ラマヌジャン
やっぱりやばいじゃん
辻順平 @tsujimotter (今⽇の)
やばい⼈
タクシー数
1729 = 13
+ 123
= 93
+ 103
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1729は、2つの3乗数の和として2通りに表すことができる最⼩の数
(by ラマヌジャン)
2
ラマヌジャンやばい︕
3
そんなの知ってるし
いやいや・・・
4
・・・なあなたに
5
Wikipedia “ラマヌジャン”
情報提供︓みずすましさん(感謝︕)
6
タクシー数とK3曲⾯
『・・・ラマヌジャンは答を導出する過程で1729と楕円曲線か
ら今⽇で⾔う K3曲⾯ を構成していたことを発⾒した。これ
はアンドレ・ヴェイユによるK3曲⾯の再発⾒と命名に30年以上
先⾏する仕事である。』
7
なんかかっこいい︕
8
9
読んでみました
10
ラマヌジャンの発⾒(1913)
オイラーのディオファントス⽅程式
の無限個の解を与えるパラメータ解︓
X3
+ Y 3
= Z3
+ W3
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(6A2
4AB + 4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A
<latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">AAAGX3icpVRLTxNRFD6gFKwKRTcaNxNrTRUhty98JCYFN+zkDQkU0hkuMGFemZkW6qR/wD/gwpUmLow/g41/wAVLl8SdmLhx4XfPVGjB1oUzmXvP/c7jO+fce0f3LDMIhTjq6790eSAxOHQlefXa9eGR1OiN5cCt+YZcMlzL9Vf1aiAt05FLoRlactXzZdXWLbmi771Q+pW69APTdRbDhicrdnXHMbdNoxoC2kxVspNTG3ltXCtOTWtjWnF6I/9go6A917IFhY9pJeDjWqmFj2nZYrv95BleivHYvhDjm6m0mBD8aBeFXEtIlwXxM+uO9tdonbbIJYNqZJMkh0LIFlUpwLtGORLkAatQBMyHZLJeUpOS8K3BSsKiCnQP4w5Way3UwVrFDNjbAIuFz4enRhnxRXwUJ+Kz+CSOxS/EynSJFuGrMaPfgzNiLpVzA7Mec0hvc+T1rYWf//SyMYe0e+bVs7aQtukJ12SiRo8RVa0R+9dfvTlZeDafie6L9+Ib6nwnjsQhKnXqP4wPc3L+bY98dOSiquzO/5IWKU+PeM5xFtvoqtOWQ5L3VAJfpwVUVYVNBNkEYuN1WZtErhpQ5XeAUVl0RlJ2qsId6GLWJmxUTmo3DO5DbK+BZ4aZ45jKIjjlaH//lyN5GsWEZdyp7icn5NgN2meuRs/ORhh9yB46ps74Qc/TFiC2zTeigW+rZetAt89V2ax1oFF9rXMeTZZ9aEM+bxK77YInoqe4ZUp7j3fJwdxsRVRR9mCp7FVHQvbzO/bY5qxVRi73xcJbRrTMaf/XOVPFvN9h0elr8FnxOUfVf51XzXNrtTLRhV2u4k/U7vU/BH7QwbOGXa5grJzLMT6H5baoGvpSQHdKPapM4xY0ec7/tWsu32z0DH/H3Pl/4UVhOT+RgzxXTJen498kDdEduktZ8DwG4wzN0hJqPKRj+k4nA18Tg4nhRCo27e9r+dykjidx+ze2P0rI</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit>
B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5AB 3B2
)3
11
両辺を 33 で割ると
A = 2, B = 1
<latexit sha1_base64="QkOJfQGjSqtT+30G/7wjdICdKjw=">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</latexit><latexit sha1_base64="QkOJfQGjSqtT+30G/7wjdICdKjw=">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</latexit><latexit sha1_base64="QkOJfQGjSqtT+30G/7wjdICdKjw=">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</latexit><latexit sha1_base64="LRo80GOIIqTUh0O4Dq87A0DZK4E=">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</latexit>
363
= ( 3)3
+ 303
+ 273
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123
= ( 1)3
+ 103
+ 93
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(6A2
4AB + 4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5A
<latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">AAAGX3icpVRLTxNRFD6gFKwKRTcaNxNrTRUhty98JCYFN+zkDQkU0hkuMGFemZkW6qR/wD/gwpUmLow/g41/wAVLl8SdmLhx4XfPVGjB1oUzmXvP/c7jO+fce0f3LDMIhTjq6790eSAxOHQlefXa9eGR1OiN5cCt+YZcMlzL9Vf1aiAt05FLoRlactXzZdXWLbmi771Q+pW69APTdRbDhicrdnXHMbdNoxoC2kxVspNTG3ltXCtOTWtjWnF6I/9go6A917IFhY9pJeDjWqmFj2nZYrv95BleivHYvhDjm6m0mBD8aBeFXEtIlwXxM+uO9tdonbbIJYNqZJMkh0LIFlUpwLtGORLkAatQBMyHZLJeUpOS8K3BSsKiCnQP4w5Way3UwVrFDNjbAIuFz4enRhnxRXwUJ+Kz+CSOxS/EynSJFuGrMaPfgzNiLpVzA7Mec0hvc+T1rYWf//SyMYe0e+bVs7aQtukJ12SiRo8RVa0R+9dfvTlZeDafie6L9+Ib6nwnjsQhKnXqP4wPc3L+bY98dOSiquzO/5IWKU+PeM5xFtvoqtOWQ5L3VAJfpwVUVYVNBNkEYuN1WZtErhpQ5XeAUVl0RlJ2qsId6GLWJmxUTmo3DO5DbK+BZ4aZ45jKIjjlaH//lyN5GsWEZdyp7icn5NgN2meuRs/ORhh9yB46ps74Qc/TFiC2zTeigW+rZetAt89V2ax1oFF9rXMeTZZ9aEM+bxK77YInoqe4ZUp7j3fJwdxsRVRR9mCp7FVHQvbzO/bY5qxVRi73xcJbRrTMaf/XOVPFvN9h0elr8FnxOUfVf51XzXNrtTLRhV2u4k/U7vU/BH7QwbOGXa5grJzLMT6H5baoGvpSQHdKPapM4xY0ec7/tWsu32z0DH/H3Pl/4UVhOT+RgzxXTJen498kDdEduktZ8DwG4wzN0hJqPKRj+k4nA18Tg4nhRCo27e9r+dykjidx+ze2P0rI</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit>
+ 4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5AB 3B2
)3
(タクシー数)
12
(6A2
4AB + 4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5A
<latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit>
+ 4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5AB 3B2
)3
A, B
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に適当な整数を⼊れる
無数の「タクシー数的な数」が得られる︕
13
やばい︕︕
14
が、まだまだこれから
15
Ono and Trebat-Leder の論⽂
1. タクシー数と楕円曲線
2. タクシー数とK3曲⾯
16
両辺 B6 で割って T = A/B とおく
(6T2
4T + 4)3
+ ( 3T2
5T + 5)3
= (4T2
4T + 6)3
+ (5T2
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T + 5)3
= (4T2
4T + 6)3
+ (5T2
5T 3)3
= k(T)
<latexit sha1_base64="NJzo+mawJQWz69GUD3DFwCLZGps=">AAAGBnicpVRLaxNRFD6NRmt8tNWN4CYYI1Wk3ETFBwhBN93ZNo8WmlBmpjfpmHkxM3k5ZC+u3Im6UnAhgjtxK7jxD7joTxCXFdy48LtnYk1SExfOMPeee17fOd+9d3TPMoNQiN2ZxKHDySNHZ4+ljp84eWpufuF0JXBbviHLhmu5/oauBdIyHVkOzdCSG54vNVu35LrevKfs623pB6brlMKeJ2u21nDMumloIVSVO+nmYunS1nxGLAl+0geF3EDIFATxs+IuJFpUpW1yyaAW2STJoRCyRRoFeDcpR4I86GoUQedDMtkuqU8pxLbgJeGhQdvE2MBqc6B1sFY5A442gGLh8xGZpqz4It6IPfFZvBVfxU/kyk7IFuFrMaI/BTNiLFVzD7MeY0hva+7x2eKPf0bZmEPa+RM1tbeQ6nSTezLRo8ca1a0Rx7cfPt0r3l7LRhfFK/ENfb4Uu+ITOnXa343Xq3LtxZR6dNSiupyMf59KlKcrPOe4ijpYdYZqSPGeSuirVERXGnwiyCY0Nl6XrSnUmoZWxXUxKo/RTMpPddiALUbtw0fVpHbDYB5i/zRwlhk5zqk8gn2M4fd/MVL7WUx4xkxNPjkh5+5Rh7F6U5mNMPqQPTCmznh36mkLkNvmG9HDtz3wdWDrcFc2Wx1YFK9trqPPsg9ryOdNYrdd4ER0C7dMWS/wLjmY+4OMKksTnspfMRJynD+yxzZXrSpymRcLbwHZsvv8V7lShdwZ8RiNNfis+Fyj4l/nVX9srVYmWNjhLn5nndz/Zei7Izib2OUaxtpYjfE5LAxlTYOXq2Dn+pQuM7gFfZ7zf2XN5ZsNzvB3zI3/Cw8KlfxSDvLqtUzhbvybpFk6R+dpETg3gLhMK1RGjw/oCT2j58lHyXfJ98kPsWtiZhBzhkae5MdfT9I5Tw==</latexit><latexit sha1_base64="NJzo+mawJQWz69GUD3DFwCLZGps=">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</latexit><latexit sha1_base64="NJzo+mawJQWz69GUD3DFwCLZGps=">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</latexit><latexit sha1_base64="DfyJg8K/ClzDQ0g7Vpg+1HgBtCs=">AAAGBnicpVQ7b9NQFD4NGEp4tIUFiSUiBBWEqpsA4iEhRbB0o22StlIbVbZ725r4Jdt5YXlHTGwImEBiQEhsiBWJhT/A0J+AGIvEwsB3j01pWhIGbPnec8/rO+e799rwbSuMhNgeyx06rB05On4sf/zEyVMTk1OnF0OvHZiyYXq2Fywbeihty5WNyIpsuewHUncMWy4ZrXvKvtSRQWh5bj3q+7Lp6JuutWGZegTV4p1Ca7p+aW2yKGYEP4WDQjkTipQ9c95Urk2rtE4emdQmhyS5FEG2SacQ7wqVSZAPXZNi6AJIFtslJZRHbBteEh46tC2Mm1itZFoXa5Uz5GgTKDa+AJEFKokv4o3YEZ/FW/FV/ESu0pBsMb42IwYjMGPGUjX3MRsphvTXJh6frf34Z5SDOaKtP1Eje4tog25yTxZ69FmjujXT+M7Dpzu12wul+KJ4Jb6hz5diW3xCp27nu/l6Xi68GFGPgVpUl8Px71OdKnSF5zJXsQFW3T015HlPJfSrVENXOnxiyBY0Dl6PrXnUWoBWxfUwKo/BTMpPdbgJW4qawEfVpHbDZB5S/wJwZhk5zak8wl2Mve//YuR3s1jwTJkafnIizt2nLmP1RzIbYwwg+2BMnfHeyNMWIrfDN6KPbz3zdWHrclcOW11YFK8driNhOYA14vMmsdsecGK6hVumrBd4l1zMSZZRZWnBU/krRiKOCwb22OGqVUUe82LjrSJbaZf/Va5UIXcHPAZjTT4rAdeo+Dd4lexbq5UFFra4i99Zh/d/GfreAM4KdrmJsbmvxvQcVvdkLYCXq2Dn+ogui7gFCc+Vv7Lm8c0GZ/g7lvf/Cw8Ki5WZMuT5a8Xq3ew/OU7n6DxNA+cGEGdpjhro8QE9oWf0XHukvdPeax9S19xYFnOGBh7t4y/spTjf</latexit>
(6A2
4AB + 4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5AB
<latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit>
4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5AB 3B2
)3
とおく
17
上の⽅程式
の2つの解
が得られた。
X3
+ Y 3
= k(T)
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(X, Y ) = (6T2
4T + 4, 3T2
5T + 5)
= (4T2
4T + 6, 5T2
5T + 3)
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Q(T)
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楕円曲線
18
上の楕円曲線
X3
+ Y 3
= k(T)
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Q(T)
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(有理数)T = ti
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を代⼊
X3
+ Y 3
= k(t1)
<latexit sha1_base64="+wf95UpuOMFK1BgT7T3Fgfu3tG4=">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</latexit><latexit sha1_base64="+wf95UpuOMFK1BgT7T3Fgfu3tG4=">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</latexit><latexit sha1_base64="+wf95UpuOMFK1BgT7T3Fgfu3tG4=">AAAGFXicpVTLbtNAFL0NBEp4tKUbJDYWIag8VE1SEA+BFIlNd/SVtqgNke1OUyt+yXaSBis/wA90wQKBhETFN7ACIXasWPQTEMsisWHBmetQ6pSEBbY8c+e+zr1nZmz4thVGQuyNZI4dz544OXoqd/rM2XNj4xPnl0OvGZiyYnq2F6waeihty5WVyIpsueoHUncMW64YjYfKvtKSQWh57lLU8WXV0euutWmZegRVbXxy9cmMdl17jPGB1piKasWrWm08L6YFP9pRodgT8mVB/Mx5E5kmrdMGeWRSkxyS5FIE2SadQrxrVCRBPnRViqELIFlsl9SlHGKb8JLw0KFtYKxjtdbTulirnCFHm0Cx8QWI1KggvohdsS8+ibfiq/iJXIUB2WJ8TUYMhmDGjKVq7mA2Egzp18aeXVj88c8oB3NEW3+ihvYW0Sbd4Z4s9OizRnVrJvGtpzv7i/cWCvEV8Up8Q58vxZ54j07d1nfz9bxceD6kHgO1qC4H4z+iJSrRDZ6LXMUmWHUP1ZDjPZXQr9MiutLhE0O2oHHwemzNoVYNWhW3jVF5pDMpP9VhHbYEtQsfVZPaDZN5SPw14MwycpJTeYQHGIff/8XIHWSx4JkwNfjkRJy7Q23G6gxlNsYYQPbBmDrj20NPW4jcDt+IDr6Nnq8LW5u7ctjqwqJ4bXEdXZYDWCM+bxK77QEnpru4Zcp6mXfJxdztZVRZGvBU/oqRiOOC1B47XLWqyGNebLxlZCsc8L/OlSrkdsojHWvyWQm4RsW/watu31qtLLCwxV38zjq4/2vQb6dw1rDLVYzVvhqTc1g+lFUDLzNg59aQLvO4BV2eS39lzeObDc7wdyz2/wuPCsul6SLk+Zv58v3kN0mjdJEu0RRwbgNxluaowqfpBb2h3exO9l32Q/Zj4poZ6cVMUurJfv4FXGQ9Kg==</latexit><latexit sha1_base64="ykjuerMu3VTM8VSBthGbhgkumco=">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</latexit>
X3
+ Y 3
= k(t2)
<latexit sha1_base64="r+eV9/Jgcm5PnvobFAIxIqbNht8=">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</latexit><latexit sha1_base64="r+eV9/Jgcm5PnvobFAIxIqbNht8=">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</latexit><latexit sha1_base64="r+eV9/Jgcm5PnvobFAIxIqbNht8=">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</latexit><latexit sha1_base64="i/IuZqrDSM+pvSfjNjNSBeKIjZU=">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</latexit>
X3
+ Y 3
= k(t3)
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· · ·<latexit sha1_base64="V/+jetJDctN/3gvM6L9+1nXb2ls=">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</latexit><latexit sha1_base64="V/+jetJDctN/3gvM6L9+1nXb2ls=">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</latexit><latexit sha1_base64="V/+jetJDctN/3gvM6L9+1nXb2ls=">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</latexit><latexit sha1_base64="UipsC/MA8iPJXmHgIQTrMrXjkIA=">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</latexit>
無数の( 上の)楕円曲線を⽣成Q
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19
楕円曲線の(⾃由部分の)⽣成元の個数 = ランク
P1
<latexit sha1_base64="dtbqckMtlTA9mto7c1KcIjiTF9U=">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</latexit><latexit sha1_base64="dtbqckMtlTA9mto7c1KcIjiTF9U=">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</latexit><latexit sha1_base64="dtbqckMtlTA9mto7c1KcIjiTF9U=">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</latexit><latexit sha1_base64="5YV5vnRz1jqmkjuP1FDOamnSIcA=">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</latexit>
P2
<latexit sha1_base64="rlDnM70TEUBzQnuZCZfum5zMiVE=">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</latexit><latexit sha1_base64="rlDnM70TEUBzQnuZCZfum5zMiVE=">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</latexit><latexit sha1_base64="rlDnM70TEUBzQnuZCZfum5zMiVE=">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</latexit><latexit sha1_base64="QhGBpMlJmukk7ttQsj2lFPFxedw=">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</latexit>
この図の場合は ランク2
20
ランク2以上はレア︕
ランク0の
( 上の)楕円曲線
50%
ランク1の
( 上の)楕円曲線
50%
ランク2の
( 上の)楕円曲線
0%
21
Q
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Q
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アンケート Q. あなたのランクはいくつですか︖
実は、ラマヌジャンの発⾒した2点は独⽴だった︕
(X, Y ) = (6T2
4T + 4, 3T2
5T + 5)
= (4T2
4T + 6, 5T2
5T + 3)
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定理(Ono and Trebat-Leder)
上の楕円曲線 は ランク2 を持つEk(T ) : X3
+ Y 3
= k(T)
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Q(T)
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22
P1
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P2
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P1
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P2
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ランク2の 上の楕円曲線を無数に⽣成できる︕︕
定理(Ono and Trebat-Leder)
上の楕円曲線 は ランク2 を持つEk(T ) : X3
+ Y 3
= k(T)
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Q(T)
<latexit sha1_base64="2h2i4PHr0WjW2bc1PQqZbKdu9g0=">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</latexit><latexit sha1_base64="2h2i4PHr0WjW2bc1PQqZbKdu9g0=">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</latexit><latexit sha1_base64="2h2i4PHr0WjW2bc1PQqZbKdu9g0=">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</latexit><latexit sha1_base64="Oo4R1SdTT+bhy61i3j39SgmSiFA=">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</latexit>
系
有限個を除くすべての に対して は ランク2以上t 2 Q
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スーパーレア
Q
<latexit sha1_base64="V8YmV6e/Qv5MOJ5CGYyJh5xFA2E=">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</latexit><latexit sha1_base64="V8YmV6e/Qv5MOJ5CGYyJh5xFA2E=">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</latexit><latexit sha1_base64="V8YmV6e/Qv5MOJ5CGYyJh5xFA2E=">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</latexit><latexit sha1_base64="UlmVpJCEVVap4y5gAknNUd8Bjew=">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</latexit>
23
Ek(t)/Q
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実は、ラマヌジャンの発⾒した2点は独⽴だった︕
やばい︕︕︕
24
K3曲⾯
25
この辺から説明が怪しくなります・・・
26
曲⾯として解釈する
X3
+ Y 3
= k(T)
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を変数と思う
の3変数を動かしてできる曲⾯を考える(X, Y, Z)
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T
<latexit sha1_base64="WWwbKhIPBcPkmOeaRBavAHuVECU=">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</latexit><latexit sha1_base64="WWwbKhIPBcPkmOeaRBavAHuVECU=">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</latexit><latexit sha1_base64="WWwbKhIPBcPkmOeaRBavAHuVECU=">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</latexit><latexit sha1_base64="zeXJVMryhBZQ7aPwsEM1Gu6DDV8=">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</latexit>
→「極⼩なもの」をとるとK3曲⾯になるらしい・・・
27
K3曲⾯って何︖
• 2次元の連結なコンパクト複素多様体(曲⾯)
• ⾃明な標準バンドルを持つ(≒ 極⼩)
• 不正則数が0
楕円曲線は(複素多様体としての次元が) 1次元だが、
それが2次元になった版みたいなやつ(だと私は認識している・・・)
28
論⽂の主定理
定理(Ono and Trebat-Leder)
に付随する滑らかな極⼩曲⾯は、
ピカール数 18 の 上の楕円K3曲⾯である
X3
+ Y 3
= k(T)
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Q
