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{
"Title": "Zeta polynomials of Fermat curves",
"Definition": "The zeta polynomial $P_{N,p}(T)$ is the numerator with constant term $1$ of the local zeta function of the smooth projective Fermat curve $C_N:X^N+Y^N=Z^N$ over $\\mathbb F_p$, as defined in CITE{formula-zeta} CITE{Otsubo2014}.",
"Parameters": {
"N": {
"type": "Z",
"title": "curve degree",
"display": "$N$",
"constraints": "$N\\geq3$"
},
"p": {
"type": "Z",
"title": "rational prime",
"display": "$p$",
"constraints": "$p$ is prime and $p\\nmid N$"
}
},
"Tags": [
"number theory",
"polynomial",
"L-function"
],
"References": {
"Otsubo2014": {
"bib": "Noriyuki Otsubo, On special values of Jacobi-sum Hecke L-functions, 2014, Sections 2.1 and 2.2.",
"arxiv": "1404.7476v1"
}
},
"Links": {
"LMFDBWeil": {
"title": "LMFDB: characteristic polynomial of an abelian variety over a finite field",
"url": "https://www.lmfdb.org/knowledge/show/av.fq.weil_polynomial"
}
},
"Data properties": {
"type": "Z[]",
"rigour": "exact",
"complete": "no",
"complete-note": "it holds every degree $3\\leq N\\leq8$ at every prime $p\\leq101$ with $p\\equiv1\\pmod N$; this initial degree range covers odd, even, prime and composite curves, while the split-prime slice gives several directly computable reductions per curve. The bounds are practical, noncanonical choices balancing breadth and expanded polynomial length, which reaches $1335$ characters. Nonsplit good primes remain uncomputed; the attached argument-selection note gives the coverage rationale",
"rigour details": "Computed in exact cyclotomic fields by the full positive Jacobi-sum product, with every coefficient required to coerce exactly to an integer. All 48 polynomials were independently recomputed from additive Gauss sums in $\\mathbb Q(\\zeta_{Np})$, using a different primitive root and explicit coefficient convolution. Sage's DirichletGroup implementation gives the same complete polynomials. Direct projective point counts over $\\mathbb F_p$ and $\\mathbb F_{p^2}$ agree with CITE{formula-counts}; the cubic counts also agree with the elliptic model $v^2=u^3-432$. The checks use nonzero controls from projective lines, conics, the cubic at $p=7$ and the quartic at $p=5$. Degree, leading coefficient, reciprocity, conjugate Jacobi norms and all character-orbit products were checked throughout. The reproducible generator, independent checks and numerical evidence are attached; the same checks are applied to the values read back from the draft."
},
"Display properties": {
"number-header": "$P_{N,p}(T)$",
"group parameters": [
[
"N"
],
[
"p"
]
]
},
"Numbers": {
"3": {
"7": "7*T^2 + T + 1",
"13": "13*T^2 - 5*T + 1",
"19": "19*T^2 + 7*T + 1",
"31": "31*T^2 + 4*T + 1",
"37": "37*T^2 - 11*T + 1",
"43": "43*T^2 - 8*T + 1",
"61": "61*T^2 + T + 1",
"67": "67*T^2 - 5*T + 1",
"73": "73*T^2 + 7*T + 1",
"79": "79*T^2 - 17*T + 1",
"97": "97*T^2 + 19*T + 1"
},
"4": {
"5": "125*T^6 + 50*T^5 + 55*T^4 + 12*T^3 + 11*T^2 + 2*T + 1",
"13": "2197*T^6 - 1014*T^5 + 39*T^4 + 60*T^3 + 3*T^2 - 6*T + 1",
"17": "4913*T^6 - 1734*T^5 + 1071*T^4 - 212*T^3 + 63*T^2 - 6*T + 1",
"29": "24389*T^6 + 8410*T^5 - 377*T^4 - 420*T^3 - 13*T^2 + 10*T + 1",
