Jacobian Varieties - James Milne

1 Jacobian MilneJanuary 4, 2018 AbstractThis is the original TEX file for my article Jacobian Varieties , published as ChapterVII of Arithmetic geometry (Storrs, Conn., 1984), 167 212, Springer, New York, table of contents has been restored, some corrections and minor improvements tothe exposition have been made, and an index and a some asides added. The numberingis ..12 The canonical maps fromCto its Jacobian variety ..53 The symmetric powers of a curve ..74 The construction of the Jacobian variety..125 The canonical maps from the symmetric powers ofCto its Jacobian Jacobian variety as Albanese variety; autoduality ..177 Weil s construction of the Jacobian variety ..218 Generalizations..249 Obtaining coverings of a curve from its Jacobian ..2610 Abelian Varieties are quotients of Jacobian Varieties ..2911 The zeta function of a curve ..3112 Torelli s theorem: statement and applications.

2 3313 Torelli s theorem: the proof ..3514 Bibliographic notes ..40 Bibliography ..44 Index ..45 This article contains a detailed treatment of Jacobian Varieties . Sections 2, 5, and 6 provethe basic properties of Jacobian Varieties starting from the definition in Section 1, while theconstruction of the Jacobian is carried out in Sections 3 and 4. The remaining sections arelargely independent of one is a companion to my article Abelian Varieties (?), which is cited as AVs . Theconventions are the same as in those listed at the start of AVs (see also the start of Section 5of AVs). In particular,kis a DEFINITIONS21 DefinitionsLetCbe a complete nonsingular curve over a fieldk. We would like to define a varietyJ,called the Jacobian variety ofC, such Unfortunately, thisis not always possible: clearly, we would want ; but ; ;and it is not always true However, this is true that for a schemeS, the ;O S/of isomorphism classesof invertible sheaves onS, and thatS7!

3 A functor from the category of schemesoverkto that of abelian a complete nonsingular curve overk. The degree of a divisorDDPiniPionCisPini . Since every invertible sheafLonCis of the divisorD, andDis uniquely determined up to linear equivalence, we can , and the Riemann-Roch theoremsays that .C;Ln/Dn g:This gives a more canonical description .C;Ln/is written as a polynomialinn, the leading coefficient. We the group of isomorphismclasses of invertible sheaves of degree zero a connected scheme overk, and letLbe an invertible sheaf onC T(by whichwe meanC Then (AVs, (b)) shows that .Ct;Lnt/, and , isindependent oft; moreover, the constant degree ofLtis invariant under base change relativeto mapsT0!T. Note that for a sheafMonT,.q M/tis isomorphic toOCtand, inparticular, has degree 0. :We may think being the group of families of invertible sheaves onCof degree0parametrized byT, modulo the trivial families.)

4 Note thatP0 Cis a functor from schemesoverkto abelian groups. It is this functor that the Jacobian attempts to exists an abelian varietyJoverkand a morphism of functors WP0C!Jsuch that ! an isomorphism a finite Galois extension ofksuch nonempty, and letGbe theGalois group ofk0overk. Then for every schemeToverk, nonempty, and so .Tk0 ! an isomorphism. ; ;Jk0 ,we see thatJrepresents the functorT7! , and this implies that the ; /is uniquely determined up to a unique isomorphism by the condition in the theorem. ThevarietyJis called theJacobian varietyof C. Note that for any fieldk0 kin whichChas arational point, defines an isomorphism ! ak-rational point, the definition takes on a more attractive form. Apointedk-schemeis a connectedk-scheme together with an Abelian varieties1 DEFINITIONS3will always be regarded as being pointed by the zero element.

5 Adivisorial correspondencebetween two pointed ; ;t/overkis an invertible sheafLonS TsuchthatLjS ftgandLjfsg Tare both ak-rational point onC. Then there is a divisorial ;P/andJsuch that, for every divisorial ;P/and a ;t/, there exists a unique morphism'WT!Jsuch that'. '/ MP the ;MP/is uniquely determined up to a unique isomorphism by thecondition in ( ). Note that each element represented by exactly one sheafMa, , and the map'WT! the uniqueasuch thatMa will be proved in 4. Here we merely show that it implies ( ). implies Theorem that there is ak-rational pointPonC. Then for anyk-schemeT, theprojectionqWC T!Thas a !.P;t//, which induces a maps !LjfPg T/! thats q Did. Consequently, / /, and identified ,LjfPg Tis assume ( ). nonempty, never empty, andJrepresents thefunctorP0 CDP0. This means that there is an (corresponding toidWJ!)

