PROOF. Let K be an algebraic number field, and let S be a finite set of primes in K.
From (4.1) and (4.2) we know that the mapC 7! .J.C /; .C //defines an injection from the set of isomorphism classes of complete nonsingular curves of genus 2to the set of isomorphism classes of principally polarized abelian varieties overK with good reduction outside S. Thus Shafarevich’s conjecture follows from the modified version of Finiteness
II. 2
PROOF(OF4.1) We are given a complete nonsingular curveC overK that reduces to a complete nonsingular curveC.v/over the residue fieldk.v/. Therefore we have Jacobian varietiesJ.C /overKandJ.C.v//overk.v/, and the problem is to show thatJ.C /reduces toJ.C.v//(and therefore has good reduction). It is possible to do this using only varieties, but it is much more natural to use schemes. LetR be the local ring corresponding to the prime ideal pv in OK. To say that C has good reduction to C.v/means that there is a proper smooth scheme C over SpecR whose general and special fibres are C and C.v/
respectively. The construction of the Jacobian variety sketched in (17) works overR(see JV,8), and gives us an abelian schemeJ.C/overSpecRwhose general and special fibres areJ.C /andJ.C.v//, which is what we are looking for. 2 PROOF(OF4.2) The original Torelli theorem applied only over an algebraically closed field and had no restriction on the genus (of course, Torelli’s original paper (1914-15) only applied over C). The proof over an algebraically closed field proceeds by a combinatorial study of the subvarieties ofC.r/, and is unilluminating (at least to me, even my own exposi-tion in JV,13). Now consider two curvesC andC0over a perfect fieldk, and suppose that there is an isomorphismˇWJ.C /!J.C0/(overk) sending the polarization.C /to.C0/.
Then the original Torelli theorem implies that there is an isomorphismWC !C0overkal. In fact, it is possible to specify uniquely (in terms ofˇ). For any 2 Gal.kal=k/, the map of curves associated withˇ is . Butˇ Dˇ(this is what it means to be defined overk), and so D, which implies that it too is defined overk(III12, for the details).2 EXERCISE4.4. Does (4.2) hold if we drop the condition thatg 2‹Hints: A curve of genus 0 over a field k, having no point in k, is described by a homogeneous quadratic equation in three variables, i.e., by a quadratic form in three variables; now apply results on quadratic forms (e.g., CFT, Chapter VIII). IfC is a curve of genus1without a point, then Jac.C /is an elliptic curve (with a point).
REMARK4.5. Torelli’s theorem (4.2) obviously holds for curvesC of genus< 2overk for whichC.k/¤ ;— a curve of genus zero withC.k/ ¤ ;is isomorphic toP1; a curve of genus one withC.k/¤ ;isits Jacobian variety.
5 Shafarevich’s Conjecture implies Mordell’s Conjecture.
In this section, we write .f / for the divisordiv.f /of a rational function on a curve. A p-adic primeof a number field is a prime dividingp; adyadic prime is a prime dividing2.
The proof that Shafarevich’s conjecture implies Mordell’s conjecture is based on the following construction (of Kodaira and Parshin).
THEOREM5.1. LetKbe a number field and letS be a finite set of primes ofKcontaining the dyadic primes. For any complete nonsingular curveC of genusg 1overK having
good reduction outsideS, there exists a finite extensionLofKwith the following property:
for each pointP 2 C.K/there exists a curveCP overLand a finite map'PWCP !C=L (defined overL) such that:
(i) CP has good reduction outsidefwjwjv2Sg; (ii) the genus ofCP is bounded;
(iii) 'P is ramified exactly at P.
We shall also need the following classical result.
THEOREM5.2 (DEFRANCHIS). LetC0 andC be curves over a field k. IfC has genus 2, then there are only finitely many nonconstant mapsC0!C.
Using (5.1) and (5.2), we show that Shafarevich’s conjecture implies Mordell’s conjec-ture. For each P 2 C.K/, choose a pair.CP; 'P/ as in (5.1). Because of Shafarevich’s conjecture, the CP fall into only finitely many distinct isomorphism classes. Let X be a curve over L. If X CP for some P 2 C.K/, then we have a nonconstant map X CP
'P
!C=Lramified exactly overP, and ifX CQ, then we have nonconstant map X !C=Lramified exactly overQ— ifP ¤Q, then the maps differ. Thus, de Franchis’s Theorem shows that map sending.CP; 'P/to the isomorphism class ofCP is finite-to-one, and it follows thatC.K/is finite.
Before proving 5.1, we make some general remarks. When isQŒp
f unramified at p ¤2? Exactly whenordp.f /is odd. This is a general phenomenon: ifKis the field of fractions of a discrete valuation ringRand the residue characteristic is¤2, thenKŒp
f is ramified if and only iford.f /is odd. (After a change of variables,Y2 f will be an Eisenstein polynomial iford.f / is odd, and will be of the formY2 uwithua unit if ord.f /is even. In the second case, the discriminant is a unit.)
Consider a nonsingular curveC over an algebraically closed field k of characteristic
¤ 2, and letf be a nonzero rational function onC. Then there is a unique nonsingular curveC0overkand finite mapC0 !C such that the corresponding mapk.C / ,!k.C0/ is the inclusion k.C / ,! k.C /Œp
f . Moreover, when we write.f / D 2DCD0 with D0having as few terms as possible, the remark in the preceding paragraph shows that' is ramified exactly at the points of support ofD0. (IfC is affine, corresponding to the ringR, thenC0is affine, corresponding to the integral closure ofRink.C /Œp
f .) For example, is finite, and is ramified exactly over the roots off1.X /.