<latexit sha1_base64="HJOWPUtsWGk42GzX/5kVFbkGxqM=">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</latexit><latexit sha1_base64="HJOWPUtsWGk42GzX/5kVFbkGxqM=">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</latexit><latexit sha1_base64="HJOWPUtsWGk42GzX/5kVFbkGxqM=">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</latexit><latexit sha1_base64="0us+3wQickmF5Bxr68l46v2MCnI=">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</latexit>
29
曲線の各点に楕円曲線がくっついたような構造をしているもの
まとめ
• ラマヌジャンが与えた 1729 に関する「やばい」公式︓
• ランク2以上の楕円曲線が無数に得られた(やばい)
• (ピカール数18の)K3曲⾯が得られた(やばい)
(6A2
4AB + 4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5
<latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">AAAGX3icpVRLTxNRFD6gFKwKRTcaNxNrTRUhty98JCaAG3byhoQWMjNcYMI8mplpoTb9A/4BF640cWH8GWz8Ay5YuiTuxMSNC797ZoQWbF04k7n33O88vnPOvXeMqm0FoRBHff2XLg8kBoeuJK9euz48khq9sRJ4Nd+Uy6Zne/6aoQfStly5HFqhLdeqvtQdw5arxt4LpV+tSz+wPHcpbFRlxdF3XGvbMvUQ0Gaqkp2c3shr41pxekYb04ozG/kHGwXtuZYtKHxMKwEf10oxPqZli+32k2d4KcIj+0KEb6bSYkLwo10UcrGQpviZ80b7a1SmLfLIpBo5JMmlELJNOgV41ylHgqrAKtQE5kOyWC+pRUn41mAlYaED3cO4g9V6jLpYq5gBe5tgsfH58NQoI76Ij+JEfBafxLH4hViZLtGa+GrM6PfgbDKXyrmB2Yg4ZHVz5PWtxZ//9HIwh7R75tWztpC26QnXZKHGKiOqWjPyr796c7L4bCHTvC/ei2+o8504Eoeo1K3/MD/My4W3PfIxkIuqsjv/S1qiPD3iOcdZbKOrblsOSd5TCbxMi6hKh00TsgXEweuxNolcNaDK7wCjsuiMpOxUhTvQRawt2Kic1G6Y3IfIXgPPLDNHMZVFcMrR/v4vR/I0igXLqFPdT07IsRu0z1yNnp1tYvQhV9ExdcYPep62ALEdvhENfFuxrQvdPlflsNaFRvW1znm0WPahDfm8Sey2B54mPcUtU9p7vEsu5lYcUUXZg6WyVx0J2c/v2GOHs1YZedwXG+8UomVO+1/mTBXzfodFp6/JZ8XnHFX/DV61zq3VykIXdrmKP1G71/8Q+EEHzzp2uYKxci7H6BxOtUXV0JcCulPqUWUat6DFc/6vXfP4ZqNn+Dvmzv8LLwor+Ykc5Pliemom/k8O0R26S1nwPAbjLM3RMmo8pGP6TicDXxODieFEKjLt74t9blLHk7j9G1MhSlg=</latexit>
+ 4B2
)3
= (3A2
+ 5AB 5B2
)3
+ (4A2
4AB + 6B2
)3
+ (5A2
5AB 3B2
)3
30
詳しくは、ブログで︕
31
さっき、公開︕︕
ラマヌジャン
やっぱりやばいじゃん
辻順平 @tsujimotter
やばい⼈
以下、おまけスライド
33
泣く泣くカットしたスライドたちです・・・
ちなみにこれは K2 という「世界で2番⽬に⾼い⼭」
こっちは北岳
34
35
代数多様体 複素多様体
楕円曲線
(3次曲線)
楕円曲線
(複素トーラス)
(複素多様体としての)
次元
1次元
イメージ(正しさは保証しません。。。)
36
代数多様体 複素多様体
楕円曲線
(3次曲線)
(︖)
楕円曲線
(複素トーラス)
K3曲⾯
(複素多様体としての)
次元
1次元
2次元
イメージ(正しさは保証しません。。。)
37
代数多様体 複素多様体
例外曲線を
持つ曲⾯
ラマヌジャンの⽅程式に付随する
楕円K3曲⾯
Xram : X3
+ Y 3
= k(T)
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Xan
ram
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ˆXan
ram
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極⼩曲⾯
ブローダウンブローアップ
ラマヌジャンの⽅程式で定義される
代数曲⾯
たぶんこういうことだと思う・・・というイメージ図
Ono and Trebat-Leder の論⽂
38
→ の数学部分の監修をされた⽅
映画、⾯⽩いのでぜひみてね︕

ラマヌジャンやっぱりやばいじゃん - 第15回 #日曜数学会

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    タクシー数 1729 = 13 +123 = 93 + 103 <latexit sha1_base64="Mv6nB7ymJNFAxSg3DzF2l2ZFXXI=">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</latexit><latexit sha1_base64="Mv6nB7ymJNFAxSg3DzF2l2ZFXXI=">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</latexit><latexit sha1_base64="Mv6nB7ymJNFAxSg3DzF2l2ZFXXI=">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</latexit><latexit sha1_base64="kb7jrfF6fwBYZST2g4gcIGPxeNc=">AAAGIHicpVQ7b9NQFD4tBEp4tIUFCQlZhCAEqLpOQSWISpFYutE2fUlNGtnubWvVL9lO2mBlY+IPMDBRiQEY4R+wdEMMDOUfIMYisTDw3WNTmpaEAVu+99zz+s757r02A8eOYiH2BgZPnMydOj10Jn/23PkLwyOjFxcivxlact7yHT9cMo1IOrYn52M7duRSEErDNR25aG4+UvbFlgwj2/fm4nYg666x7tlrtmXEUDVGruoTpbI2qekr49ptTS9hmtTK6UKsjDdGCmJM8KMdF/RMKFD2TPujg02q0Sr5ZFGTXJLkUQzZIYMivMukk6AAujol0IWQbLZL6lAesU14SXgY0G5iXMdqOdN6WKucEUdbQHHwhYjUqCg+i9diX+yKt+Kr+IlcxR7ZEnxNRgz7YCaMpWpuYzZTDBk0hp9drv74Z5SLOaaNP1F9e4tpje5zTzZ6DFijurXS+NaT5/vVB7PF5IbYEd/Q50uxJz6gU6/13Xo1I2df9KnHRC2qy974j2mOSnSHZ52rWAOr3qEa8rynEvoaVdGVAZ8Esg2Ni9dnax61atCquG2MyqM7k/JTHa7DlqJ24KNqUrthMQ+pvwacKUZOcyqP6ADj8Pu/GPmDLDY8U6Z6n5yYc7dpi7HafZlNMIaQAzCmzvh239MWIbfLN6KNbzXz9WDb4q5ctnqwKF5bXEeH5RDWmM+bxG77wEmojFumrNd5lzzMnSyjyrIJT+WvGIk5LuzaY5erVhX5zIuDt4JsxQP+a1ypQt7q8uiOtfishFyj4t/kVefIWq1ssLDBXfzO2rv/W9Bvd+EsY5frGOtHakzPYeVQVg28jIOde326LOAWdHgu/ZU1n282OMPfUT/6LzwuLJTGdMgzdwuVh9l/coiu0DW6CZwJIE7RNM2jx6f0ht7R+9xObjf3MfcpdR0cyGIuUdeT+/ILNiM+og==</latexit> 1729は、2つの3乗数の和として2通りに表すことができる最⼩の数 (by ラマヌジャン) 2
