{"version":2,"work":"hilbert12-ambient-dual-slice","title":"Hilbert 12: ambient dual slice","source_revision":"13c5e7243e0945f18e8f5d18ff050cbab16f152b","sources":[{"repository_path":"research/hilbert12-ambient-dual-slice/H12_Ambient_Dual_Slice_Research_Note.md","file_url":"documents/files/c7a16593696d13ad074d24a8c67cf01d3875e7fb/H12_Ambient_Dual_Slice_Research_Note.md","url":"read/hilbert12-ambient-dual-slice/h12-ambient-dual-slice-research-note-1318b075a9a8.html","sha256":"2e372de30326863f96fe21735389e760ca40c781baadf45a57d5ce52edad050a","git_blob_sha1":"c7a16593696d13ad074d24a8c67cf01d3875e7fb","size":11497,"source_url":"read/hilbert12-ambient-dual-slice/h12-ambient-dual-slice-research-note-1318b075a9a8-source.html","passages":126,"macros":{}}],"pdfs":[],"passages":[{"line":5,"start":221,"end":239,"expression":"K=\\mathbb Q(\\beta)","context":0,"kind":"use","source":0},{"line":5,"start":245,"end":254,"expression":"\\beta^3=7","context":0,"kind":"use","source":0},{"line":13,"start":1297,"end":1316,"expression":"\\omega^2+\\omega+1=0","context":1,"kind":"use","source":0},{"line":13,"start":1322,"end":1333,"expression":"J=2+3\\omega","context":1,"kind":"use","source":0},{"line":13,"start":1339,"end":1356,"expression":"\\bar J=-1-3\\omega","context":1,"kind":"use","source":0},{"line":13,"start":1367,"end":1377,"expression":"J+\\bar J=1","context":1,"kind":"use","source":0},{"line":13,"start":1383,"end":1392,"expression":"J\\bar J=7","context":1,"kind":"use","source":0},{"line":15,"start":1404,"end":1465,"expression":"\nL=\\mathbb Z(\\beta^2+2\\beta)+\\mathbb Z(-\\beta-2)+15\\mathbb Z\n","context":2,"kind":"use","source":0},{"line":19,"start":1481,"end":1499,"expression":"b_0=\\beta^2+2\\beta","context":3,"kind":"use","source":0},{"line":19,"start":1505,"end":1517,"expression":"b_1=-\\beta-2","context":3,"kind":"use","source":0},{"line":19,"start":1523,"end":1529,"expression":"b_2=15","context":3,"kind":"use","source":0},{"line":19,"start":1553,"end":1569,"expression":"\\epsilon=2-\\beta","context":3,"kind":"use","source":0},{"line":21,"start":1586,"end":1656,"expression":"\nM=\\begin{pmatrix}0&1&0\\\\-4&2&15\\\\-1&0&4\\end{pmatrix},\\quad \\det M=1.\n","context":4,"kind":"use","source":0},{"line":25,"start":1689,"end":1710,"expression":"\\lambda(x)=[\\beta^2]x","context":5,"kind":"use","source":0},{"line":25,"start":1744,"end":1753,"expression":"\\beta^3=7","context":5,"kind":"use","source":0},{"line":25,"start":1775,"end":1820,"expression":"\\operatorname{Tr}_{K/\\mathbb Q}(x/(3\\beta^2))","context":5,"kind":"use","source":0},{"line":25,"start":1842,"end":1865,"expression":"H_{ij}=\\lambda(b_i b_j)","context":5,"kind":"use","source":0},{"line":27,"start":1874,"end":1974,"expression":"\nH=\\begin{pmatrix}4&-4&15\\\\-4&1&0\\\\15&0&0\\end{pmatrix},\\qquad \\det H=-225,\\qquad M^{\\mathsf T}H=HM.\n","context":6,"kind":"use","source":0},{"line":31,"start":1985,"end":1997,"expression":"(M-2I)^3=-7I","context":7,"kind":"use","source":0},{"line":31,"start":2013,"end":2024,"expression":"\\mathbb F_7","context":7,"kind":"use","source":0},{"line":31,"start":2030,"end":2036,"expression":"N=M-2I","context":7,"kind":"use","source":0},{"line":31,"start":2086,"end":2107,"expression":"u=(1,2,4)^{\\mathsf T}","context":7,"kind":"use","source":0},{"line":31,"start":2113,"end":2134,"expression":"v=(0,1,2)^{\\mathsf T}","context":7,"kind":"use","source":0},{"line":31,"start":2140,"end":2161,"expression":"w=(0,0,1)^{\\mathsf T}","context":7,"kind":"use","source":0},{"line":31,"start":2172,"end":2176,"expression":"Nu=0","context":7,"kind":"use","source":0},{"line":31,"start":2182,"end":2186,"expression":"Nv=u","context":7,"kind":"use","source":0},{"line":31,"start":2192,"end":2196,"expression":"Nw=v","context":7,"kind":"use","source":0},{"line":33,"start":2220,"end":2314,"expression":"\n[u\\ v\\ w]^{\\mathsf T}H[u\\ v\\ w]\\equiv\n\\begin{pmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\end{pmatrix}\\pmod7.\n","context":8,"kind":"use","source":0},{"line":38,"start":2327,"end":2328,"expression":"u","context":9,"kind":"use","source":0},{"line":38,"start":2344,"end":2367,"expression":"\\beta^2+56\\equiv\\beta^2","context":9,"kind":"use","source":0},{"line":38,"start":2373,"end":2374,"expression":"v","context":9,"kind":"use","source":0},{"line":38,"start":2390,"end":2410,"expression":"28-\\beta\\equiv-\\beta","context":9,"kind":"use","source":0},{"line":38,"start":2420,"end":2421,"expression":"w","context":9,"kind":"use","source":0},{"line":38,"start":2437,"end":2446,"expression":"15\\equiv1","context":9,"kind":"use","source":0},{"line":38,"start":2454,"end":2482,"expression":"\\mathbb F_7[\\beta]/(\\beta^3)","context":9,"kind":"use","source":0},{"line":38,"start":2540,"end":2547,"expression":"\\beta^2","context":9,"kind":"use","source":0},{"line":38,"start":2629,"end":2643,"expression":"u^{\\mathsf T}H","context":9,"kind":"use","source":0},{"line":38,"start":2660,"end":2693,"expression":"\\ker N^2=\\operatorname{span}(u,v)","context":9,"kind":"use","source":0},{"line":38,"start":2716,"end":2717,"expression":"w","context":9,"kind":"use","source":0},{"line":44,"start":2933,"end":2945,"expression":"A=E_\\omega^3","context":10,"kind":"use","source":0},{"line":44,"start":2951,"end":2987,"expression":"E_\\omega=\\mathbb C/\\mathbb Z[\\omega]","context":10,"kind":"use","source":0},{"line":44,"start":3092,"end":3093,"expression":"H","context":10,"kind":"use","source":0},{"line":44,"start":3131,"end":3147,"expression":"X=(a+\\omega b)/7","context":10,"kind":"use","source":0},{"line":44,"start":3153,"end":3169,"expression":"Y=(c+\\omega d)/7","context":10,"kind":"use","source":0},{"line":46,"start":3182,"end":3239,"expression":"\ne_{7,H}(X,Y)=\\zeta_7^{a^{\\mathsf T}Hd-b^{\\mathsf T}Hc}.