\uff08z\uff09\u3002<br \/>\n\u4e0d\u53ea\u5982\u6b64\uff0c\u6211\u5011\u9084\u6709\uff1a<\/p>\n<p><strong><u>\u5b9a\u7406 F<\/u><\/strong>\uff1a\u51fd\u6578 f\uff08z\uff09=h\uff08e<sup>2\u03c0iz<\/sup>\uff09\u6eff\u8db3\u65b9\u7a0b\u5f0f\uff1a<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_6.jpg\" alt=\"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818\" width=\"350\" height=\"100\" class=\"aligncenter size-full wp-image-31250\" srcset=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_6.jpg 350w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_6-300x86.jpg 300w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><\/p>\n<p>\u5b9a\u7406 F \u5b8c\u6574\u7684\u63cf\u8ff0\u662f\uff1a<\/p>\n<p><strong><u>\u5b9a\u7406 G<\/u><\/strong>\uff1a\u5c0d\u65bc\u4efb\u610f\u6574\u6578 a, b, c, d \u6eff\u8db3 ad \u0096 bc = 1\uff0c\u4e14 23 \u6574\u9664 c \u548c d\u00961\uff0c\u6211\u5011\u6709\uff1a<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_7.jpg\" alt=\"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818\" width=\"350\" height=\"100\" class=\"aligncenter size-full wp-image-31251\" srcset=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_7.jpg 350w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_7-300x86.jpg 300w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><br \/>\n\uff08\u5b9a\u7406 F \u662f\u5b9a\u7406 G \u5728 a=d=1, b=0, c=23 \u6642\u7684\u7279\u4f8b\u3002\uff09<\/p>\n<p>\u6eff\u8db3\u5b9a\u7406 G \u7684\u51fd\u6578\u7a31\u70ba\u4e00\u500b\u6a21\u5f62\u5f0f\uff08modular form\uff09\u3002\u6a21\u5f62\u5f0f\u6709\u9ad8\u5ea6\u7684\u5c0d\u7a31\u6027\u3002\u628a\u95dc\u65bc\u8b8a\u6578 z \u7684\u5c0d\u7a31\u6027\u756b\u51fa\u4f86\uff0c\u5927\u6982\u50cf\u9019\u6a23\uff1a<\/p>\n<div id=\"attachment_21652\" style=\"width: 609px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-21652\" src=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_4.jpg\" alt=\"\u3010\u5c08\u6b04\u3011\u6578\u5b57\u8207\u65b9\u7a0b\u5f0f\u7684\u5c0d\u7a31\u6027\u2014\u2014Langlands \u7db1\u9818\" width=\"599\" height=\"301\" class=\"aligncenter size-full wp-image-31252\" srcset=\"https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_4.jpg 599w, https:\/newsletter.sinica.edu.tw\/wp-content\/uploads\/2023\/05\/3-1_4-300x151.jpg 300w\" sizes=\"(max-width: 599px) 100vw, 599px\" \/><p id=\"caption-attachment-21652\" class=\"wp-caption-text\">\u25b2\u50c5\u70ba\u793a\u610f\u5716\uff1b\u4e0a\u5716\u662f\u6c92\u6709\u8003\u616e 23 \u7684\u60c5\u6cc1\u3002\u4f5c\u8005\u8868\u793a\uff1a\u300c23 \u9019\u500b\u6578\u5b57\u6709\u9ede\u5927\uff0c\u6211\u4e0d\u6703\u756b\u3002\u300d<\/p><\/div>\n<p>\u73fe\u5728\u6211\u5011\u53ef\u4ee5\u8ddf\u5927\u5bb6\u5206\u4eab\u6a19\u984c\u88e1\u7684 Langlands \u7db1\u9818\u662f\u4ec0\u9ebc\u3002\u9019\u662f\u6578\u8ad6\u8fd1\u4f86\u4f86\u6700\u91cd\u8981\u7684\u9818\u57df\u4e4b\u4e00\u3002\u4f60\u53ef\u4ee5\u8aaa Langlands \u7db1\u9818\u662f\u4e00\u5957\u54f2\u5b78\uff0c\u5b83\u8868\u8ff0\u7684\u662f\uff1a<\/p>\n<p><strong>\u300c\u6bcf\u500b\u591a\u9805\u5f0f\u6574\u6578\u65b9\u7a0b\u5f0f\uff0c\u90fd\u5c0d\u61c9\u5230\u67d0\u500b\u5b9a\u7406 G \u9019\u6a23\u6709\u6975\u5927\u5c0d\u7a31\u6027\u7684\u51fd\u6578\u3002\u300d<\/strong><\/p>\n<p>\u4e0a\u9762\u9019\u53e5\u8a71\u6709\u5f88\u591a\u6a21\u7cca\u4e4b\u8655\uff0c\u6bd4\u5982\u300c\u5c0d\u61c9\u300d\u5230\u5e95\u662f\u8ab0\u5c0d\u61c9\u8ab0\uff0c\u4ee5\u53ca\u600e\u6a23\u7b97\u662f\u300c\u6709\u6975\u5927\u5c0d\u7a31\u6027\u7684\u51fd\u6578\u300d\uff1b\u9019\u4e9b\u90fd\u6709\u5177\u9ad4\u7684\u9673\u8ff0\uff0c\u4f46\u7b46\u8005\u6c92\u6709\u80fd\u529b\u5728\u5e7e\u5343\u5b57\u7684\u7bc7\u5e45\u89e3\u91cb\u3002\u5728\u6bd4\u8f03\u7c21\u55ae\u7684\u60c5\u6cc1\uff0c\u597d\u6bd4\u5b9a\u7406 B \u548c\u5b9a\u7406 D\uff0c\u6211\u5011\u7684\u300c\u51fd\u6578\u300d\u975e\u5e38\u76f4\u63a5\uff1a\u9664\u4ee5 4 \u53d6\u9918\u6578\uff0c\u6216\u662f\u9664\u4ee5 p \u53d6\u9918\u6578\uff1b\u5b9a\u7406 E \u7684\u51fd\u6578 f\uff08z\uff09=h\uff08e<sup>2\u03c0iz<\/sup>\uff09 \u5247\u76f8\u7576\u8907\u96dc\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u300c\u591a\u9805\u5f0f\u6574\u6578\u65b9\u7a0b\u5f0f\u300d\u7684\u89e3\u7684\u6982\u5ff5\u53ef\u4ee5\u505a\u76f8\u7576\u7684\u63a8\u5ee3\u3002\u9019\u9700\u8981\u7528\u5230\u62bd\u8c61\u4ee3\u6578\u88e1\u9762\u300c\u57df\u64f4\u5f35\u300d\uff08field extension\uff09\u548c\u300c\u4f3d\u7f85\u74e6\u7fa4\u300d\uff08Galois group\uff09\u7684\u6982\u5ff5\u3002<\/p>\n<p>\u5728 Langlands 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