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APPEARED IN BULLETIN OF THE

arXiv:math/9307231v1 [math.NT] 1 Jul 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 29, Number 1, July 1993, Pages 14-50ON THE PASSAGE FROM LOCALTO GLOBAL IN NUMBER THEORYB. MazurIntroductionWould a reader be able to predict the branch of mathematics that is the subjectof this article if its title had not included the phrase “in Number Theory”?

Thedistinction “local” versus “global”, with various connotations, has found a home inalmost every part of mathematics, local problems being often a stepping-stone tothe more difficult global problems.To illustrate what the earthy geometric terms local and global signify in NumberTheory, let us consider any Diophantine equation, say,Xn + Y n + Zn = 0(n > 2) .Determining the rational (or equivalently, integral) solutions of such equations canbe difficult! In contrast, to determine its solutions in integers modulo m (at least,for any fixed modulus m > 0) is a finite problem—a comparatively easy problem.Moreover, this easy problem, thanks to the Chinese Remainder Theorem, can bereplaced by the problem of finding its solutions modulo the prime powers pν dividingm.

Given an integral solution, i.e., an + bn + cn = 0, for a, b, c ∈Z, by viewing(a, b, c) as a triple of integers modulo pν for any prime number p and any ν ≥0,we get, of course, a solution modulo pν.Ever since (implicitly) the work of Kummer and (explicitly) the work of Hensel,we know that it is useful to note that an integral solution to such a Diophantineequation yields more: for each prime number p, and for positive integers ν =1, 2, 3, . .

., if we choose to view the integral solution, successively, as a solutionmodulo pν, we then get, for every prime number p, a system of solutions modulopν for each ν = 1, 2, . .

. , the system being “coherent” in the sense that reduction ofthe (ν + 1)st solution modulo pν yields the νth solution.

Of course, such a systemis nothing more than a solution in the projective limit Zp := Lim(Z/pνZ), the ringof p-adic integers (Zp being considered as a topological ring, given the “profinitetopology”, i.e., the natural topology that it inherits as a projective limit of thefinite discrete rings Z/pνZ).For more flexibility, one can work in Qp, the field of fractions of Zp (most eco-nomically obtained from Zp by adjoining the inverse of the single element p). Thefield Qp contains Q and can be regarded as the topological (in fact, metrized) field1991 Mathematics Subject Classification.

Primary 11-02, 14-02, 14-G05, 14G20, 14G25, 14J20,14K15.Received by the editors October 1, 1992 and, in revised form, February 17, 1993c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2B. MAZURobtained by completing the field Q with respect to the “p-adic” metric1.

Evidentlythen, if a polynomial equation with Q-coefficients fails to have a solution over somep-adic field Qp, it cannot have a solution over Q.The formal relationship between the field Q and the collection of completions Qp(for p ranging through all prime numbers) is closely analogous to the relationshipbetween the field K of global rational functions on, say, a smooth algebraic curveS over C in the affine plane, and the collection of fields Ks of formal Laurentseries with finite order poles, centered at points s ∈S. The distinction of globalv.

local is evident here: any global meromorphic function f ∈K gives rise to acollection of local data, namely, the collection of Laurent expansions of f at eachof the points s ∈S. Our affine plane curve S is not compact, and therefore thecollection of fields {Ks}fs∈S does not reflect “the whole local story”.

To get “thewhole story”, we usually augment our collection {Ks} by including the completionsof K corresponding to the “missing” points, the points at infinity. It is only whenone takes account of all the completions {Ks}s for all points s in the smoothcompactification of S that certain global constraints on the collections of localdata hold, the most basic being that the number of zeros of a global function(zeros counted with appropriate multiplicity) is equal to the number of its poles;equivalently, if {fs}s is a collection of Laurent series all coming from the sameglobal meromorphic function f, then(1)Xords(fs) = 0,where ords(fs) is the order of zero (or, if negative, pole) of the Laurent series fs ats.In analogy, it has long been understood (cf.

[W]) that there is one “missingfield” which should be considered along with the collection of p-adic completionsof Q; namely, the field R, of real numbers, i.e., the Ur-completion of Q (withrespect to the standard absolute value metric, which we denote by | |∞, to bring itnotationally into line with the p-adic metrics). The analogue of (1) is the following(easily proved) formula2: If r ∈Q, then(2)Xplog |r|p = 0,provided the summation subscript p ranges over all prime numbers and (“the infiniteprime”) ∞.One weakness in the analogy between the collection {Ks}s∈S for a compactRiemann surface S and the collection {Qp, for prime numbers p, and R} is that thefields Ks for all points s are very much alike; they are even isomorphic.

In contrast,no manner of squinting seems to be able to make R the least bit mistakable for anyof the p-adic fields, nor are the p-adic fields Qp isomorphic for distinct p. A majortheme in the development of Number Theory has been to try to bring R somewhat1Two rational numbers are very close in the p-adic metric if their difference, expressed as afraction in lowest terms, has the property that its numerator is divisible by a high power of p.The standardly normalized p-adic metric | |p is characterized by the fact that it is multiplicative,that |p|p = 1/p, and that if a is an integer not congruent to 0 mod p, then |a|p = 1.2Exponentiating both sides of this equation, one gets the somewhat more familiar “productformula”.

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY3more into line with the p-adic fields; a major mystery is why R resists this attemptso strenuously.Here are two types of questions in Number Theory one might want to pursue bypassing from local to global:Type (A). Questions about rational points.

Given a Diophantine equationor a system of Diophantine equations with coefficients in Q, when does knowledgeabout its rational solutions over the collection of local fields Qp for all prime num-bers p, and over R, give us some palpable information about its solutions overQ?Here is one of the simplest examples of such a question:For which classes of systems of Diophantine equations with coefficients in Q isit the case that the existence of solutions over Qp for all prime numbers p and overR guarantees the existence of a solution over Q?The question of existence of solutions of systems of Diophantine equations overR or over Qp is “certifiably easy”; at least it is a decidable question in the sense ofmathematical logic. In fact, all first-order theories over R (or over Qp) are decidable,this being a result due to Tarski (and Ax-Kochen), a concise treatment of whichcan be found in [Coh].

Whether or not the question of existence of solutions ofsystems of Diophantine equations over Q is a decidable question is still up in theair3, but, as the example which we trotted out at the beginning of this article wasmeant to convince, decidable or not, even specific polynomials with coefficients inQ can require quite elaborate theory before we can say that all their Q-rationalpoints are known4.Type (B). Passing from knowledge about local “structures” to knowledgeabout global “structures”.

There is a wealth of literature on various aspectsof questions of type (A)5. One cannot, really, effectively “separate” questions oftype (A) from questions of type (B), but the central aim of this article is to discussan exciting development (finiteness of certain Tate-Shafarevich groups6), which isconveniently expressed in the language of (B).In 1986 at the MSRI, Larry Washington told Karl Rubin of some new ideas ofThaine, suspecting that they might be helpful in Rubin’s work.

These ideas, asformulated by Thaine7, had been brought to bear on questions concerning the idealclass groups of cyclotomic fields [Th]. Rubin refashioned Thaine’s ideas to serve as atool in the study of the arithmetic of complex multiplication elliptic curves.

By theirmeans, Rubin was able, among other things, to prove that some Tate-Shafarevichgroups of elliptic curves over number fields were finite.This provided us withthe first examples (in the number field case8) where the finiteness of some Tate-3Matijasevic has established the algorithmic undecidability of the question of whether anygiven Diophantine equation has an integral solution [Mat].4Anyone who doubts this should try to read published accounts of the proof of “Fermat’s LastTheorem” for exponent n = 37.5For example, for a recent article which contains, among other things, an extensive review andbibliography concerning current activities concerning the Hasse principle and “weak approxima-tion” (of local solutions by global ones) in some classes of varieties, see [C-T–S-D].6We will be explaining this.7Thaine says that at least partial inspiration for these ideas came from a close reading of apaper of Kummer.8There had already been some function field examples [Mi1–3].

4B. MAZURShafarevich groups was actually proved, despite the fact that (all) Tate-Shafarevichgroups had been conjectured, three decades ago, to be finite.Soon thereafter, working independently, V. A. Kolyvagin devised his approachto these finiteness questions (Kolyvagin’s method of “Euler Systems” resembles themethod of Thaine-Rubin but is more flexible and is applicable to a much broaderrange of problems).Kolyvagin’s ideas (and those of Rubin and Thaine) represent something of arevolution9.My plan in Part I is to try to explain to a general audience some of the importanceof Finiteness of the Tate-Shafarevich Group.

Anything discussed in Part I thatsmacks of being too technical is tucked into footnotes or into Part III.Part II is an analytic continuation of Part I in which I try to give a bit of theflavor of how one goes about showing such finiteness. It is written in more technicallanguage than Part I, but still no proofs are given here.

I hope that it may beuseful to some readers as an accompaniment to some of the published articles onthe subject or as a companion to some of the other surveys; see, for example, therecent [Ne2].Part III provides proofs of some things that seemed natural to bring up in anexpository treatment of this sort, results that are, very likely, well known to theexperts but for which there is no reference in the literature.ContentsPart I1. Local-to-global principles (an example of a question of type (A): zeroes ofquadratic forms)2.

Local-to-global principles (an example of a question of type (B): isomor-phisms of projective varieties)3. First examples: projective space, quadrics, and curves of genus zero4.

Curves of genus ≥1. The “local-to-global principle up to finite obstruction”5.

General projective varieties6. The Tate-Shafarevich group7.

Implications of the Tate-Shafarevich Conjecture in the direction of the gen-eral local-to-global principle (up to finite obstruction)9The impressive number of otherwise inaccessible problems that these methods successfullytreat is one measure of the revolutionary aspect of this work; the bibliography at the end of thesenotes gives a small sample of the activity involved here. For example, thanks to Kolyvagin andRubin we now have: (1) a significantly shorter proof of the “Main Conjecture over Q” (originallyproved by Andrew Wiles and me); (2) the one- and two-variable “Main Conjecture” over quadraticimaginary fields; and (3) (most of) the Birch Swinnerton-Dyer Conjecture for modular ellipticcurves whose L function does not have a multiple zero at s = 1, and in particular, that (for theseelliptic curves) the Tate-Shafarevich group is finite (see Theorem 3).But even more telling is that, after all this, we now think of the subject in quite a different way,and a range of new arithmetic problems seems nowadays to be within reach.

For example, thereis Nekov´aˇr’s recent study of Selmer groups attached to higher (even) weight to gain informationabout algebraic cycles on Kuga-Sato varieties [Ne1]; there is Flach’s finiteness theorem for certainSelmer groups attached to automorphic forms arising as the symmetric square of classical modularforms which yields information about the deformation space of the Galois representations attachedto classical forms [F]; there is Darmon’s attack on the “refined conjectures” of Birch Swinnerton-Dyer type [D1, D1] (see also [BD]); there is Kato’s recent exciting announcement concerning theconstruction of Kolyvagin-type (“Euler”) systems of classes in algebraic K-groups.

The passage from local to global in number theory58. Examples of finiteness of the Tate-Shafarevich group9.

Returning to Selmer’s curvePart II10. The Tate-Shafarevich group and the Selmer group11.

Local control of the Selmer group12. Class field theory, duality, and Kolyvagin test classes13.

Rational points in extension fields yielding “test classes”14. Heegner pointsPart III15.

The cohomology of locally algebraic group schemes16. Discrete locally algebraic group schemes17.

Finiteness theorems18. The mapping of the Tate-Shafarevich group of A/K to S(A/K)19.