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    ラマヌジャンの発⾒(1913) オイラーのディオファントス⽅程式 の無限個の解を与えるパラメータ解︓ X3 + Y 3 =Z3 + W3 <latexit 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sha1_base64="ivcMEGKOck8RDdl/8PMK+UL99cs=">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</latexit><latexit sha1_base64="ivcMEGKOck8RDdl/8PMK+UL99cs=">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</latexit><latexit sha1_base64="vfi5F3ixY7dJ9l2aCXYGFFkYavQ=">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</latexit> (6A2 4AB + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A <latexit 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sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit> B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 11
  • 12.
    両辺を 33 で割ると A= 2, B = 1 <latexit 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sha1_base64="LRo80GOIIqTUh0O4Dq87A0DZK4E=">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</latexit> 363 = ( 3)3 + 303 + 273 <latexit 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sha1_base64="V5K2VRsWpe5Hw8rqp4Nvxl5lhZs=">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</latexit> 123 = ( 1)3 + 103 + 93 <latexit 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(6A2 4AB + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5A <latexit 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sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit> + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 (タクシー数) 12
  • 13.
    (6A2 4AB + 4B2 )3 =(3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5A <latexit 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sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit> + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 A, B <latexit 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無数の「タクシー数的な数」が得られる︕ 13
  • 14.
  • 15.
  • 16.
    Ono and Trebat-Lederの論⽂ 1. タクシー数と楕円曲線 2. タクシー数とK3曲⾯ 16
  • 17.
    両辺 B6 で割ってT = A/B とおく (6T2 4T + 4)3 + ( 3T2 5T + 5)3 = (4T2 4T + 6)3 + (5T2 <latexit 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T + 5)3 = (4T2 4T + 6)3 + (5T2 5T 3)3 = k(T) <latexit 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sha1_base64="DfyJg8K/ClzDQ0g7Vpg+1HgBtCs=">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</latexit> (6A2 4AB + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB <latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit 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sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit> 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 とおく 17
  • 18.
    上の⽅程式 の2つの解 が得られた。 X3 + Y 3 =k(T) <latexit 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(6T2 4T + 4, 3T2 5T + 5) = (4T2 4T + 6, 5T2 5T + 3) <latexit 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Q(T) <latexit 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  • 19.
