{"id":560,"date":"2026-02-27T21:44:00","date_gmt":"2026-02-27T13:44:00","guid":{"rendered":"https:\/\/wo.city\/?p=560"},"modified":"2026-02-27T21:44:00","modified_gmt":"2026-02-27T13:44:00","slug":"rational-points-algebraic-curves-uniform-bound","status":"publish","type":"post","link":"https:\/\/wo.city\/?p=560","title":{"rendered":"\u66f2\u7ebf\u4e0a\u7684\u6709\u9650\u661f\u8fb0"},"content":{"rendered":"

\u4e00\u6761\u7ebf\u4e0a\u80fd\u85cf\u591a\u5c11\u79d8\u5bc6<\/h2>\n

x\u00b2 + y\u00b2 = 1\uff0c\u4e00\u4e2a\u5706\u3002\u5706\u4e0a\u6709\u65e0\u7a77\u591a\u4e2a\u70b9\uff0c\u4f46\u5982\u679c\u53ea\u5141\u8bb8\u5750\u6807\u662f\u6574\u6570\u6216\u5206\u6570\u2014\u2014\u6240\u8c13”\u6709\u7406\u70b9”\u2014\u2014\u60c5\u51b5\u5c31\u53d8\u5f97\u5fae\u5999\u4e86\u3002(1, 0) \u662f\u4e00\u4e2a\uff0c(3\/5, 4\/5) \u4e5f\u662f\u4e00\u4e2a\uff0c\u800c (1\/\u221a2, 1\/\u221a2) \u867d\u7136\u7a33\u7a33\u843d\u5728\u5706\u4e0a\uff0c\u5374\u6c38\u8fdc\u65e0\u6cd5\u88ab\u5199\u6210\u4e24\u4e2a\u5206\u6570\u7684\u7ec4\u5408\u3002\u53e4\u5e0c\u814a\u4eba\u5c31\u5df2\u7ecf\u7740\u8ff7\u4e8e\u8fd9\u4ef6\u4e8b\uff1a\u4e00\u6761\u7531\u65b9\u7a0b\u5b9a\u4e49\u7684\u66f2\u7ebf\uff0c\u7a76\u7adf\u80fd\u5bb9\u7eb3\u591a\u5c11\u4e2a\u6709\u7406\u70b9\uff1f<\/p>\n

\u8fd9\u4e2a\u95ee\u9898\u770b\u8d77\u6765\u50cf\u521d\u4e2d\u6570\u5b66\u7684\u5ef6\u4f38\uff0c\u5b9e\u9645\u4e0a\u5b83\u5728\u6570\u8bba\u7684\u6838\u5fc3\u4f4d\u7f6e\u76d8\u8e1e\u4e86\u4e24\u5343\u5e74\u30022026\u5e742\u67082\u65e5\uff0c\u4e09\u4f4d\u4e2d\u56fd\u6570\u5b66\u5bb6\u5728\u9884\u5370\u672c\u4e2d\u7ed9\u51fa\u4e86\u4e00\u4e2a\u6b64\u524d\u4ece\u672a\u5b58\u5728\u8fc7\u7684\u4e1c\u897f\u2014\u2014\u4e00\u4e2a\u9002\u7528\u4e8e\u6240\u6709\u66f2\u7ebf\u7684\u6709\u7406\u70b9\u6570\u91cf\u4e0a\u754c\u516c\u5f0f\u3002\u54c8\u4f5b\u5927\u5b66\u6559\u6388 Barry Mazur \u7684\u8bc4\u4ef7\u5f88\u514b\u5236\uff0c\u4f46\u5206\u91cf\u4e0d\u8f7b\uff1a”\u8fd9\u4e00\u4e2a\u547d\u9898\u5c31\u7ed9\u4e86\u6211\u4eec\u4e00\u6574\u7247\u7406\u89e3\u7684\u89c6\u91ce\u3002”<\/p>\n

\u4ece\u65e0\u7a77\u5230\u6709\u9650\u7684\u60ac\u5d16<\/h2>\n

\u8981\u7406\u89e3\u8fd9\u4e2a\u7ed3\u679c\u4e3a\u4ec0\u4e48\u91cd\u8981\uff0c\u9700\u8981\u5148\u770b\u6e05\u4e00\u9053\u5206\u754c\u7ebf\u3002<\/p>\n