The local-to-global principle for quadricsBibliographyPart I1.Local-to-global principles (an example of a question of type (A)10:zeros of quadratic forms). The Theorem of Hasse-Minkowski [H] guaranteesthata quadratic form with coefficients in Q admits a nontrivial zero over Q if and onlyif it does so over Qp for all prime numbers p and over R.To get a closer view of the sort of information that this classical result gives us,let us think about it in the following special case: Consider a diagonal quadraticform in three variables(1)F(X, Y, Z) = a · X2 + b · Y 2 −c · Z2with a, b, c positive integers.The fact that the coefficients in (1) are not all of the same sign guarantees thatour form has a nontrivial zero over R. The Theorem of Chevalley (which says thata polynomial in more variables than its degree, with integer coefficients modp andno constant term has a nontrivial zero modp) gives us that (1) has a nontrivialzero modulo p for any prime number p. Moreover, for odd prime numbers p notdividing a · b · c, Hensel’s Lemma then tells us that the nontrivial zero modp givento us by Chevalley’s theorem lifts to a nontrivial p-adic zero.

Thus, in this case,the Hasse-Minkowski Theorem assures us that, if(2)a · X2 + b · Y 2 −c · Z2 = 0has a nontrivial p-adic solution for the finite set of primes p dividing 2·a·b·c, thenit has a rational solution.There is no difficulty in stipulating congruence conditions on a, b, c under which(2) will have p-adic solutions for this finite set of prime numbers p.10Type (A) refers to the distinction alluded to in the discussion in the introduction.

6B. MAZURBut even in a concrete instance of (2), where the Hasse-Minkowski Theoremguarantees the existence of a rational solution, there remains much yet to ponderabout; for the proof of the Hasse-Minkowski Theorem does not produce any specificnontrivial rational solution easily, despite the fact that the problem of finding suchsolutions is perfectly effective (e.g., in Mordell’s book [Mo] it is shown that if (2)has a nontrivial rational solution, it has such a solution with|X| ≤√b · c,|Y | ≤√c · a,|Z| ≤√a · b .For other, more general bounds see [Ca1, 6.8, Lemma 8.1]).As Victor Miller pointed out, it would be good to give explicit bounds for theadelic version of this problem, i.e., given a finite set of prime numbers p and specificp-adic points for these p, find rational points of prescribed closeness to these p-adicpoints.Moreover, it would be of interest to specify fast algorithms (e.g., are there algo-rithms which are of polynomial time in max(log a, log b, log c)?) for the determina-tion of these rational points.For recent results concerning the equidistribution properties of the sets Sg(n)of integral solutions of g(x, y, z) = n (for specified integers n) where g(x, y, z) is apositive-definite ternary quadratic form (equidistribution of the sets Sg(n), that is,when they are scaled back to the unit ellipsoid g(x, y, z) = 1 in R3), see [Du, GF,D–S-P]11.There are also distributional types of estimates coming from the circle methodconcerning rational zeros of diagonal forms (in four or more variables).

For example,Theorem 4 of Siegel’s 1941 paper [Si] treats the special case of a nondegeneratequadratic form in four variables with integral coefficients,F(X, Y, Z, T ) = aX2 + bY 2 + cZ2 + dT 2,whose determinant is the square of an integer. Explicitly, fix such a quadratic formF, and for S = (X, Y, Z, T ) a four-tuple of integers, put∥S∥:= |a| · X2 + |b| · Y 2 + |c| · Z2 + |d| · T 2 .Theorem 4 of loc.

cit. gives a positive constant κ such thatXSexp(−∥S∥/λ) = κ · λ log λ + o(λ log λ)(as λ →∞),the summation being extended over all (nontrivial) integral zeros S of F, assumingthat F has nontrivial integral zeros (equivalently, assuming that F is indefinite andhas a nontrivial zero in Qp for all prime numbers p).2.Local-to-global principles (an example of a question of type (B)12:isomorphisms of projective varieties).

One says that two quadratic forms (inthe same number of variables) over a field K are equivalent if one can pass from11The case of three variables is particularly difficult; these equidistribution results are basedon delicate upper bounds for the absolute values of the Fourier coefficients of modular forms ofhalf-integral weight obtained by Iwaniec [I].12Type (B) refers to the distinction alluded to in the discussion in the introduction.

The passage from local to global in number theory7one of the quadratic forms to the other (and from the other to the one) by suitablelinear substitutions of variables, where the matrix describing the linear change ofvariables has coefficients in K. Questions of equivalence of quadratic forms overlocal fields are quite manageable; for example, by “Sylvester’s Law”, quadraticforms over R are determined by three integers (their rank, nullity, and index).Here is a variant of the Hasse-Minkowski theorem (cf. [Has, Si]) expressed asthe answer to a question of type (B):Two quadratic forms with coefficients in Q which are equivalent over Qp for allprime numbers p and over R are equivalent over Q.In this type (B) spirit, let us prepare for a more supple and more general local-to-global question.Let V be a projective variety defined over any field k. We can think of V as“given by” some finite collection of homogeneous forms in N variables (for someN) with coefficients drawn from k. The set V (L), of L-rational points of V , for Lany field extension of k, is the common “locus of zeros” (in projective N-space withhomogeneous coordinates in L) of the collection of homogeneous forms defining V .Assume that V is smooth13.The notion of isomorphism between two (smooth) varieties V and W over k (thetechnical term being biregular isomorphism14 defined over k) will be important tous.

But here, in contrast to the notion of equivalence of quadratic forms, one allowschanges of variables more fluid than the mere linear changes of variables allowedin the definition of equivalence in the theory of quadratic forms.15 We admit thepossibility that V and W may be given as loci of zeros of systems of homogeneousforms in different dimensional projective spaces and that the systems for V and forW may comprise different numbers of homogeneous forms and of different degrees.16Now let V be a smooth variety over Q. Embedding Q in its various completions,we may forget that our V is defined over Q and think of V as defined over Qp,where p runs through all prime numbers, or over R, by simply regarding the definingcollection of homogeneous forms of V as having their coefficients in Qp or in R. To13Why not? “Smoothness” is defined, as in multivariable calculus, by the requirement thatthe appropriate jacobian determinant does not vanish at k-rational points of V .14See Lecture 2 of [Harr] for the notion of “regular morphism” over algebraically closed fieldsand [Hart] for the foundations for the more general concept of “morphism of schemes”.15Nevertheless, in the special case of (smooth) quadrics, i.e., varieties given as the locus ofzeros of a single (nondegenerate) quadratic form, the quadratic forms are similar (i.e., equivalentup to multiplication by a nonzero scalar) over K if and only if the varieties are isomorphic overK.

The point here is that (setting aside the case of quadric surfaces, for the moment), if V is asmooth quadric of dimension 1 or ≥3, then Pic(V ) = Z. Therefore, any automorphism of thevariety V must preserve the isomorphism class of L, the (unique) generating ample line bundle,and consequently must extend to projective linear automorphisms of the projective spaces comingfrom the linear spaces of global sections of L⊗n for any n. Since the initial embedding of V inprojective space (via the defining quadratic form) is the projective space of global sections of L⊗2if dim V = 1, and of L if dim V ≥3, one easily sees that similarity between such quadratic formsis the same as isomorphism between their corresponding varieties.

The same conclusion holds forsmooth quadric surfaces V with a slightly different argument (since the Picard group of V overK is Z ⊕Z).16When the field k is C, using the faithfulness of the functor which passes from a smoothprojective varieties V over C to its underlying complex analytic manifold V (C) [Se1], one seesthat to give a biregular morphism from V to W over C is equivalent to giving a complex-analyticisomorphism from V (C) to W (C).

8B. MAZURindicate that we have done this, we shall sometimes refer to the “original” varietyV , defined over Q as V/Q; and if we wish to think of it as defined over Qp or overR, we refer to it as V/Qp or V/R.Question.

To what extent do the “local” varieties{V/Qp, for all p, and V/R}determine the “global” variety V/Q (up to isomorphism over Q)?To have vocabulary for this, let us say that a (smooth projective) variety V ′/Qis a companion to V/Q if their corresponding local varieties are isomorphic,17 i.e.,if V ′/Qp is isomorphic to V/Qp (as varieties over Qp) for all prime numbers p, andV ′/R is isomorphic to V/R (as varieties over R).Denote by S(V ) the set of isomorphism classes over Q of varieties V ′/Q whichare companions to V .18One can think of the cardinality of S(V ) as roughly analogous to a class number,i.e., a measure of the extent to which local data (in this case, the isomorphismclasses of V/Qp for all p, and V/R) determine or fail to determine global data (theisomorphism class of V/Q). One might say that the local-to-global principle holdsfor a class of varieties V if S(V ) consists of the single isomorphism class {V } foreach member V of V.3.

First examples: projective space, quadrics, and curves of genus zero.Projective space. The local-to-global principle holds for any variety V/Q which isisomorphic (over C) to projective N-space for some N. This statement,19 which isone of the many manifestations of the local-to-global principle for elements of theBrauer group 20 of Q, might be viewed as an arithmetic addendum to the largebody of literature (e.g., differential geometric and algebraic geometric) establishingvarious ways in which projective space is rigid.

The analogous statements holdmore generally with Q replaced by any number field.Quadrics. The local-to-global principle holds for any smooth quadric variety V overQ (and, again, more generally over any number field).

I am thankful to Colliot-Th´el`ene for providing me with a sketch of a proof of this (see Part III, §19).Curves of genus 0. The local-to-global principle holds for any (smooth, projective)curve of genus 0 which is defined over Q (or over any number field).

This followsfrom either of the two previously cited examples, since:(i) any smooth curve of genus 0 over an algebraically closed field is isomorphicto P1, and17Companion is a neologism; twisted Q-form is the term sometimes used, but I prefer a newword here, since the term “twisted form” occurs in a number of different contexts.Moreover,we will want to reserve the term “twist” (below) for abelian varieties which are twists of otherabelian varieties.18So as not to get sidetracked in technical things, we leave for Part III a discussion of thecohomological interpretation of S(V ).19Contained in the 1944 Ph.D. thesis of Chˆatelet, for a historical discussion of which see [C-T,§1].20For the Brauer group and the relationship between Brauer groups over local fields and globalones, see [Se3].

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY9(ii) any smooth curve of genus zero over any field k is canonically isomorphicto a plane conic curve over k.214. Curves of genus ≥1.

The “local-to-global principle up to finite ob-struction”. That the local-to-global principle does not always hold for curves ofgenus 1 was seen early on, the first explicit examples being given by Lind and Re-ichart in the early forties.

In fact, this principle fails either in its variants type(a) or type (B), for the failure of these two variants are rather close in the contextof curves of genus 1: If a curve of genus 1 possesses a Qp-rational point for allprime numbers p and a real point, and it does not have a Q-rational point, then ithas at least one nonisomorphic companion. Conversely, if it has a nonisomorphiccompanion, then at least one of its companions has no Q-rational point (yet hasQp-rational points for all p and real points).That the local-to-global principle fails even for smooth plane cubic curves22 wasshown by Selmer (1951).

Here is his most widely quoted example:The curve C : 3x3 + 4y3 + 5z3 = 0 has nonisomorphic companions. This equationpossesses nontrivial solutions over Qp for all prime numbers p and over R, but itpossesses no nontrivial solutions over Q.When I was preparing these notes, I initially intended to quote this Selmerexample and go on.

But then it occurred to me that after the recent work whichis the subject of this article, we are actually in a position to explore the questionof companions to Selmer’s curve (and others) with some precision. Working outsuch an example seems to me to be a good way of demonstrating the power of therecent results of Rubin and Kolyvagin.

So, the following is an explicit list of all thecompanions (worked out with the help of some e-mail communications of Rubin):Theorem 1. Selmer’s curve C : 3x3 + 4y3 + 5z3 = 0 has, counting itself, preciselyfive companions :3x3 + 4y3 + 5z3 =0,12x3 + y3 + 5z3 =0,15x3 + 4y3 + z3 =0,3x3 + 20y3 + z3 =0,60x3 + y3 + z3 =0 .Commentary.

(1) This innocuous-sounding result lies quite deep. The fact that allits companions are of degree 3 may be misleading; there is, a priori, no guaranteethat the companion curves to V are isomorphic to smooth plane curves over Q,and, in fact, there is a priori no upper bound to their degree in projective space.21This isomorphism is given as the projective embedding associated to the anticanonical linebundle on the smooth curve of genus 0.

Serre suggested that I also point out the fact that thecategory of quaternionic division algebras over any field k of characteristic ̸= 2 is equivalent tothe category of smooth curves of genus 0 over k, an equivalence in one direction being given asfollows: If D is a quaternionic division algebra over k, let VD denote the plane conic over k givenby the equations Tr(x) = 0 and N(x) = 0 in D, where Tr is the trace and N is the quaternionicnorm. Exercise (Serre).

Find an explicit equivalence of categories going in the other direction.22These are all of genus 1; smooth curves of degree d in the projective plane are of genusg = (d −1)(d −2)/2.

10B. MAZUREstablishing finiteness of S(C), something we could not even begin to do beforethe work of Rubin and Kolyvagin, is tantamount to establishing an upper boundto their degree.

(2) All five equations on this list have nontrivial rational solutions over Qp forall prime numbers p and over R. The first four equations on the list possess nonontrivial rational solutions.The fifth equation possesses a nontrivial rationalsolution (0, 1, −1), and this solution is unique up to scalar multiplication (cf. [Ca2,§18]).

If we take this point as “origin” of the projective curve E defined by theequation60x3 + y3 = z3 = 0,then E is an elliptic curve over Q isomorphic to the jacobian of all five curves onthe list. (3) In analogy with the discussion in §2 of the Hasse-Minkowski Theorem, givingthe full list of companions as we have just done does not complete in a satisfactorymanner the discussion of passage from local to global for this example: the demand-ing reader might ask how, if we are given some projective smooth curve over Qtogether with the data that exhibits it as a companion to Selmer’s C, we may finda specific isomorphism over Q between it and one of the members of our list.23In view of this type of example, why not weaken somewhat the local-to-globalprinciple?

Say that the local-to-global principle holds, up to finite obstruction for aclass of varieties V, if each member V of V has only a finite number of nonisomorphiccompanions, i.e., if S(V ) is a finite set for all V of V.Despite the fact that we now, thanks to Rubin and Kolyvagin, can produceexamples of some curves of genus 1 for which the local-to-global principle holds upto finite obstruction, this question has not yet been resolved for all curves of genus1. In contrast, consider:Curves of genus ≥2.

Here the local-to-global principle holds up to finite obstruc-tion; moreover, this is a rather easy result (see the discussion in §5 and Part III).Summary. The one outstanding unresolved case remaining for curves is the case ofgenus 1.5.

General projective varieties.Conjecture 1. The local-to-global principle holds, up to finite obstruction for all(smooth) projective varieties ; that is, any projective variety over Q has only a finitenumber of nonisomorphic companions.Notes.

As will be discussed in Part III, the set S(V ), of companions to V , dependsonly upon the nature of the Gal(Q/Q)-module of automorphisms of V/Q, where Q23I am thankful for e-mail correspondence with Rubin in which he has proposed a direct andsuccinct recipe for this. To put it in slightly different terms, suppose that you are given a curve ofC of genus 1 over Q as a projective variety in PN and you are merely told that C has a rationalpoint, but you are also given a basis of the Mordell-Weil group of its jacobian E. Here is how youcan actually find a rational point on C. You first find any algebraic point P on C of some degreed.

Then find an algebraic point τ on E such that d · τ is the class of the divisor (of degree 0) onC : ΣP σ −d·P , where the summation is over all Q-conjugates of P . Let E denote a representativesystem for the group of points on E whose dth multiple lies in E(Q), modulo E(Q).

Then E is afinite set of points of E. Running through the (finite) set of translates of P by e + τ where e isdrawn from E, you are guaranteed that one of these is Q-rational.

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY11is an algebraic closure of Q. For a general projective variety, this Gal(Q/Q)-moduleconsists of the Q-rational points of a “locally algebraic group” Aut(V ).In specific instances when the group Aut(V ) is easy to deal with, we can proveConjecture 1.

For example, the proof of Conjecture 1 is particularly easy if Aut(V )is finite. This accounts for the fact that we know the local-to-global principle up tofinite obstruction for curves of genus ≥2.

For the same reason24, we know the local-to-global principle up to finite obstruction for all varieties V/Q of general type.25For almost the same reason the local-to-global principle up to finite obstructionholds for all smooth hypersurfaces of dimension ≥3, or more generally, for allsmooth complete intersections of dimension ≥3 (see Part III, §17, Corollary 4).A question, which arose in a conversation with N. Katz on these matters, is thefollowing:Is it within the range of the present “state-of-the-art” to showfiniteness of the part of S(V ) represented by projective varietieswhose numerical invariants are “limited” in various ways?Forexample, fixing an integer d define S(V )d to be the subset of S(V )represented by projective varieties which are companions to V andwhich admit a projective imbedding of degree ≤d. Is S(V )d finitefor every d?6.

The Tate-Shafarevich group. In this section, we will be considering abelianvarieties over Q; i.e., smooth, geometrically connected, projective varieties overQ which are also algebraic groups, the group law A × A →A defined over Q.Such algebraic groups are necessarily abelian; since A(C), the complex points of A,form a connected compact and commutative Lie group, A(C) is isomorphic, as atopological group, to a product of circles.If A/Q is an abelian variety, then its Tate-Shafarevich group, denoted X(A/Q),may be defined, first as a set, to be the set of isomorphism classes over Q ofcompanions of A endowed with principal homogeneous group action by A. Moreprecisely, X(A/Q) is the set of isomorphism classes over Q of pairs (W, α) whereW is a projective variety defined over Q which is a companion of A and α: A ×W →W is a mapping of projective varieties, defined over Q, which is a principalhomogeneous group action of A on W. (Note.

To guarantee that the action α isa principal homogeneous action, one need only check that it induces a principalhomogeneous action of the Lie group A(C) on the topological space W(C). )The set X(A/Q) is given a (commutative) group structure via the natural Baer-24Cf.

[KO].25General type.Let r be a positive integer.Consider global tensors τ on a (smooth) n-dimensional variety V which for any system of local coordinates z1, . .

. , zn can be expressed ash(z)(dz1 ∧dz2 ∧· · · ∧dzn)⊗r where h is some holomorphic function.

If, for some choice of r,there are “enough” linearly independent such tensors τ0, . .

. , τν—enough in the sense that therule which sends a point v ∈V to the point in ν-dimensional projective space with homogeneouscoordinates [τ0(v), .

. .

, τν(v)] is almost everywhere defined and gives a birational imbedding of Vto ν-dimensional projective space—then one says that V is of general type. The adjective generalsignifies that, by some reckonings, many varieties have this property.

For example, for (smooth)curves, “general type” is equivalent to genus > 1; for surfaces, the varieties that are not of generaltype are either rational, ruled, elliptic surfaces, abelian surfaces, K3 surfaces, or Enriques surfaces,for smooth hypersurfaces V of degree d in projective ν-space, if d ≥ν + 2 then V is of generaltype.

12B. MAZURsum construction for principal homogeneous spaces.26Forgetting the principalhomogeneous action (W, α) 7→W gives a natural map (of pointed sets)X(A/Q) →S(A/Q) .For an analysis of this mapping, see Part III, §18, Theorem 5.

In particular, in§17 we will see that there are a finite set A of abelian varieties A′/Q and naturalmappingsX(A′/Q) →S(A/Q)for each A′ ∈A such that the images of these mappings produce a partition ofthe set S(A/Q) indexed by A and the image of X(A′/Q) in S(A/Q) may beidentified with the orbit-space of X(A′/Q) under the natural action of the groupof K-automorphisms of A′.From this description (cf.explicitly, Part III, §18,Corollary 2 to Theorem 5) one sees that Conjecture 1 restricted to the class ofabelian varieties over Q is equivalent toConjecture 2 (Tate-Shafarevich). For any abelian variety A over Q, X(A/Q) isfinite.The relation between X(A/Q) and S(A/Q) has a resonance, in the classicaltheory of Gauss and Lagrange, in the distinction between the problems of classifyingintegral binary quadratic forms up to strict equivalence and equivalence.

The notionof equivalence of integral binary quadratic forms is more intuitive than and ishistorically prior to the notion of strict equivalence, but only with the latter notiondoes one get a natural group structure on classes of forms.What is known, in general, about the group structure of X(A/Q) is that it isa torsion abelian group with the property that the kernel of the homomorphismgiven by multiplication by any nonzero integer n on it is finite.7. Implications of the Tate-Shafarevich Conjecture in the direction ofthe general local-to-global principle (up to finite obstruction).

In prepar-ing this article I have been tantalized by the urge to show that the Tate-ShafarevichConjecture (i.e., Conjecture 2 for abelian varieties over Q) is equivalent to the local-to-global principle up to finite obstruction (i.e., Conjecture 1) for all projectivevarieties over Q. This I have not done, but after some conversations and corre-spondence with Yevsey Nisnevich and with Ofer Gabber, Theorem 2 has emerged.For its proof, see Part III, §17, Corollary 2 to Theorem 4.

Before we state theresult, however, we must talk a bit about the group of connected components ofautomorphism groups.Let V/Q be an arbitrary projective variety and Aut(V ) its locally algebraic groupof automorphisms. Let us denote by Γ(V ) the group of connected components of thelocally algebraic group Aut(V )/Q, endowed with its natural continuous Gal(Q/Q)-action.

Let ∆(V ) denote the quotient of Gal(Q/Q) which acts faithfully on Γ(V ),and let Γ ⋊∆denote the semidirect product constructed via this action.26Baer-sum. If W1 and W2 are two principal homogeneous spaces for A, let A act on W1 × W2by the antidiagonal action, i.e., the action of a in A on (w1, w2) is (w1 + a, w2 −a), where wehave written the action of A on W1 and on W2 additively.

Then the Baer-sum of the principlehomogeneous spaces W1 and W2 is the quotient variety of W1×W2 with respect to the antidiagonalaction, this quotient variety being viewed as principle homogeneous space for A by an actioninduced from the diagonal action of A on W1 × W2.

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY13What is known about the structure of Γ(V )? As will be discussed in Part III,there is a finitely generated abelian group NS(V ) with a continuous Gal(Q/Q)-action on which the group Γ(V ) acts with finite kernel (and in a manner compatiblewith its natural Gal(Q/Q)-action).

In particular, a quotient of Γ(V ) by a finitegroup is isomorphic to a subgroup of GL(n, Z) for some positive integer n, andtherefore there is an upper bound to the orders of finite subgroups of Γ(V ). It seemsthat little else is known about the structure of Γ(V )!

Is Γ(V ) finitely generated?Is it finitely presented? Is it an arithmetic group?

Conversely, what arithmeticgroups occur as Γ(V )’s?One can show (Part III, §17, Corollary 2):Theorem 2. Let V/Q be a projective variety.

Suppose that:(a) the group Γ(V ) is finitely presented;(b) there are only a finite number of distinct Γ-conjugacy classes of finite sub-groups in the semidirect product Γ⋊∆(i.e., the group Γ(V ) endowed with Gal(Q/Q)-action is decent in the sense defined in Part III, §16); and(c) the Tate-Shafarevich Conjecture holds27.Then S(V/Q) is finite (i.e., Conjecture 1 holds for V ).Some notes. If Γ(V ) is finitely generated, then ∆(V ) is finite.

Ofer Gabber hassketched a proof that the conclusion of Theorem 2 holds even if one weakens con-ditions (a) to the requirement that Γ(V ) be finitely generated.Are there anyprojective varieties V for which Γ(V ) does not satisfy (a) and (b), i.e., for whichΓ(V ) is not decent?288.Examples of finiteness of the Tate-Shafarevich group. Let E/Q bea modular elliptic curve.

That is, E is an elliptic curve, defined over Q, whoseunderlying Riemann surface admits a nonconstant holomorphic mapping from acompactification of the quotient of the upper half plane under the action of acongruence subgroup of SL(2, Z).29 The conjecture of Taniyama-Weil asserts thatevery elliptic curve over Q is modular.We must now introduce the L-function of E/Q. For most of our purposes, wemay deal with a crude version of it, which we will call LS(E, s), where, if we havefixed a model(3)y2 = x3 + ax + bfor E over Q, with a, b ∈Z, the subscript S will be any finite set of prime numberswhich includes all the prime numbers dividing the discriminant of this model.

If pis a prime number not in S, define the local factorLp(E, s) := (1 −app−s + +p1−2s)−1,27Or, somewhat more specifically, that it holds for all twists of certain subabelian varieties ofPic0(V ) over Q.28Is it true that the automorphism groups of K3 surfaces, for example, are decent? They arefinitely generated.Are they arithmetic?

I have begun to pester experts about this. Relevanthere are [P-S–S, §7, Proposition and Theorem 1] which give somewhat explicit descriptions of theautomorphism groups of K3 surfaces: If V is a K3 surface over Q, NS its N´eron-Severi lattice,and Γ its automorphism group, then Γ is commensurate with O(NS)/W (NS) where O(NS) is theorthogonal group of the lattice NS, and W (NS) is the subgroup generated by reflections comingfrom elements in NS of square −2.

For a finer study of which Γ’s occur, see Nikulin [Nik1, 2].29This is, in fact, equivalent to the more usual formulation of “modularity”, see [Ma2].

14B. MAZURwhere ap is the integer defined by the formula 1 + p −ap = the number of rationalpoints of the projective curve (over the prime field Fp) determined by reducing theabove model modulo p. Then LS(E, s) is defined (initially) as the Dirichlet seriesYp/∈SLp(E, s),where p runs through all prime numbers not in S. Using either elementary, or notso elementary, estimates on the size of the ap, one sees that the Dirichlet seriesLS(E, s) converges to yield an analytic function of s in a right half plane.

Underthe hypothesis that E is modular, this analytic function LS(E, s) extends to anentire function in the s-plane.Remark. There is, in fact, a “good way” of extending the definition of the localfactors Lp(E, s) for prime numbers p in S. The Lp(E, S) are polynomials in p−s ofdegree ≤2 (cf.

[Ta]) explicitly given in terms of a “minimal model” for the ellipticcurve E over Z, and the “good” L function L(E, s) is defined to be the product ofthese local factors Lp(E, s) for all prime numbers p.The methods of Kolyvagin, aided by results of Gross and Zagier, Waldspurger,Bump, Friedberg, and Hoffstein, and R. Murty and K. Murty, yield the followingextraordinary result (which is an important piece of the Conjecture of Birch andSwinnerton and Dyer).Theorem 3. If E is a modular elliptic curve for which the Hasse-Weil L functionLS(E, s) either does not vanish at the point s = 1 or has a simple zero at s = 1,then X(E/Q) is finite.In the former case, i.e., if LS(E, 1) ̸= 0, then E possesses only a finite number ofQ-rational points.

If LS(E, s) has a simple zero, then the group E(Q) of Q-rationalpoints of E is of rank 1.The order of zero of LS(E, s) at s = 1 is independent of the choice of set ofprimes S and would be the same if we dealt with the “good” L function L(E, s).For refinements of the above theorem, see [Coa, K1, K2, R1–8].9. Returning to Selmer’s curve.

Briefly, this is what goes into the proof ofTheorem 1: the last of the five curves in the list (given in the statement of Theorem1) has a Q-rational point and can be identified with the jacobian, call it E, ofall five curves (cf. [Ca2, Chapters 18, 20] where it is shown that the cubic curveax3+by3+cz3 = 0, for nonzero rational numbers a, b, c has, as jacobian, the ellipticcurve abcx3 + y3 + z3 = 0).

This elliptic curve E has complex multiplication, anda computation gives that the ratio L(E, 1)/Ωis equal to 9, where L(E, s) is the(good) L-function of E and Ωis the real period of E. It follows then from the recentwork of Rubin [R5] that, if X = X(E/Q) is the Tate-Shafarevich group of E overQ, then X is finite and of order equal to a power of 2 times a power of 3. Checkingprior computations of the 2- and 3-primary components of X in tables of Stephens[St], one then has that X is a product of two cyclic groups of order 3.

By Part III,§18, Corollary 1 to Theorem 5 the set of companions S(C) may be identified withthe quotient-set of X under the involution x 7→−x and, therefore, has cardinality= 1 + (9 −1)/2 = 5. One is left with the chore of finding representatives for the 5companions; this being done by the list presented in the statement of the theorem,it remains only to check that all five curves on our list possess rational points over

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY15Qp for all prime numbers p, and over R, which they do, and that they are pairwisenonisomorphic over Q, which they are30.Part II10.The Tate-Shafarevich group and the Selmer group. Let Q be thealgebraic closure of Q in C, and for each prime number p fix an embedding of Q inQp, an algebraic closure of Qp.

Let G denote the Galois group Gal(Q/Q), and foreach p let Gp := Gal(Qp/Qp) and G∞:= Gal(C/R), these being viewed as closedsubgroups of G (after our choices of embeddings).Fix an elliptic curve E/Q.If K is a field extension of Q, the group of K-rational points of E is denoted E(K). Since E is defined over Q, there is a natural(continuous) action of G on E(Q), of Gp on E(Qp), and of G∞on E(C).

Thecohomological definition of X(E/Q) is as follows.X = X(E/Q) :=\ker{H1(G, E(Q)) →H1(Gp, E(Qp))}where the intersection is taken over all “finite” primes p and p = ∞. It is useful tolighten the notation somewhat and to convene thatH1(G, E) :=H1(G, E(Q)),H1(Gp, E) :=H1(Gp, E(Qp)),H1(G∞, E) :=H1(G∞, E(C)) .If A is an abelian group, and n an integer, let A[n] denote the kernel of mul-tiplication by n, so that we have the “Kummer” exact sequences of continuousG-modules,0 →E(Q)[n] →E(Q)n→E(Q) →0,and of continuous Gp-modules,0 →E(Qp)[n] →E(Qp)n→E(Qp) →0,each yielding long exact sequences on cohomology, pieces of which, following ourconvention, can be written as the horizontal exact sequences occurring in the com-mutative diagram:0 −−−−−→E(Q)/n · E(Q)−−−−−→H1(G, E[n])j−−−−−→H1(G, E)[n] −−−−−→0yyy0 −−−−−→E(Qp)/n · E(Qp) −−−−−→H1(Gp, E[n]) −−−−−→H1(Gp, E)[n] −−−−−→0Visibly, X[n], the kernel of multiplication by n in X, is the intersection for allp ≤∞of the kernels of the right-hand vertical arrows in the above diagrams (forall p).30Here are some hints about this latter assertion.Let E[3] be the kernel of multiplicationby 3 in E, and note that the natural mapping H1(Q, E[3]) →H1(Q, E) is injective (since E(Q)vanishes).Now if C is any one of our five curves ax3 + by3 + cz3, let β, γ ∈Q be such thatβ3 = b, γ3 = c. The cohomology class corresponding to C in H1(Q, E) is the image of the classin H1(Q, E[3]) represented by the 1-cocycle which sends an element σ ∈Gal(Q/Q) to the point(0, 1, (β/γ)σ−1) in E[3].

These 1-cocycles lie in distinct cohomology classes, for our five curves C.

16B. MAZURDefinition.

The n-Selmer group (of the elliptic curve E/Q), denoted S[n], is thesubgroup of H1(G, E[n]) defined as the full inverse image of X[n] ⊂H1(G, E)[n]under the map labelled j above.For any finite field extension K of Q in Q, making the base change from Q to K,we have the Tate Shafarevich group of E/K and the analogously defined n-Selmergroup of E/K, which we will denote XK and SK[n], respectively. Essentially bydefinition these fit into an exact sequence0 →E(K)/n · E(K) →SK[n] →XK[n] →0 .The importance of the Selmer group is that, as the above exact sequence displays,it packages information concerning both the Mordell-Weil group and the Tate-Shafarevich group; but the Selmer group is, in many instances, curiously moretractable than either of them.

This is rather like the phenomenon met in the studyof number fields, where, at times, the product of class number times regulator ismore readily computed than either factor. In any event, to study elements of ordern in XK, we pass to the study of elementss ∈SK[n] ⊂H1(GK, E[n]),where GK is the closed subgroup of G which is the identity on K.11.

Local control of the Selmer group. At this point we are ready to discussa step in Kolyvagin’s program that may seem modest but is of crucial importancein setting the stage for what is to come.

The “abstract” group E[n] is isomorphicto a product of two cyclic groups of order n and consequently its full automorphismgroup is isomorphic to the finite group GL2(Z/nZ). The natural action of GK onE[n] gives us, up to equivalence, a representation GK →GL2(Z/nZ), whose kernelwe will denote GM, and, to be consistent, we use the letter M to denote the finiteextension field of K in Q comprised of the elements in Q fixed by GM.

So, letting∆n denote Gal(M/K) = GK/GM, we have the natural inclusion ∆n ⊂GL2(Z/nZ).Let us now say that the integer n is good if the “restriction mapping” on coho-mology, described below, is injective:H1(GK, E[n]) →H1(GM, E[n])∆n = Hom∆n(GM, E[n]) .The usefulness of this definition of goodness is that:(a) invoking the known richness of action of the Galois groups of number fieldson n-torsion in elliptic curves, the hypothesis of goodness is not a strong hypothesison n (and even when n is not good, the kernel of the “restriction mapping”, for nreplaced by powers of n, is finite, of order bounded independent of the power). (b) any element h ∈H1(GK, E[n]) (and consequently, also, any element s in theSelmer group) gives rise, by the “restriction homomorphism” above, to a continuous∆n-equivariant homomorphism; if n is good, then triviality of the ∆n-equivarianthomomorphism h: GM →E[n] implies triviality of the element h ∈H1(GK, E[n])giving rise to it.From now on we assume n to be good, and we will freely identify elements ofH1(GK, E[n]) with the ∆n-equivariant homomorphisms GM →E[n] to which theygive rise.

This allows us to “study” elements s ∈SK[n] by local means, in thefollowing sense.

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY17Let v run through all places of K, both nonarchimedean and archimedean, andlet Kv denote the completion of K at v, Kv an algebraic closure containing Q, andGKv := Gal(Kv/Kv). Then we have the exact sequence0 →E(Kv)/n · E(Kv) →H1(GKv, E[n]) →H1(GKv, E)[n] →0,and, directly from the definition of the Selmer group, we have that the imageof an element s ∈SK[n] under the natural homomorphism H1(GK, E[n]) →H1(GKv, E[n]) maps to an element, call it sv, in the subgroupE(Kv)/n · E(Kv) ⊂H1(GKv, E[n]) .Since n is good, and since a homomorphism from GK is trivial if it is trivial onall decomposition groups, we have something that could be called the principle oflocal control:If the “local ” elements sv ∈E(Kv)/n · E(Kv) vanish for all places v, then s van-ishes.12.

Class field theory, duality, and Kolyvagin test classes. The GK-moduleE[n] admits a (nondegenerate) self-pairing, called the “Weil-pairing”E[n] × E[n] →µnwhere µn is the GK-module of nth roots of unity.The Weil pairing induces apairing on local cohomologyH1(GKv, E[n]) × H1(GKv, E[n]) →H2(GKv, µn) ⊂Q/Z(x, y) 7→⟨x, y⟩vwhere the inclusion of H2(GKv, µn) in Q/Z is given by Local Class Field Theory.Important for us will be the fact31 that if both x and y map to zero under thehomomorphism to H1(GKv, E), then ⟨x, y⟩v = 0.The Weil pairing also induces a pairing on global cohomologyH1(GK, E[n]) × H1(GK, E[n]) →H2(GK, µn) ⊂MvH2(GKv, µn) ⊂MvQ/Z(x, y) 7→M⟨xv, yv⟩vwhere xv and yv are the images of x and y in the local cohomology H1(GKv, E[n]),the direct sum on the right being taken over places v of K. The inclusion on theright is that given by Global Class Field Theory, which embeds H2(GK, µn) intothe kernel of the sum mapping Lv Q/Z →Q/Z.31One might add that the pairing E(Kv)/n · E(Kv) × H1(GKv , E)[n] →Q/Z induced by the(Weil) self-pairing on H1(GKv , E[n]) is compatible, up to sign, with the pairing coming from TateLocal Duality (for abelian varieties).

Cf. the related discussion [Mi4, Chapter I, §3, pp.

54–55].

18B. MAZURKolyvagin’s basic strategy.

With our data, E, K, n, fixed and understood, let wbe a place of K. By a Kolyvagin test class for w let us mean a cohomology classc ∈H1(GK, E[n]), which has the property that c goes to 0 under the compositionof natural mappings H1(GK, E[n]) →H1(GKv, E[n]) →H1(GKv, E) for all placesv ̸= w of K and does not go to 0 for v = w.Let us first see that any Kolyvagin test class imposes a local condition on everyelement of the Selmer group. For, if s ∈SK[n] and c is a Kolyvagin test class, wehave that ⟨cv, sv⟩v = 0 for v ̸= w, since for places different from w, both cv and svmap to zero in H1(GKv, E).On the other hand, by the consequence of Global Class Field Theory cited above,we haveX⟨cv, sv⟩v = 0,the summation being taken over all places v of K.Putting these together, we get a local condition at w, which is satisfied for anyelement s in the Selmer group SK[n]; namely,(1)⟨cw, sw⟩w = 0 .Kolyvagin’s strategy, roughly put, is to produce a large and systematic collectionof test classes c so as to obtain enough conditions of the type (1) on the localcomponents of elements s to provide the tightest bounds on the size of the Selmergroup.For example, in one series of applications, n is a prime number > 2, the field Kis a quadratic imaginary field, and, letting the superscript ± denote ± eigenspacefor the nontrivial automorphism of K (we are in a situation where each of thetwo eigenspaces {E(Kw)/n · E(Kw)}± is cyclic of order n).In this setup, themere existence of a Kolyvagin test class for w in the ± eigenspace guarantees (viacondition (1)) that, for any Selmer class s ∈SK[n]± (same sign), we have sw = 0.Naturally, one might (and in fact one does) wish, at times, to loosen the require-ment on a test class c that goes to 0 in H1(GKv, E) for all but one place v. Fixinga finite set S of places, one can make similar good use of classes c that go to 0 inH1(GKv, E) for all v outside the finite set S.13.

Rational points in extension fields yielding “test classes”. The basicmechanism that links rational points to Kolyvagin test classes is encapsulated bythe diagram displayed below.

Here, L/K is a finite Galois extension with Galoisgroup G, and we assume that L/K is unramified outside the single place w of Kand that the “restriction mapping”, labelled Res below, is an isomorphism:0yH1(G, E)[n]yInf0 −−−−−→E(K)/n · E(K)−−−−−→H1(GK, E[n])−−−−−→H1(GK, E)[n]y∼=yResy0 −−−−−→{E(L)/n · E(L)}G −−−−−→{H1(GL, E[n])}G⊂H1(GL, E[n]) GNow let P ∈E(L) be a rational point satisfying the following two conditions—the first being the “serious” condition, and the second a minor one:

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY19(a) For all σ ∈G, P σ −P is divisible by n in E(L). (b) For each place ν of L which does not lie over the place w of K, the special-ization of P to the group of connected components of the N´eron fiber of E at ν isof order relatively prime to n.For N´eron models, see Artin’s exposition (in [A2]) of N´eron’s original paper [N´e];see also Grothendieck’s Exp.

IX in [Grot3] and [BLR].Let us call rational points P ∈E(L) satisfying (a), (b) above Kolyvagin rationalpoints.By (a), P gives rise to a class [P] in {E(L)/n · E(L)}G.Definition. The class c = cP ∈H1(GK, E[n]) is the unique homology class suchthat Res(c) is equal to the image of [P] in {H1(GL, E[n])}G.There is a unique homology class d ∈H1(G, E)[n] such that Inf(d) is the imageof c in H1(GK, E)[n].

Using (b) one can show that the restriction of the class d todecomposition groups attached to every place v ̸= w of K vanishes, i.e., that cPis a Kolyvagin test class for w provided its image in H1(GKw, E) does not vanish.Clearly a necessary requirement for this nonvanishing to occur is that [P] be a“new” class in {E(L)/n · E(L)}G, i.e., a class not in the image of E(K)/n · E(K).Summary. We have started with an elliptic curve E/Q.

We have fixed a “good”nonzero integer n. We have somewhat silently passed to a finite extension fieldK/Q. Let us call this K the first auxiliary field extension: as already hinted, itwill be chosen to be a suitable quadratic imaginary field extension of Q.

For anyplace w of K, we now wish to find a suitable Galois extension field L/K unramifiedoutside w (call L/K the second auxiliary field extension) and also a “Kolyvagin”rational point P in E(L) which yields a “new” class [P] in {E(L)/n · E(L)}G.The construction of a suitable “first” auxiliary field extension K/Q and then anample and systematic supply of “second” auxiliary field extensions L/K togetherwith rational points P is the punch line of Kolyvagin’s program.For then wewould have a significant supply of Kolyvagin test classes cP which would imposea significant number of conditions on the local components of elements of the n-Selmer group. By the “principle of local control”, one then would have control ofthe n-Selmer group itself.14.

Heegner points. For any quadratic imaginary field extension K/Q (an ap-propriate one of these will be our choice of “first auxiliary field extension”) and forany positive integer f, let Kf denote the ring class field of conductor f over K (forsuitable f, Kf/K will be our choice of “second auxiliary field extension” L/K).We have a tower of field extensions, with Galois Groups as marked below, andwhere the full Galois group Gal(Kf/Q) is equal to a “generalized dihedral group”,i.e., an extension of a cyclic group of order 2 by the abelian group Gf := Gal(Kf/K),with action of the nontrivial element in the cyclic group on Gf given by multipli-cation by −1.Kf| }(OK/f · OK)∗/(Z/fZ)∗K1| } the ideal class group of KK| } cyclic of order twoQ

20B. MAZURHere, OK denotes the ring of integers of K. If OK,f denotes the order Z+f·OK ⊂OK, we have that Gal(Kf/K) is isomorphic to the group of isomorphism classes oflocally free OK,f -modules of rank one.As previously mentioned, the Taniyama-Weil conjecture asserts that any ellipticcurve E/Q can be realized as a quotient curve (over Q) of the modular curve X0(N)for a suitable positive integer N. Noncuspidal points x of X0(N) may be “identified”with pairs (E, C) up to isomorphism, where E is an elliptic curve and C is a cyclicsubgroup of E of order N. The point x is rational over a given field extension of Qif and only if the pair (E, C) is definable over that number field.

Suppose then thatwe have a nonconstant mapping, defined over Q, ϕ: X0(N) →E.We now choose a quadratic imaginary field K ⊂C such that all prime factors ofN split in OK. Under this hypothesis, there is an ideal N ⊂OK such that OK/Nis cyclic of order N; we choose such an ideal.For any positive integer f relatively prime to N, the ideal Nf := N ∩OK,f isan invertible OK,f-module such that OK,f/Nf is cyclic of order N. Let Ef be theelliptic curve C/OK,f, and let Cf ⊂Ef be the cyclic subgroup of order N givenby N −1f/OK,f.

Let xK,f ∈X0(N) be the point corresponding to the pair (Ef, Cf),and put yK,f = ϕ(xK,f).Then yK,f is a Kf -rational point of E. We refer to it as the Heegner point ofE (for the choices: K ⊂C, N, f, and the parametrization ϕ). The collection ofHeegner points for a fixed K and all positive integers f which are relatively primeto N satisfy the axioms of what Kolyvagin calls a “Euler system”, these axiomsbeing the essence of what is needed to produce a supply of “Kolyvagin points”.To avoid getting at all into the technicalities of the passage from Heegner pointsto Kolyvagin points, and yet to give some small, but honest, hint of the flavor ofit, consider the following:We restrict attention to the case where n is a prime number.

Now choose f to bea prime number which is inert (and unramified) in K, which does not divide N · n,and such that the Frobenius element at (f ) in GQ acts as a nonscalar involutionon the group of n-torsion points of the elliptic curve E.Then Gal(Kf/K1) iscyclic of order f + 1. Fix a generator γ ∈Gal(Kf/K1) ⊂Gal(Kf/K), and fix arepresentative system S ⊂Gal(Kf/K) modulo Gal(Kf/K1) so that every elementof Gal(Kf/K) is in γi · S for a unique integer i in the range 0 ≤i ≤f.

PutPK,f :=Xi · γi · s · yK,f ∈E(Kf),the summation being taken over all s ∈S and i in the above range.Then (see [Gros] for a very neatly written proof of this) taking our “secondauxiliary choice” L/K to be Kf/K, the point PK,f is a “Kolyvagin point”; i.e.,it satisfies (a), (b) above, and, moreover, the class [PK,f] in {E(L)/nE(L)}G isindependent of the choice of representative system S; it is independent up to scalingby (Z/nZ)∗of the choice of generator γ.Part III3215. The cohomology of locally algebraic group schemes.

In Part I, for easeof notation, the base field was taken to be Q, but here it is more natural to allow32I am very thankful to Ofer Gabber and Yevsey Nisnevich for the crucial help they gave to mewhileIwaswritingthis.

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY21arbitrary number fields K (of finite degree over Q). For v a place of K, we let Kvdenote the completion of K with respect to v. So, if V is a projective variety overK, we have the analogous definition of companion variety to V , i.e., a projectivevariety V ′ over K which is isomorphic to V over every completion Kv.

We alsohave S(V/K), the set of isomorphism classes of companions (over K).Let X denote the spectrum of the ring of integers in K, and we reserve the letterS for finite sets of places of K, containing all infinite places. Let X −S be the opensubscheme in X which is the complement of the closed points corresponding to thenonarchimedean places of S.We shall be studying group schemes over X−S which are (separated and) locallyof finite type but not necessarily of finite type.

To signal this, we shall refer to suchgroup schemes as locally algebraic group schemes (extending the terminology in [BS]where locally algebraic groups are over fields rather than over more general baseschemes). We shall refer to a group scheme which is of finite type as an algebraicgroup scheme.Let G, then, be a smooth locally algebraic group scheme over X −S.

Thatis, G is a group scheme (separated, and), locally of finite type over X −S, forwhich G0, the connected component containing the identity section, is a smoothconnected open subgroup scheme of G, of finite type over X −S. If Z →X −Sis any morphism of schemes, the pullback of our group scheme G to the base Z isdenoted G/Z.

Whenever we write H1(Z, G) we can mean, equivalently, ´etale or flat(fppf) cohomology of G/Z over Z (compare [Grot]). Given a G-valued 1-cocyclec for the ´etale topology on Z, we can use the conjugation action of G on G to“twist” the locally algebraic group scheme G/Z by c (an “inner twist”; cf.

[Gi])and the isomorphism class of the resulting locally algebraic group scheme over Z isdependent only upon the cohomology class γ in H1(Z, G) of our cocycle c; we shalldenote this “inner twist” by Gγ/Z.Since our groups are, in the main, noncommutative, we are referring here tononcommutative one-dimensional cohomology, given the structure only of pointedset. Giving the concepts “Q” and “ker” the evident meanings in the context ofpointed sets, we letX(G) := ker(H1(K, G) →YvH1(Kv, G)),the product being taken over all places v of K, andXS(G) := ker(H1(X −S, G) →Yv∈SH1(Kv, G)).Examples.

(1) by a “locally constant” group scheme G/S we mean a group schemewhich is locally constant for the ´etale topology, i.e., such that there is a finite ´etalecover S′ →S for which the pullback G/S′ is a constant group scheme. Let G/Kbe a locally constant group over the number field K, and let L/K be the (minimal)finite splitting field for the action of Gal(K/K) on G(K).

Then X(G) is isomorphictoker(H1(L/K, G) →YvH1(Kv, G)).

22B. MAZURThe reason for this is the following.

First, if the action of Gal(K/K) on G(K)is trivial, then X(G) is trivial, because in that case H1(K, G) is identified withconjugacy classes of continuous homomorphisms from Gal(K/K) to G(K), andif a homomorphism restricts to the trivial homomorphism from Gal(Kv/Kv) toG(Kv), for all places v, then it is trivial. In the general situation, then, an elementof X(G) ∈H1(K, G) must map to the trivial element in H1(L, G) and thereforecome from H1(L/K, G), as stated.Corollary.

Let G be a locally constant group whose Galois action is split by afinite cyclic extension of K. Then X(G) is trivial.Proof. Let L/K be the finite cyclic extension which splits the Galois action of G.By the Cebotarev Theorem, there is a place w of L lying over a place v of K (whichis unramified, and) for which Gal(Lw/Kv) = Gal(L/K).

It then follows that forthis v, the mapping H1(L/K, G) →H1(Kv, G) is injective.□(2) Let G be a locally algebraic group over K which is an extension33 of a locallyconstant group which is an arithmetic group (whose continuous Gal(K/K)-actioncomes from an action of an ambient linear algebraic group over Q) by a linearalgebraic group (such groups are said to be of “type ALA” in [BS]). Then X(G)is finite and, moreover, the mappingH1(K, G) →YvH1(Kv, G)is proper, i.e., has finite fibers [BS, Theorem 7.1].If G is a connected, simply connected, semisimple algebraic group over K, thenthis mapping is now known to be an isomorphism (if one either includes or ne-glects to include the factors on the right corresponding to nonarchimedean v, sinceH1(Kv, G) = 0 for finite v, and connected, simply connected, semisimple algebraicgroups G).

This fact, going under the heading The Hasse Principle for connected,simply connected, semisimple algebraic groups, has been the fruit of a long devel-opment (cf. [Se2, Kne] for a discussion of the work on this problem for the classicalsemi-simple groups; Harder’s paper [Hard] for a proof of the above assertion forall connected, simply connected, semisimple algebraic groups G/K which do nothave a factor of type E8; Platonov’s survey [Pl] for a neat discussion of where theproblem stood circa 1982; and Chernousov’s resolution of what remained to be donewith respect to E8 [Ch], thereby finishing the problem).16.

Discrete locally algebraic group schemes. Fix K an algebraic closure ofK, and consider the category of discrete Gal(K/K)-groups, i.e., “abstract groups”(viewed as discrete topological groups) endowed with continuous Gal(K/K)-action.This category is equivalent to the category of discrete locally algebraic groups overK, the equivalencediscrete loc alg gp G/K ֌ discrete Gal(K/K)- gp Γbeing given by taking Γ to be the group G(K) with its natural Gal(K/K) Galoisaction.

The discrete locally algebraic group G/K is locally constant (in the sense33We keep to the convention that a group G is an extension of A by B if B is the normalsubgroup of G and A is the quotient group G/B.

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY23discussed in the previous section, i.e., locally constant for the ´etale topology) if andonly if the action of Gal(K/K) on Γ factors through a finite quotient group (thisbeing automatic if Γ is finitely generated).Let Γ be a discrete Gal(K/K)-group. Denote by ∆= Gal(L/K) the quotient ofGal(K/K) which acts faithfully on Γ; i.e., L ⊂K is the splitting field of the actionon Γ.

Let Γ ⋊∆denote the semidirect product constructed via the action of ∆onΓ.If Γ is finitely generated, then ∆is finite. In this case, if S denotes a (finite)set of places including all archimedean places and all places ramified in the finiteextension L/K, and if G/K is the discrete, locally algebraic group over K associatedto Γ, then G/K is the generic fiber of a (unique) locally algebraic group schemeG/X −S which is locally constant for the ´etale topology over X −S.Definition.

A discrete Gal(K/K)-group Γ (and its associated discrete, locallyalgebraic group G/K) is called decent if(a) the group Γ is finitely presented;(b) the group Γ⋊∆(notation as above) has only a finite number of Γ-conjugacyclasses of finite subgroups.We will also call a discrete locally algebraic group scheme over X −S decent ifits generic fiber is so.A sub-Gal(K/K)-group of finite index in a decent Gal(K/K)-group is againdecent. An inner twist of a decent group is decent.

An arithmetic group Γ endowedwith trivial Galois action is decent.Lemma (a). If G = G/X −S is a decent (discrete) locally constant group scheme,then H1(X −S, G) is finite and (therefore) so is XS(G).Proof.

Let Γ be the associated discrete Gal(K/K)-group. Letting Y be the spec-trum of the ring of integers of L, the splitting field of the natural Galois action onΓ, and T the inverse image of S in Y , we have that Y −T is a “Galois” extensionof X −S, with group ∆= Gal(L/K).

We may suppose that Y −T is a finite ´etaleGalois extension of X −S.The set H1(Y −T, G) is finite, for we may identify it with the orbit-space ofHomcont(π1(Y −T ), Γ) under the action of Γ-conjugation, where Homcont means“continuous homomorphism”. Since the image of π1(Y −T ) in Γ under any con-tinuous homomorphism is a finite subgroup of Γ, by “decency” of Γ, we have thatthere are at most a finite number of Γ-conjugacy classes of finite subgroups of Γ,and our finiteness assertion follows.Consequently there is a finite ´etale Galois extension Y ′ −T ′ →X −S admittingY −T →X −S as an intermediate extension, such that every class in H1(Y −T, G)is trivialized in Y ′ −T ′.

Denote by ∆′ the Galois group of Y ′ −T ′ over X −Sand by ψ: ∆′ →∆the natural surjection. Then H1(X −S, G) = H1(∆′, Γ) = {Γ-conjugacy classes of homomorphisms ϕ: ∆′ →Γ⋊∆which lift ψ: ∆′ →∆}.

Since,by the decency hypothesis, the image of ϕ has only a finite number of possibilitiesup to Γ-conjugacy, so does ϕ.□17. Finiteness theorems.

In this section we let G be a smooth locally algebraicgroup scheme over X −S and G0 ⊂G the open subgroup scheme which is the“connected component” of G containing the identity section. The quotient G/G0viewed as sheaf for the ´etale topology over X−S is representable, in general, only as

24B. MAZURan algebraic space group over X −S (i.e., group object in the category of algebraicspaces over X −S; cf.

[A1, Knu]); even if it is representable as a group scheme,the group scheme does not have to be separated. Denote by G/X −S the algebraicspace group G/G0 over the base X −S.

As usual, we let G/K denote its genericfiber viewed as (discrete, locally algebraic) a group over K.From now on we shall assume that two further properties hold for G:(A) The algebraic space group G/X −S = G/G0 is representable as a locallyconstant group (for the ´etale topology) over X −S and is decent (cf. §16).

(B) The smooth algebraic group scheme G0/X −S is an extension of an abelianscheme A/X −S by a connected linear affine group scheme.The second property above is assumed only to tidy up the mental picture wehave of our locally algebraic group scheme: it is not really necessary for what wewill do and can always be achieved by suitable augmentation of the finite set S.Under our assumptions, then, our locally algebraic group scheme G is a successiveextension of the following three basic “building blocks”:(a) a discrete, decent, locally constant group scheme G/X −S,(b) an abelian scheme A/X −S,(c) a connected (smooth) linear affine group scheme B/X −S.We refer below to the subquotient A of G as the abelian scheme part of G. LetA0 ⊂A denote the “connected component” of the N´eron model of A, i.e., the opensubgroup scheme all fibers of which are connected.Lemma (b). If A/Q is an abelian variety and S contains all the primes of badreduction of A/Q (so that A extends to an abelian scheme over X −S), thenXS(A/X −S) is isomorphic to the image of H1(X, A0) in H1(X −S, A).

Thegroup XS(A/X −S) is finite if X(A) is finite.Proof. One has a commutative diagram, in which the horizontal rows are exact:H1(X, A0)−−−−−→H1(X −S, A0) −−−−−→Qv∈SH1(Kv, A0)y=y=0 −−−−−→XS(A/X −S) −−−−−→H1(X −S, A) −−−−−→Qv∈SH1(Kv, A)the middle arrow being an equality because all primes of bad reduction for A areassumed to be in S. Exactness of the top horizontal row can be seen, for example,from the discussion in the Appendix of [Ma1, p. 263]; exactness of the bottomhorizontal row is simply by the definition of XS(A/X −S).

It follows that thereis a natural surjectionα: H1(X, A0) ։ XS(A/X −S) ⊂H1(X −S, A) .Let Σ denote the image of H1(X, A0) in H1(X, A). By the appendix of [Ma1] Σcontains X(A) as a subgroup of finite index (in fact, this index is a power of two).By the above discussion, there is a natural surjection of Σ onto XS(A/X −S).This proves the lemma.□Lemma (c).

If B/X −S is a connected (smooth) linear affine group scheme, thenH1(X −S, B) is finite.Proof. This is [Nis1, Theorem 3.7]; compare also the discussion around [Nis2, The-orem 3.10].□

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY25To deal with groups which are extensions of these building blocks, consider anexact sequence of locally algebraic smooth group-schemes over a scheme Y ,(1)1 →U →Gp→W →1 .If γ ∈H1(Y, G) is a one-dimensional (noncommutative) cohomology class, let Uγbe the group scheme U twisted by the class γ, where G acts on U by conjugation.Letπ: H1(Y, G) →H1(Y, W)be the mapping induced from p on one-dimensional (noncommutative) cohomology.If ω ∈H1(Y, W), let H1(Y, G)ω denote the fiber of π above ω. If γ ∈H1(Y, G)and ω = π(γ), by the “long exact sequence” for noncommutative cohomology [Gi,Chapter III, 3.3.5]34, there is a natural surjection of setsσ: H1(Y, Uγ) →H1(Y, G)ω(where the image of the trivial element in H1(Y, Uγ) is the element γ in H1(Y, G)ω).Now suppose that Y = X −S.

We have the diagram1 −−−−→XS(G) −−−−→H1(X −S, G) −−−−→Qv∈SH1(Kv, G)yπyyπ1 −−−−→XS(W) −−−−→H1(X −S, W) −−−−→Qv∈SH1(Kv, W)Extension Lemma. Given the exact sequence (1), suppose that(i) XS(W) is finite;(ii) The XS(Uγ) are all finite (for all twisting classes γ ∈H1(Y, G));and either(iii1) The group scheme U is commutative and the maps H1(Kv, Uγ) →H1(Kv, G)ωare all proper (for all v ∈S, and all twisting classes γ ∈H1(X −S, G), whereω = π(γ)),or(iii2) The H1(X−S, Uγ) are all finite (for all twisting classes γ ∈H1(X−S, G)).Then XS(G) is finite.Proof.

If ω is an element in the image of XS(G) in XS(W) and if γ ∈XS(G) ⊂H1(X −S, G) is such that ω = π(γ), consider the diagram1 −−−−→XS(Uγ) −−−−→H1(X −S, Uγ) −−−−→Qv∈SH1(Kv, Uγ)yσyy1 −−−−→XS(G)ω −−−−→H1(X −S, G)ω −−−−→Qv∈SH1(Kv, G)where XS(G)ω is defined so as to make the lower sequence exact.34For the Galois-cohomological version, compare [Se2, Chapter VII, Appendix, Proposition 2].

26B. MAZURSince, by (i), XS(W) is finite, our lemma will follow if we show XS(G)ω tobe finite for each element ω in the image of XS(G) in XS(W).

Fix, then, such aω ∈XS(W), and an element γ ∈XS(G) such that ω = π(γ).From (ii) and (iii1) we see that the “diagonal mapping” of the right-hand rec-tangleH1(X −S, Uγ) →Yv∈SH1(Kv, G)has finite kernel. By surjectivity of σ we see that this kernel maps onto XS(G)ωproving the lemma in the case when (iii1) holds.Now suppose (iii2), giving us finiteness of H1(X −S, Uγ).

But since σ: H1(X −S, Uγ) →H1(X −S, G)ω is surjective, finiteness of XS(G)ω is immediate.□Theorem 4. Let G be a locally algebraic (smooth) group scheme over X −S sat-isfying our two properties (A) and (B).

Suppose, further, that the Tate-ShafarevichConjecture holds for abelian varieties over K. Then XS(G) is finite.Remark. If S ⊂S′ is an inclusion of finite sets of places of K, and if G is a locallyalgebraic (smooth) group scheme over X −S satisfying properties (A) and (B), thenthe restriction of G to X −S′ also satisfies these properties, so we might amplifythe conclusion of our theorem to say that XS′ also satisfies these properties, sowe might amplify the conclusion of our theorem to say that XS′(G) is finite for S′any finite set of places of K containing S.Proof.

We do this in two steps.First we shall assume that our locally algebraic group scheme G fits into an exactsequence of locally algebraic group schemes over X −S,(1)1 →U →Gp→W →1where W is a discrete, decent, locally constant group scheme and U is an abelianscheme. Let us show that, in this situation, XS(G) is finite.For this, we use the Extension Lemma, noting that property (i) holds by Lemma(a), property (ii) holds by Lemma (b), and our assumption of finiteness of theTate-Shafarevich groups (of abelian varieties which are “twists” of A).

We nowshow that property (iii1) holds. We use the long exact sequence of noncommu-tative cohomology groups and are significantly helped here by the fact that U isan abelian scheme and, in particular, commutative.

Properness of the mappingϕ: H1(Kv, Uγ) →H1(Kv, G)ω (for all twisting classes γ) is equivalent to show-ing that the kernels are finite. Let Lw/Kv be a finite local field extension overwhich the discrete locally constant group W is constant, i.e., such that the actionof Gal(Lw/Lw) on W is trivial.

We have a commutative squareH1(Kv, Uγ)ϕ−−−−→H1(Kv, G)ωyyH1(Lw, Uγ)ϕ′−−−−→H1(Lw, G)ωand a coboundary mappingδ: H0(Lw, W) →H1(Lw, Uγ)

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY27which is a homomorphism of groups and whose image is the kernel of ϕ′. To seethat ker ϕ is finite, it suffices to show that the kernel of the diagonal mappingψ: H1(Kv, Uγ) →H1(Lw, G)ω in the square above is finite.

Since the kernel of theleft-vertical map H1(Kv, Uγ) →H1(Lw, Uγ) is finite, finiteness of ψ will follow ifwe show that the image of the homomorphism δ is finite. But since W is a constantdecent group over Lw, the domain of δ, H0(Lw, W), is finitely generated.

Therange of δ is an abelian torsion group, and therefore the image of δ is finite. Thisconcludes step 1.Now we suppose that our locally algebraic groups scheme G is an extension (1)where W is any locally algebraic group scheme with XS(W) is finite, and where Uis a connected linear affine group scheme.

In this case, by assumption we have (i)and by Lemma (c) we have (iii2) (which already implies (ii)) so that the ExtensionLemma applies again to yield our theorem.□Corollary 1. Let G be a locally algebraic group over K whose group of connectedcomponents is decent.

Suppose, further, that the Tate-Shafarevich Conjecture holds.Then X(G) is finite.Proof. We begin with a lemma on prolongations of locally algebraic groups over Kto (smooth) locally algebraic group schemes over X −S (some S).Lemma.

Let G be a locally algebraic group over K whose group of connected com-ponents G/K is decent. Then G admits an extension to a smooth locally algebraicgroup scheme, G/X −S, satisfying (A) and (B) over X −S, for some finite set ofplaces S.Proof.

Let G0/K denote the connected component of the identity in G = G/K, sothat G/K = G/G0 is decent, by our hypothesis. We must, for a suitable finite set ofplaces S, construct a “prolongation of G/K”, i.e., a smooth locally algebraic groupscheme G/X −S with the property that its generic fiber is G/K and its quotientby G0/X −S, the connected component in G/K containing the identity, is locallyconstant (with generic fiber isomorphic to G).

We then can achieve (A) and (B) byfurther augmentation of S.Let Γ = G(K) be the associated discrete Gal(K/K)-group. To construct ourprolongation G/X −S we shall not use the full hypothesis of decency of G—onlythat Γ is finitely presented35.

For simplicity of notation, let us first consider thecase where Γ is constant rather than only locally constant, when viewed as sheaf forthe ´etale topology over K. Let R →F →Γ →1 be a finite presentation of Γ, withF a free group generated by elements x1, . .

. , xn and R a free group generated byelements y1, .

. .

, ym. By some abuse of notation we may view the natural projectionG →G as giving us a homomorphism G →Γ, and let G♯denote the fiber-productof G →Γ and F →Γ, so that we have a commutative diagram1 −−−−→G0 −−−−→G♯−−−−→F −−−−→1=yyy1 −−−−→G0 −−−−→G −−−−→Γ −−−−→135Ofer Gabber has sketched a construction of such a “prolongation” in the case where G =Aut(V/K) for V a projective smooth variety over K which requires only that Γ be finitelygenerated.

28B. MAZURwhere the rows are exact.

For any element s in F (or in Γ) let G0s denote theG0-bitorsor in G♯(or in G) which is the full inverse image of s. To say that ascheme Y is a bitorsor for a group scheme H means that Y has the structure of aleft- and a right-H-torsor, these actions commuting (cf. [Grot3, Exp.

VII] and [Br]for a treatment of this notion).First note that G0 itself does prolong to a smooth group scheme over X −S (forsome finite S). Fix such a prolongation; call it G0/X −S.

If S′ is any finite setcontaining S, let G0/X −S′ denote the restriction of G0/X −S to X −S′.We shall now extend this prolongation of G0 to a prolongation of G♯to a smoothlocally algebraic group scheme over X −S (possibly enlarging S). For this, considerthe G0-bitorsors G0x1, G0x2, .

. .

, G0xn for the set of generators x1, . .

. , xn of F, theseall being bitorsors over the base Spec K. Each one of these bitorsors G0xi prolongsto a bitorsor for G0/X −Si for some finite set Si containing S. Replacing S by theunion of the Si’s, we have bitorsors G0xi/X −S for G0/X −S (for i = 1, .

. .

, n).For any word w in x1, x2, . .

. , xn and their inverses in the free group F, lettingG0w/X −S be the appropriate (“contraction-product”36) bitorsor for G0/X −Sbuilt from the G0xi/X −S one checks that the disjoint union ` G0w/X −S takenover all elements w in F (together with multiplication induced by the contraction-product construction) constitutes a prolongation of G♯to a locally algebraic groupscheme over X −S with the desired properties.

Call it G♯/X −S.To “descend” this prolongation to a prolongation of G one sees that it sufficesto choose trivializations of the G0/X −S bitorsors G0yj/X −S for each of thegenerators yj of the group of relations R, such that these chosen trivializations are“consistent” in the sense that the induced trivialization on G0y/X −S for any y ∈Ris well defined, i.e., independent of how y is expressed in terms of the generatorsyj. But since G0 is separated, “consistency” of these trivializations over K implies“consistency” over X −S.

We can find such a “consistent” choice of trivializations(for our generating set yj) over K, and then, after a possible further enlargementof S to a finite set of places of K, we may extend this choice to X −S.We have then taken care of the case of constant G, i.e., discrete Gal(K/K)-groupsΓ with trivial Galois action. In general though, we have that Gal(K/K) acts on Γcontinuously, and therefore, recalling the notation of §16, the action factors througha finite quotient, ∆= Gal(L/K).

We then begin the procedure as above, i.e., bychoosing a prolongation G0/X −S as before, then making the base change to Y −Twhere Y is the spectrum of the ring of integers in L and T is the inverse image ofS, and continuing the construction over Y −T appropriately “equivariantly” forthe action of Gal(L/K), and then finally descending back to X −S.□Let G denote the extended group scheme given by the preceding lemma. Weknow by Theorem 4 that XS(G) is finite.

Let us define an “intermediate group”(call it X(S; G)) which will allow us to compare XS(G) and X(G), namely,X(S, G) := ker(H1(X −S, G) →YvH1(Kv, G)),where the summation over v is over all places v of K. Now, by definition, X(S, G)is contained in XS(G) and is consequently also finite. Moreover, if S ⊂T is an36This is the noncommutative version of the Baer-sum: one contracts the product of bitorsorsby the group action which is the left-action on the right bitorsor and the right-action on the leftbitorsor.

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY29inclusion of finite sets of places of v, the natural mapping X(S, G) →X(T, G) iseasily seen to be surjective. Since X(G) = lim−→X(T, G), the limit taken over thedirected system of all finite sets of places containing S, the corollary follows.□Now let V/K be a smooth projective variety.

The automorphism group G =Aut(V/K) is a locally algebraic group over K. Let G = G/G0 be its (discrete)group of connected components and Γ(V ) := G(K) the group of K-valued pointsof G, i.e., the group of connected components of the locally algebraic group G/K =Aut(V/K).Very little seems to be known, in general, about the groups Γ(V ). Are theyfinitely generated for all V ?

Finitely presented? decent?

Are they all arithmeticgroups?The group Γ(V ) admits a natural mapping (with finite kernel) to an arithmeticgroup.37 To see this, one uses the natural representation of G on NS(V ), the N´eron-Severi group of V which we view as finitely generated abelian locally constant groupover K, whose full automorphism group is an arithmetic group. One then provesthat the subgroup G1 of G which is the identity on N´eron-Severi is an algebraicgroup and, in particular, has only a finite number of components.Let us recall, briefly, how G1 is analyzed.

First, the action of G1 on V givesus an action on Pic0(V ). If L is any line bundle and g ∈G1, then [g∗L ⊗L−1] =[(g−1)∗L ⊗L−1] determines a class in Pic0(V ), giving us a 1-cocycle ϕL on thegroup scheme G1 with values in Pic0(V ) defined by g 7→[g∗L ⊗L−1].

Fixing L tobe the very ample line bundle on V coming from a chosen projective embedding,denote by G1,L ⊂G1 the “kernel” of the 1-cocycle ϕL. Define G2,L to be the group(representing the functor) of paired automorphisms of (V, L), i.e., automorphismsof V together with compatible automorphism of L. We have a surjection of locallyalgebraic groups G2,L →G1,L (in fact, G2,L is a central extension of G1,L withkernel isomorphic to Gm).

But G2,L is clearly a closed subgroup of the generallinear group of automorphisms of H0(V, L). It follows that G1 is of finite type, andtherefore its image in Aut(Pic0(V )) is finite.Collecting things, then, we can say that G = Aut(V/K) is a “successive exten-sion” of a subgroup of Aut(NS(V )) by a subgroup of Aut(Pic0(V )) by a subgroupscheme in Pic0(V ) by a linear group.Remark.

The “abelian variety part” of G is trivial if Pic0(V ) vanishes.Compare [Ra] where the connected component containing the identity in Aut(V )is discussed.By [BS, Theorem 2.6], or by the more general “descent theory” (cf. [Grot]) theset S(V/K) may be identified with X(G).

Hence, we haveCorollary 2. Let V/K be a projective variety such that the locally constant groupG(V/K) of components of Aut(V/K) is decent.

Suppose, further, that the Tate-Shafarevich Conjecture holds (for all twists of abelian subvarieties of Pic0(V/K)defined over K). Then S(V/K) is finite, i.e., the local-to-global principle holds forV/K, up to finite obstruction.Corollary 3.

Let V/K be a projective variety such that Pic(V/K) = Z. ThenS(V/K) is finite, i.e., the local-to-global principle holds for V/K, up to finite ob-struction.37Under the hypothesis that Γ(V ) is finitely generated, Ofer Gabber has sketched a proof thatΓ(V ) can be embedded in an arithmetic group.

30B. MAZURProof.

This follows directly from Corollary 2. For if Pic(V/K) = Z, then G(V/K)is a finite locally constant group scheme and, therefore, is decent.Also, sincePic0(V/K) vanishes, the hypothesis involving finiteness of the Tate-Shafarevichgroup of twists of subabelian varieties of Pic0(V ) is certainly satisfied.□Corollary 4.

Let V/K be a smooth variety of dimension ≥3 which is a hypersur-face (or more generally, a complete intersection) in projective space. Then S(V/K)is finite, i.e., the local-to-global principle holds for V/K, up to finite obstruction.Proof.

For such varieties, Pic(V/K) = Z [Grot2, Corollary 3.7 of Exp. XII] andtherefore Corollary 3 applies.□I am thankful to O. Gabber, J. Harris, and J.-P. Serre who told me the proofof the following proposition, which is a result due to Jordan [J] and which impliesthat, for smooth hypersurfaces V of dimension ≥3 and of degree ≥3, Aut(V ) is,in fact, finite.

Therefore, for such varieties the conclusion of Corollary 4 followsdirectly.Proposition. Let n ≥2, and let V be a smooth hypersurface in Pn/C of degree≥3.

Let Φ denote the subgroup of PGLn+1(C) which stabilizes V . Then Φ is finite.Proof.

Let V be defined by the homogeneous form F(X0, . .

. , Xn) of degree d. Thegroup Φ is algebraic; it suffices to show that its Lie algebra vanishes.

Suppose, then,that A = (aij) (i, j = 0, . .

. , n) is an (n + 1) × (n + 1) matrix in the Lie algebra ofGLn+1 which stabilizes V , i.e., such thatXi,jaij · Xj · ∂F/∂Xi = λ · Ffor λ ∈C∗.Writing λ · F = λ · d−1 P δij · Xj · ∂F/∂Xi where (δij) is the identity matrix, wehave(2)Xi,j(aij −λd−1δij)Xj · ∂F/∂Xi = 0 .Since V is smooth, the forms ∂F/∂Xi only have the origin in Cn+1 as a commonintersection, i.e., the sequence(∂F/∂X0, ∂F/∂X1, .

. .

, ∂F/∂Xn)is a “regular sequence”; therefore, the ideal of relations of this (regular) sequenceis generated by the evident ones∂F/∂Xi · ∂F/∂Xj −∂F/∂Xj · ∂F/∂Xi = 0 .In other words, the kernel R of the homomorphism of the C[X0, . .

. , Xn]-algebraC[X0, .

. .

, Xn, Y0, . .

. , Yn] to C[X0, .

. .

, Xn] which sends the variable Yj to ∂F/∂Xj(j = 0, . .

. , n) is generated by the elements∂F/∂Xj · Yi −∂F/∂Xi · Yj(i, j = 0, .

. .

, n) .

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY31Since these elements are homogeneous of degree d −1 ≥2 in the Xk’s, it followsthat if ρj(X0, . .

. , Xn) are linear forms in the Xk’s for j = 0, .

. .

, n, such thatXjρj(X0, . .

. , Xn) · ∂F/∂Xj = 0then each of the forms ρj(X0, .

. .

, Xn) vanish identically. Applying this to (2) wehave that aij = λd−1δij, i.e., that A gives 0 in the Lie algebra of PGLn+1.□Remarks.

Applying this proposition to smooth hypersurfaces of dimension ≥3 anddegree ≥3, and using (as in the proof of Corollary 4) the fact that, for such varieties,Pic(V ) = Z, we see that Aut(V ) is finite. See the discussion of this proposition on[B, pp.

41–42].Exercise (Serre). What happens when the base field is algebraically closed ofcharacteristic p, and p divides the degree d of V ?What analogous results are there for smooth hypersurfaces in multiprojectivespaces?18.

The mapping X(A/K) →S(A/K). If A is an abelian scheme over a baseS, to give an automorphism of A/S as abelian scheme is equivalent to giving an au-tomorphism of A viewed merely as scheme over S, which fixes the zero-section.

LetAut•(A) denote the group of such automorphisms. Fixing A/K, an abelian variety,let Aut•(A/K) denote the locally algebraic group of automorphisms of A preserv-ing zero-section, i.e., of the abelian variety A.

Let A denote the set of isomorphismclasses of abelian varieties A′/K which are isomorphic to A/Kv as abelian varietiesover Kv, for all places v of K. The set A is finite [BS, Corollary 7.11]. In fact, Acan be identified with the set X(Aut•(A/K)), and, since Aut•(A/K) is a locallyconstant group of “type ALA” in the terminology of [BS], the required finitenessfollows.Theorem 5.

The set S(A/K) has a natural partition into a finite number of (dis-joint) subsets indexed by A,S(A/K) =aα∈AS(A/K)α,and, for any α ∈A, if A′/K is an abelian variety in the isomorphism class α, thenthere is a natural surjection(3)X(A′/K) →S(A/K)αwhich identifies S(A/K)α with the orbit-space of X(A′/K) under the action of thegroup of automorphisms of A′ defined over K.Proof. First, let V be a “companion” to A, i.e., a projective variety over K, isomor-phic to A over the completions Kv where v ranges over all places of K. For a modelof the “Albanese variety”, AV , of V let us take AV = Pic0(Pic0(V )).

Thus AV is anabelian variety over K isomorphic, as abelian variety, to A over every completionof K and therefore the isomorphism class of the abelian variety AV over K is anelement of A. There is a natural principal homogeneous action of AV on V (defined

32B. MAZURover K).

Since V is also a companion to AV over K, it (taken with its structure asprincipal homogeneous space for AV ) determines an element in X(AV ).Next, for α ∈A, choose an abelian variety A′/K in the isomorphism class α,and define S(A/K)α to be the subset of S(A/K) represented by varieties V/K suchthat AV ∼= A′ (the isomorphism being as abelian varieties over K). This clearlygives a partition of S(A/K).

For each α ∈A, we have the required surjection (3),and this mapping visibly factors through the quotient to the orbit-space under thegroup of K-automorphisms of A′.To conclude the theorem, we consider two elements in X(A′/K) which representthe same element in S(A/K)α. Equivalently, on the same projective variety V , weare given two principal homogeneous space structures under the abelian variety A′.We must show that one can bring one of these structures to the other by a K-automorphism α of A′. But this is easy: We get a natural morphism over K fromthe projective variety V to the (locally constant) group Aut•(A′/K) by the rulewhich associates to each point v of V the mapping ϕv : A′ →A′ given as follows: toany a ∈A′, the translate of v by a via the first principal homogeneous structure isequal to the translate of v by ϕv(a) via the second principal homogeneous structure.Since V is connected and Aut•(A′/K) is locally constant, the mapping v 7→ϕv is,in fact, constant, and therefore its image is an automorphism α of A′, defined overK.□When is the mapping X(A′/K) →S(A/K) surjective?

From the above analysis,this will happen if and only if X(Aut•(A/K)) is trivial. Here are two instances:Corollary 1.

X(A′/K) →S(A/K) is surjective if either:(a) there is a finite cyclic extension L/K such that every automorphism of Aover K is already defined over L, or(b) A = E is an elliptic curve.Proof. Since Aut•(A/K) is discrete, case (a) follows immediately from the corollaryto Example (1) in §15.

Case (b) follows from case (a), for if A is an elliptic curveover a number field K, the splitting field over K for the action of Gal(K/K) onAut•(A) is of degree ≤2.□Corollary 2. For K a fixed number field, the Tate-Shafarevich Conjecture holds forall abelian varieties over K if and only if S(A/K) is finite for all abelian varietiesA over K.Proof.

Recall that A is a finite set. In view of the above theorem, then, it suffices toshow that, for any abelian variety, A/K, X(A/K) is finite if and only if its imagein S(A/K) is finite, or, equivalently, if and only if there are only a finite number oforbits of X(A/K) under the action of the K-automorphism group of A.

But, asTate remarked to me, this is easy to see: there are only a finite number of elementsin X(A/K) of any fixed order, and automorphisms of groups preserve the order ofelements.□19.The local-to-global principle for quadrics. I am thankful to Colliot-Th´el`ene for showing me this proof of the local-to-global principle for any smoothquadric V over a number field K.Let W/K be a smooth projective variety which is potentially quadric in the sensethat its base change to K is isomorphic to a quadratic hypersurface in projective

THE PASSAGE FROM LOCAL TO GLOBAL IN NUMBER THEORY33space. Putting W = W ×Spec K Spec K we have that Pic(W) is isomorphic eitherto Z or Z ⊕Z (the latter only if W is of dimension 2).

Call an element ξ of Pic(W )of quadric type if a line bundle L over W representing ξ has the property that L isvery ample and the embedding of W in projective space associated to the completelinear system of L identifies W with a quadric hypersurface over K. There is onlyone element ξ “of quadric type” in Pic(W), and that element ξ is fixed under theaction of Gal(K/K). We have the exact sequence(4)0 →Pic(W) →Pic(W)Gal(K/K) →δK Br(K)where Br(K) is the Brauer group of K and δK is the natural “coboundary” mappingwhose definition is as follows.

For a line bundle l on W whose isomorphism class isfixed by Gal(K/K) and for each σ ∈Gal(K/K) choose an isomorphism ισ : l ∼= lσ.Denote by c the 2-cocycle on Gal(K/K) with coefficients in K∗defined by theformula c(σ, τ) = (ιστ)−1 · (ισ)τ · ιτ. Then if [l] denotes the isomorphism class of lin Pic(W)Gal(K/K), δK([l]) is defined to be the cohomology class of c in the Brauergroup Br(K) = H2(Gal(K/K); K∗); this cohomology class is seen to be dependentonly upon the isomorphism class of [l] and independent of the choice of the ισ’s.The element δK(ξ) ∈Br(K) may be thought of as the obstruction to W/K beingisomorphic to a quadric hypersurface over K, for its vanishing is a necessary andsufficient condition for the line bundle L corresponding to ξ on W to come froma line bundle L on W/K whose linear system would identify W with a quadrichypersurface in projective space over K.If V is a smooth quadric over K and W a companion to V over K, then V isisomorphic to W over K, so W is potentially quadric.

Since V/Kv is isomorphicW/Kv for all places v of K, W is, in fact, quadric over Kv for all v; therefore, itfollows from the version of (4) over Kv that δKv(ξ) = 0 in Br(Kv) for all v. Sincethe global Brauer group injects into the direct sum of the local Brauer groups,δK(ξ) = 0; therefore, the companion W is a quadric variety over K.38 Now let qVand qW denote the two quadratic forms over K corresponding to V and W. SinceV and W are companions, it follows (from the argument alluded to in the footnoteto §2 of Part I) that the corresponding forms qV,v and qW,v over the completionsKv for each place v of K are similar. Using [O] we then see that qV and qW aresimilar over K.□AcknowledgmentI am thankful to M. Artin, P. Diaconis, J.-L. Colliot-Th´el`ene, O. Gabber, J. Har-ris, N. Katz, Y. Nisnevich, K. Rubin, and J.-P. Serre for their generous help and tothe audiences at the MSRI39, the University of Toronto, and Penn State40, for theircomments, suggestions, and patience during my talks on material corresponding toPart I of this article.38A similar argument shows that a smooth projective variety W/K of dimension ≥3 whichis a companion to a hypersurface of degree d defined over K is itself isomorphic over K to ahypersurface of degree d.39At the conference in June 1992 commemorating the 10th anniversary of the MSRI and inhonor of Irving Kaplansky on the occasion of his retirement as director.40The Weisfeiler Lecture, November 1992.

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