When in this last example, we replace the algebraically closed field k with Q, one additional complication occurs: f might be constant, say f D r, r 2 Q. Then C0 ! SpecmQis the composite
CQŒpr!C !SpecmCQŒpr
5. SHAFAREVICH’S CONJECTURE IMPLIES MORDELL’S CONJECTURE. 147
— hereC0is not geometrically connected. This doesn’t happen ifordP.f /is odd for some pointP ofC.
Next fix a pair of distinct points P1, P2 2 A1.Q/, and let f 2 Q.X / be such that .f /DP1 P2. Construct theC0corresponding tof. Where doesC0have good reduction?
Note we can replacef withcf for anyc2Qwithout changing its divisor. If we wantC0 to have good reduction on as large a set as possible, we choose
f D.X P1/=.X P2/ rather than, say, (*)
f Dp.X P1/=.X P2/:
The curve
Y2 D.X P1/=.X P2/
has good reduction at any prime whereP1andP2remain distinct (except perhaps2). After these remarks, the next result should not seem too surprising.
LEMMA5.3. LetC be a complete nonsingular curve over a number fieldK, and consider a principal divisor of the form
P1 P2C2D; P1; P22C.K/:
Choose anf 2K.C /such that.f / DP1 P2C2D, and let'WC0 !C be the finite covering of nonsingular curves corresponding to the inclusionK.C / ,!K.C /Œp
f . With a suitable choice off, the following hold:
(a) The map'is ramified exactly atP1andP2.
(b) LetSbe a finite set of primes ofKcontaining thosevat whichC has bad reduction, thosevat whichP1 andP2 become equal, and all primes dividing2. If the ring of S-integers is a principal thenC0has good reduction at all the primes inS.
PROOF. (a) We have already seen this—it is really a geometric statement. (b) (Sketch.) By assumptionC extends to smooth curveCoverSpec.R/, whereRis the ring ofS-integers.
The Zariski closure of D0 Ddf P1 P2C2D inC is a divisor onC without any “vertical components”, i.e., without any components containing a whole fibreC.v/ofC!Spec.R/.
We can regardf as a rational function onCand consider its divisor as well. Unfortunately, as in the above example (*), it may have vertical components. In order to remove them we have to replacef with a multiple by an elementc 2 K having exactly the correct value ordp.c/ for every prime idealpinR. To be sure that such an element exists, we have to
assume thatRis principal. 2
REMARK5.4. (Variant of the lemma.) Recall that the Hilbert class fieldKHCF ofK is a finite unramified extension in which every ideal inK becomes principal. Even if the ring ofS-integers is not principal, there will exist anf as in the theorem inKHCF.C /.
PROPOSITION5.5. LetAbe an abelian variety over a number fieldKwith good reduction outside a set of primesS. Then there is a finite extensionLofKsuch thatA.K/2A.L/.
PROOF. The Mordell-Weil Theorem implies thatA.K/=2A.K/is finite, and we can choose Lto be any field containing the coordinates of a set of representatives forA.K/=2A.K/.
[In fact, the proposition is more elementary than the Mordell-Weil Theorem — it is proved
in the course of proving the Weak Mordell-Weil Theorem. 2
PROOF(OF5.1) If C.K/ is empty, there is nothing to prove. Otherwise, we choose a rational point and use it to embed C into its Jacobian. The map 2JWJ ! J is ´etale of degree22g(see 7.2). When we restrict the map to the inverse image ofC, we get a covering 'WC0 ! C that is ´etale of degree22g. I claim thatC0has good reduction outsideS, and that each point of ' 1.P /has coordinates in a fieldL that is unramfied overK outside S. To see this, we need to use that multiplication by2is an ´etale mapJ !J of abelian schemes overSpecRS (RS is the ring ofS-integers inK). The inclusionC ,!J extends to an inclusionC,!J of schemes smooth and proper overSpecRS, and fibre product of this with 2WJ !J gives an ´etale mapC0 Ddf CJ J ! C. ThereforeC0 ! SpecRS is smooth (being the composite of an ´etale and a smooth morphism), which means thatC0has good reduction outsideS. The pointP defines anRS-valued point Spec.RS/ ! C, and the pull-back ofC0!Cby this is a scheme finite and ´etale overSpec.RS/whose generic fibre is' 1.P /— this proves the second part of the claim. For anyQ2' 1.P /,ŒK.Q/W K22g. Therefore, according to Theorem 3.1, there will be a finite field extensionL1of K such that all the points of' 1.P /are rational overL1for allP 2C.K/. Now choose two distinct pointsP1andP2 lying overP, and consider the divisorP1 P2. According to (5.5), for some finite extension L2 of L1, every element of J.L1/lies in 2J.L2/:In particular, there is a divisorDonC=L2such that2DP1 P2. Now replaceL2with its Hilbert class field L3. Finally choose an appropriatef such that .f / D P1 P2 2D, and extract a square root, as in (5.3,5.4). We obtain a map'P
CP !C=L3!' C=L3
overL3of degree222gthat is ramified exactly overP. Now the Hurwitz genus formula 2 2g.CP/D.2 2g.C //deg'C X
Q7!P
.eQ 1/
shows thatg.CP/is bounded independently ofP. The fieldL3 is independent ofP, and (by construction)CP has good reduction outside the primes lying overS. Thus the proof
of Theorem 5.1 is complete. 2
PROOF(OF5.2) The proof uses some algebraic geometry of surfaces (Hartshorne, Chapter V). Consider a nonconstant map'WC0 ! C of curves. Its graph ' C0C Ddf X is a curve isomorphic toC0, and is therefore of genusg0D genusC0. Note that
' .fP0g C /D1; ' .C0 fPg/D#' 1.P /Dd; d Ddeg.'/:
The canonical class ofX is
KX .2g 2/.C0 fPg/C.2g0 2/.fP0g C / and so
'KX D.2g 2/d C.2g0 2/: