Faltings, G.,
Endlichkeitss¨atze f ¨ur abelsche Variet¨aten ¨uber Zahlk¨orpern. [Finiteness Theorems for Abelian Varieties over Number Fields],
Invent. Math. 73 (1983), 349-366; Erratum,ibid. (1984), 75, 381.
The most spectacular result proved in this paper is Mordell’s famous 1922 conjecture: a nonsingular projective curve of genus at least two over a number field has only finitely many points with coordinates in the number field. This result is in fact obtained as a corollary of finiteness theorems concerning abelian varieties which are themselves of at least equal significance. We begin by stating them. Unless indicated otherwise, K will be a number field, the absolute Galois groupGal.K=K/ofK,Sa finite set of primes ofK, andAan
8. THE COMPLETION OF THE PROOF OF FINITENESS I. 159 abelian variety overK. For a prime numberl,TlAwill denote the Tate group ofA(inverse limit of the groups ofln-torsion points onA) andVlAD Ql ˝Zl TlA. The paper proves the following theorems.
THEOREM3. The representation of onVlAis semisimple.
THEOREM4. The canonical mapEndK.A/˝ZZl !End.TlA/ is an isomorphism.
THEOREM5.For givenSandg, there are only finitely many isogeny classes of abelian varieties overKwith dimensiongand good reduction outsideS.
THEOREM 6. For givenS; g, andd, there are only finitely many isomorphism classes of polarized abelian varieties overKwith dimensiong, degree (of the polarization)d, and good reduction outsideS.
Both Theorem 3 and Theorem 4 are special cases of conjectures concerning the ´etale co-homology of any smooth projective variety. The first is sometimes called the Grothendieck-Serre conjecture; the second is the Tate conjecture. Theorem 6 is usually called Shafare-vich’s conjecture because it is suggested by an analogous conjecture of his for curves (see below).
In proving these theorems, the author makes use of a new notion of the heighth.A/of an abelian variety: roughly, h.A/is a measure of the volumes of the manifoldsA.Kv/, v an Archimedean prime ofK, relative to a N´eron differential onA. The paper proves:
THEOREM 1. For giveng andh, there exist only finitely many principally polarized abelian varieties over K with dimensiong, height h, and semistable reduction every-where.
THEOREM 2. LetA.K/.l/be thel-primary component ofA.K/, some prime number l, and letG be anl-divisible subgroup ofA.K/.l/stable under . LetGndenote the set of elements ofG killed byln. Then, fornsufficiently large,h.A=Gn/is independent ofn.
THEOREM (*) LetA be an abelian variety overK with semistable reduction every-where; then there is an N such that for every isogeny A ! B of degree prime to N, h.A/Dh.B/.
The proof of Theorem 4 is modelled on a proof of J. T. Tate for the case of a finite field K [same journal 2 (1966), 134–144; MR 34#5829]. There, Tate makes use of a (trivial) analogue of Theorem 6 for a finite field to show that a special element ofEnd.TlA/ lies in the image of the map. At the same point in the proof, the author applies his Theorems 1 and 2. An argument of Yu. G. Zarkhin [Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 272–277; MR 51#8114] allows one to pass from the special elements to a general element.
Theorem 3 is proved simultaneously with Theorem 4.
From Theorem 4 in the case of a finite field, it follows that the isogeny class of an abelian variety over a finite field is determined by the characteristic polynomial of the Frobenius element. By making an adroit application of the Chebotarev density theorem (and Theorems 3 and 4), the author shows the following: givenSandg, there exists a finite set T of primes of K such that the isogeny class of an abelian variety over K of dimen-sion g with good reduction outsideS is determined by the characteristic polynomials of the Frobenius elements at thevinT. (This in fact seems to give an algorithm for deciding when two abelian varieties over a number field are isogenous.) Since the known properties of these polynomials (work of Weil) imply there are only finitely many possibilities for each prime, this proves Theorem 5.
In proving Theorem 6, only abelian varieties B isogenous to a fixed abelian variety Aneed be considered (because of Theorem 5), and, afterK has been extended,Acan be
assumed to have semistable reduction everywhere. The definition of the height is such that e.B=A/Ddf exp.2ŒK WQ.h.B/ h.A//
is a rational number whose numerator and denominator are divisible only by primes di-viding the degree of the isogeny between AandB. Therefore (*) shows that there exists an integer N such that e(A/B) involves only the primes dividing N. The isogenies whose degrees are divisible only by the primes dividingN correspond to the -stable sublattices ofQ
ljNTlA. From what has been shown aboutTlA, there exist only finitely many iso-morphism classes of such sublattices, and this shows that the set of possible valuesh.B/is finite. Now Theorem 1 can be applied to prove Theorem 6.
The proof of Theorem 1 is the longest and most difficult part of the paper. The basic idea is to relate the theorem to the following elementary result: given h, there are only finitely many points in Pn.K/with height (in the usual sense) h. The author’s height defines a function on the moduli space Mg of principally polarized abelian varieties of dimensiong. IfMg is embedded inPnK by means of modular forms rational overK, then the usual height function onPndefines a second function onMg. The two functions must be compared. Both are defined by Hermitian line bundles onMg and the main points are to show (a) the Hermitian structure corresponding to the author’s height does not increase too rapidly as one approaches the boundary of Mg (it has only logarithmic singularities) and (b) by studying the line bundles on compactifications of moduli schemes over Z, one sees that the contributions to the two heights by the finite primes differ by only a bounded amount. This leads to a proof of Theorem 1. (P. Deligne has given a very concise, but clear, account of this part of the paper [“Preuve des conjectures de Tate et de Shafarevitch”, Seminaire Bourbaki, Vol. 1983/84 (Paris, 1983/84), no. 616; per revr.].)
The proofs of Theorems 2 and (*) are less difficult: they involve calculations which reduce the questions to formulas of M. Raynaud [Bull. Soc. Math. France 102 (1974), 241–280; MR 547488]. (To obtain a correct proof of Theorem 2, one should replace theA of the proof in the paper byA=Gn, some n sufficiently large.)
Torelli’s theorem says that a curve is determined by its canonically polarized Jacobian.
Thus Theorem 6 implies the (original) conjecture of Shafarevich: givenSandg, there exist only finitely many nonsingular projective curves over K of genusg and good reduction outside S. An argument of A. N. Parshin [Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1191–1219; MR 411740] shows that Shafarevich’s conjecture implies that of Mordell: to each rational pointP on the curveX one associates a covering'P W XP ! X ofX; the curve XP has bounded genus and good reduction outside S; thus there are only finitely many possible curvesXP, and a classical theorem of de Franchis shows that for eachXP
there are only finitely many possible'P; as the associationP 7!.XP; 'P/is one-to-one, this proves that there are only finitely manyP.
Before this paper, it was known that Theorem 6 implies Theorems 3 and 4 (and Mordell’s conjecture). One of the author’s innovations was to see that by proving a weak form of The-orem 6 (namely TheThe-orem 1) he could still prove TheThe-orems 3 and 4 and then could go back to get Theorem 6.
Only one misprint is worth noting: the second incorrect reference in the proof of Theo-rems 3 and 4 should be to Zarkhin’s 1975 paper [op. cit.], not his 1974 paper.
James Milne.
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Index
semisimplicity, 131
Shafarevich’s conjecture, 132 Tate’s conjecture, 132 Wedderburn’s, 79 Zarhin’s trick, 67 variety, v
abelian, 8 group, 7 Jacobian, 86 pointed, 86 Weil pairing, 15, 57 Weil q-integer, 79