"37": "50653*T^6 + 2738*T^5 + 3959*T^4 + 140*T^3 + 107*T^2 + 2*T + 1",
"41": "68921*T^6 - 50430*T^5 + 17343*T^4 - 3460*T^3 + 423*T^2 - 30*T + 1",
"53": "148877*T^6 - 39326*T^5 - 1961*T^4 + 1260*T^3 - 37*T^2 - 14*T + 1",
"61": "226981*T^6 + 37210*T^5 + 5063*T^4 + 220*T^3 + 83*T^2 + 10*T + 1",
"73": "389017*T^6 + 95922*T^5 + 23871*T^4 + 2844*T^3 + 327*T^2 + 18*T + 1",
"89": "704969*T^6 - 237630*T^5 + 50463*T^4 - 6340*T^3 + 567*T^2 - 30*T + 1",
"97": "912673*T^6 - 508086*T^5 + 122511*T^4 - 16308*T^3 + 1263*T^2 - 54*T + 1",
"101": "1030301*T^6 + 20402*T^5 + 30199*T^4 + 396*T^3 + 299*T^2 + 2*T + 1"
},
"5": {
"11": "1771561*T^12 + 483153*T^11 - 351384*T^10 - 26620*T^9 + 78045*T^8 + 4488*T^7 - 7131*T^6 + 408*T^5 + 645*T^4 - 20*T^3 - 24*T^2 + 3*T + 1",
"31": "887503681*T^12 + 944761983*T^11 + 504242466*T^10 + 190066580*T^9 + 56405895*T^8 + 13479048*T^7 + 2649259*T^6 + 434808*T^5 + 58695*T^4 + 6380*T^3 + 546*T^2 + 33*T + 1",
"41": "4750104241*T^12 - 3128117427*T^11 + 1288547016*T^10 - 390782070*T^9 + 96396945*T^8 - 19423422*T^7 + 3305729*T^6 - 473742*T^5 + 57345*T^4 - 5670*T^3 + 456*T^2 - 27*T + 1",
"61": "51520374361*T^12 + 2533788903*T^11 + 3821452116*T^10 + 165696130*T^9 + 136356045*T^8 + 4920138*T^7 + 2840869*T^6 + 80658*T^5 + 36645*T^4 + 730*T^3 + 276*T^2 + 3*T + 1",
"71": "128100283921*T^12 - 102841073007*T^11 + 42844094166*T^10 - 12104550020*T^9 + 2559794595*T^8 - 423542832*T^7 + 56029839*T^6 - 5965392*T^5 + 507795*T^4 - 33820*T^3 + 1686*T^2 - 57*T + 1",
"101": "1061520150601*T^12 - 914378743587*T^11 + 381485430066*T^10 - 102484040470*T^9 + 19759285995*T^8 - 2877443742*T^7 + 324275199*T^6 - 28489542*T^5 + 1936995*T^4 - 99470*T^3 + 3666*T^2 - 87*T + 1"
},
"6": {
"7": "282475249*T^20 + 161414428*T^19 - 46118408*T^18 - 65883440*T^17 + 1176490*T^16 + 15798580*T^15 + 3346994*T^14 - 2224012*T^13 - 949963*T^12 + 144340*T^11 + 182860*T^10 + 20620*T^9 - 19387*T^8 - 6484*T^7 + 1394*T^6 + 940*T^5 + 10*T^4 - 80*T^3 - 8*T^2 + 4*T + 1",
"13": "137858491849*T^20 + 42417997492*T^19 + 32629228840*T^18 + 23342448324*T^17 + 6921644106*T^16 + 3559957284*T^15 + 1524986034*T^14 + 395433636*T^13 + 161188989*T^12 + 50890528*T^11 + 11400076*T^10 + 3914656*T^9 + 953781*T^8 + 179988*T^7 + 53394*T^6 + 9588*T^5 + 1434*T^4 + 372*T^3 + 40*T^2 + 4*T + 1",
"19": "6131066257801*T^20 - 2581501582232*T^19 + 67934252164*T^18 + 57207791296*T^17 - 94091762*T^16 + 2713804504*T^15 + 1457770706*T^14 - 981714952*T^13 + 58990649*T^12 + 12938392*T^11 + 180436*T^10 + 680968*T^9 + 163409*T^8 - 143128*T^7 + 11186*T^6 + 1096*T^5 - 2*T^4 + 64*T^3 + 4*T^2 - 8*T + 1",
"31": "819628286980801*T^20 + 423033954570736*T^19 + 305334991403878*T^18 + 108729850966672*T^17 + 45764127310765*T^16 + 12666452535232*T^15 + 3931635765704*T^14 + 888889081664*T^13 + 220326842066*T^12 + 41512246816*T^11 + 8499564964*T^10 + 1339104736*T^9 + 229268306*T^8 + 29837504*T^7 + 4257224*T^6 + 442432*T^5 + 51565*T^4 + 3952*T^3 + 358*T^2 + 16*T + 1",
"37": "4808584372417849*T^20 + 6758010469344004*T^19 + 4776972057332560*T^18 + 2229380202591372*T^17 + 763852672129026*T^16 + 201903252080340*T^15 + 42040913421138*T^14 + 6916871585508*T^13 + 898455595605*T^12 + 97624393144*T^11 + 12336924604*T^10 + 2638497112*T^9 + 656286045*T^8 + 136554036*T^7 + 22431858*T^6 + 2911620*T^5 + 297714*T^4 + 23484*T^3 + 1360*T^2 + 52*T + 1",