\n","context":11,"kind":"use","source":0},{"line":50,"start":3274,"end":3286,"expression":"7\\nmid\\det H","context":12,"kind":"use","source":0},{"line":50,"start":3331,"end":3335,"expression":"A[7]","context":12,"kind":"use","source":0},{"line":50,"start":3345,"end":3354,"expression":"A^\\vee[7]","context":12,"kind":"use","source":0},{"line":50,"start":3378,"end":3391,"expression":"H:A\\to A^\\vee","context":12,"kind":"use","source":0},{"line":52,"start":3512,"end":3547,"expression":"\ne_{7,H}(MX,M^{-1}Y)=e_{7,H}(X,Y).\n","context":13,"kind":"use","source":0},{"line":56,"start":3560,"end":3565,"expression":"P=u/J","context":14,"kind":"use","source":0},{"line":56,"start":3571,"end":3582,"expression":"P'=u/\\bar J","context":14,"kind":"use","source":0},{"line":56,"start":3592,"end":3602,"expression":"Q=w/\\bar J","context":14,"kind":"use","source":0},{"line":56,"start":3635,"end":3653,"expression":"1/J=(-1-3\\omega)/7","context":14,"kind":"use","source":0},{"line":56,"start":3662,"end":3684,"expression":"1/\\bar J=(2+3\\omega)/7","context":14,"kind":"use","source":0},{"line":58,"start":3696,"end":3744,"expression":"\ne_{7,H}(P,Q)=\\zeta_7^3,\\qquad e_{7,H}(P,P')=1.\n","context":15,"kind":"use","source":0},{"line":62,"start":3784,"end":3794,"expression":"Y=a/\\bar J","context":16,"kind":"use","source":0},{"line":62,"start":3850,"end":3861,"expression":"Ma=2a\\pmod7","context":16,"kind":"use","source":0},{"line":62,"start":3872,"end":3873,"expression":"a","context":16,"kind":"use","source":0},{"line":62,"start":3895,"end":3896,"expression":"u","context":16,"kind":"use","source":0},{"line":62,"start":3909,"end":3919,"expression":"\\ker(M-2I)","context":16,"kind":"use","source":0},{"line":62,"start":3957,"end":3971,"expression":"e_{7,H}(P,Y)=1","context":16,"kind":"use","source":0},{"line":64,"start":4220,"end":4238,"expression":"Q=(w+su+tv)/\\bar J","context":17,"kind":"use","source":0},{"line":64,"start":4250,"end":4251,"expression":"u","context":17,"kind":"use","source":0},{"line":64,"start":4277,"end":4280,"expression":"u,v","context":17,"kind":"use","source":0},{"line":64,"start":4349,"end":4361,"expression":"\\zeta_7^{3r}","context":17,"kind":"use","source":0},{"line":64,"start":4367,"end":4389,"expression":"r\\in\\mathbb F_7^\\times","context":17,"kind":"use","source":0},{"line":66,"start":4454,"end":4457,"expression":"H_+","context":18,"kind":"use","source":0},{"line":66,"start":4467,"end":4486,"expression":"M^\\mathsf TH_+=H_+M","context":18,"kind":"use","source":0},{"line":66,"start":4508,"end":4509,"expression":"M","context":18,"kind":"use","source":0},{"line":66,"start":4554,"end":4578,"expression":"\\chi_M(X)=X^3-6X^2+12X-1","context":18,"kind":"use","source":0},{"line":66,"start":4609,"end":4614,"expression":"-1323","context":18,"kind":"use","source":0},{"line":66,"start":4775,"end":4776,"expression":"H","context":18,"kind":"use","source":0},{"line":70,"start":4923,"end":4983,"expression":"\\chi(1),\\ldots,\\chi(6)=(1,\\omega,\\omega^2,\\omega^2,\\omega,1)","context":19,"kind":"use","source":0},{"line":70,"start":4992,"end":5029,"expression":"G_r=\\sum_{k=1}^{6}\\chi(k)\\zeta_7^{rk}","context":19,"kind":"use","source":0},{"line":70,"start":5050,"end":5055,"expression":"r\\ne0","context":19,"kind":"use","source":0},{"line":72,"start":5062,"end":5100,"expression":"\nG_r=\\chi(r)^{-1}G_1,\\qquad G_r^3=7J.\n","context":20,"kind":"use","source":0},{"line":76,"start":5284,"end":5329,"expression":"\\mathbb Z[\\omega,z]/(\\omega^2+\\omega+1,z^7-1)","context":21,"kind":"use","source":0},{"line":76,"start":5340,"end":5355,"expression":"(G_r/\\beta)^3=J","context":21,"kind":"use","source":0},{"line":78,"start":5572,"end":5603,"expression":"f(k)=e_{7,H}(kP,Q)=\\zeta_7^{3k}","context":22,"kind":"use","source":0},{"line":78,"start":5617,"end":5622,"expression":"G_3/6","context":22,"kind":"use","source":0},{"line":80,"start":5629,"end":5698,"expression":"\n(\\Pi_{\\chi^{-1}}f)(1)=\\frac16\\sum_{k\\ne0}\\chi(k)f(k)=\\frac{G_3}{6}.\n","context":23,"kind":"use","source":0},{"line":86,"start":5794,"end":5811,"expression":"p=(\\tau,\\sigma,1)","context":24,"kind":"use","source":0},{"line":88,"start":5822,"end":5948,"expression":"\n\\tau=\\frac{\\beta^2+2\\beta}{15},\\quad\\sigma=-\\frac{\\beta+2}{15},\\quad\n\\delta=\\frac{\\tau+2\\sigma+4}{7}=\\frac{\\beta^2+56}{105}.\n","context":25,"kind":"use","source":0},{"line":93,"start":5976,"end":5985,"expression":"\\pi(v)=pv","context":26,"kind":"use","source":0},{"line":93,"start":6023,"end":6038,"expression":"\\pi(u/7)=\\delta","context":26,"kind":"use","source":0},{"line":93,"start":6044,"end":6052,"expression":"\\pi(w)=1","context":26,"kind":"use","source":0},{"line":95,"start":6063,"end":6150,"expression":"\nP+P'=\\frac{u}{7},\\quad\\pi(P)=\\bar J\\delta,\\quad\\pi(P')=J\\delta,\\quad\\pi(P+P')=\\delta.