crystalline_obstruction

This package computes an approximation the crystalline obstruction map on its space of Tate classes. In practice, this gives rigorous upper bounds on the:

middle geometric Picard number of a given projective hypersurface;
the geometric Picard number of a Jacobian; or
the dimension of the geometric endomorphism algebra of a Jacobian.

See examples below.

For more details see:

"Effective obstruction to lifting Tate classes from positive characteristic" by Edgar Costa and Emre Can Sertoz. (arXiv:2003.11037)

To compute a p-adic approximation for the Frobenius matrix we rely on the libraries controlledreduction, and (hypellfrob). At the moment we support:

Jacobians hyperelliptic curves with a Weierstrass point over Qp (hypellfrob)
Jacobians of quartic curves (controlledreduction)
quartic and quintic surfaces (controlledreduction)

Installing

sage -pip install --upgrade git+https://github.com/edgarcosta/crystalline_obstruction.git@master#egg=crystalline_obstruction

If you don't have permissions to install it system wide, please add the flag --user to install it just for you.

sage -pip install --user --upgrade git+https://github.com/edgarcosta/crystalline_obstruction.git@master#egg=crystalline_obstruction

Examples

Jacobians of hyperelliptic curves with a Weierstrass point over Qp

Example 5.1

sage: from crystalline_obstruction import crystalline_obstruction
sage: f = ZZ['x,y']('x^5 - 2*x^4 + 2*x^3 - 4*x^2 + 3*x - 1 -y^2')
sage: crystalline_obstruction(f=f, p=31, precision=3) # bounding dim Pic
(1,
 {'dim Li': [1],
  'dim Ti': [2],
  'factors': [(t - 1, 2)],
  'p': 31,
  'precision': 3,
  'rank T(X_Fpbar)': 2})

Bounding the geometric dimension of Endomorphism algebra

sage: from crystalline_obstruction import crystalline_obstruction
sage: f = ZZ['x,y']('x^5 - 2*x^4 + 2*x^3 - 4*x^2 + 3*x - 1 -y^2')
sage: crystalline_obstruction(f=f, p=31, precision=3, tensor=True) # bounding dim End
(1,
 {'dim Li': [1],
  'dim Ti': [4],
  'factors': [(t - 1, 4)],
  'p': 31,
  'precision': 3,
  'rank T(X_Fpbar)': 4})

Example 5.2

With prec = 100 takes about 3 minutes

sage: f = ZZ['x,y']('x^5 - 2*x^4 + 7*x^3 - 5*x^2 + 8*x + 3 -y^2')
sage: crystalline_obstruction(f=f, p=4999, precision=20) # bounding dim Pic
(2,
 {'dim Li': [2],
  'dim Ti': [2],
  'factors': [(t - 1, 2)],
  'p': 4999,
  'precision': 20,
  'rank T(X_Fpbar)': 2})

Hyperelliptic curve given in a non-Weierstrass format

sage: f = ZZ['x,y']('(2*x^6+3*x^5+5*x^4+6*x^3+4*x^2+x) -y*(x^4+x^3+x) -y^2')
sage: crystalline_obstruction(f=f, p=59, precision=3)
(3,
 {'dim Li': [1, 2],
  'dim Ti': [3, 6],
  'factors': [(t - 1, 3), (t^2 + t + 1, 3)],
  'p': 31,
  'precision': 3,
  'rank T(X_Fpbar)': 9})

Jacobians of quartic plane curves

Example 5.3

sage: f = ZZ['x,y,z']('x*y^3 + x^3*z - x*y^2*z + x^2*z^2 + y^2*z^2 - y*z^3')
sage: crystalline_obstruction(f, p=31, precision=3) # bounding dim Pic
(1,
 {'dim Li': [1],
  'dim Ti': [3],
  'factors': [(t - 1, 3)],
  'p': 31,
  'precision': 3,
  'rank T(X_Fpbar)': 3})

Example 5.4

sage: f=ZZ['x,y,z']('x^3*z+2*x^2*z^2-x*z^3-y^4')
sage: crystalline_obstruction(f,p=43,precision=4,tensor=True)
(9,
 {'precision': 4,
  'p': 43,
  'rank T(X_Fpbar)': 18,
  'factors': [(t + 1, 4), (t - 1, 6), (t^2 + 1, 4)],
  'dim Ti': [4, 6, 8],
  'dim Li': [2, 3, 4]})
sage: crystalline_obstruction(f,p=43,precision=4,tensor=False)
(5,
 {'precision': 4,
  'p': 43,
  'rank T(X_Fpbar)': 7,
  'factors': [(t + 1, 2), (t - 1, 3), (t^2 + 1, 1)],
  'dim Ti': [2, 3, 2],
  'dim Li': [1, 2, 2]})

Product of 3 elliptic curves over x^3 - 3*x - 1

sage: f=ZZ['x,y,z']('x^3*z + x^2*y*z + x^2*z^2 - x*y^3 - x*y*z^2 - x*z^3 + y^2*z^2')
sage: crystalline_obstruction(f=f, p=31, precision=3) # bounding dim Pic
(3,
 {'dim Li': [1, 2],
  'dim Ti': [3, 6],
  'factors': [(t - 1, 3), (t^2 + t + 1, 3)],
  'p': 31,
  'precision': 3,
  'rank T(X_Fpbar)': 9})

Another gennus 3 plane quartic

sage: f = ZZ['x,y,z']('x^4+x^2*y^2+2*x^2*y*z-x^2*z^2-6*y^4+16*y^3*z-12*y^2*z^2-16*y*z^3-6*z^4')
sage: crystalline_obstruction(f=f, p=5003, precision=3) # bounding dim Pic
(6,
 {'dim Li': [2, 2, 2],
  'dim Ti': [2, 3, 4],
  'factors': [(t + 1, 2), (t - 1, 3), (t^2 + 1, 2)],
  'p': 5003,
  'precision': 3,
  'rank T(X_Fpbar)': 9})
sage: crystalline_obstruction(f=f, p=5003, precision=3, tensor=True) # bounding dim End
(9,
 {'dim Li': [2, 3, 4],
  'dim Ti': [4, 6, 8],
  'factors': [(t + 1, 4), (t - 1, 6), (t^2 + 1, 4)],
  'p': 5003,
  'precision': 3,
  'rank T(X_Fpbar)': 18})

Quartic surfaces

Example 5.6

sage: f = ZZ['x,y,z,w']("x^4 + y^4 + z^4 + w^4 + 101^3*x*y*z*w")
sage: crystalline_obstruction(f, p=101, precision=3)
(20,
 {'dim Li': [1, 7, 12],
  'dim Ti': [1, 7, 12],
  'factors': [(t - 1, 1), (t - 1, 7), (t + 1, 12)],
  'p': 101,
  'precision': 3,
  'rank T(X_Fpbar)': 20})
sage: crystalline_obstruction(f=f, p=101, precision=4)
(19,
 {'dim Li': [1, 6, 12],
  'dim Ti': [1, 7, 12],
  'factors': [(t - 1, 1), (t - 1, 7), (t + 1, 12)],
  'p': 101,
  'precision': 4,
  'rank T(X_Fpbar)': 20})

Example 5.7

sage: f = ZZ['x,y,z,w']("y^4 - x^3*z + y*z^3 + z*w^3 + w^4")
sage: crystalline_obstruction(f=f, p=89, precision=3)
(4,
 {'dim Li': [1, 0, 3, 0],
  'dim Ti': [1, 1, 4, 4],
  'factors': [(t - 1, 1), (t + 1, 1), (t - 1, 4), (t^4 + 1, 1)],
  'p': 89,
  'precision': 3,
  'rank T(X_Fpbar)': 10})

Example 5.8

sage: f = ZZ['x,y,z,w']("x^4 + 2*y^4 + 2*y*z^3 + 3*z^4 - 2*x^3*w- 2*y*w^3")
sage: crystalline_obstruction(f=f, p=67, precision=3)
(3,
 {'dim Li': [1, 2],
  'dim Ti': [1, 3],
  'factors': [(t - 1, 1), (t + 1, 3)],
  'p': 67,
  'precision': 3,
  'rank T(X_Fpbar)': 4})

Quintic surfaces

It takes about 1 day of CPU time to compute the zeta function of a quintic surface. We may still demonstrate the functionality of crystalline_obstruction by providing the characteristic polynomial and an approximation of the Frobenius matrix.

Example 5.10

sage: # f doesn't play a role
sage: f =  ZZ['x,y,z,w']('10*x^4*y + 3*x^4*w - 10*x*w^4 + 3*y^4*w - 3*z^5 - 23*x^3*y*z')
sage: p, prec, cp, frob_matrix = [23, 6, [1, 0, 0, 0, -85169, 0, 0, 0, -48851843370, 0, 0, 0, 603386141590105, 0, 0, 0, 3460669831806921011036, 0, 0, 0, 39592132843636527277692216, 0, 0, 0, -116969749152102735961006849109758, 0, 0, 0, 32732931572473581734064117661723788478, 0, 0, 0, -867646721730951228138863823200631295521336, 0, 0, 0, -5939048712179129622337175952133368350994228438236, 0, 0, 0, -81091389047171959500149106085257035712245669586120505, 0, 0, 0, 514141964489056211156037329282607472395213736740030063102570, 0, 0, 0, 70195024966720298936256907286265573899667849017556809742662099729, 0, 0, 0, -64542751082767918430897798773372060387158551764172664212787858136578721], [[43145601836, 35420286327, 27049816172, 20137969355, 2878219241, 11345072, 1555741252, 193785856, 2949727069, 965882079, 2037967003, 583118034, 2617496370, 658203581, 1366351294, 494709990, 1861415346, 2179594195, 919758109, 986893200, 1625552186, 2663839001, 1823113699, 2592775464, 2517487149, 1703787744, 1691432420, 1578490644, 2272546119, 2586889603, 2305183947, 1279774096, 1644135450, 2195528963, 148322262, 145807327, 1127363653, 413106703, 166922569, 340504788, 1870742237, 492979907, 943523687, 604499524, 20213918, 2038789828, 424238082, 513674962, 22472441, 54796591, 87871245, 27553740], [76351266693, 1653383152, 48979429706, 30551797230, 2636662270, 1914377860, 29116505, 181840553, 3219683244, 1951847206, 2161625514, 291127261, 2291446668, 755413587, 38054259, 434794875, 1225215129, 2944381248, 833134244, 3113042134, 323142939, 2660911193, 124099007, 1717189729, 5609125, 1402127081, 1545409583, 498455609, 2530088228, 3085028755, 1016328945, 887194111, 12243774, 1465801983, 1004836374, 1119255923, 2886081492, 24995549, 1141201120, 1886194810, 382762665, 1289586425, 3372476936, 213972473, 419694064, 399097380, 3121904126, 2489215595, 134715794, 109018138, 48155237, 141777182], [64957489594, 14744045718, 25852998108, 23745728724, 928551262, 2119591818, 833133002, 2597680053, 3144649631, 1739765931, 2861686956, 3271970478, 2476173123, 312642036, 151268378, 1700741716, 442552982, 1232672419, 782790717, 955956360, 1457689561, 3255048343, 1096779518, 670769700, 2875290559, 3012001133, 1949365782, 484734522, 183394916, 1937948168, 1757310124, 3342437878, 1426436241, 2154195870, 2057816808, 1406984566, 1301411116, 1658399590, 2256316836, 3026127020, 1357059639, 2432209106, 2126293604, 845754758, 1621494940, 2426584755, 2583759441, 75874355, 96373936, 12502618, 84684835, 79999103], [76939530034, 65245824747, 43832676029, 67810880151, 2335073137, 195753943, 2427015269, 3107385744, 1228381401, 1748677695, 508431491, 3226892364, 1325045249, 2343760697, 142545720, 2449986128, 1743282723, 1547980822, 1841846187, 613774918, 1274850877, 2797907542, 2415674015, 739347972, 1692546770, 58500684, 763010556, 2957306742, 2748952433, 115257554, 2778897789, 3086519293, 2943013254, 2741293088, 1519886575, 2560541470, 2621715536, 125521580, 3029973218, 1415617179, 3370806009, 2002478578, 1748434539, 956679664, 529020010, 2854357247, 86807198, 3066548669, 94520432, 31193870, 95605391, 58612354], [53105791051, 24076203488, 46505406209, 53244168458, 2583523622, 3144995896, 1182030145, 436692766, 1182976388, 956286663, 2564597221, 1799216676, 1699444424, 3130929993, 2911434806, 458322242, 2530153962, 2752381595, 2887021663, 690154882, 363981670, 2822543624, 3040579875, 258869991, 1975750324, 2578555553, 3342858364, 2018853405, 3124487509, 1420980043, 937696890, 2201760882, 2621837344, 2213450402, 1084670409, 2368721792, 34076869, 1698495628, 600955477, 997708007, 424868397, 2555080603, 2524548885, 1855741062, 1125757586, 525548988, 1351402927, 1344973369, 110164245, 72444915, 127864098, 38561519], [73116281986, 34596178387, 44219376616, 67215120880, 1424905522, 2044453049, 1389802646, 2895937429, 323472184, 1570197465, 3341285946, 3245268145, 2123043014, 2943437374, 2394530299, 1939293461, 1631163657, 2452635015, 1812651459, 3127706221, 679033692, 2876922777, 2965502102, 319388327, 1867262866, 2300658122, 2087304073, 1049447036, 157861742, 595560919, 2109811160, 1807585157, 1057946778, 74660530, 2505579267, 293647762, 2042485905, 2711731970, 1416122535, 2762143485, 2805277524, 1442502224, 410663689, 1136024602, 449972920, 2071799129, 3006445184, 557154070, 121564134, 110515084, 147767506, 61160560], [57888016647, 68548901805, 37281031131, 48912683660, 1452094167, 1811347957, 1242004186, 3192343213, 2974267724, 3075404244, 2670991610, 2369541788, 3404535141, 1186237282, 96965585, 1334745246, 2283126625, 536564976, 2538009566, 1856539967, 355747049, 1209052822, 474999565, 1901824230, 589784952, 10648678, 1436496211, 1340840844, 3150491171, 2067453946, 2950978522, 1692599210, 754101782, 3281361815, 1160314212, 1212925516, 1486435766, 2750992809, 2903904169, 890518140, 1377750301, 974405166, 2346248469, 1678059760, 3336730037, 3071567062, 961731982, 712060979, 19378646, 7970338, 42973401, 31378617], [20717392048, 63960583804, 16206189551, 70835291118, 890179787, 246369882, 2039339275, 1188532544, 3029210078, 840577113, 1348038326, 1256893650, 970216797, 976423968, 1900928196, 3138635246, 3066272370, 788699854, 2349683657, 1103540046, 3322743737, 3325096522, 909995575, 1683898494, 472054208, 1487587146, 1727077222, 1017381034, 2613949080, 3187769893, 1621446042, 1428755860, 1130699780, 789568035, 2587463315, 2799514460, 260450229, 1393730517, 944443871, 2667736167, 214685082, 515011078, 3186846259, 1403515591, 3833617, 305321044, 2548041913, 487299206, 57269297, 88516079, 86712302, 87305688], [35075027071, 31521865792, 55167271782, 58612187494, 1248360765, 985926303, 3182491692, 626025684, 3321887424, 554719911, 2641618218, 1558876612, 2991831490, 2333342065, 1474437080, 2562146387, 1255130010, 587532493, 2736526292, 2884975606, 2653150280, 426805204, 1138235868, 2629132806, 2834211501, 1072821752, 2506099757, 1008464900, 1339909505, 846034599, 1820232581, 3391486229, 1782743984, 2362858356, 1882488981, 241108724, 900599385, 1860824637, 1906209065, 2948893411, 2974734233, 2964874064, 3307688834, 2901920603, 2262363145, 965947445, 3222508794, 3034953822, 128500822, 8377444, 119699238, 42880020], [12179982718, 69259063145, 9011722897, 45667839451, 2393330021, 1724353654, 1650735576, 3010162513, 3235663529, 1851237041, 2985071629, 2925568904, 762733912, 494438245, 1727395427, 2799533297, 2224849685, 2013221234, 2172566108, 1536963293, 3044227491, 2163876846, 3096098494, 2189538406, 1257086252, 478001847, 1129789072, 186142933, 181073388, 724940772, 2410126438, 2938292895, 174411300, 3317716213, 1479159900, 3219607459, 2363722512, 2935695620, 1791012162, 2599629602, 2492614190, 760430209, 1369613844, 2108400064, 173124151, 1623779323, 1236277163, 785100952, 45800614, 104499655, 108755733, 5199384], [53278722209, 23310891782, 71068225159, 20445874863, 2173798724, 2172262761, 2023609736, 1627220376, 2812933557, 3281603752, 2834278385, 2316630541, 80917128, 677442391, 1926789741, 1083632534, 663335548, 508890065, 364562926, 275257261, 895061445, 2397962957, 1198035201, 1988769865, 948822565, 3403151231, 3105887685, 3274670931, 