6 Junder /such that, for , there is a uniquemorphism'WT!Jsuch '/ M L. In particular, for each invertible sheafLonCof degree 0, there is a thatMa L. After ta/ Mfor a , we can assume thatM0is trivial, and therefore thatMis a divisorial correspondence ;P/andJ. It is clear thatMhas the universalproperty required by ( ). ;MP/be a pair having the universal property in ( ) relative tosome pointPon C. Show thatJis the Jacobian next make some remarks concerning the relation betweenP0 CandJin the case thatCdoes not have ak-rational allk-schemesT, . ! injective. The proof of thisis based on two observations. Firstly, becauseCis a complete ;OC/Dk,and this holds universally: for anyk-schemeT, the canonical mapOT!q OC Tis anisomorphism. Secondly, for any morphismqWX!Tof schemes such thatOT !q OX,the functorM7!q Mfrom the category of locally freeOT-modules of finite-type to thecategory of locally freeOX-modules of finite-type is fully faithful, and the essential imageis formed of those modulesFonXsuch thatq Fis locally free and the canonical mapq.

7 Q F/!Fis an isomorphism. (The proof is similar to that of AVs, )Now letLbe an invertible sheaf onC Tthat has degree 0 on the fibres and whichmaps to zero ; we have to show thatL q Mfor some invertible DEFINITIONS4 Letk0be a finite extension ofksuch thatChas ak0-rational point, and letL0be the inverseimage T/k0. ThenL0maps to zero , and so (by definition ofJ/wemust haveL0 q M0for some invertible sheafM0onTk0. Thereforeq L0is locally freeof rank one onTk0, and the canonical mapq .q L0/!L0is an isomorphism. Butq L0is the inverse image ofq LunderT0!T(see AVs, ), and elementary descent theory(cf. below) shows that the properties ofL0in the last sentence descend toL; thereforeL q MwithMDq is sometimes possible to compute the cokernel to ! Thereis always an exact sequence0! ! ! the Brauer group ofk. Whenkis a finite extension ofQp, ,and it is known (see?))

8 , p. 130) that the image 1Z=Z, whereP(theperiodofC/is the greatest common divisor of the degrees of thek-rational divisors a presheaf on the large etale site overC; then the preciserelation betweenJandP0 Cis thatJrepresents the sheaf associated withP0C(see?, 5).Finally we show that it suffices to prove ( ) after an extension of the base field. Forreference, we first state a result from descent theory. Letk0be a finite Galois extensionof a fieldkwith Galois groupG, and letVbe a variety overk0. Adescent datumforVrelative tok0=kis a collection of isomorphisms' W V!V, one for each 2G, suchthat' D' ' for all and . There is an obvious notion of a morphism of varietiespreserving the descent data. Note that for a varietyVoverk,Vk0has a canonical descentdatum. IfVis a variety overkandV0 DVk0, then a descent datum on anOV0-moduleMisa family of isomorphisms' W M!Msuch that' D' ' for all and.

9 A finite Galois extension with Galois groupG.(a)The map sending a varietyVoverktoVk0endowed with its canonical descent datumdefines an equivalence between the category of quasi-projective Varieties overkandthe category of quasi-projective Varieties overk0endowed with a descent datum.(b)LetVbe a variety overk, and letV0 DVk0. The map sending anOV-moduleMtoM0 DOV0 Mendowed with its canonical descent datum defines an equivalencebetween the category of coherentOV-modules and that of coherentOV0-modulesendowed with a descent datum. Moreover, ifM0is locally free, then so also , V, 20, or?, 17. (For the final statement, note that being locally free isequivalent to being flat for a coherent module, and thatV0is faithfully flat overV.) a finite separable extension ofk; if ( ) is true forCk0, thenit is true possibly enlargingk0, we may assume that it is Galois overk(with GaloisgroupG, say) and nonempty.

10 LetJ0be the Jacobian ofCk0. ThenJ0representsP0Ck0, and so there is a For any 2G, M2P0C. J0/, and sothere is a unique map' W J0!J0such ' / MD M(inP0C. J0//. One2 THE CANONICAL MAPS FROMCTO ITS Jacobian VARIETY5checks directly that' D' ' ; in particular,' ' 1D'id, and so the' areisomorphisms and define a descent datum onJ0. We conclude from ( ) thatJ0has amodelJoverksuch that the ! for allk-schemesT. In particular, for allT, there is a ! ! To seethat the map is an isomorphism nonempty, we have to show that in this ! an isomorphism. ; then (cf. the proof of ( )), we the set of isomorphism classes of ; /whereLis an invertiblesheaf onC Tk0whose fibres are of degree 0 and is an isomorphismOTk0 !.s;1/ pairs are rigid they have no automorphisms and so each such pair fixed underGhas a canonical descent datum, and therefore arises from an invertible sheaf onC The canonical maps fromCto its Jacobian varietyThroughout this section,Cwill be a complete nonsingular curve, andJwill be its Jacobianvariety (assumed to exist).)