ON A CERTAIN LIE RING DEFINED ON THE DIVISOR
CLASS GROUP OF A COMPLETE NON−SINGULAR
AIGEBRAIC CURVE AND DECOMPOSITION
OF IT’S JACOBIAN VARIETY
BY
SEISHI WADA
introduction． For an elliptic curve， that is， fbr a complete non−singular algebraic curve of genus 1， there is an abelian structure and it’s law of compositio11． For genus＞1 case， we may cOnsider linear transfbrmation on the divisor class group defined by a certain cycle or divisor of the double or triple product of the curve・ The purpose of this paper is to study the relation between the dir㏄t「sum d㏄om・ position of the divisor class group and the decompositio皿of Jacobian variety． At first we construct a certain Lie ring on the divisor dass group． 1． On a certain type of Lie ring． Let Z，（！and C be the rational integer血g， the rational number 6eld and the complex number field resp㏄tively． Let C be a complete non−singular algebraic curve defined on the丘eld丘， which is finitely gen− erated over the prime 6eld． Let g be the divisoエclass group of C， modulo by the groug of aU divisors linearly equivalent to zero． And let gκbe the subgroup of g consisting of a皿classes that contain、K−rational divisor， where K is丘nitely generated over k． Then we know gκis麺tely generated over Z． Let rc be a divisorial correspondence class between C×Cand C， and let、9 be it’s representative divisor of C×C×C． For two divisors 2I and 80f C， we can se1㏄t such 20fκthat Prc8［2・（班×田×C）］is defined and we write it、9（2t， S）． Herea丘er we denote C×C×C as C1×C2×C3．皿en at血st we have the next le㎜a； LEMMA 1． For a function∫and a divisor蕩of C，9（（∫），9t）and⑫（21，（∫））are l口1eady equivalent to zero， if they are defined・ PRooF． As 9（（∫），別）and 2（9t，（∫））are defined， any component of 2 does notcontain any component of（f）×班×Cand gt×（f）×C、 Therefore any component
of9 does not contain any component of C× Ut×C and so any component of C×田×」P， where P is a generic point of C over some field k， and any component of 2，衷 and（f）is k、−rationa1． If（∫）一Σ±吾and瓢＝Σ：tρi，・9 inters㏄ts properly wi血 C×（∼∫×Cand C×Oj×Pon C×C×C． We put fbr simplicity 2・（C×Qi×C）＝ρi ［33］34
S．WADA
×9（ω，then〆（ωis in C 1×C3． T血en we see a皿terms are de血ed in the next calculations； qPi・ρ∫）−Pr・・［ρ・（Pi×ρ’×（Dコーft・・［2・｛（C×0∫×C）・（民×CxC）｝コ ＝Prc8［｛9・（C×《2∫×C）｝・（民×C×C）］ −Pr・・［Pr・・xc・｛（0∫×ρ”（ω）・（R−i×C×の｝コーPr。・［2’（0∫）・（民xC）］． Hence we have Prc8［Σa’（（∼∫）・（（の×C）］＝9（（の，2〔）． If 9’（（≧∫）has a component of s the fbm磐×C， where磐is a divisor of C，（賜×C）・（（の×C）is defmed as in the abo鴨equations and so O b㏄ause of田・（の＝0． If 9’（（2∫）has a component of the fbrm C×⑮， whe鰺〔s． is also a divisor of C， then 9’（（｝ゴ）・（（の×C）has a component （∫）×E，and the a190braic projection of（∫）×Eto C3（＝C）is O because of degree of（の＝0． If we take a fUnction乃of C×C defined by∫（P）＝乃（P，0）fbr two indepelldent generic points P，ρof C oves k，，血en by Tlieorem 40f chapter g of We皿’s Foundation dle賠are㎞ctions Fj on C such that Prc8［〆（ρjXh）］＝（吟）． It is b㏄cause Prc・…［9．・（C×（｝i×P）］−Prc・・c・［｛ρ・（C×ρ，×C）｝・（C×CxP）］ ．。 −9’（ρ」）・（C×P），
whe■all terms aオe de血ned． If s is a triVial diviSoria1 COrTespondenoe class betwee耳 C×C．and C， then all Fi are◎ons迦ts． This◎ompletes血e proo￡ For the next， we have to state the thθolem of Squa鵡．THEoREM oF SQUARE． For points P1， P2 andα，020f C and a divisor g of