2888660896, 2726974392, 589886083, 622912887, 90433907, 1849297405, 779251316, 2857385726, 602764381, 420693920, 1379585172, 106086120, 2099829758, 1103681013, 2692724678, 1618481779, 2898690874, 784420129, 2016255279, 2385211297, 116972459, 77335372, 90287523, 142330198], [64871454621, 57326935971, 61823488871, 15658121746, 932282322, 1013928228, 2273343874, 1511453648, 1113270150, 3324083602, 2398943056, 531689505, 962939390, 1279003021, 1039158515, 1577746962, 3256722858, 3092492416, 2388047059, 1286945220, 1766272879, 2294918219, 1440110247, 1376015181, 1535446972, 3131523416, 2964546107, 3297413791, 956611377, 2738720331, 2739052060, 1573461648, 2171911850, 601236284, 1637009245, 2545193708, 1307635468, 2724228560, 856386761, 3351777419, 3328800465, 2531954356, 77859853, 1530723117, 1440033381, 2247702347, 3215229294, 2370841426, 66530325, 27998380, 99675288, 20986277], [49736139872, 72237041666, 71146475368, 62242323563, 1807199654, 200124495, 1861750180, 1907715611, 1743601319, 1287067994, 2943193229, 2820562335, 3178385111, 1840005014, 132372981, 2273796951, 318936423, 489636052, 2329646724, 2123574590, 2504866681, 62474348, 17128399, 2842504496, 2341222555, 1510167603, 1923177131, 3394786798, 341649866, 3338723953, 2891399667, 1616953245, 2675788191, 2854996026, 3284664201, 100489829, 446689808, 1325050815, 434867739, 1402852869, 490515802, 920045517, 84683677, 36877234, 2127362414, 1955956156, 614917098, 617872069, 6078916, 12266003, 37972796, 35779620], [7399384855, 77298324284, 74774063773, 78046242999, 1562270101, 2176398230, 3230893398, 1026577607, 2864188666, 1821012994, 2447450539, 948410566, 3123856481, 273531548, 1104025829, 1882728549, 2830241701, 3167543946, 2730965559, 671863902, 3326534551, 133010817, 3053187302, 451705878, 851526884, 2781203838, 1765917552, 2044844532, 2176970769, 3218528506, 1958342659, 1570821317, 518029161, 278279231, 424468956, 3191718464, 2753760744, 3249370540, 3139489926, 1994988789, 1430176271, 1698296609, 3257103140, 2832371754, 619036076, 2058080664, 2347002639, 2791873791, 25167521, 143882096, 40343003, 49993338], [1156822490, 50304973043, 65364284775, 39321505272, 1207160175, 1494871959, 1783328138, 100876298, 319092501, 861246868, 1418314021, 2402613994, 2059945389, 569955433, 3198830524, 2770357291, 291045220, 1794915239, 225402783, 1962995260, 2537373754, 164391235, 784336685, 1511535137, 2616889561, 1805956642, 1469023225, 1481058274, 2078282231, 1909209829, 1082848096, 2419955304, 610894099, 2900797628, 2422895624, 1044241354, 893826759, 87530502, 2187425718, 1921854792, 1182613862, 2395799462, 975009008, 3077205903, 1049577722, 1088501864, 620844727, 367900341, 31981472, 37463578, 110991774, 106975139], [11340704645, 61765606220, 14659972277, 5265987103, 170900764, 965053596, 2749316224, 2699033486, 1879695861, 1409000493, 1322977480, 2325591755, 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276314571, 9061563, 1193079805, 1646045991, 573010891, 1299773516, 1897238996, 488238779, 3154430289, 1500477979, 2599066010, 923288540, 1542548084, 74438040, 94098730, 87830656, 81047525], [39978686733, 35185884311, 24247113866, 60811935600, 688094266, 526776038, 3167190229, 1834094083, 2687642276, 732943645, 1017424688, 1227056555, 3073071929, 1579719764, 2026139368, 1989644210, 542111725, 2399570726, 2638342075, 3069725751, 1085841017, 1496325329, 2424861296, 1098290066, 348120709, 2392060559, 1582254456, 1692933630, 3236035370, 2902707686, 2987140226, 2583939784, 2450973311, 2359543113, 1364999538, 1546954930, 2214388618, 2190690982, 2828969088, 3305816266, 2950364514, 2424424802, 3115138101, 2732357887, 1914850648, 3039993605, 1654618114, 2731158943, 75541291, 80067391, 26906169, 13571600], [1269409031, 29005176858, 36081847379, 76288066534, 104366249, 2481506547, 3313418295, 785833709, 1108535485, 109603602, 2839325528, 2345999586, 403079554, 479210497, 1093637304, 436357909, 1429331274, 632358389, 421890978, 1121403157, 126552417, 2684023916, 1536501982, 1177836325, 1100801781, 3092856276, 946281502, 2940230760, 538440695, 1812472036, 2563916582, 1648801736, 1248653187, 335426158, 1515950424, 3039612403, 965082737, 2255317095, 1159019726, 2657287175, 765130765, 969260641, 2925080568, 1999532554, 1319441322, 2144920407, 1827634487, 2781162921, 75406158, 56427149, 78827984, 39695386], [29682091077, 8592119039, 45584334685, 69834798179, 1824058286, 2970701367, 336275065, 650569881, 1395574243, 3337823664, 3145572069, 395560141, 428159283, 976740034, 2721610194, 3200770321, 993309280, 1691802467, 669307889, 783612829, 3272472545, 970597861, 1082214860, 3092830102, 1010084445, 2742835744, 2382672971, 366241857, 180909582, 3059231449, 2370042038, 1111784695, 1934604520, 1280547678, 1776190088, 122342658, 1824602949, 3224426741, 2859641152, 848956680, 991157170, 2214979120, 1250488495, 1430347161, 50747223, 2272308092, 1154183861, 3349387949, 98119460, 28800808, 82675308, 41752351], [24378140818, 31136158667, 41034811957, 16448544553, 2194168536, 3339224778, 1271919481, 123460941, 2023266254, 2068050382, 146449947, 2477745955, 3306140014, 1789922491, 674006973, 2465329520, 3360625519, 996489513, 2941252121, 2208356914, 3175522577, 410745408, 170710462, 996954090, 427079180, 1210505686, 1933231144, 2298384457, 2177171881, 506552000, 1132767066, 2365538408, 3189658354, 188994404, 3279280591, 1098952581, 1647100081, 1201746435, 2124177397, 274828449, 1193864312, 794269097, 3116700560, 1371947723, 1095810574, 1336871665, 200882437, 1170776866, 28631297, 43726749, 89315762, 29414102], [37402184827, 50409273328, 69367993767, 4357308462, 1309386458, 1382176858, 3102550293, 703630812, 854799370, 2003378108, 1143783077, 2623868129, 1323279953, 1487205875, 2424612137, 531408790, 334162722, 1055413581, 1709290816, 688647738, 1150753457, 1390238174, 2362086246, 628094419, 2480759622, 150608738, 695909321, 1289102758, 838420196, 2918007102, 2994587511, 3224654142, 2594426887, 2733981020, 540420834, 2066973246, 2490173959, 2344325554, 2087133459, 1318283686, 2234213491, 2830555743, 3146903378, 1522002345, 327529430, 2223573001, 556803481, 1264894568, 74989499, 102218923, 100113825, 104382464], [12544689601, 49210145121, 14715495059, 72207555735, 1525784787, 1148920955, 2639887560, 1691595030, 326332372, 764307733, 1242622058, 2400248697, 1223944080, 1247454266, 2479902021, 1007199992, 3184880426, 300045948, 45547015, 935721213, 2934525219, 1555154821, 734350831, 2630308428, 2214627496, 1882853531, 1919646815, 2691733539, 1869968931, 1058671002, 1886022701, 1552854292, 1241417502, 1844222524, 3340509972, 582498874, 3303930404, 1834755586, 2097105753, 2059229192, 1485597577, 156571143, 2143536382, 3385729099, 1346515335, 3263083163, 1347623682, 1641595123, 67016319, 111968416, 4216429, 34616012], [16696504310, 58174313563, 54039350149, 20063422675, 954805923, 2389350929, 2868915695, 1756726269, 3112027765, 482627124, 2340547620, 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sage: crystalline_obstruction(None, p, prec, cp=ZZ['T'](cp), frob_matrix=matrix(frob_matrix), shift=1)
(1,
 {'precision': 6,
  'p': 23,
  'rank T(X_Fpbar)': 5,
  'factors': [(t - 1, 1), (t - 1, 1), (t + 1, 1), (t^2 + 1, 1)],
  'dim Ti': [1, 1, 1, 2],
  'dim Li': [1, 0, 0, 0]})

Example 5.10

sage: # f doesn't play a role
sage: f = ZZ['x,y,z,w']('3*x^4*z + 9*x*y^4 + 9*y^2*z^3 + z^5 + 5*w^5')
sage: p, prec, cp, frob_matrix = [31, 6, [1, -50, 1452, -78116, 2321986, -40307502, 1906053166, -11171982492, -230889485210, -46463852390356, 1022477903317523, -49641284766618410, 2911562749301475837, -31630618937964180114, 1120399902799562715761, -20083673210179063376992, -2115596073582206419866076, 54740915682760522743399988, -1121590321854386635690836935, 70848900184596578675160353810, 6872693654267406372651488488, 4849628113451439548007546839660, -1301384083974162208410945269944517, 3268802689260334562010701613725640, -1704339289519299940348514141687753010, 95392689820377535587415105215080269670, -1648118163971196864446307190441090159924, 91672374917382811699505916111692139152870, -1573993124996153400210600128645615347548210, 2901074419181246091076020443574145624080840, -1109938821489928668170426767342450900639660997, 3974892383122137044267325691380592570645367660, 5413365015846166068173386436794750287506651368, 53628645311793937109461707184255025635872452773010, -815870733244748194617159391892361411291869179745735, 38266835270096239552058525066271147853504496689357108, -1421237405039364562023237580343205232021196116643187676, -12965832610082491520272303206906341202281203416259388512, 695110285041395279318564258259398495205858314435516084081, -18858699041220700504796517862291225880088579208184329814034, 1668220991586525252720170152539903538634478318472771909000317, -27333410766265432700922015376550666602104614598787148463402410, 541038454672851524169165213002721966047700900469015482515766803, -23627229803324070371704332604903825996040464394726192674223665876, -112830141677901665752197721953714192901947046995168296826877882010, -5246559616888568632410536919581193848905619557698946655009299535772, 860206586798100164968002237461960092955845352143879520297981444344366, -17481432806887543949976999223811177021113136918628928243723134680379822, 967774390739217006844241355155973280576184016956324151876974366866051906, -31288007937606976680344158900651775007624986641817120761085072083150492196, 558892035074946518823354878559069442827841835992186484595176394116568546732, -18495015348038002912852756139644136864929614476187713901376188524312065200050, 355474194989290415985029973003960310543947190232327861184450343437277893144961], [[791205227773, 315606997590, 565469833501, 578557400357, 10092875530, 1538779860, 3915753189, 13342654448, 18520623105, 271807163, 25186898104, 3669759137, 26911843318, 23877977943, 25135788838, 6237155391, 7130481740, 9872243291, 25707080708, 23139261645, 6307963049, 37954757, 10917007957, 19265913387, 3228193711, 24740328937, 17587989314, 20845291901, 26152411983, 25037118442, 23452832194, 20983311434, 3144769022, 23952960929, 5743303412, 2975338894, 14928854047, 8433924790, 12673637527, 5302123843, 24686658420, 5737347909, 19949226578, 980205306, 10957471668, 16497243774, 20160680957, 5889034505, 300904776, 780250744, 823662214, 623656], [163895925509, 651002798587, 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25412244730, 19533756394, 4524198683, 20285140935, 16795289554, 10236074822, 22091676878, 3726042117, 5415910273, 4916607750, 787934967, 17453668856, 261787343, 23403457657, 21269912789, 3864649596, 24856481628, 25606636368, 6645798910, 3087924663, 25037777378, 1594752282, 11921948287, 25684438804, 11797570924, 10079143398, 12927748619, 24766835177, 4071288404, 12987941877, 3878266811, 12037104731, 26731800805, 23102565705, 15589451244, 26304278565, 249349740, 11770607496, 23502971997, 25642013940, 19873357395, 22663999095, 15168302945, 21406185751, 242214291, 560942446, 628640707, 368957802], [368673883675, 670069935200, 400303518847, 81125867136, 10745439914, 5780032181, 1767879470, 42162697, 10246327514, 10390260514, 8446329905, 23824877299, 10670204774, 17619338777, 6788197286, 13015624040, 21418212046, 19397199441, 24963226935, 9950178903, 19730050006, 27399557328, 10225677980, 5148128553, 20646100068, 27379544689, 26921074343, 530606168, 24882158525, 26942088530, 21484899618, 6556733647, 2202836688, 9850314480, 13862294893, 11976654297, 15045879543, 12840607309, 11555662207, 7246343682, 1564923834, 480539277, 10574908201, 1768193965, 10251206821, 1205460203, 20957651370, 19610763711, 243350809, 414852066, 521697079, 873837196], [671510222418, 125190633771, 813362902907, 333370418539, 11908392111, 26683034798, 18944207942, 7776015324, 1878871529, 9455578584, 15401073761, 10581728294, 8009287720, 3444414123, 18584766011, 13445722442, 20204826848, 7886250332, 18358307942, 12403744366, 21503358320, 18291571855, 22960635522, 23843373015, 13178049160, 3858383039, 7005297478, 26331831148, 14050590970, 12311852213, 9224015094, 17226895889, 16366207766, 10656881811, 17948843481, 9537592804, 23080321376, 9000239362, 6968132322, 16076729921, 25249424701, 2855114167, 15092662077, 25125206709, 9864646586, 25174574209, 11651271353, 5132738324, 164725465, 365366414, 122938822, 380440573], [373323478443, 384301104451, 145220512677, 336204900532, 4052674671, 24843684983, 18885168287, 24225676733, 26181232373, 12296557278, 24752565598, 10380385805, 23982413099, 19756752414, 10969055810, 13198028319, 390039520, 17603198100, 23843554148, 24848689530, 16914489576, 17316534528, 6167264620, 15331181502, 16067991083, 4451623157, 1719208323, 6829665831, 4681019561, 25263537017, 17815087192, 4842446047, 15774164868, 19973117782, 1105351562, 15448551501, 10273192052, 20789398188, 15544214153, 24530656717, 14030707136, 18593536438, 8840116240, 26605433676, 8059651343, 16641146921, 23722504077, 19542138453, 691200696, 668540204, 36483819, 147189896], [774014642587, 336933985285, 800807986638, 823742777305, 17771088489, 24982956079, 24042454953, 7913613474, 16909832353, 4854813497, 13489664942, 17804346188, 15703511683, 19265176176, 22776043852, 8489084702, 25451300049, 14895812511, 8001839319, 9723370658, 6187079045, 7111819864, 25389407154, 9806309205, 9263139047, 27418386015, 18253714717, 23135324769, 6250487406, 11009631214, 11199447934, 5214166489, 23475601973, 22933124107, 17776714369, 26114672366, 8583716542, 24898092308, 22735407874, 18208478153, 17461181427, 21545701096, 13050900614, 9527447775, 19907151053, 3769839537, 9219735482, 22625061669, 718461975, 154987717, 415301024, 128389746], [416261089139, 822930715007, 589012936207, 458367437718, 22542210480, 12392986467, 25421592873, 14047586946, 17237053876, 19320274491, 25026749345, 17592818339, 15344339297, 7641355106, 15893944497, 20545827778, 10347616046, 8962959971, 13548933873, 6354561753, 3055000710, 8985611950, 940932243, 6537855577, 22569696754, 9083975105, 17799227623, 4129067382, 6444914384, 4933242040, 16833860777, 3690282935, 11057788133, 13288363156, 15413079441, 9442002661, 5522371465, 12534264348, 10669725297, 24004706222, 9540500077, 10393797273, 15159838457, 4740472477, 417625583, 21694178572, 27082266593, 13735890564, 647719800, 543736300, 312282009, 717365730], [164690618693, 847471799876, 310874457951, 795033245382, 26539974634, 7093319033, 26625157736, 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sage: crystalline_obstruction(None, p, prec, cp=ZZ['T'](cp), frob_matrix=matrix(frob_matrix), shift=1)
(1,
 {'precision': 6,
  'p': 31,
  'rank T(X_Fpbar)': 1,
  'factors': [(t - 1, 1)],
  'dim Ti': [1],
  'dim Li': [1]})
sage: p, prec, cp, frob_matrix = [29, 20, [1, 0, -2842, 0, 6013150, 0, -10071827774, 0, 15254991663062, 0, -20413558347816132, 0, 25183725078986957040, 0, -28930070889860844230928, 0, 31354850260650911077304990, 0, -32210910187460313693800838218, 0, 31469749051369186051984935167815, 0, -29487887771743951016409382448309010, 0, 26375254206671919727420037524201218102, 0, -22687771924747537276536393100397652851152, 0, 18654716170549122056729371560154561740400662, 0, -14751210083611446627308483619671774252597078610, 0, 11134462438140703573965162950721453584523122115415, 0, -8060666688446319677418837207062523136249075778277578, 0, 5549638432165421174581815213181323814007419569332900990, 0, -3621608054910425904455783818875359831667388368853669213968, 0, 2229789730011501301782431770242470466232536768280068005971440, 0, -1278364339274865965744914151333054799649434908505466913889814212, 0, 675678187351849649046322496707080304502427854168152833458887250102, 0, 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sage: crystalline_obstruction(None, p, prec, cp=ZZ['T'](cp), frob_matrix=matrix(frob_matrix), shift=1)
(3,
 {'precision': 20,
  'p': 29,
  'rank T(X_Fpbar)': 5,
  'factors': [[(29*T - 1, 1)], (29*T - 1, 2), (29*T + 1, 2)],
  'dim Ti': [[1], [2], [2]],
  'dim Ki': [[1], [1], [1]]})
sage: p, prec, cp, frob_matrix = [23, 16, [1, 0, 0, 0, -1329377, 0, 0, 0, 944707116511, 0, 0, 0, -475048924246822985, 0, 0, 0, 186554727206404860527516, 0, 0, 0, -61591830166121444161069480130, 0, 0, 0, 18295662732925011468096778914628840, 0, 0, 0, -5119876554844468134243670708248649214440, 0, 0, 0, 1349761826171331226472102726373233629065721730, 0, 0, 0, -320156982958304705872194217790579355890918360850716, 0, 0, 0, 63843655790670841419910707994522381723827254660386815785, 0, 0, 0, -9942584337524400988781250471688178784684095736164873856654271, 0, 0, 0, 1095652781002286405135488248511710479564380773795778030331117121857, 0, 0, 0, -64542751082767918430897798773372060387158551764172664212787858136578721], [[0, 0, 0, 0, 2931369991417881028711, 0, 45671654465898100062831, 0, 0, 0, 34599003473583763402644, 0, 0, 0, 18038971750047685691957, 0, 0, 0, 0, 97394571556004121963191, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125909835698288480587686, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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sage: crystalline_obstruction(None, p, prec, cp=ZZ['T'](cp), frob_matrix=matrix(frob_matrix), shift=1)
(1,
 {'precision': 16,
  'p': 23,
  'rank T(X_Fpbar)': 5,
  'factors': [(t - 1, 1), (t - 1, 1), (t + 1, 1), (t^2 + 1, 1)],
  'dim Ti': [1, 1, 1, 2],
  'dim Li': [1, 0, 0, 0]})

Citing this code

Please cite the following preprint if this code has been helpful in your research:

"Effective obstruction to lifting Tate classes from positive characteristic" by Edgar Costa and Emre Can Sertoz. (arXiv:2003.11037)

Name		Name	Last commit message	Last commit date
Latest commit History 67 Commits
crystalline_obstruction		crystalline_obstruction
.gitignore		.gitignore
GPLv2		GPLv2
GPLv3		GPLv3
LICENSE		LICENSE
Makefile		Makefile
README.md		README.md
VERSION		VERSION
setup.py		setup.py

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crystalline_obstruction

Installing

Examples

Jacobians of hyperelliptic curves with a Weierstrass point over Qp

Example 5.1

Bounding the geometric dimension of Endomorphism algebra

Example 5.2

Hyperelliptic curve given in a non-Weierstrass format

Jacobians of quartic plane curves

Example 5.3

Example 5.4

Product of 3 elliptic curves over x^3 - 3*x - 1

Another gennus 3 plane quartic

Quartic surfaces

Example 5.6

Example 5.7

Example 5.8

Quintic surfaces

Example 5.10

Example 5.10

Citing this code

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