\u66f2\u7ebf\u7531\u591a\u9879\u5f0f\u65b9\u7a0b\u5b9a\u4e49\uff0c\u591a\u9879\u5f0f\u7684”\u6b21\u6570”\uff08degree\uff09\u2014\u2014\u4e5f\u5c31\u662f\u53d8\u91cf\u88ab\u4e58\u5230\u7684\u6700\u9ad8\u5e42\u6b21\u2014\u2014\u51b3\u5b9a\u4e86\u66f2\u7ebf\u7684\u590d\u6742\u7a0b\u5ea6\u3002\u6b21\u6570\u4e3a2\u7684\u66f2\u7ebf\uff0c\u6bd4\u5982\u5706\u3001\u692d\u5706\u3001\u53cc\u66f2\u7ebf\uff0c\u884c\u4e3a\u5f88\u6e29\u987a\uff1a\u5b83\u4eec\u8981\u4e48\u4e00\u4e2a\u6709\u7406\u70b9\u90fd\u6ca1\u6709\uff0c\u8981\u4e48\u6709\u65e0\u7a77\u591a\u4e2a\u3002\u6b21\u6570\u4e3a3\u7684\u66f2\u7ebf\u2014\u2014\u692d\u5706\u66f2\u7ebf\u662f\u5176\u4e2d\u6700\u8457\u540d\u7684\u4ee3\u8868\u2014\u2014\u60c5\u51b5\u5f00\u59cb\u5206\u88c2\uff0c\u6709\u4e9b\u6709\u65e0\u7a77\u591a\u4e2a\u6709\u7406\u70b9\uff0c\u6709\u4e9b\u53ea\u6709\u6709\u9650\u4e2a\u3002\u800c\u692d\u5706\u66f2\u7ebf\u4e0a\u7684\u6709\u7406\u70b9\u7ed3\u6784\uff0c\u540e\u6765\u50ac\u751f\u4e86\u4e00\u6574\u4e2a\u5bc6\u7801\u5b66\u5206\u652f\uff0c\u5982\u4eca\u4fdd\u62a4\u7740\u4f60\u6bcf\u4e00\u6b21\u7f51\u4e0a\u652f\u4ed8\u3002<\/p>\n

1922\u5e74\uff0c\u82f1\u56fd\u6570\u5b66\u5bb6 Louis Mordell \u63d0\u51fa\u4e86\u4e00\u4e2a\u731c\u60f3\uff1a\u4e00\u65e6\u6b21\u6570\u8fbe\u52304\u6216\u66f4\u9ad8\uff0c\u6709\u7406\u70b9\u7684\u6570\u91cf\u5c31\u4e00\u5b9a\u662f\u6709\u9650\u7684\u3002\u4e0d\u7ba1\u65b9\u7a0b\u957f\u4ec0\u4e48\u6837\uff0c\u4e0d\u7ba1\u7cfb\u6570\u591a\u5927\u591a\u602a\uff0c\u6709\u7406\u70b9\u6c38\u8fdc\u53ea\u6709\u6709\u9650\u4e2a\u3002<\/p>\n

\u8fd9\u4e2a\u731c\u60f3\u60ac\u5728\u90a3\u91cc\u7b49\u4e86\u516d\u5341\u4e00\u5e74\u30021983\u5e74\uff0c\u5fb7\u56fd\u6570\u5b66\u5bb6 Gerd Faltings \u8bc1\u660e\u4e86\u5b83\uff0c\u5e76\u56e0\u6b64\u83b7\u5f97\u4e86\u83f2\u5c14\u5179\u5956\u2014\u2014\u6570\u5b66\u754c\u7684\u6700\u9ad8\u8363\u8a89\u3002\u4f46 Faltings \u7684\u5b9a\u7406\u53ea\u8bf4\u4e86”\u6709\u9650”\uff0c\u6ca1\u8bf4”\u6709\u591a\u5c11”\u3002\u5c31\u50cf\u6709\u4eba\u544a\u8bc9\u4f60\u8fd9\u4e2a\u623f\u95f4\u91cc\u7684\u4eba\u6570\u662f\u6709\u9650\u7684\uff0c\u5374\u4e0d\u80af\u544a\u8bc9\u4f60\u5230\u5e95\u662f\u4e09\u4e2a\u8fd8\u662f\u4e09\u5343\u4e2a\u3002<\/p>\n

\u4ece\u90a3\u4ee5\u540e\uff0c\u6570\u5b66\u5bb6\u4eec\u4e00\u76f4\u5728\u5bfb\u627e\u4e00\u4e2a\u516c\u5f0f\uff0c\u80fd\u628a\u8fd9\u4e2a”\u6709\u9650”\u53d8\u6210\u4e00\u4e2a\u53ef\u4ee5\u8ba1\u7b97\u7684\u6570\u5b57\uff0c\u6216\u8005\u81f3\u5c11\u7ed9\u51fa\u4e00\u4e2a\u4e0a\u754c\u3002\u95ee\u9898\u5728\u4e8e\uff0c\u66f2\u7ebf\u7684\u4e16\u754c\u6d69\u5982\u70df\u6d77\uff0c\u7531\u65e0\u6570\u4e2a\u65e0\u7a77\u65cf\u7ec4\u6210\uff0c\u6bcf\u4e00\u65cf\u6709\u81ea\u5df1\u7684\u7ed3\u6784\u548c\u813e\u6c14\u3002\u4e4b\u524d\u7684\u6240\u6709\u5c1d\u8bd5\uff0c\u8981\u4e48\u53ea\u5bf9\u67d0\u4e9b\u7279\u5b9a\u7c7b\u578b\u7684\u66f2\u7ebf\u6709\u6548\uff0c\u8981\u4e48\u4f9d\u8d56\u4e8e\u5b9a\u4e49\u66f2\u7ebf\u7684\u65b9\u7a0b\u4e2d\u7684\u5177\u4f53\u7cfb\u6570\u2014\u2014\u6362\u4e00\u6761\u66f2\u7ebf\uff0c\u516c\u5f0f\u5c31\u4f5c\u5e9f\u4e86\u3002<\/p>\n

\u7b2c\u4e00\u4e2a”\u4e00\u89c6\u540c\u4ec1”\u7684\u516c\u5f0f<\/h2>\n

\u65b0\u8bc1\u660e\u7684\u6838\u5fc3\u7a81\u7834\u5728\u4e8e”uniform”\u8fd9\u4e2a\u8bcd\u3002\u8fd9\u4e2a\u516c\u5f0f\u4e0d\u6311\u66f2\u7ebf\u3002\u5b83\u4e0d\u5173\u5fc3\u4f60\u7684\u65b9\u7a0b\u7cfb\u6570\u662f\u4ec0\u4e48\uff0c\u4e0d\u8981\u6c42\u4f60\u5148\u628a\u65b9\u7a0b\u5316\u7b80\u6210\u67d0\u79cd\u6807\u51c6\u5f62\u5f0f\u3002\u5b83\u5bf9\u6570\u5b66\u5b87\u5b99\u4e2d\u7684\u6bcf\u4e00\u6761\u66f2\u7ebf\u90fd\u6210\u7acb\u3002<\/p>\n

\u8fd9\u4e2a\u516c\u5f0f\u53ea\u4f9d\u8d56\u4e24\u6837\u4e1c\u897f\u3002\u7b2c\u4e00\u6837\u662f\u591a\u9879\u5f0f\u7684\u6b21\u6570\u2014\u2014\u6b21\u6570\u8d8a\u9ad8\uff0c\u4e0a\u754c\u8d8a\u5bbd\u677e\uff0c\u8fd9\u7b26\u5408\u76f4\u89c9\uff0c\u56e0\u4e3a\u9ad8\u6b21\u66f2\u7ebf\u7684\u51e0\u4f55\u7ed3\u6784\u66f4\u590d\u6742\u3002\u7b2c\u4e8c\u6837\u662f\u4e00\u4e2a\u53eb\u505a”Jacobian variety”\u7684\u5bf9\u8c61\uff0c\u8fd9\u662f\u4e00\u4e2a\u53ef\u4ee5\u4ece\u4efb\u610f\u66f2\u7ebf\u6784\u9020\u51fa\u6765\u7684\u7279\u6b8a\u66f2\u9762\u3002Jacobian variety \u672c\u8eab\u5c31\u662f\u4ee3\u6570\u51e0\u4f55\u4e2d\u7684\u91cd\u8981\u7814\u7a76\u5bf9\u8c61\uff0c\u800c\u65b0\u516c\u5f0f\u4e3a\u7814\u7a76\u5b83\u4eec\u4e5f\u6253\u5f00\u4e86\u4e00\u6761\u901a\u9053\u3002<\/p>\n

\u667a\u5229\u5929\u4e3b\u6559\u5927\u5b66\u7684\u6570\u5b66\u5bb6 Hector Pasten \u6ca1\u6709\u53c2\u4e0e\u8fd9\u9879\u5de5\u4f5c\uff0c\u4f46\u4ed6\u7684\u8bc4\u4ef7\u76f4\u63a5\uff1a”\u8fd9\u771f\u7684\u662f\u4e00\u4e2a\u60ca\u4eba\u7684\u7ed3\u679c\uff0c\u5b83\u4e3a\u6211\u4eec\u5e94\u8be5\u671f\u5f85\u4ec0\u4e48\u8bbe\u7acb\u4e86\u65b0\u7684\u6807\u51c6\u3002”\u8fd9\u53e5\u8bdd\u7684\u6f5c\u53f0\u8bcd\u662f\uff1a\u4ee5\u524d\u5927\u5bb6\u751a\u81f3\u4e0d\u786e\u5b9a\u8fd9\u79cd\u7edf\u4e00\u516c\u5f0f\u662f\u5426\u53ef\u80fd\u5b58\u5728\u3002<\/p>\n

\u9700\u8981\u8bf4\u6e05\u695a\u7684\u662f\uff0c\u8fd9\u4e2a\u516c\u5f0f\u7ed9\u51fa\u7684\u662f\u4e0a\u754c\uff0c\u4e0d\u662f\u7cbe\u786e\u6570\u5b57\u3002\u5b83\u544a\u8bc9\u4f60”\u8fd9\u6761\u66f2\u7ebf\u4e0a\u7684\u6709\u7406\u70b9\u4e0d\u4f1a\u8d85\u8fc7\u8fd9\u4e48\u591a”\uff0c\u4f46\u5b9e\u9645\u6570\u91cf\u53ef\u80fd\u8fdc\u4f4e\u4e8e\u8fd9\u4e2a\u4e0a\u9650\u3002\u8fd9\u5c31\u50cf\u4e00\u4e2a\u7c97\u7cd9\u4f46\u575a\u56fa\u7684\u56f4\u680f\u2014\u2014\u5b83\u7b2c\u4e00\u6b21\u628a\u6240\u6709\u66f2\u7ebf\u90fd\u5708\u5728\u4e86\u91cc\u9762\uff0c\u4f46\u56f4\u680f\u5185\u7684\u7cbe\u786e\u5730\u5f62\u8fd8\u6709\u5f85\u52d8\u6d4b\u3002<\/p>\n

\u66f2\u7ebf\u4e4b\u540e\uff0c\u662f\u66f4\u9ad8\u7ef4\u7684\u4e16\u754c<\/h2>\n

\u66f2\u7ebf\u53ea\u662f\u5f00\u59cb\u3002<\/p>\n

\u591a\u9879\u5f0f\u65b9\u7a0b\u5982\u679c\u5305\u542b\u66f4\u591a\u53d8\u91cf\u2014\u2014\u4e0d\u53ea\u662f x \u548c y\uff0c\u8fd8\u6709 z\u3001w\u2026\u2026\u2014\u2014\u5c31\u4f1a\u751f\u6210\u66f4\u590d\u6742\u7684\u51e0\u4f55\u5bf9\u8c61\uff1a\u66f2\u9762\uff0c\u4ee5\u53ca\u5b83\u4eec\u7684\u9ad8\u7ef4\u63a8\u5e7f\uff0c\u6570\u5b66\u5bb6\u79f0\u4e4b\u4e3a”\u6d41\u5f62”\uff08manifold\uff09\u3002\u6d41\u5f62\u662f\u73b0\u4ee3\u6570\u5b66\u7684\u4e2d\u5fc3\u821e\u53f0\uff0c\u4e5f\u662f\u7406\u8bba\u7269\u7406\u7528\u6765\u63cf\u8ff0\u65f6\u7a7a\u7ed3\u6784\u7684\u57fa\u672c\u8bed\u8a00\u3002\u6240\u6709\u5173\u4e8e\u6709\u7406\u70b9\u7684\u95ee\u9898\uff0c\u5728\u90a3\u4e9b\u66f4\u9ad8\u7ef4\u7684\u5bf9\u8c61\u4e0a\u540c\u6837\u5b58\u5728\uff0c\u800c\u4e14\u66f4\u96be\u3002<\/p>\n

Pasten \u548c\u6570\u5b66\u5bb6 Jerson Caro \u57282023\u5e74\u7684\u4e00\u7bc7\u8bba\u6587\u4e2d\uff0c\u5df2\u7ecf\u4e3a\u67d0\u4e9b\u7279\u5b9a\u66f2\u9762\u4e0a\u7684\u6709\u7406\u70b9\u6570\u91cf\u7ed9\u51fa\u4e86\u4e0a\u754c\u3002\u65b0\u7ed3\u679c\u8ba9 Pasten \u770b\u5230\u4e86\u5728\u66f4\u5e7f\u9614\u9886\u57df\u53d6\u5f97\u8fdb\u5c55\u7684\u5e0c\u671b\u3002\u4ece\u66f2\u7ebf\u5230\u66f2\u9762\u518d\u5230\u9ad8\u7ef4\u6d41\u5f62\uff0c\u8fd9\u662f\u4e00\u6761\u6f2b\u957f\u7684\u8def\uff0c\u800c\u8fd9\u4e2a\u516c\u5f0f\u662f\u7b2c\u4e00\u4e2a\u771f\u6b63\u7ad9\u7a33\u7684\u7acb\u8db3\u70b9\u3002<\/p>\n

\u6709\u7406\u70b9\u7684\u95ee\u9898\u4e4b\u6240\u4ee5\u8ba9\u6570\u5b66\u5bb6\u5982\u6b64\u7740\u8ff7\uff0c\u4e0d\u4ec5\u56e0\u4e3a\u5b83\u53e4\u8001\uff0c\u66f4\u56e0\u4e3a\u5b83\u5904\u5728\u6570\u8bba\u3001\u4ee3\u6570\u51e0\u4f55\u548c\u7b97\u672f\u51e0\u4f55\u7684\u4ea4\u53c9\u53e3\u3002\u692d\u5706\u66f2\u7ebf\u4e0a\u7684\u6709\u7406\u70b9\u7ed3\u6784\u50ac\u751f\u4e86\u5bc6\u7801\u5b66\uff0cFaltings \u7684\u5b9a\u7406\u6df1\u523b\u5f71\u54cd\u4e86 Diophantine \u51e0\u4f55\u7684\u53d1\u5c55\u65b9\u5411\u3002\u6bcf\u4e00\u6b21\u5bf9\u6709\u7406\u70b9\u7684\u65b0\u7406\u89e3\uff0c\u90fd\u53ef\u80fd\u5728\u610f\u60f3\u4e0d\u5230\u7684\u5730\u65b9\u4ea7\u751f\u56de\u54cd\u3002<\/p>\n

\u4e00\u4e2a\u6b63\u5728\u52a0\u901f\u7684\u6545\u4e8b<\/h2>\n

\u8fd9\u7bc7\u9884\u5370\u672c\u5e76\u4e0d\u662f\u5b64\u7acb\u4e8b\u4ef6\u3002\u6700\u8fd1\u51e0\u5e74\uff0c\u5173\u4e8e\u66f2\u7ebf\u4e0a\u6709\u7406\u70b9\u7684\u65b0\u7ed3\u679c\u5bc6\u96c6\u6d8c\u73b0\uff0c\u5f62\u6210\u4e86\u4e00\u80a1\u53ef\u8fa8\u8bc6\u7684\u6d6a\u6f6e\u3002Mazur \u7528\u4e86\u4e00\u4e2a\u8c28\u614e\u4f46\u5174\u594b\u7684\u8bf4\u6cd5\uff1a”\u8fd9\u662f\u4e00\u4e2a\u6fc0\u52a8\u4eba\u5fc3\u7684\u3001\u5feb\u901f\u63a8\u8fdb\u7684\u9886\u57df\u3002\u73b0\u5728\u6b63\u5728\u53d1\u751f\u67d0\u79cd\u5927\u4e8b\u3002”<\/p>\n

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