    上の楕円曲線 X3 + Y 3 =k(T) <latexit 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<latexit 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k(t2) <latexit 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sha1_base64="r+eV9/Jgcm5PnvobFAIxIqbNht8=">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</latexit><latexit 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k(t3) <latexit 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  • 20.
    楕円曲線の(⾃由部分の)⽣成元の個数 = ランク P1 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sha1_base64="5YV5vnRz1jqmkjuP1FDOamnSIcA=">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</latexit> P2 <latexit 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sha1_base64="rlDnM70TEUBzQnuZCZfum5zMiVE=">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</latexit><latexit sha1_base64="rlDnM70TEUBzQnuZCZfum5zMiVE=">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</latexit><latexit sha1_base64="QhGBpMlJmukk7ttQsj2lFPFxedw=">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</latexit> この図の場合は ランク2 20
  • 21.
    ランク2以上はレア︕ ランク0の ( 上の)楕円曲線 50% ランク1の ( 上の)楕円曲線 50% ランク2の (上の)楕円曲線 0% 21 Q <latexit 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sha1_base64="V8YmV6e/Qv5MOJ5CGYyJh5xFA2E=">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</latexit><latexit 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sha1_base64="V8YmV6e/Qv5MOJ5CGYyJh5xFA2E=">AAAGCnicpVS7bhNBFL0JGIJ5JIEGicbCGCGEorEB8RCFJZp0JHGcRNhWtLsZ2yvvS7vrF6v9AUSJRIFAAokC0dEhSij4AYp8AqIMEg0FZ+6aYDvYKdjVzty5r3PvmZnVPcsMQiF2Z2aPHE0dOz53In3y1Okz8wuLZzcCt+0bsmy4lutv6VogLdOR5dAMLbnl+VKzdUtu6q37yr7ZkX5gus562PdkzdYajlk3DS2E6mHV1sKmrker8fZCViwJfjIHhfxAyBYF8bPiLs62qUo75JJBbbJJkkMhZIs0CvBWKE+CPOhqFEHnQzLZLimmNGLb8JLw0KBtYWxgVRloHaxVzoCjDaBY+HxEZignvoq3Yk98Ee/EN/ELuXITskX42ozoT8GMGEvV3MesJxjS255/fL7089AoG3NIzb9RU3sLqU63uScTPXqsUd0aSXzn0bO90t21XHRZvBbf0ecrsSs+oVOn88N4syrXnk+pR0ctqsvJ+A9onQp0jec8V1EHq85QDWneUwl9lUroSoNPBNmExsbrsjWNWjPQqrgeRuUxmkn5qQ4bsCWoMXxUTWo3DOYh8c8AZ5mRk5zKI9jHGH7/FyO9n8WEZ8LU5JMTcu4+dRmrP5XZCKMP2QNj6oz3pp62ALltvhF9fDsDXwe2Lndls9WBRfHa4Tpiln1YQz5vErvtAieiO7hlynqJd8nBHA8yqiwteCp/xUjIcf7IHttctarIZV4svEVky+3zX+VKFXJ3xGM01uCz4nONin+dV/HYWq1MsNDkLv5kndz/Veh7IzgV7HINY22sxuQcFoeyZsDLdbBzc0qXWdyCmOfCP1lz+WaDM/wd8+P/woPCRmEpD3n1RrZ4L/lN0hxdoIt0BTi3gLhMK1RGjw49pRf0MvUk9T71IfUxcZ2dGcSco5En9fk330U8Fw==</latexit><latexit sha1_base64="UlmVpJCEVVap4y5gAknNUd8Bjew=">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</latexit> アンケート Q. あなたのランクはいくつですか︖
  • 22.
    実は、ラマヌジャンの発⾒した2点は独⽴だった︕ (X, Y )= (6T2 4T + 4, 3T2 5T + 5) = (4T2 4T + 6, 5T2 5T + 3) <latexit 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sha1_base64="mowuJ1ehuoSL/vH13rY4VXs7anU=">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</latexit> 定理(Ono and Trebat-Leder) 上の楕円曲線 は ランク2 を持つEk(T ) : X3 + Y 3 = k(T) <latexit 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Q(T) <latexit 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sha1_base64="5YV5vnRz1jqmkjuP1FDOamnSIcA=">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</latexit> P2 <latexit sha1_base64="rlDnM70TEUBzQnuZCZfum5zMiVE=">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</latexit><latexit 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  • 23.
    ランク2の 上の楕円曲線を無数に⽣成できる︕︕ 定理(Ono andTrebat-Leder) 上の楕円曲線 は ランク2 を持つEk(T ) : X3 + Y 3 = k(T) <latexit 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sha1_base64="/Du86Sm15yZiqczwTLqDheKkvzE=">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</latexit><latexit 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Q(T) <latexit 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sha1_base64="2h2i4PHr0WjW2bc1PQqZbKdu9g0=">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</latexit><latexit sha1_base64="2h2i4PHr0WjW2bc1PQqZbKdu9g0=">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</latexit><latexit sha1_base64="Oo4R1SdTT+bhy61i3j39SgmSiFA=">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</latexit> 系 有限個を除くすべての に対して は ランク2以上t 2 Q <latexit 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実は、ラマヌジャンの発⾒した2点は独⽴だった︕
  • 24.
  • 25.
  • 26.
  • 27.
    曲⾯として解釈する X3 + Y 3 =k(T) <latexit 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の3変数を動かしてできる曲⾯を考える(X, Y, Z) <latexit 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sha1_base64="zeXJVMryhBZQ7aPwsEM1Gu6DDV8=">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</latexit> →「極⼩なもの」をとるとK3曲⾯になるらしい・・・ 27
  • 28.
    K3曲⾯って何︖ • 2次元の連結なコンパクト複素多様体(曲⾯) • ⾃明な標準バンドルを持つ(≒極⼩) • 不正則数が0 楕円曲線は(複素多様体としての次元が) 1次元だが、 それが2次元になった版みたいなやつ(だと私は認識している・・・) 28
  • 29.
    論⽂の主定理 定理(Ono and Trebat-Leder) に付随する滑らかな極⼩曲⾯は、 ピカール数18 の 上の楕円K3曲⾯である X3 + Y 3 = k(T) <latexit 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sha1_base64="HJOWPUtsWGk42GzX/5kVFbkGxqM=">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</latexit><latexit sha1_base64="0us+3wQickmF5Bxr68l46v2MCnI=">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</latexit> 29 曲線の各点に楕円曲線がくっついたような構造をしているもの
  • 30.
    まとめ • ラマヌジャンが与えた 1729に関する「やばい」公式︓ • ランク2以上の楕円曲線が無数に得られた(やばい) • (ピカール数18の)K3曲⾯が得られた(やばい) (6A2 4AB + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5 <latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">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</latexit><latexit sha1_base64="EdDBpHj5FZ3dQ6zU42YC9DEtnv4=">AAAGX3icpVRLTxNRFD6gFKwKRTcaNxNrTRUhty98JCYFN+zkDQkU0hkuMGFemZkW6qR/wD/gwpUmLow/g41/wAVLl8SdmLhx4XfPVGjB1oUzmXvP/c7jO+fce0f3LDMIhTjq6790eSAxOHQlefXa9eGR1OiN5cCt+YZcMlzL9Vf1aiAt05FLoRlactXzZdXWLbmi771Q+pW69APTdRbDhicrdnXHMbdNoxoC2kxVspNTG3ltXCtOTWtjWnF6I/9go6A917IFhY9pJeDjWqmFj2nZYrv95BleivHYvhDjm6m0mBD8aBeFXEtIlwXxM+uO9tdonbbIJYNqZJMkh0LIFlUpwLtGORLkAatQBMyHZLJeUpOS8K3BSsKiCnQP4w5Way3UwVrFDNjbAIuFz4enRhnxRXwUJ+Kz+CSOxS/EynSJFuGrMaPfgzNiLpVzA7Mec0hvc+T1rYWf//SyMYe0e+bVs7aQtukJ12SiRo8RVa0R+9dfvTlZeDafie6L9+Ib6nwnjsQhKnXqP4wPc3L+bY98dOSiquzO/5IWKU+PeM5xFtvoqtOWQ5L3VAJfpwVUVYVNBNkEYuN1WZtErhpQ5XeAUVl0RlJ2qsId6GLWJmxUTmo3DO5DbK+BZ4aZ45jKIjjlaH//lyN5GsWEZdyp7icn5NgN2meuRs/ORhh9yB46ps74Qc/TFiC2zTeigW+rZetAt89V2ax1oFF9rXMeTZZ9aEM+bxK77YInoqe4ZUp7j3fJwdxsRVRR9mCp7FVHQvbzO/bY5qxVRi73xcJbRrTMaf/XOVPFvN9h0elr8FnxOUfVf51XzXNrtTLRhV2u4k/U7vU/BH7QwbOGXa5grJzLMT6H5baoGvpSQHdKPapM4xY0ec7/tWsu32z0DH/H3Pl/4UVhOT+RgzxXTJen498kDdEduktZ8DwG4wzN0hJqPKRj+k4nA18Tg4nhRCo27e9r+dykjidx+ze2P0rI</latexit><latexit sha1_base64="9Nlc+QgyxIgZRSq039MrkUTJvaw=">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</latexit> + 4B2 )3 = (3A2 + 5AB 5B2 )3 + (4A2 4AB + 6B2 )3 + (5A2 5AB 3B2 )3 30
  • 31.
  • 32.
  • 33.
  • 34.
  • 35.
  • 36.
  • 37.
    37 代数多様体 複素多様体 例外曲線を 持つ曲⾯ ラマヌジャンの⽅程式に付随する 楕円K3曲⾯ Xram :X3 + Y 3 = k(T) <latexit 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Xan ram <latexit 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sha1_base64="xqFsPRBHZFSfggb/jDwhHNoFwzI=">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</latexit> 極⼩曲⾯ ブローダウンブローアップ ラマヌジャンの⽅程式で定義される 代数曲⾯ たぶんこういうことだと思う・・・というイメージ図
  • 38.
    Ono and Trebat-Lederの論⽂ 38 → の数学部分の監修をされた⽅ 映画、⾯⽩いのでぜひみてね︕