"43": "21611482313284249*T^20 - 16082963581978976*T^19 + 7270060572667822*T^18 - 2252832648854816*T^17 + 580990156470541*T^16 - 131888918654336*T^15 + 28568622522728*T^14 - 5680651755136*T^13 + 1041509782562*T^12 - 172596498752*T^11 + 27122718868*T^10 - 4013872064*T^9 + 563282738*T^8 - 71448448*T^7 + 8356328*T^6 - 897152*T^5 + 91909*T^4 - 8288*T^3 + 622*T^2 - 32*T + 1",
"61": "713342911662882601*T^20 - 514542428084702204*T^19 + 141096582365998816*T^18 - 14444046074352516*T^17 - 501087161035086*T^16 + 68787301138644*T^15 + 57580560508290*T^14 - 13278129741660*T^13 + 1185830475765*T^12 - 46955806520*T^11 + 1365987580*T^10 - 769767320*T^9 + 318685965*T^8 - 58498860*T^7 + 4158690*T^6 + 81444*T^5 - 9726*T^4 - 4596*T^3 + 736*T^2 - 44*T + 1",
"67": "1822837804551761449*T^20 + 435304550340719152*T^19 - 66595099119289124*T^18 - 17066963880589568*T^17 + 2151281244743158*T^16 + 349066745664208*T^15 - 63378298213150*T^14 - 5033824615024*T^13 + 1311843709553*T^12 + 35283232336*T^11 - 20640392828*T^10 + 526615408*T^9 + 292235177*T^8 - 16736848*T^7 - 3145150*T^6 + 258544*T^5 + 23782*T^4 - 2816*T^3 - 164*T^2 + 16*T + 1",
"73": "4297625829703557649*T^20 + 941945387332286608*T^19 + 80646009189408100*T^18 - 41228891273270004*T^17 - 8799479921800194*T^16 - 702024964253520*T^15 + 169208135970882*T^14 + 34674700408584*T^13 + 2589973215585*T^12 - 378664484852*T^11 - 73710548204*T^10 - 5187184724*T^9 + 486014865*T^8 + 89134152*T^7 + 5958402*T^6 - 338640*T^5 - 58146*T^4 - 3732*T^3 + 100*T^2 + 16*T + 1",
"79": "9468276082626847201*T^20 + 479406383930473276*T^19 + 534022301087109472*T^18 + 3994413069121072*T^17 + 13076646582296674*T^16 - 517302413574284*T^15 + 215735540140370*T^14 - 17731869561004*T^13 + 3257586192461*T^12 - 306724612844*T^11 + 45266926588*T^10 - 3882590036*T^9 + 521965421*T^8 - 35964436*T^7 + 5538770*T^6 - 168116*T^5 + 53794*T^4 + 208*T^3 + 352*T^2 + 4*T + 1",
"97": "73742412689492826049*T^20 - 60818484692365217360*T^19 + 26365126611484096804*T^18 - 7596654706632184260*T^17 + 1590055262376938526*T^16 - 248103923333358768*T^15 + 28228207333542738*T^14 - 2036030114927928*T^13 + 17267947950993*T^12 + 18485040574588*T^11 - 2756157412556*T^10 + 190567428604*T^9 + 1835258577*T^8 - 2230842936*T^7 + 318857298*T^6 - 28891824*T^5 + 1908894*T^4 - 94020*T^3 + 3364*T^2 - 80*T + 1"
},
"7": {
"29": "8629188747598184440949*T^30 - 2678024094082195171329*T^29 + 1231275445555032262680*T^28 - 222903313419445495830*T^27 - 4611792691436803362*T^26 + 10672501094359498968*T^25 - 6731141647051505572*T^24 + 1201436307297891129*T^23 - 122221652604241476*T^22 - 18320223401270277*T^21 + 16964965045191156*T^20 - 3088045199320956*T^19 + 466751873415982*T^18 + 10657727794014*T^17 - 23882463650193*T^16 + 4658590961810*T^15 - 823533229317*T^14 + 12672684654*T^13 + 19137802838*T^12 - 4366079676*T^11 + 827109444*T^10 - 30799437*T^9 - 7085364*T^8 + 2401689*T^7 - 463988*T^6 + 25368*T^5 - 378*T^4 - 630*T^3 + 120*T^2 - 9*T + 1",