っ C×C×C，we haveΣ（−1）i’i9（Pi， Pi）∼0， if a皿terms砲de血ed． ‘，∫ PRooF． The「proof may 1⊃e refered to Lang’s Abelian va亘ety． By Lemma l and Theorem of Squa爬， we can define a Lie血1g stmctule on the divisor dass group of C folloWingly． Let X andγbe two classes of the divisor dass gro叩of C， and let瓢，覧8 be their’representative divisors， then we de6ne［X， y］byαass［9（衷，磐）−9（磐，助］for such g ofκas it bdng de血ed． Then we ロustミh・w．that this is a sUitable de血riti・n’as the symb・1・n the（liVis・r dass group of C． But we can show that（1）fbr two 9， and 220f rc，9，（21，58）alld 92（21，18） are血learly equivalent， if both of them are defined． T垣s iS proved in the appendix of Lang’s A13dian variety；（2）fbr伽o血mctions∫and g of C， we can ta1【e such910fxthat
g1（21，》8）， g，（衷1，磐i）， g，（8， gり， g1（S，，211）are all de丘ned when we Put衷1＝班十（の and S1＝8十（8）；then gl（21，覧8）∼9，（衷1， W8i）and 91（6，班）∼91（賜i，2〔1）． Because we can take such 9，0f x that a皿of ’ ・9・（o「，8），．9・（乳（f）），’ρ、（（9），Pt），ρ、（（∫），（9）） ρ・便，O「），ρ・（＄，（f）｝，ρ、（（9）， Ut），9、㈹，（の） 9・e d・血・d・・bef・・e・Th・n加m t』Lemm・1，ρ・（Ut，＄）a・d・9・（Ut・， S・）are血副y equivalent・ From（1）and（2），・血e de血nition of［，］b㏄comes suitable， and 1田Uqudy 《1etermined by血e non−trivial divisuai correspond㎝ce class rc betWeen C×CJ and C． Hereafter we denote the da8s of 9I by班． 、ON A CERTAIN LIE RING DEFINED ON THE DIVISOR CLASS GROUP
35 Now we want to show that the divisor『class group of C becomes Lie ring over Z by this symbol［・，］． We can easily show that（1）［X， Yコ＝一［】∼X］．（2）bllin・ earity． So we have only to prove that（3）［X，口∼Z］］十［y，［Z， X］］十［Z，［萬 γ］］＝0． （Proof of（3））We take the trnsformation on C×C×C that chang｛rs P×0×R to O×P×R． If・we take’2e＝2−t2 where t2 is the trans丘）rm of 2 by it，、and if we take』2 and 2’such that all the terms in the fb皿owi皿g calculatiolls・・are defined， then we‘see at丘rst 20（（］，（1）∼0．．Therefore by the theorem of Square， we have the next rdatien（0） ． … （0）：2。（P；ρ）∼、Ωo（P，．R）一．20（（2，R） ・ ∫ Ibr any points P，ρ．and R；if all terms defined．’Here we w口te 2。（P，R）＝・．SPi−．SP！ and ’・．∵ 、 2。（（2，R）・・．xρi一ΣρS’． Then We sQe that fbr．sUitable』γ 20’（R，ρ0（P，0））∼20’（R，、Ω0（P，R））−20’（R，90（0， R）） ． ∼Σ［・Ωol（R， Pi）−20’（R， P‘）］一Σ［20’（R，0∫）−20’（R，0ゴ’）］・ ・ さ Moreover we have 』 2。’（、R， Pi）eY2。’（R， t）一ρ。’（Pi，の 』20’（R，’Pi∼）∼20’（R，’）−20’（P！， t） for some point t． Theref（〕re we have ．2。’（R，2。（P，R））∼Σ2。’（Pi’，’）一Σ、9。’（P‘， t） s ： ∼−2。’（’，2。（R，P））， − an（i． we． have alsb ． 』 ’．． ． 2。’（R，2。（αR））∼ρ。’（オ’，⑫。（ρ，R））− by the same discussions， fbr some point〆of C． Therefbre we have ［R，［P，ρ］］＋［〆，［2，R］］＋［’，［R，戸］］−O for the class P，．ρand R’of points P，（∼and R． ． ． 、 If we take〆rP and t＝0， then we have Jacobi’s equation fbr the classes of：P， ρ and R．’lherefore by bilinearity of the symbol［，］We can prove（3）．（Q．E．D．） Morθover， we have． the皿portant relation ．（4） （degree（s．）［班，9｝］十（degree gt）［磐，（s．］十（degree田）［（s．，衷］＝O by the relation（0）and b皿inearity of the symbo1［，］．1 Here we de6ne the degree of the class by the degree of it’s represe皿tative divisofピ In（4）， we use the letters−，．X， Z etc． fbr gt，18， E etc． By（1）…（4）， the divisor class group g b㏄omes Lie血g over the rational integer ．亘ngZ． ；’ DEspecially， by（4）， this血g is a solvable Lie血g． Because；if we assume血（4） degr㏄X．」．degree y＝O and degr㏄Z』1， then We have［x，Y］＝0；and， as．we easiy see， degree of［P， e］＝O fbr any two poi皿ts P and（｝of C and therefore、degree ot ［エY］＝Ofbr any’two divisor classes X and｝「；we have ’ ．