\n","context":27,"kind":"use","source":0},{"line":99,"start":6194,"end":6203,"expression":"z+k\\delta","context":28,"kind":"use","source":0},{"line":99,"start":6266,"end":6273,"expression":"k(P+P')","context":28,"kind":"use","source":0},{"line":99,"start":6318,"end":6321,"expression":"\\pi","context":28,"kind":"use","source":0},{"line":99,"start":6362,"end":6395,"expression":"A=\\mathbb C^3/\\mathbb Z[\\omega]^3","context":28,"kind":"use","source":0},{"line":101,"start":6530,"end":6552,"expression":"\\beta\\delta=-8\\sigma-1","context":29,"kind":"use","source":0},{"line":103,"start":6587,"end":6645,"expression":"\n\\epsilon\\delta=2\\delta+8\\sigma+1,\\qquad\\epsilon=2-\\beta.\n","context":30,"kind":"use","source":0},{"line":107,"start":6711,"end":6727,"expression":"k\\mapsto2k\\pmod7","context":31,"kind":"use","source":0},{"line":107,"start":6787,"end":6797,"expression":"k\\mapsto4k","context":31,"kind":"use","source":0},{"line":111,"start":6974,"end":7057,"expression":"F(z)=\\Gamma_{\\mathrm{ell}}(z;\\tau,\\sigma)^5/\\Gamma_{\\mathrm{ell}}(5z;5\\tau,5\\sigma)","context":32,"kind":"use","source":0},{"line":111,"start":7066,"end":7085,"expression":"f_k(z)=F(z+k\\delta)","context":32,"kind":"use","source":0},{"line":111,"start":7132,"end":7159,"expression":"P_7(z)=\\prod_{k=0}^6 f_k(z)","context":32,"kind":"use","source":0},{"line":113,"start":7216,"end":7238,"expression":"\\chi(k)=a_k+\\omega b_k","context":33,"kind":"use","source":0},{"line":115,"start":7276,"end":7350,"expression":"\n(a_1,\\ldots,a_6)=(1,0,-1,-1,0,1),\\quad\n(b_1,\\ldots,b_6)=(0,1,-1,-1,1,0).\n","context":34,"kind":"use","source":0},{"line":122,"start":7403,"end":7466,"expression":"\nA(z)=\\frac{f_1f_6}{f_3f_4},\\qquad\nB(z)=\\frac{f_2f_5}{f_3f_4}.\n","context":35,"kind":"use","source":0},{"line":129,"start":7524,"end":7622,"expression":"\nD_\\chi(z)=\\sum_{k=1}^6\\chi(k)\\frac{f'_k(z)}{f_k(z)}\n=\\frac{A'(z)}{A(z)}+\\omega\\frac{B'(z)}{B(z)}\n","context":36,"kind":"use","source":0},{"line":134,"start":7689,"end":7712,"expression":"f_k\\mapsto f_{2k\\bmod7}","context":37,"kind":"use","source":0},{"line":136,"start":7753,"end":7841,"expression":"\n(A,B)\\longmapsto(B/A,1/A),\\qquad\nT=\\begin{pmatrix}-1&1\\\\-1&0\\end{pmatrix},\\quad T^3=I.\n","context":38,"kind":"use","source":0},{"line":141,"start":7866,"end":7867,"expression":"T","context":39,"kind":"use","source":0},{"line":141,"start":7876,"end":7891,"expression":"\\omega,\\omega^2","context":39,"kind":"use","source":0},{"line":141,"start":7901,"end":7907,"expression":"D_\\chi","context":39,"kind":"use","source":0},{"line":141,"start":7926,"end":7934,"expression":"\\omega^2","context":39,"kind":"use","source":0},{"line":141,"start":8042,"end":8060,"expression":"z\\mapsto\\epsilon z","context":39,"kind":"use","source":0},{"line":143,"start":8185,"end":8190,"expression":"(A,B)","context":40,"kind":"use","source":0},{"line":143,"start":8264,"end":8291,"expression":"\\operatorname{ord}_\\tau F=4","context":40,"kind":"use","source":0},{"line":143,"start":8301,"end":8307,"expression":"\\delta","context":40,"kind":"use","source":0},{"line":143,"start":8335,"end":8389,"expression":"\\Lambda_5=\\mathbb Z\\tau+\\mathbb Z\\sigma+(1/5)\\mathbb Z","context":40,"kind":"use","source":0},{"line":143,"start":8398,"end":8412,"expression":"z=\\tau-2\\delta","context":40,"kind":"use","source":0},{"line":143,"start":8435,"end":8438,"expression":"f_2","context":40,"kind":"use","source":0},{"line":145,"start":8462,"end":8526,"expression":"\n\\operatorname{Res}_{z=\\tau-2\\delta}D_\\chi(z)=4\\chi(2)=4\\omega.\n","context":41,"kind":"use","source":0},{"line":149,"start":8627,"end":8649,"expression":"4\\omega\\notin\\mathbb Z","context":42,"kind":"use","source":0},{"line":149,"start":8715,"end":8721,"expression":"D_\\chi","context":42,"kind":"use","source":0},{"line":155,"start":9037,"end":9065,"expression":"F(z+\\sigma)/F(z)=q_t(z)^{-1}","context":43,"kind":"use","source":0},{"line":155,"start":9077,"end":9084,"expression":"t=-\\tau","context":43,"kind":"use","source":0},{"line":155,"start":9093,"end":9131,"expression":"q_t(z)=\\theta_0(z;t)^5/\\theta_0(5z;5t)","context":43,"kind":"use","source":0},{"line":155,"start":9151,"end":9163,"expression":"k=1,\\ldots,6","context":43,"kind":"use","source":0},{"line":157,"start":9170,"end":9275,"expression":"\nF(\\epsilon(z+k\\delta))=\nF(\\epsilon z+2k\\delta)\n\\prod_{j=0}^{8k-1}q_t(\\epsilon z+2k\\delta+j\\sigma)^{-1}.