"43": "3177070365797955661914307*T^30 + 5541401800810387782408675*T^29 + 4608384381325103886021726*T^28 + 2426268862748189113915918*T^27 + 907533397267764556092888*T^26 + 257558471197593669211332*T^25 + 58411513391159444323514*T^24 + 11275972753664176109709*T^23 + 2040134299166772393222*T^22 + 384372601207710194019*T^21 + 76845924002116456614*T^20 + 15036905383535760480*T^19 + 2667165026922242812*T^18 + 421354663118240166*T^17 + 61892600563704483*T^16 + 9176447378293162*T^15 + 1439362803807081*T^14 + 227882457067734*T^13 + 33546291860116*T^12 + 4398297936480*T^11 + 522731364498*T^10 + 60805335531*T^9 + 7505498946*T^8 + 964731309*T^7 + 116220398*T^6 + 11917668*T^5 + 976584*T^4 + 60718*T^3 + 2682*T^2 + 75*T + 1",
"71": "5873205959385493353867330551*T^30 - 4218781745474086775313152931*T^29 + 531279888410986901282186616*T^28 + 325765021767201382636952332*T^27 - 95964749156435950319302452*T^26 - 7471350361945553075623974*T^25 + 5609905725929592825029012*T^24 - 183793373665469783610777*T^23 - 188906142519814842266766*T^22 + 18411580514336103001365*T^21 + 4121829722959503692676*T^20 - 670881738931141571694*T^19 - 56879819938502744792*T^18 + 15605138096758489860*T^17 + 338899844976249381*T^16 - 257705907076333098*T^15 + 4773237253186611*T^14 + 3095643343931460*T^13 - 158921687063272*T^12 - 26400525763374*T^11 + 2284537562076*T^10 + 143727866565*T^9 - 20770054626*T^8 - 284618457*T^7 + 122357452*T^6 - 2295174*T^5 - 415212*T^4 + 19852*T^3 + 456*T^2 - 51*T + 1"
},
"8": {
"17": "69091933913008732880827217*T^42 + 24385388439885435134409606*T^41 + 33709213431606336803448573*T^40 + 9562897427406052993886120*T^39 + 8492448528525133221041082*T^38 + 2138951350086812870950836*T^37 + 1540928201319279135845690*T^36 + 357614298272980445084520*T^35 + 223333268824621158479193*T^34 + 48355304713920973820302*T^33 + 27067421023077955689117*T^32 + 5509860932802481519392*T^31 + 2825141290539357615640*T^30 + 543868982974565471088*T^29 + 258802167290987747160*T^28 + 47249585501556558752*T^27 + 21038767469650201890*T^26 + 3647510089340895084*T^25 + 1529489186848959450*T^24 + 252002805221074224*T^23 + 99927768288088188*T^22 + 15636339735318968*T^21 + 5878104016946364*T^20 + 871982024986416*T^19 + 311314713382650*T^18 + 43671772241004*T^17 + 14817525616770*T^16 + 1957512187808*T^15 + 630703816920*T^14 + 77965581168*T^13 + 23823188120*T^12 + 2733074208*T^11 + 789784749*T^10 + 82995982*T^9 + 22548489*T^8 + 2123880*T^7 + 538330*T^6 + 43956*T^5 + 10266*T^4 + 680*T^3 + 141*T^2 + 6*T + 1",
"41": "7386879101213258109624223688656841*T^42 - 4684362356866944167078775997684826*T^41 + 2342181178433472083539387998842413*T^40 - 890871274202124191570008376258552*T^39 + 320663676733250094142839759977146*T^38 - 100358499912529517826806677047948*T^37 + 29567808938748896356391444901066*T^36 - 7944957844895484474548708261560*T^35 + 2050500877704164261970135139329*T^34 - 494508704471195240429487362322*T^33 + 114812693729210403890877846877*T^32 - 25233700370897101070604668256*T^31 + 5378653160332284253006164632*T^30 - 1094649395510335728943840656*T^29 + 216379103329638628822135960*T^28 - 41049682251478081275630816*T^27 + 7590515322364408348222674*T^26 - 1351783934736676883105108*T^25 + 234724850353332013189498*T^24 - 39330691654582698372496*T^23 + 6436625440069911449340*T^22 - 1017474399394172079560*T^21 + 156990864391949059740*T^20 - 23397199080655977616*T^19 + 3405708715098910538*T^18 - 478378721603375828*T^17 + 65516694461303874*T^16 - 8641848719268576*T^15 + 1111036482115160*T^14 - 137089498536336*T^13 + 16429291281112*T^12 - 1879933013856*T^11 + 208625544197*T^10 - 21916321362*T^9 + 2216510649*T^8 - 209467960*T^7 + 19013466*T^6 - 1574028*T^5 + 122666*T^4 - 8312*T^3 + 533*T^2 - 26*T + 1",