3．6 ．S．、WADA ’ …．一 ’．．tt ．． ［［9，9］，［9，9］］−0． For the ’nextj．we Consider to extend the coe丘icient血teger血g Z to the丘eld￠ We constr叫ct the．tens6r’product of g and the rational number．field（20ver Z． We write this tensor product as g⑧〔2． For two elements X and Y， we can de丘ne the symbol［X，γ］as fbllowing．「If X− Xo⑧a， y＝1偏⑧b fbr Xo，］陥of．g，．we de丘11e ［X，γ］＝［Xo， Yo］⑧ab． Then we can easily prove the relations（1）…（4）for g⑧0． If（x，，晃，・）is a complete set of independent generators over Z of．the diVisor dass group g， then（Xl⑧1，品⑧1，…）is a basis of g⑧（1，0ver Q． For simplicity we write them hereafter Xl，晃，…and X⑧a as aX． A1）ove de丘nition of［X， Y］． fbr X，γof g⑧〔2 is that［X， Y］一Σaゴゐ5［尤， Xi］， if X＝Σo汲， r＝Σb∫X｝． Now let A be any fidd extended over仏then we can extend the Lie ring structure f士oin g⑧（2to g⑧．4 which is a tensor product of g〈9）C‘and A over O as usually． And we de丘ne the degree of X of g⑧A by degr㏄X〒Σゐ（deg尤）f（）r X＝Σλぷ． Then we have the extended（4）easily， by the de丘垣tions of extended symbol and extended degr㏄． Then we have．・the皿ext Lemma 2． L壼MMA 2． If degree of Xo is 1， th㎝Xo is a regular element of g⑧、4． ［，］ PRooF． Lgt gb＝（、Hlad”Xo・H＝0， fbr someη＞0−of Z）， gb is a module， because for th and H2 ofg， we have a sUitable integer〃＞O such that a〃Xo・（Hl十H，）＝0． And． as．we have genera1Uy from（4） ［Hl， H，］＝（deg．砥）［Xe， H2］一（deg H，）［Xo，正ll］， therefbre we have fbr Hl and H， of g （adカXo）・［Hl， H2］＝（deg Hl）adn＋1Xo・H，＝（deg H，）ad蕗＋1Xo・1＝O fbr someπ＞0． This proves gb is a sub血g of g⑧．4． For the llext，1et ge be lhe normalizer of鍋〕， that is， 、喜b＝（MladH・H／is．in gb fbr all H of gb）． Then we have ad”Xo・［H，m］＝O fbr some〃＞O and any H of gb． （of．courseπ depends on 1五） 、’
撃??窒??盾窒?@we have ．． ．
（deg H）adn＋IXo・W−（deg ”ノ）ad兄＋12叱b・H＝0． If we take here Ko as耳we．s㏄．．αゴカ＋IXo・〃」O and． W iSjn gO． So gb is equal｝to ξ． For the last， we must prov6 that gb is n皿potent． If we take Hl，、H2 and H3 丘om gb， and apply the rdation（4）to Xo，疏and［H2， H3］， then we have・．． ［石r，，［H，，H3］］＝（deg』五）［Xo，［H2， H3］］ ．． ・ ．’ 』 ＝（deg．昆）（deg．疏）［Xo，［石r2，．El3］］ 一（deg 1五）（deg H3）［Xle，［X，〕，石r2］］． By the same manner， we．can prgve that gb is nilpotent． This compk旗5’the pr60￡ ∫：、Hereafter、“re Write the．s）血bol［，］as （ad・げ・）五）r simplicitY：；・・．And．as“Xo’． we take al dass R． c of k： rational pOint Pθand write it asεagain． ・ ．．’． Now we take fbr the field．4 the complex number field C、 and take g⑧CI LetoN A cERTAIN L肥RING DEFINED ON THE DlvlsoR− cLAss GRouP
37
ebe a regular element as above， and let K be a．． field血亘tdy gen｛rated．over the de血iti・n㈱た・（ther・f・re・K i・al・・a興dy gCn・・at・d・v・・th・p血・e丘dd・）・nd let gK be the subgroup of g constituting of the−・K−rational diViS（∼r dasses・ Now we d㏄ompose gば⑧C by the Cartan subalgebra go detem血ied、by e． That is， gκ⑧C＝gb十＆十gθ十…， this is permitted fbr gK（9）C， too． 、馳 Hereα（H）is a root of Lie血g and翫is de丘ned by 翫一（X。1（ad・H一α（H））・・Xa−O f・…me．・＞0．and・all・H・r g・）．Then we have the next Lemma 3． ． ’
LEMMA 3． α（H）＝ca deg H， ca is an algebraic hlteger． P・…．L・tα（H）b・n・n−zer・…t・f gK⑧C・：’．−D國
x．D’ Then there is a non−zero element Xa in gk⑧C sUch that ad丑・Xa＝α（H）Xa fbr au H in gb， then we see that deg xa＝o．becauseα（H）キO fbr some H of gb． AIldfbr亮we have丘om（4）and丘om degree Xa＝0，
（deg H）α（e）・Xa＝（deg H）αdθ・Xa＝（degθ）α4」V・Xa＝α（H）Xa・・…ew・h・ve＞
ｿ（の一隠豊一・・n…n・㊨鵬・・9・・…e・d・gree・・聡
not O． For H of degr㏄Owe haveα（H）＝0． T』reR）re y爬haveα（五）＝ca deg H fbr all H of gb． If we take the expression of Xa一Σλ必using the ．basis（X・， X2，…）befbre men− i tioned， and if ad e・尤＝Σ砲苦where dii are all rational integers， we haveΣλ，・dii コ ‘ ＝cαλ5and（λ1，λ2，…）キ（0，0，…）． Therefbre we have the algQbraic equation 砺 00．．・．：．・．0 一 This means that cαis an algebraic．1nteger． d21……・…・……・…・……… （d22−Ca）・……diゴ・…・……・…… d23−・・（d33−Ca）・…・…・…・・ ・ ． dii…・．………・……（d。。−Ca） ． ． ＝0． This completes the proof． LEMMA 4．＆一（X。 l fbr some n＞0，（ad・e−ea）カ・Xa−0）・ PRooF． We have脇＝（Xal（α岨一cαdeg． H）π・Xa＝O for some int《）gerπ＞O and fbr all H of gb）：1．．ド’．』’ 1．． ’1． （ad・e−cα）外・Xa＝0． th・n w・hav・（一1MCa”Xa一Σ（−1 i）・・（：）曜一・・d・e・Xa←Ya）・・nd・d・g・Y・一…㎝・・ deg Xa ＝・O by cαキ0． So， if we write∬＝λθ十Ho by Ho∈gb such that deg H＝‘λ and deg」70＝＝O， then （ad H−ea deg 17）カ・Xa＝（adλe十」Uo−Zcα）n・Xa” ． 一λ・＠le一励・Xa−e； 一∴
because ad He・Ha＝O by’deg Ho’ ＝ deg X．＝0． The hlverse is dとar． This completeSthe proo￡ 「『’、． ． ・
38
S．WADA
LEMMA 5． All the co皿jugates cαa of ea are again the root coeMcients． PRooF． We takeaan automorphismσof C1ρ． For X of g⑧C， we can take’Xl’ of g⑧O and a of C such as X＝Xl⑧a， hence we extendσof』Cκ2 to the auto− morphisim Of g⑧C by Xu＝X，⑧m． Then fbr 1」Xl⑧a and γ＝Y，⑧b，［瓦γ］σ 一［Xi⑧a， Y，Xb］・一（［x，，γi］⑧abアー［x、， Y、］⑧〆〃一［x，⑧〆，｝∼⑧〃］一［Xa，珂． Then， we have（ad e−caσ）弼・Xaσ＝0． Thus cασis a root coe丘icient again． ThiS completes the proo￡・ To the next， we consider the extention of the sp㏄ialization of divisors． Let ◇陥，X2，…）and（Yl， Y，，・…）be two sets of independent generaters over Z of g， and（班1，？12，… ）and（wa、， M2，…）be their representative divisors resp㏄tively． Let ， 9 ｝ （2〔i’，》t2，，… ） be the specialization the specialiZation of（軌1，班2， in 9⑧A where A is a field …）over some丘eld k，． We define f（）r尤⑧1＝班《（⑧）1，it’s sp㏄iaUzation over the specialization（瓢1’，》t2’， … ）over k， by⊇OfollQwingly．
’ At first we define … ）’of（班1， Ut2，． ．（Xi⑧1）』Ut！⑧1 he酷e we Wfite fbr simpliCity（濫⑧1）s as濫s． For the validity and suitab血ty of this き ド ヒ ． de丘1亘tion『we”Write later． Hereafter we say（xl・， x2・，…）is a basis speCializatiOn of （X，，X2，…）over k， in g⑧A． For general X＝Σλぷof g⑧Aプwe de丘ne the ま Sp㏄ialization｝． Xs・oVer the basis specialization， by X』ΣλiXis． Let 1）∫be the tor− i siOn divisors and・磐i”＝磐」’十1）」． Vゾhen the speciali2ation（Ut、’， Ut2t，…）of（班1，212， …）and the specialiZation・（8、’，∼82’，…）of（磐，，磐2，㌦・・）over ki’are su6h that層（班1’， W2｛，…；磐1’，，182”，…）is the specialization of（班i，Ut2，…．；磐、，82， …）over k，， we say two generators of g over Z have the co血patible specializations． When that is so， any X of g⑧、4 hqs the same sp㏄ializatiohS fbr tWo specialiZations of generators over Z of g・ Because fbr X＝Σλ‘Xli＝Σ1μ∫1つ， Ve haVe 】ノ］’＝ZaゴkXk十1膓 and t ジ グ き8ゴ∼ΣajkUt鳶＋島（7｝and島are torsion dass and torsion、 divisor．）， and磐ゴ”∼ΣaikPtk’＋Ei’鳶Bv。， th。、p㏄i。1セ。tiO。（Ut、’， Ut2’，…）。f（ut、， ut、，＿）’翌?B，e幻、 th。、：㏄i。1i−
zatioll of Ej over it and also a torsion． And we have therefbre 鰺』（Yi⑧1）・・一（Σα‘ゴ㌢⑧1）・』Σ砺蛎メ（多1一否ノ’⑧1 E可6’⑧1＝（｝’li⑧1）・2＝’｝7・・． 3 豆 Conclttsively we have XS・＝・・ ZλヵXkS・＝Σ（Σμ∫α∫鳶）品s・一Σμ∫｝㌃i・一Σμ∫rs・rXS・， み h ゴ ゴ ゴ as we haveλ浩＝Σμ∫α」先． Therefore the de丘nition of the specialization in g⑧A be一 み ㊤「e・qefine・1・・is uniq・dy d・t・輌碩佃any・rt・・9f・・mP・tiblg・p㏄i・』ti・n gfg・n− erators over Z of g． For the specialization in g（8）、4， we have the・11ext lemma． ’シ ．’ 國 』 『 ・ ， ， ．・ LEMMA 6． For X andγof 9⑧A， we have［Xiy］s＝［Xs， Ys］； PRooF． Let X＝Σλ汲and r＝Σμ5X｝． Then we have［X，γ］＝ΣZipt」diikXk． i グ あ On the other hand， we can ．take a k−rational’parameter divisor such that all’2（鵯，瓢∫） and⑫慨ノ，班／）are de血ed and not empty． Then we know 2（9ti’，班／）is a．’ uniquely・
ON A CERTAIN HE RING DEFINED ON THE DIVISOR CLASS GROUP
39
determ血ed specialization of 2（鵯，2Ij）over k1（here the丘eld kl conta桓k．）over the specia血zation（211’，衷2’，…）of（班1，班2，…）over k1． And we have 2（9ti，9ti）−9（別ノ，班i）∼Σ砲概汁D‘∫（1）‘∫torsion）， k therefbre．we have 、ρ（2ti’，鵯’）−9（班∫’，瓢q’）∼Σ砲概ノ＋1），ノ （D￡／also torsion）． 力 Therefbre、ve have ［尤，Xi］』［鵯⑧1， Uti⑧1］』｛［Uti，馬］⑧1｝』Σ砲轟（班為⑧1）s一Σd，ノ（Utk’⑧1） ゐ ゐ ＝［Utit，9t！］（8）1＝［？Ii’⑧1，衷／⑧1］＝［XiS， XiS］．Therefbre we haveΣdijkXks＝［澱㌧XiS］and
み ［x，Y］s＝（Σz，ttidiihX，）』Σλψ∫晦鳶品』Σ∼しμ」［XiS， x｝s］＝［Xs，珂・ This completes the proof． From this Lemma 6， we have the important resUlt， that iS， for Xa of ga such． that （ad e−eα）カ・Xa＝0， we have（ad e−eα）カ・』らs＝O by induction fbrπ＞0． Here we change the symbol g⑧c to gL⑧C where L is also a field麺tely gen− erated over k， and take care of the fact that go，翫and gβare detemmi皿ed by only one elementθor（ad e）as the linear transformation on gL⑧C． So we consider gL⑧C 仕om this point of view． Then the linear transfbr【nation（ed e）has a ’minimal poly− nomial Fωof variable t． All cαof gエ⑧C are the roots of Fω＝0．1We d㏄com’…eFω一〇（’一げ・andp・・凡ω一（恐。・…n・fw…k・・h・・d・A・」
0（￠的cβ，…）fbr all cα， then a11」Fa（，）are relatively prime， and have so皿e polynomials Gα（’）such thatΣ瓦（’）・Gαω＝1． All of Fa（t）and Ga（t）are in AL［’］． Then we αhave
8L，a＝（璽α（adの・XlX∈9L⑧C）， fbr Ya（t）−Fa（t）・Gαω． （This is an usual theory of the eigen value↓） Then we have nt ロ 9L，α一（Σ（璽「α（adの・尤）⑧CIX・， X・．，…are basis of gL⑧の． i＝1Hence we have
が Σ（アα（adの・Xi）⑧AL⊇（9L，aUgL⑧AL）and Σ（翫，α∩9L⑧！鍾L）⊇9L⑧メ1 ‘＝1 a But if we put m 蕊＝Σ（Va（ad e）・Xi）⑧メL， ’＝1we have
蕊⑧C＝9L，αand d㎞鯉蕊＝d㎞cgL，α．Hence we have
gL⑧AL＝Σ麿＝Σ（9L，a∩9LOPAL） ロコ a and non−zero．wa of gL，α∩gL⑧AL． LetΣmean∫（cの＝O fbr the irreducible poly一 ア a 皿omial of cα． And we d㏄ompose X bf gL⑧O in gL⑧AL as X一ΣXa． Then if let a GL be the Galois group of、4L／ρ， fbrσofσL we have Xa＝X， Xaσb｛：longs to caσ， amd m the dir㏄t Sum docomposition of gL⑳4L we have40
’『「PS．WADA「
・ ’．．．’t Σ石・∈Σ9a， Xf・ ΣXa・＝ΣXaσ． ア∋α ア∋a ア∋a ア∋α AsエーΣλα，泥fbr Zα， i of ’AL，’Xf一Σ（Σλα， i）Xi a耳d by qboVe；we have t ； a Xf・一Σ（Σλα；i）・Xi一Σ（Σλα， i）及．》「 ち a t This means（1．λαi）鰍＝（ΣRαi）f〈）r all o of Gb and，易∈gL（⑧，e． As We、can do the a same discussions fbr all Xi of the basis of’ №k⑧0， we can take non−zero Xf such that品＝Σ毛fbr non−zero Xai because if皿ot−so，加m、）he丘rst sudl ca． does not ア∋α1 ． ＿ appear． Therefbre there is a L−rational divisor襲｝such．．that rXf ・一・ ％⑧1 fbr some suitable rational integer r． Now we take a gene〔ic point P．．of C over k a皿d the field、K−k（P）． Thep we can．prpve that a！l Ca of g⑧C akCady appear in gx⑧C． To Prove． it，；we tak’e sbme丘dd L and asSume that’gL⑧C has other’e3・than that of gx⑧C． We may take ． L to be丘Pttely generated over’K・． ket（濫， X2・．…；Xn） be the set of independent generators of gx over Z． We decompose戸⑭1 in gκ⑧C as戸⑧1一ΣXa by Xa一ΣaiXli of 9a． Now we．take a L−rational divisor田一ΣOi a ドコ such that蕩⑧1＝rXg fbr a irreducible polynomia1．g（x）．of cβ， as’above mentioned． And we take a specialiZation over k丘o血、P to（2i， and extend it to the speciaUza− tion in g⑧C． That is， we take the specialization over k，（P， X，， X，，…）→（（1∫， Xls， X2s，’…）and over it the specializatio皿 Xas of． Xa． Then、as．i皿the Le皿ma 6， if （ad e−ca）兄・Xa＝0， then（ad e−ea）＃・瓦s＝0． That is to say， Xas bdongs again．ga of 9⑭C．Hence e；・⑧1一ΣXa・is坤Σ＆， in the sum of 9a of g⑧C correspondi皿g to a a ca of g⑧C， and蕩⑧1 is also in the sum of ga correspo皿ding cαof gκ⑧C． This is a contra（iiction． Therefore all ca of g⑧C already appear in gκ⑧C． In the proof above， if we take a丘dd L’which contain L and is瓠so丘nitdy generated over L｝ and on which allρj and aU澄are rational， then we can discuss i l the specializa− ti6ns in gL・⑧C， as to毎⑧1 at Ieast． This completes the proof of the next Theorem L THEoREM 1・Let P be the generiC po血t of C oveT”apd K＝k（・P）・． Then fbr the丘dd L extended finitely oveLK， gL⑧C has no other cαthan（cα）appearh｝g m9κ⑧Calready．
§2．1）ecompoぷition oゾ’乃θ直y∫ぷo’α〃d Jacobia〃v・arieリノoゾC We take a generic point P of C over丘and put，K＝ん（P）． Then by the Theorem 1， the number of the root coe伍cients cαand their加《…ducible polynominals∫（x），9（x），… 《io not increase， if we extend gx⑧C to g⑧C． Therefbre we call take the field ．4＝（2（cα，cβ，…）dependent of only C and the corresponding dassκbetween C×C and C．（We have to notice that the direct sum decomposition of g⑧C does not depend on the sd［㏄tion ofθ．） 、As befbre mentioned， we have 、 9⑧A−（9⑧A）∩（9⑧C）一Σ（＆∩9⑧・4）． び WC rewrite毎∩9⑧メas脇and g⑧A一Σ＆． Let G be， dle Galois． group of Al② ぴ If we d㏄ompose X of g⑧O to the fbml X＝Σ吃，’thgn． applingσof G to元we αON A CERTAIN L肥Rl［NG DEFINED ON THE DIVISOR CLASS GROUP、
411『have Xc、i X．．But we haveΣ垢is．contained again inΣga， whereΣmeans
ア∋a ア∋a ア∋a ゾ（ctr）．O．J， Therefore we− haveΣXa ・・ΣXa・because of the uniqueness’of・the・manner ア∋a ア∋a of direct sum d㏄omposition of X in g⑧C． TherefoteΣXd of one fてx）is in g⑧0， ・nd・h・・e・・．・di・・・・・…su・h・h・・頭・⑧1一搭⇔・s／・脇血…1era・i・nql・’・…g・r r・ Thus we have the llext theoretn． ．THEoREM．2． Every． divisor gt of C is d㏄omposed in the fbrm瑚∼Σ衷∫＋蘂where ’ ＿ ア r is a sUitablg．rational integer， and句 ・is a torsion， alld衷f⑧l is h1Σga．．Now we’ ア a
know Sq 9ives the isomorphism from’the伽sor．class． group of degr㏄Oto the
addtive group J， h ereψis a canonical map from C to J． Let班be a divisor of degree O of C corresponding to the g㎝eric pomt x of J over k． Then価m ．above there is some rational ．integer r such that r班∼Σ賜＋S Where Ut∫⑧1∈Σ9a，⑤ ∫ ： プ∋α torslon． So we may write r班∼Σ賜、 fbr a largeア． Put xl・＝5「ψ（ア班）＝rx and xf＝助（賜）． ．ゾ Then rx is a generic point of J 6ver k agam． And asΣ9a∩9⑧O is not 6mpty， ＿ プ∋a there is a divisor班プ。 such as馬。⑧1∈Σga and not Zero．．Put xf。−5φ（馬。）， then プ∋α xf。十〇in J． By the same way， if dim go＞1， there are馬。 of degr㏄Osuch that Sq（況プ。）＝xf。キO fbr aU irre（luCible polynominals f（x）bf（cα）． But if dim go＝1， theh there is no戎。 of dCgree O such that玩。⑧1∈go． ．1 LEMMA 7． For∫（x）キx，．we have dimk xf＞0， and if dim 90＞1 we．have dim， x、＞0． PR°°F・・lf・dim・・x・〒°．f°「f（be）ヤ・then f「・m・・＝ヲ警・ちるζ・i・蝸・ g㎝eric point’of J over k． Therefore we have a sp㏄cializati（）n oyer．k加m兄to xf。． Therefbre if we ext㎝d this sp㏄ializatibn to the speda血Lati6n over k． （s，Utf， Utg）マ（Xf。， Utf’，賜〃），．．． then we have xf。一ΣSq（Utg’）and班∫。∼Σ班ξ’． So we have班ノ。⑧1一Σ’U．’⑧1， and 9キデ げ≒プ 9キプ this contradicts to the uniqueneSs of the d㏄omposition， b㏄ause班g⑧l does not belong toΣga． If dim go＞1， then for f（x）we have the same proof as above． ∫∋a・ ． ．． ． 1 ． 』 ． （Q．E．D．） Now wg extend the definition field k of C， J to瓦，．algebraiC dosure．of k， dnd assume x is．a generic point’of J’ over k：By TheOrem 1，（ca） does not inc’rease in this situation． Now let！1プbe the locus of x／over k． And let 7f be a generic pomt of面oy⊇，．丘eC’of x∫over元． And we． ass㎜eωof all∫（x）． are reli− tively independent over・k；and xf and yアare independent．generic points of Af over ＿ 大・Put y＝ｱア・・’thep．愉e．a「e th・・peci・li輪・・f「・mア・t・ x・・f・・all f（・）・・nd 丘om the ass㎜ption．’we can take these specialiZations．together， therefore x1 is a specializatiQn of．ア．over k， andアis a generic point of J over k． Then z＝xl十アis ageneric p6int of J oVer元． Here we take a specialiZation from xl to z， and ex− tend it to the specialiZation（κ1， xf）→（z， zf）over E． Let gtf，磐プand（s．f be the．divi− sors such that xf−SiP（i場），．γf t SaP（8プ）and zf＝SiP（Ef）． Then we have z一Σzf and ∫42
S．WADA
zf ・＝ ・2（xf＋γ∫）；and衷∫⑧1，…5ア（8）1 and Ef⑧l are inΣ翫＝9f． By the uniqueness of ∫ デ∋α ＿ ＿ ＿ the manner of the direct sum decomposition in g⑧A；we have〔ら⑧1＝別∫⑧1十Mf⑧1． Therefore we have Ef∼鵬十％f十1）∫， where 1）f is a torsion， and fbr some large rational integer a we have’＝is．f∼aUtf十aU8f． So if we take the locus A了of axプover瓦， Af is on abelian subvariety of J， be− cause axf and 4アプare indqpendent over k and azf is a specialization of axf over k． As before mentioned， we have dimk xf＞O a皿d we can prove dimk axf＞O by notic− ing that axl is also a generic point of J overえi． Therefbre d㎞∠4∫＞O f（）rノ『（x）キx； and if dim go＞1， dim Ax＞O for f（x）’＝x． Thus we have the next theorem． THEoREM．3． If C’is a complete non−singular algebraic curve， de丘ned over the field finitely generated over the prime fiel［d， then it’s Jacobian variety J has the next d㏄omposition by the abdian subvarieties Aプthat bdollg to some irreducible poly− nominals f（x）；J＝ ZAf， d㎞∠1プ＞O fbr f（x）キx and dim AfキO if and o㎡y if dim プ 90＞1． The set of f（x）depends only of the non−trivial correspondence d［ass rc be・ tween C×Cand C．、 Af and Aεhave皿ot the㎞tersection other than some torsion points， and so genus of C＝Σdim Aプ． プ §3．On each Af． We assume k−k， and consider Af of d㎞・4プ＞0． Let x be ageneric pol皿t of J over k， and let it correspond to a divisor班of degr㏄0． Let Ut⑧1 ・＝ ZXa be the d㏄omposition垣g⑳4 where A is a丘eld used in the proof of a the Theorem 3， here we writeぷin．the Thcorem 2 as衷fbr simplicity． Then for each Xa， there iS a rationai integer nヂsuch that（ad e−ea）nf・Xa＝O but （ad c−eα）鴨∫−1・Xa十〇． The generic point xプof Af over k corresponds to the diVisor 瓢∫of C， degree O， and Sq（馬）＝xf． And if Gプis a Galois group of∫（x） of ca， then Xal of衷∫⑧1一Σ為correspondS to one alld only oneτof Gf and Xal ＝ Xa・， ア∋〆because from
（Utf（911）・＝（班∫⑧1）Σ工’一Σ石τ プ∋α’ プ∋a．・ XaτbelongS to caτand cal corresponds to only oneτof G∫． Here we put Xai＝ （ad e−ca）i・Xa for nf≧i≧1 and X｝i ＝ 2Xai． Clearly Xai is contained in ga， apd each アDa Xalε of ca’of f（x）is equal to Xaiτfbr． oneτof G∫． And moreover Xfi is contained in g⑧（2 as followingly proved． Let Go be the set of all 60f a Galois group G of A／（］such that do not change（ea）プ∋αand GτbeτGo forτof Gプ． Then q（9」プ⑧i）一Σ（班∫⑧1ア＝Σ（ΣXd・）z．rΣXd・＋Σ工〆＋… σ・∋・ G・∋・Gプ∋τ 』∼G・∋・ G・∋・’ where q is the order of G。， and we have qXd・＝＝ ’ Q Xati’， asΣXd・’belongs to car． Gτ∋a’ Gτ∋σ’Then
（ad・−c∂・・Xa−z｝dd e一ρ）i・Xd・一Σ壱〈ade−c・・）㌔（Σ・￥d〆） Σ f∋a Gプ∋・ ’ τ Gτ∋σ’ ’ 一Σ［〉（ad e−cα・）・・Xde・一÷Σ・陥… G∋σ G∋aON A CERTAIN LIE RING DEFINFD ON THE DIVISOR CLASS GROUP
43 Therefore we have ’・ 乃・一Σエ’・一Σ石・”
G∋a f∋a is in 9⑧0． Therefore there is a divisor Utfi such that W∫‘⑧1＝〆ΣXai for some rational integer 〆．Of’B・u・、e d。9ree。f馬、 i、0．一 ・．． f∋a
Let晒ε＝為， then ．we have the next lemma． LEMMA 8． Dimk Xfi＞0． PRooF． We take the sp㏄ialization O of xf over k， and extend it to the specia1血一 tion・over・k，（xf， Utf， Xa）→（0， Ut〆， Xas）． Then we have Sφ（Ut／）−O gnろand賜’⑧1 一ΣX．s−O， and therefore all Xas−0． Therefore the specialization品is of Xai over a above sp㏄ialization is O and such that gtfi’⑧1＝0． So xノ！ ＝＝ Sφ（賜‘’） is a torsion of J and k−rational． But xプ《is not a torsion， therefbre d㎞・為＞0． This completbS the proof． We take now the locus Z∫《of為over k． Let yfi be a generic point oM∫i over k，independent to xfi over k． And we take the sp㏄ialization f士om xfi to〃《， alld extend it to the specia且zation over k： （Xfi， Utf，馬‘， Xa）一→（γfi，8f，磐プi， Ya）． Then we have γfi＝Sφ（Sfi），8f，（9）1 一 2〆（ad e−Ca）i・Ya， and磐∫⑧1＝Σγ乙． プ∋α ∫∋α If we put yf＝Sq（磐f）， then zプ＝xプ十〃is the specialiZation of xf over k． Hereafter we mainly deal with this specialization from xf to zデover．k． Let膨プbe the divisor of C sucb that Sq（（s．f）＝＝ zf， and〔S∫⑧1＝Σ1乏， thell we have Za＝Xa十】㌦by the uniqueness of the d㏄omposition． Now we take a extended sp㏄ialization over聡 （xf，馬，班ノi， Xa）→（zf，〔匡∫’，（s．fi， Zaノ）． Then zノ＝Sq（（S／）and therefore（s．f∼（S．〆． A血d moreover厄∫⑧1＝琶／⑧1 and琶〆⑧1一ΣL’． By the uniqueness of the d㏄omposi− tion aga㎞， we have Za ・＝ Za’． Therefore Za is the speCialiZation of xa over． the specialiZation from xf to zf over． k． Therefore we conclude that（ad e−ca）i・Za and 〆Σ（ad e−ca）i・Za are the specialiZations of（ad e−cα）i・泓and〆Σ（ad e−ca）i・Xa respec−一 ア∋a ＿ ∫∋a tively， here〆is such a rational．integer that we have before 9ノ！⑧1−〆Σ（ad e−ca）i・Zat f∋a over the above extended sp㏄ialization． Therefbre we have 琶プi’⑧li． r！Σ（ad e−ea）ε・Za ＝＝〆Σ（ad e−cα）1・（Xa＋Ya） f∋α 方∋α 一（Utfi十田fi）⑧1． Therefore we have（s．fi’∼9tfi 十 Sfi 十 Dfi， where句ノ《is a torsion． So we can take a rational integer〆’such that f，（転’∼〆’賜6十．r”磐fi． Now we Ntake the 1㏄us of〆’xfi over k， then this locuc A∫i is an abelian subvariCty of J and だof Af， because〆’xプi and〆’7fi are independent generic points of Afi over k， and （〆’xfi十〆’yプi）is a specialization of〆’xfi over k． Thus we have the next thcorem．覗 寸’‘ttF ・．．L．．：・， ．’ r．．WADA i’・t”’． THEoREM 4． Each abelian subva亘ety Aプhas the series of abelian・su1）varietieS …h也・t栂λ・？・…・A・f−・⊇0 w』〃〆i・an・mb…f（ade−ca）”・一’・X・・VO・nd （ade−Ca）nプ・Xa＝O fbr a Xa of％プ⑧1．．rΣ品and Sφ（賜）＝＝ Xf． N・．．PROOFi Let．物．and“巧ξ．、Pe．祖de雀）ep（lent generic pohlts of・4プand／1プ《of aUごover k】respectively． And put叉＝1”xf十xfl十…十xfヵプ＿1．・Th旬、 x is also a generic point Of Af over k． Now we take a SpecializaSion over k， xプ→0， X｝‘→0，：；・．． One after one． Then we have a sedes of abelian subvarieties of．4プsuch that Aプ⊇Af・⊇Af・⊇…⊇Af．f．・⊇0．・＼∵．．・ ト・・ 耳頃瀧伍k・g・n嘩Pgi・t・、xi・nd蚕・・f A・and 4・・reSp・if ti・dy 9・e・k・・nd th・歯 9・亨r輌gdi碗・ρ「≒1衷三・、望・・i・f d・g・ee『0・F・・ril・pliCity・：W・頭tC加r9輌ピ