\n","context":44,"kind":"use","source":0},{"line":167,"start":9583,"end":9608,"expression":"U_\\Gamma=U_{\\mathrm{alg}}","context":45,"kind":"use","source":0},{"line":167,"start":9695,"end":9742,"expression":"\\operatorname{Im}(C_{\\mathrm{pol}}U_\\Gamma^3)=0","context":45,"kind":"use","source":0}],"entries":[{"id":"symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","occurrences":[2,3,5,7,8,29,42,43,54,62,67,68,70,77,83,87,88,89,93,97,99,102,113,119,123]},{"id":"symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","occurrences":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,20,21,22,23,24,25,26,37,39,40,42,43,44,49,50,51,52,53,54,55,56,57,61,62,68,70,73,74,76,78,79,81,82,83,84,85,86,87,91,92,93,96,97,98,99,100,101,102,104,111,113,114,116,119,120,121,122,123,124,125]},{"id":"symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","occurrences":[39,46,47,48,91,101,102,104,110]},{"id":"symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","occurrences":[42,44,56,57,58,99,100]},{"id":"symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","occurrences":[102]},{"id":"symbol-76b949b0834efe6d","tex":"a^{\\mathsf T}","aliases":"a^ T mathsf","display":"a^{\\mathsf T}","kind":"use","occurrences":[44]},{"id":"symbol-c3b0f6b395c9c99c","tex":"A^\\vee","aliases":"A^vee vee","display":"A^\\vee","kind":"use","occurrences":[47,48]},{"id":"symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","occurrences":[100]},{"id":"symbol-4f9a23e6ec67096e","tex":"a_6","aliases":"a_6","display":"a_6","kind":"use","occurrences":[100]},{"id":"symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","occurrences":[99]},{"id":"symbol-59a706af3e94a284","tex":"\\mathrm{alg}","aliases":"alg mathrm","display":"\\mathrm{alg}","kind":"use","occurrences":[124]},{"id":"symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","occurrences":[101,102,104,110]},{"id":"symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","occurrences":[8,9,10,16,42,44,99,100]},{"id":"symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","occurrences":[102]},{"id":"symbol-8d0c91c11bff49a2","tex":"b^{\\mathsf T}","aliases":"b^ T mathsf","display":"b^{\\mathsf T}","kind":"use","occurrences":[44]},{"id":"symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","occurrences":[8]},{"id":"symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","occurrences":[9,100]},{"id":"symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","occurrences":[10]},{"id":"symbol-5435c46c9f88e741","tex":"b_6","aliases":"b_6","display":"b_6","kind":"use","occurrences":[100]},{"id":"symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","occurrences":[16]},{"id":"symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","occurrences":[16]},{"id":"symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","occurrences":[99]},{"id":"symbol-bb3cdc779ccbade8","tex":"\\bar J","aliases":"bar J bar","display":"\\bar J","kind":"use","occurrences":[4,5,6,51,52,54,56,62,87]},{"id":"symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","occurrences":[103]},{"id":"symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","occurrences":[125]},{"id":"symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","occurrences":[40,91]},{"id":"symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","occurrences":[43,44]},{"id":"symbol-e2fdab254f7c2081","tex":"\\mathbb C^3","aliases":"C^3 mathbb","display":"\\mathbb C^3","kind":"use","occurrences":[91]},{"id":"symbol-c52f7a54c072ec1b","tex":"C_{\\mathrm{pol}}","aliases":"C_pol mathrm","display":"C_{\\mathrm{pol}}","kind":"use","occurrences":[125]},{"id":"symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","occurrences":[102,107,116,118]},{"id":"symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","occurrences":[43,44]},{"id":"symbol-61af2e48dff62660","tex":"D_\\chi","aliases":"D_χ chi","display":"D_\\chi","kind":"use","occurrences":[102,107,116,118]},{"id":"symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","occurrences":[12,17,45]},{"id":"symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","occurrences":[39,40]},{"id":"symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","occurrences":[44,49,55,61,79]},{"id":"symbol-20000cb9b2dd1efb","tex":"e_{7,H}","aliases":"e_7,H","display":"e_{7,H}","kind":"use","occurrences":[44,49,55,61,79]},{"id":"symbol-207db81da6a3101d","tex":"E_\\omega","aliases":"E_ω omega","display":"E_\\omega","kind":"use","occurrences":[40]},{"id":"symbol-505a8aab02afef64","tex":"E_\\omega^3","aliases":"E_ω^3 omega","display":"E_\\omega^3","kind":"use","occurrences":[39]},{"id":"symbol-fad271e883a47357","tex":"\\mathrm{ell}","aliases":"ell mathrm","display":"\\mathrm{ell}","kind":"use","occurrences":[96]},{"id":"symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","occurrences":[96,97,111,119,123]},{"id":"symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","occurrences":[19,34,66]},{"id":"symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","occurrences":[79,81,97,98,101,102,103,115]},{"id":"symbol-b6a9755de225fef9","tex":"f'_k","aliases":"f'_k","display":"f'_k","kind":"use","occurrences":[102]},{"id":"symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","occurrences":[101]},{"id":"symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","occurrences":[101,115]},{"id":"symbol-5a06733f80278422","tex":"f_{2k\\bmod7}","aliases":"f_2kbmod7 bmod","display":"f_{2k\\bmod7}","kind":"use","occurrences":[103]},{"id":"symbol-4fac6558c4f0f59e","tex":"f_3","aliases":"f_3","display":"f_3","kind":"use","occurrences":[101]},{"id":"symbol-8f053787a8e5d3cf","tex":"f_4","aliases":"f_4","display":"f_4","kind":"use","occurrences":[101]},{"id":"symbol-3374f149cfeb50af","tex":"f_5","aliases":"f_5","display":"f_5","kind":"use","occurrences":[101]},{"id":"symbol-8275a44e3c107b0c","tex":"f_6","aliases":"f_6","display":"f_6","kind":"use","occurrences":[101]},{"id":"symbol-23af4eb7208f9cad","tex":"\\mathbb F_7","aliases":"F_7 mathbb","display":"\\mathbb F_7","kind":"use","occurrences":[19,34]},{"id":"symbol-f28730c126e1e18a","tex":"\\mathbb F_7^\\times","aliases":"F_7^× mathbb times","display":"\\mathbb F_7^\\times","kind":"use","occurrences":[66]},{"id":"symbol-2f6b5c0f84584144","tex":"f_k","aliases":"f_k","display":"f_k","kind":"use","occurrences":[97,98,102,103]},{"id":"symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","occurrences":[74,76,78,80,81]},{"id":"symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","occurrences":[76]},{"id":"symbol-736fb9ac4210e748","tex":"G_3","aliases":"G_3","display":"G_3","kind":"use","occurrences":[80,81]},{"id":"symbol-3f8720ad3f121275","tex":"G_r","aliases":"G_r","display":"G_r","kind":"use","occurrences":[74,76,78]},{"id":"symbol-7eb857d1584a837b","tex":"G_r^3","aliases":"G_r^3","display":"G_r^3","kind":"use","occurrences":[76]},{"id":"symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","occurrences":[16,17,27,36,41,44,45,48,49,55,61,67,68,72,79]},{"id":"symbol-6bfe1e61875f685a","tex":"H_+","aliases":"H_+","display":"H_+","kind":"use","occurrences":[67,68]},{"id":"symbol-f74f6a57337ee481","tex":"H_{ij}","aliases":"H_ij","display":"H_{ij}","kind":"use","occurrences":[16]},{"id":"symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","occurrences":[18,20,60,104]},{"id":"symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","occurrences":[16]},{"id":"symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","occurrences":[125]},{"id":"symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","occurrences":[3,5,6,50,53,76,78,87]},{"id":"symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","occurrences":[16,123]},{"id":"symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","occurrences":[0,15]},{"id":"symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","occurrences":[74,79,81,88,89,94,95,97,98,99,102,103,122,123]},{"id":"symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","occurrences":[37,60]},{"id":"symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","occurrences":[7]},{"id":"symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","occurrences":[12,17,18,20,49,57,60,68,69,70]},{"id":"symbol-9905e7d4d709aef7","tex":"M^\\mathsf","aliases":"M^ mathsf","display":"M^\\mathsf","kind":"use","occurrences":[68]},{"id":"symbol-a2127b3e9668951d","tex":"M^{\\mathsf T}","aliases":"M^ T mathsf","display":"M^{\\mathsf T}","kind":"use","occurrences":[17]},{"id":"symbol-c693ce1e8e9e894c","tex":"M^{-1}","aliases":"M^-1","display":"M^{-1}","kind":"use","occurrences":[49]},{"id":"symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","occurrences":[20,24,25,26,37]},{"id":"symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","occurrences":[37]},{"id":"symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","occurrences":[45]},{"id":"symbol-fd71dfd389a79ac5","tex":"\\operatorname{ord}","aliases":"ord operatorname","display":"\\operatorname{ord}","kind":"use","occurrences":[111]},{"id":"symbol-903f67cd60746bcf","tex":"\\operatorname{ord}_\\tau","aliases":"ord_τ operatorname tau","display":"\\operatorname{ord}_\\tau","kind":"use","occurrences":[111]},{"id":"symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","occurrences":[50,51,55,61,79,87,89,98]},{"id":"symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","occurrences":[82,84]},{"id":"symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","occurrences":[51,55,87,89]},{"id":"symbol-74df595de253199d","tex":"P_7","aliases":"P_7","display":"P_7","kind":"use","occurrences":[98]},{"id":"symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","occurrences":[27,57,94]},{"id":"symbol-5964bd51960ac9c2","tex":"\\mathrm{pol}","aliases":"pol mathrm","display":"\\mathrm{pol}","kind":"use","occurrences":[125]},{"id":"symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","occurrences":[52,55,62,79]},{"id":"symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","occurrences":[0,15]},{"id":"symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","occurrences":[119,121,123]},{"id":"symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","occurrences":[119,121,123]},{"id":"symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","occurrences":[65,66,74,75,76,78]},{"id":"symbol-1ecf96b50772dd74","tex":"\\operatorname{Res}","aliases":"Res operatorname","display":"\\operatorname{Res}","kind":"use","occurrences":[116]},{"id":"symbol-0e2d62a66cd8a3e6","tex":"\\operatorname{Res}_{z=\\tau-2\\delta}","aliases":"Res_z=τ-2δ operatorname tau delta","display":"\\operatorname{Res}_{z=\\tau-2\\delta}","kind":"use","occurrences":[116]},{"id":"symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","occurrences":[62]},{"id":"symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","occurrences":[37]},{"id":"symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","occurrences":[68,104,105]},{"id":"symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","occurrences":[17,21,22,23,27,36,44]},{"id":"symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","occurrences":[62,119,120,121,123]},{"id":"symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","occurrences":[104]},{"id":"symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","occurrences":[15]},{"id":"symbol-3f5afdf091510edb","tex":"\\operatorname{Tr}_{K/\\mathbb Q}","aliases":"Tr_K/ Q operatorname mathbb","display":"\\operatorname{Tr}_{K/\\mathbb Q}","kind":"use","occurrences":[15]},{"id":"symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","occurrences":[124,125]},{"id":"symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","occurrences":[21,24,25,27,28,36,37,50,51,59,62,63,64,85,87]},{"id":"symbol-a257f9d1ed0e54ad","tex":"u^{\\mathsf T}","aliases":"u^ T mathsf","display":"u^{\\mathsf T}","kind":"use","occurrences":[36]},{"id":"symbol-fd49da06a993b46d","tex":"U_{\\mathrm{alg}}","aliases":"U_alg mathrm","display":"U_{\\mathrm{alg}}","kind":"use","occurrences":[124]},{"id":"symbol-f3b9ec843ee07186","tex":"U_\\Gamma","aliases":"U_Γ Gamma","display":"U_\\Gamma","kind":"use","occurrences":[124]},{"id":"symbol-349daf8c6dd8d0cb","tex":"U_\\Gamma^3","aliases":"U_Γ^3 Gamma","display":"U_\\Gamma^3","kind":"use","occurrences":[125]},{"id":"symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","occurrences":[22,25,26,27,30,37,62,64,84]},{"id":"symbol-a675678fa6689935","tex":"\\vee","aliases":"vee","display":"\\vee","kind":"use","occurrences":[47,48]},{"id":"symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","occurrences":[23,26,27,32,38,52,62,86]},{"id":"symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","occurrences":[42,44,49,70]},{"id":"symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","occurrences":[13,15]},{"id":"symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","occurrences":[70]},{"id":"symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","occurrences":[70]},{"id":"symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","occurrences":[43,44,49,56,61]},{"id":"symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","occurrences":[7,40,77,91,113,117]},{"id":"symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","occurrences":[77,88,96,97,98,101,102,109,114,116,119,121,123]},{"id":"symbol-82a6f2d2c744dfea","tex":"z^7","aliases":"z^7","display":"z^7","kind":"use","occurrences":[77]},{"id":"symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","occurrences":[66]},{"id":"symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","occurrences":[0,1,7,8,9,11,13,14,15,29,31,34,35,78,83,92,93]},{"id":"symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","occurrences":[7,8,13,15,29,35,83]},{"id":"symbol-1210911b063d7dfc","tex":"\\beta^3","aliases":"β^3 beta","display":"\\beta^3","kind":"use","occurrences":[1,14,34]},{"id":"symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","occurrences":[96,124,125]},{"id":"symbol-38a0212391c9e2e9","tex":"\\Gamma_{\\mathrm{ell}}","aliases":"Γ_ell Gamma mathrm","display":"\\Gamma_{\\mathrm{ell}}","kind":"use","occurrences":[96]},{"id":"symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","occurrences":[83,85,87,88,92,93,97,112,114,116,123]},{"id":"symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","occurrences":[11,93,109,123]},{"id":"symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","occurrences":[44,55,65,74,79]},{"id":"symbol-c09239290921b6d4","tex":"\\zeta_7^3","aliases":"ζ_7^3 zeta","display":"\\zeta_7^3","kind":"use","occurrences":[55]},{"id":"symbol-76738c7c39098112","tex":"\\zeta_7^{3k}","aliases":"ζ_7^3k zeta","display":"\\zeta_7^{3k}","kind":"use","occurrences":[79]},{"id":"symbol-a0e51cfac9f120cf","tex":"\\zeta_7^{3r}","aliases":"ζ_7^3r zeta","display":"\\zeta_7^{3r}","kind":"use","occurrences":[65]},{"id":"symbol-fdca4f874947d601","tex":"\\zeta_7^{a^{\\mathsf T}Hd-b^{\\mathsf T}Hc}","aliases":"ζ_7^a^ THd-b^ THc zeta mathsf mathsf","display":"\\zeta_7^{a^{\\mathsf T}Hd-b^{\\mathsf T}Hc}","kind":"use","occurrences":[44]},{"id":"symbol-9b05e0da34666219","tex":"\\zeta_7^{rk}","aliases":"ζ_7^rk zeta","display":"\\zeta_7^{rk}","kind":"use","occurrences":[74]},{"id":"symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","occurrences":[121]},{"id":"symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","occurrences":[121]},{"id":"symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","occurrences":[113]},{"id":"symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","occurrences":[13,16]},{"id":"symbol-64100f915b9d565e","tex":"\\Lambda_5","aliases":"Λ_5 Lambda","display":"\\Lambda_5","kind":"use","occurrences":[113]},{"id":"symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","occurrences":[81]},{"id":"symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","occurrences":[84,85,86,87,90]},{"id":"symbol-74f9f064f393eb05","tex":"\\Pi_{\\chi^{-1}}","aliases":"Π_χ^-1 Pi chi","display":"\\Pi_{\\chi^{-1}}","kind":"use","occurrences":[81]},{"id":"symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","occurrences":[82,83,92,93,96,113,119,123]},{"id":"symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","occurrences":[82,83,96,111,113,114,116,120]},{"id":"symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","occurrences":[70,73,74,76,81,99,102,107,116,118]},{"id":"symbol-7c934e3eb93a78aa","tex":"\\chi^{-1}","aliases":"χ^-1 chi","display":"\\chi^{-1}","kind":"use","occurrences":[81]},{"id":"symbol-2e32874fc399e98b","tex":"\\chi_M","aliases":"χ_M chi","display":"\\chi_M","kind":"use","occurrences":[70]},{"id":"symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","occurrences":[2,3,4,39,40,42,43,53,54,73,77,91,99,102,106,108,116,117]},{"id":"symbol-f2f53155f7e59bd3","tex":"\\omega^2","aliases":"ω^2 omega","display":"\\omega^2","kind":"use","occurrences":[2,73,77,106,108]},{"id":"symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","occurrences":[73,100,122]},{"id":"symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","occurrences":[48]},{"id":"symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","occurrences":[94,95,103,109]},{"id":"symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","occurrences":[66]},{"id":"symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","occurrences":[117]},{"id":"symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","occurrences":[98,123]},{"id":"symbol-d2c602c786fba304","tex":"\\prod_{j=0}^{8k-1}","aliases":"∏_j=0^8k-1 prod","display":"\\prod_{j=0}^{8k-1}","kind":"use","occurrences":[123]},{"id":"symbol-1cfbf8cb0a797584","tex":"\\prod_{k=0}^6","aliases":"∏_k=0^6 prod","display":"\\prod_{k=0}^6","kind":"use","occurrences":[98]},{"id":"symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","occurrences":[74,81,102]},{"id":"symbol-913ac2d9fb8649e2","tex":"\\sum_{k=1}^6","aliases":"∑_k=1^6 sum","display":"\\sum_{k=1}^6","kind":"use","occurrences":[102]},{"id":"symbol-edb8cb451c5ccc6b","tex":"\\sum_{k=1}^{6}","aliases":"∑_k=1^6 sum","display":"\\sum_{k=1}^{6}","kind":"use","occurrences":[74]},{"id":"symbol-982415998cf9e445","tex":"\\sum_{k\\ne0}","aliases":"∑_k≠0 sum ne","display":"\\sum_{k\\ne0}","kind":"use","occurrences":[81]},{"id":"symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","occurrences":[75,81]},{"id":"symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","occurrences":[27,29,31,33]},{"id":"symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","occurrences":[104]}],"contexts":["**Author:** Komorebi P.  \n**Research status:** New exact organization and checks of finite algebra. The analytic comparison is open.  \n**Working case:** \\(K=\\mathbb Q(\\beta)\\), \\(\\beta^3=7\\), with the selected complex embedding.","Put \\(\\omega^2+\\omega+1=0\\), \\(J=2+3\\omega\\), \\(\\bar J=-1-3\\omega\\). Then \\(J+\\bar J=1\\), \\(J\\bar J=7\\). With","\\[\nL=\\mathbb Z(\\beta^2+2\\beta)+\\mathbb Z(-\\beta-2)+15\\mathbb Z\n\\]","and basis \\(b_0=\\beta^2+2\\beta\\), \\(b_1=-\\beta-2\\), \\(b_2=15\\), multiplication by \\(\\epsilon=2-\\beta\\) has matrix","\\[\nM=\\begin{pmatrix}0&1&0\\\\-4&2&15\\\\-1&0&4\\end{pmatrix},\\quad \\det M=1.\n\\]","The coefficient functional \\(\\lambda(x)=[\\beta^2]x\\), applied after reduction via \\(\\beta^3=7\\), is equivalently \\(\\operatorname{Tr}_{K/\\mathbb Q}(x/(3\\beta^2))\\). The Gram matrix \\(H_{ij}=\\lambda(b_i b_j)\\) is","\\[\nH=\\begin{pmatrix}4&-4&15\\\\-4&1&0\\\\15&0&0\\end{pmatrix},\\qquad \\det H=-225,\\qquad M^{\\mathsf T}H=HM.\n\\]","Also \\((M-2I)^3=-7I\\). Thus over \\(\\mathbb F_7\\), \\(N=M-2I\\) has one nilpotent block of length three. Set \\(u=(1,2,4)^{\\mathsf T}\\), \\(v=(0,1,2)^{\\mathsf T}\\), \\(w=(0,0,1)^{\\mathsf T}\\). Then \\(Nu=0\\), \\(Nv=u\\), \\(Nw=v\\) modulo seven, and","\\[\n[u\\ v\\ w]^{\\mathsf T}H[u\\ v\\ w]\\equiv\n\\begin{pmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\end{pmatrix}\\pmod7.\n\\]","Indeed \\(u\\) represents \\(\\beta^2+56\\equiv\\beta^2\\), \\(v\\) represents \\(28-\\beta\\equiv-\\beta\\), and \\(w\\) represents \\(15\\equiv1\\) in \\(\\mathbb F_7[\\beta]/(\\beta^3)\\). The codifferent pairing extracts the coefficient of \\(\\beta^2\\). It is perfect, and pairs the socle against the top quotient. In particular, \\(u^{\\mathsf T}H\\) vanishes on \\(\\ker N^2=\\operatorname{span}(u,v)\\) and is nonzero on \\(w\\).","On \\(A=E_\\omega^3\\), \\(E_\\omega=\\mathbb C/\\mathbb Z[\\omega]\\), use the product principal Weil pairing composed with the integral symmetric isogeny represented by \\(H\\). For seventh-torsion coordinates \\(X=(a+\\omega b)/7\\), \\(Y=(c+\\omega d)/7\\), write","\\[\ne_{7,H}(X,Y)=\\zeta_7^{a^{\\mathsf T}Hd-b^{\\mathsf T}Hc}.\n\\]","The pairing is perfect since \\(7\\nmid\\det H\\). Its source is the canonical pairing of \\(A[7]\\) with \\(A^\\vee[7]\\). The symmetric map \\(H:A\\to A^\\vee\\) need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance","\\[\ne_{7,H}(MX,M^{-1}Y)=e_{7,H}(X,Y).\n\\]","Define \\(P=u/J\\), \\(P'=u/\\bar J\\), and \\(Q=w/\\bar J\\). The coefficient expansions \\(1/J=(-1-3\\omega)/7\\) and \\(1/\\bar J=(2+3\\omega)/7\\) yield","\\[\ne_{7,H}(P,Q)=\\zeta_7^3,\\qquad e_{7,H}(P,P')=1.\n\\]","**Fixed-point no-go theorem.** If \\(Y=a/\\bar J\\) is an opposite-twist fixed direction, equivalently \\(Ma=2a\\pmod7\\), then \\(a\\) is a multiple of \\(u\\) because \\(\\ker(M-2I)\\) is one-dimensional. Consequently \\(e_{7,H}(P,Y)=1\\). Thus this pairing cannot recover the Gauss phase from opposite fixed eigendirections alone. One must use the generalized eigenchain, a covariantly transported dual, or an equivalent extension datum.","The nontrivial phase remains unchanged for \\(Q=(w+su+tv)/\\bar J\\), since \\(u\\) pairs trivially with \\(u,v\\) modulo seven. Other nonzero top-quotient choices produce phases \\(\\zeta_7^{3r}\\), \\(r\\in\\mathbb F_7^\\times\\).","There cannot be a positive definite real symmetric matrix \\(H_+\\) with \\(M^\\mathsf TH_+=H_+M\\): that would make \\(M\\) similar to a real symmetric matrix, but \\(\\chi_M(X)=X^3-6X^2+12X-1\\) has negative discriminant \\(-1323\\) and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to \\(H\\) modulo seven still give the same finite pairing, but do not preserve the global self-adjoint identity.","Let \\(\\chi(1),\\ldots,\\chi(6)=(1,\\omega,\\omega^2,\\omega^2,\\omega,1)\\) and \\(G_r=\\sum_{k=1}^{6}\\chi(k)\\zeta_7^{rk}\\). Then for every \\(r\\ne0\\),","\\[\nG_r=\\chi(r)^{-1}G_1,\\qquad G_r^3=7J.\n\\]","The first identity follows by index substitution and the second by the classical Gauss and Jacobi identities. All six cubes were also checked by exact coefficient convolution in \\(\\mathbb Z[\\omega,z]/(\\omega^2+\\omega+1,z^7-1)\\). Thus \\((G_r/\\beta)^3=J\\), independent of the chosen nonzero dual direction. The individual cube-root branch is not fixed by this statement.","Equivalently, the primitive multiplicative-character projector applied to the additive function \\(f(k)=e_{7,H}(kP,Q)=\\zeta_7^{3k}\\) produces \\(G_3/6\\):","\\[\n(\\Pi_{\\chi^{-1}}f)(1)=\\frac16\\sum_{k\\ne0}\\chi(k)f(k)=\\frac{G_3}{6}.\n\\]","In the original analytic period row \\(p=(\\tau,\\sigma,1)\\), let","\\[\n\\tau=\\frac{\\beta^2+2\\beta}{15},\\quad\\sigma=-\\frac{\\beta+2}{15},\\quad\n\\delta=\\frac{\\tau+2\\sigma+4}{7}=\\frac{\\beta^2+56}{105}.\n\\]","The period projection \\(\\pi(v)=pv\\) on the universal cover satisfies \\(\\pi(u/7)=\\delta\\), \\(\\pi(w)=1\\), and","\\[\nP+P'=\\frac{u}{7},\\quad\\pi(P)=\\bar J\\delta,\\quad\\pi(P')=J\\delta,\\quad\\pi(P+P')=\\delta.\n\\]","Thus the exact principal-seven shifts \\(z+k\\delta\\) in the earlier gamma-divisor paper are the projections of \\(k(P+P')\\). This holds **on the universal cover**; \\(\\pi\\) does not automatically descend from \\(A=\\mathbb C^3/\\mathbb Z[\\omega]^3\\) to the original triptic quotient because of the extra Eisenstein periods. The missing deck multipliers cannot be discarded.","From \\(\\beta\\delta=-8\\sigma-1\\) follows the exact unit carry","\\[\n\\epsilon\\delta=2\\delta+8\\sigma+1,\\qquad\\epsilon=2-\\beta.\n\\]","Hence unit multiplication acts on the seven cover labels by \\(k\\mapsto2k\\pmod7\\), with explicit period shifts. The inverse unit acts by \\(k\\mapsto4k\\). These are precisely the finite orbits present in the earlier divisor-transport calculations.","Let \\(F(z)=\\Gamma_{\\mathrm{ell}}(z;\\tau,\\sigma)^5/\\Gamma_{\\mathrm{ell}}(5z;5\\tau,5\\sigma)\\) and \\(f_k(z)=F(z+k\\delta)\\). The inherited principal-seven product is \\(P_7(z)=\\prod_{k=0}^6 f_k(z)\\), the trivial character sector.","The cubic character \\(\\chi(k)=a_k+\\omega b_k\\) has integer coefficient vectors","\\[\n(a_1,\\ldots,a_6)=(1,0,-1,-1,0,1),\\quad\n(b_1,\\ldots,b_6)=(0,1,-1,-1,1,0).\n\\]","\\[\nA(z)=\\frac{f_1f_6}{f_3f_4},\\qquad\nB(z)=\\frac{f_2f_5}{f_3f_4}.\n\\]","\\[\nD_\\chi(z)=\\sum_{k=1}^6\\chi(k)\\frac{f'_k(z)}{f_k(z)}\n=\\frac{A'(z)}{A(z)}+\\omega\\frac{B'(z)}{B(z)}\n\\]","is unambiguous and meromorphic. Under formal unit relabeling \\(f_k\\mapsto f_{2k\\bmod7}\\) the functions transform exactly as","\\[\n(A,B)\\longmapsto(B/A,1/A),\\qquad\nT=\\begin{pmatrix}-1&1\\\\-1&0\\end{pmatrix},\\quad T^3=I.\n\\]","The eigenvalues of \\(T\\) are \\(\\omega,\\omega^2\\), and \\(D_\\chi\\) transforms by \\(\\omega^2\\). This relabeling statement is exact, but it is **not** by itself an analytic functional equation under \\(z\\mapsto\\epsilon z\\): the period carries and the gamma unit cocycle must be included.","There is a concrete obstruction to replacing the pair \\((A,B)\\) by a single character-power product. The prior divisor formula gives \\(\\operatorname{ord}_\\tau F=4\\), and \\(\\delta\\) has order seven modulo \\(\\Lambda_5=\\mathbb Z\\tau+\\mathbb Z\\sigma+(1/5)\\mathbb Z\\). At \\(z=\\tau-2\\delta\\), exactly the term \\(f_2\\) has a zero, hence","\\[\n\\operatorname{Res}_{z=\\tau-2\\delta}D_\\chi(z)=4\\chi(2)=4\\omega.\n\\]","A logarithmic derivative of a single-valued meromorphic function has an integer residue. Since \\(4\\omega\\notin\\mathbb Z\\), **no single meromorphic scalar** has logarithmic derivative \\(D_\\chi\\). The two-component meromorphic realization avoids this obstruction and carries an integral order-three monomial action.","The inherited shift law is \\(F(z+\\sigma)/F(z)=q_t(z)^{-1}\\), where \\(t=-\\tau\\) and \\(q_t(z)=\\theta_0(z;t)^5/\\theta_0(5z;5t)\\). Therefore for \\(k=1,\\ldots,6\\),","\\[\nF(\\epsilon(z+k\\delta))=\nF(\\epsilon z+2k\\delta)\n\\prod_{j=0}^{8k-1}q_t(\\epsilon z+2k\\delta+j\\sigma)^{-1}.\n\\]","The analytic-to-arithmetic equality \\(U_\\Gamma=U_{\\mathrm{alg}}\\) is still unproved. In the later polarized presentation the remaining statement is \\(\\operatorname{Im}(C_{\\mathrm{pol}}U_\\Gamma^3)=0\\), given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here."]}