"73": "1348279907365869037210940254745047725673*T^42 + 110817526632811153743364952444798443206*T^41 + 345355990533760787350897625769748573005*T^40 + 26867375569448679046055079481831410888*T^39 + 42613316709883766775291842261828773338*T^38 + 3134418146436355090488985946426592180*T^37 + 3368857294899905734930259688828517034*T^36 + 233955145520042182453322906664993288*T^35 + 191442052178339408895376452246137793*T^34 + 12530098185616359875601860367285390*T^33 + 8315767991947637136878051947094493*T^32 + 511842599535198122543376045414816*T^31 + 286618806207228311934765058024728*T^30 + 16546167639837969165276908256624*T^29 + 8031200810234753907022432295448*T^28 + 433438866048911702422708020768*T^27 + 185930384142147823529441096850*T^26 + 9344430668547606010327925484*T^25 + 3594445688920922771620094778*T^24 + 167427992670930556430036208*T^23 + 58403956094073296254245564*T^22 + 2506791012239852343773112*T^21 + 800054193069497208962268*T^20 + 31418275975029190547952*T^19 + 9239816483395128674634*T^18 + 329049629114268239724*T^17 + 89688356528528160450*T^16 + 2864116576121927712*T^15 + 726976653947232984*T^14 + 20517032158376304*T^13 + 4868542232903256*T^12 + 119098921082784*T^11 + 26506409478309*T^10 + 547116928590*T^9 + 114509141721*T^8 + 1916957832*T^7 + 378128762*T^6 + 4819380*T^5 + 897546*T^4 + 7752*T^3 + 1365*T^2 + 6*T + 1",
"89": "86534669543385676516186776267386878120889*T^42 - 9722996577908502979346828794088413272010*T^41 - 5189239746636560578864880536170782364275*T^40 + 692307798250270885033658810739283560840*T^39 + 154396941554954228416563011449158971834*T^38 - 25126710267413631114598751968886216620*T^37 - 3193283292041354535706578767625360406*T^36 + 641118462784575016535556625226290120*T^35 + 57239685231113575298727160214455889*T^34 - 13635003540401244601380092442369090*T^33 - 941937314414884333019690970601763*T^32 + 259037940275843684624460698296480*T^31 + 13242532571618035794172016990104*T^30 - 4398463662831730815350346833680*T^29 - 152836364395852280184076896488*T^28 + 66593565244834310305518155040*T^27 + 1515884115459531290255166002*T^26 - 918566364038535092225503540*T^25 - 12600525818233602398836038*T^24 + 11736759800234506923751280*T^23 + 54868982360979259945468*T^22 - 138038759818475275541640*T^21 + 616505419786283819612*T^20 + 1481727029445083565680*T^19 - 17873872210315066902*T^18 - 14640318060021717940*T^17 + 271466328269660098*T^16 + 133996121093540640*T^15 - 3455386656469672*T^14 - 1117328702571280*T^13 + 37797318477656*T^12 + 8307365368480*T^11 - 339415701067*T^10 - 55204588290*T^9 + 2603917081*T^8 + 327701320*T^7 - 18339494*T^6 - 1621420*T^5 + 111946*T^4 + 5640*T^3 - 475*T^2 - 10*T + 1",
"97": "527480512638944839726474542256547576188897*T^42 + 293648945180443519022985827647974939321654*T^41 + 80560048534611939067588895442943869378621*T^40 + 14855659252406325332653975338552032215656*T^39 + 2325115803991969623991649345067442565658*T^38 + 364594678639082031220478975802481773012*T^37 + 55232509405650525843384220042065031034*T^36 + 7240151369140127154358983132835571496*T^35 + 844920152224980114667576741573722921*T^34 + 102164953392949427800161741428704254*T^33 + 13260633346290850660517426206792605*T^32 + 1614837999795049571795984562723360*T^31 + 179514289815867854384980347432600*T^30 + 20794839542222220741222023780592*T^29 + 2610840309211310503676290476120*T^28 + 307692584388420925504239354528*T^27 + 32192361313205494098228046146*T^26 + 3345554163676028690149605324*T^25 + 374359545037526805907998042*T^24 + 40438896540372081947202480*T^23 + 3912425120267380773531324*T^22 + 376864719206645717749368*T^21 + 40334279590385368799292*T^20 + 4297895264148377292720*T^19 + 410179270163055996954*T^18 + 37790368631549472204*T^17 + 3748816321440597378*T^16 + 369391267134659232*T^15 + 32313066126033240*T^14 + 2653271545055472*T^13 + 236131223227800*T^12 + 21898361348640*T^11 + 1853852556285*T^10 + 147245194494*T^9 + 12554028873*T^8 + 1109030184*T^7 + 87220538*T^6 + 5935572*T^5 + 390234*T^4 + 25704*T^3 + 1437*T^2 + 54*T + 1"
}
},
"Comments": {
"comment-model": "The affine coordinates are $x=X/Z$ and $y=Y/Z$, so the equation is $x^N+y^N=1$. The sign of $Z^N$ is fixed, including at even degree. The genus is $g_N=(N-1)(N-2)/2$.",
"comment-characters": "For $p\\equiv1\\pmod N$, choose the least positive primitive root $h$ modulo $p$ and the character $\\chi(h)=\\zeta_N=\\exp(2\\pi i/N)$, with $\\chi(0)=0$. Put $I_N=\\{(a,b):1\\leq a,b<N,\\ a+b\\ne N\\}$. The product in CITE{formula-jacobi} uses all ordered pairs in $I_N$, including imprimitive pairs and both coordinate orders, with their multiplicities. Replacing $\\chi$ by another character of exact order $N$ permutes the factors.",
"comment-sign": "The positive-sum convention $J_p(a,b)$ in CITE{formula-jacobi} agrees with HREF{Jacobi_sums_of_pairs_of_Dirichlet_characters_modulo_a_prime}[the Jacobi-sum table]. Otsubo uses $j_N^{a,b}=-J_p(a,b)$, which is the Frobenius eigenvalue CITE{Otsubo2014}. Thus the numerator factors are $1+J_p(a,b)T$.",
"comment-periods": "HREF{T665}[Holomorphic periods of Fermat curves] gives complex cycle integrals for the same equation and the holomorphic characters $a+b<N$. These periods and the finite-field eigenvalues describe two realizations of Fermat cohomology CITE{Otsubo2014}; they are not asserted to be equal numerical values. The attachment records the full character-orbit factors without asserting that each factor is simple.",
"comment-reversal": "HREF{Weil_polynomials_of_abelian_varieties_over_prime_fields}[The Weil-polynomial table] stores the characteristic polynomial of Frobenius, which is the reverse of $P_{N,p}(T)$ CITE{LMFDBWeil}. For the cubic curve, the reductions at $p=7$ and $p=13$ have the class labels HREF{Weil_polynomials_of_abelian_varieties_over_prime_fields#1,7,1.7.b}[$1.7.b$] and HREF{Weil_polynomials_of_abelian_varieties_over_prime_fields#1,13,1.13.af}[$1.13.af$], respectively.",
"comment-coverage": "The definition includes all good primes. The current computation covers only $p\\equiv1\\pmod N$, the primes that split completely in $\\mathbb Q(\\zeta_N)$. Nonsplit good primes have not been computed here; an omitted prime does not signify a local factor equal to $1$."
},
"Formulas": {
"formula-zeta": "$\\displaystyle Z(C_N/\\mathbb F_p,T)=\\exp\\!\\left(\\sum_{r\\geq1}\\#C_N(\\mathbb F_{p^r})\\frac{T^r}{r}\\right)=\\frac{P_{N,p}(T)}{(1-T)(1-pT)}$.",
"formula-jacobi": "$\\displaystyle J_p(a,b)=\\sum_{x\\in\\mathbb F_p}\\chi^a(x)\\chi^b(1-x),\\qquad P_{N,p}(T)=\\prod_{(a,b)\\in I_N}(1+J_p(a,b)T)$ for $p\\equiv1\\pmod N$ CITE{Otsubo2014}.",
"formula-reciprocity": "Writing $P_{N,p}(T)=\\sum_{k=0}^{2g_N}c_kT^k$, one has $c_0=1$ and $c_{2g_N-k}=p^{g_N-k}c_k$ for $0\\leq k\\leq g_N$.",
"formula-counts": "$\\#C_N(\\mathbb F_p)=p+1+c_1$ and $\\#C_N(\\mathbb F_{p^2})=p^2+1-c_1^2+2c_2$.",
"formula-reverse": "$W_{N,p}(U)=U^{2g_N}P_{N,p}(1/U)$ is the characteristic polynomial of Frobenius on the Jacobian of $C_N$ CITE{LMFDBWeil}."
},
"Similar tables": [
{
"table": "HREF{T665}[Holomorphic periods of Fermat curves]",
"relation": "gives complex periods of the same global curves and their holomorphic character eigenspaces"
},
{
"table": "HREF{Jacobi_sums_of_pairs_of_Dirichlet_characters_modulo_a_prime}[Jacobi sums of pairs of Dirichlet characters modulo a prime]",
"relation": "gives the positive character sums whose negatives are the split-prime Frobenius eigenvalues"
},
{
"table": "HREF{Weil_polynomials_of_abelian_varieties_over_prime_fields}[Weil polynomials of abelian varieties over prime fields]",
"relation": "uses the reversed polynomial convention and indexes finite-field isogeny classes rather than global Fermat models"
},
{
"table": "HREF{Frobenius_angles_of_abelian_varieties_over_prime_fields}[Frobenius angles of abelian varieties over prime fields]",
"relation": "records arguments of normalized Frobenius eigenvalues for finite-field isogeny classes"
}
],
"Keywords": [
"Fermat curve",
"local zeta function",
"Hasse-Weil zeta function",
"L-polynomial",
"Jacobi sum",
"Frobenius",
"split prime"
],
"Programs": {
"program-sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.integer_ring import ZZ\nfrom sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing\nfrom sage.modular.dirichlet import DirichletGroup\n\ndef standard_sage(N,p):\n \"\"\"Return the zeta numerator for N >= 3 and split primes p = 1 mod N.\"\"\"\n N,p = ZZ(N),ZZ(p)\n if N < 3 or not p.is_prime(proof=True) or (p-1) % N:\n raise ValueError('This algorithm requires N >= 3 and a prime p = 1 mod N')\n D=DirichletGroup(p)\n chi=D.gen()**((p-1)//N)\n S=PolynomialRing(D.base_ring(),'T')\n T=S.gen()\n P=S.one()\n for a in range(1,N):\n for b in range(1,N):\n if a+b != N:\n P *= 1+(chi**a).jacobi_sum(chi**b)*T\n return PolynomialRing(ZZ,'T')(P)\n\nstandard_sage(3, 7)"
}
}
}