Diophantine geometry as Galois theory in the mathematis of
Serge Lang
Minhyong Kim
Marh 4, 2006
Lang's oneption of Diophantine geometry is rather ompatly modelled by the following elebrated onjeture [29℄:
Let V be a subvariety of a semi-abelian variety A, G A a nitely generated subgroup,
and Div(G) the subgroup of A onsisting of the division points of G. Then V \ Div(G)
is ontained in a nite union of subvarieties of V of the form Bi + xi , where eah Bi is a
semi-abelian subvariety of A and xi 2 A.
There is a wealth of literature at this point surveying the various ideas and tehniques employed in
its resolution, making it unneessary to review them here in any detail [10, 39℄. However, it is still
worth taking note of the valuable generality of the formulation, evidently arising from a profound
instint for the plausible strutures of mathematis. To this end, we remark merely that it was exatly
this generality that made possible the astounding interation with geometri model theory in the 90's
[3℄. That is to say, analogies to model-theoreti onjetures and struture theorems would have been
far harder to detet if attention were restrited, for example, to situations where the intersetion is
expeted to be nite. Nevertheless, in view of the sparse subset of the omplex net of ideas surrounding
this onjeture that we wish to highlight in the present artile, our intention is to fous exatly on the
ase where A is ompat and V does not ontain any translate of a onneted non-trivial subgroup.
The motivating example, of ourse, is a ompat hyperboli urve embedded in its Jaobian. Compare
then the two simple ases of the onjeture that are amalgamated into the general formulation:
(1) V
(2) V
\ A[1℄, the intersetion between V
\ G is nite.
and the torsion points of A, is nite.
Lang expeted onjeture (1) to be resolved using Galois theory alone. This insight was based upon
work of Ihara, Serre, and Tate ([32℄, VIII.6), dealing with the analogous problem for a torus, and omes
down to the onjeture, still unresolved, that the image of the Galois representation in Aut(A[1℄) '
^ ) ontains an open subgroup of the homotheties Z
^ . Even while assertion (1) is already a
GL2g (Z
theorem of Raynaud [44℄, signiant progress along the lines originally envisioned by Lang was made
in [2℄ by replaing A[1℄ with A[p1 ℄, the points of p-power torsion, and making ruial use of p-adi
Hodge theory.
It is perhaps useful to reet briey on the overall ontext of Galois-theoreti methods in Diophantine geometry, of ourse without attempting to do justie to the full range of interations and
impliations. Initially, that Galois theory is relevant to the study of Diophantine problems should
surprise no one. After all, if we are interested in X (F ), the set of rational points of a variety X over
a number eld F , what is more natural than to observe that X (F ) is merely the xed point set of
:= Gal(F =F ) ating on X (F )? Sine the latter is an objet of lassial geometry, suh an expression
might be expeted to niely irumsribe the subset X (F ) of interest. This view is of ourse very
naive and the ation on on X (F ) is notoriously diÆult to use in any diret fashion. The ation on
torsion points of ommutative group varieties on the other hand, while still diÆult, is onsiderably
more tratable, partly beause a nite abelian subgroup behaves relatively well under speialization.
1
Suh an arithmeti variation exerts tight ontrol on the elds generated by the torsion points, shaping
Galois theory into a powerful a tool for investigations surrounding (1).
For onjeture (2), where the points to be studied are not torsion, it is not at all lear that Galois
theory an be as useful. In fat, my impression is that Lang expeted analyti geometry of some sort
to be the main input to onjetures of type (2). This is indiated, for example, by the absene of any
referene to arithmeti in the formulation. We might even say that impliit in the onjeture is an
important idea that we will refer to as the analyti strategy:
(a) replae the diÆult Diophantine set V (F ) by the geometri intersetion V \ G;
(b) try to prove this intersetion nite by analyti means.
In this form, the strategy appears to have been extraordinarily eÆient over funtion elds, as in
the work of Buium [4℄. Even Hrushovski's [15℄ proof an interpreted in a similar light where passage
to the ompletion of suitable theories is analogous to the move from algebra to analysis (sine the
theory of elds is not good enough). These examples should already suÆe to onvine us that it
is best left open as to what kind of analyti means are most appropriate in a given situation. The
proof over number elds by Faltings [9℄, as well as the urve ase by Vojta [47℄, utilizes rather heavy
Arhimedean analyti geometry. Naturally, the work of Vojta and Faltings draws us away from the
realm of traditional Galois theory. However, in Chabauty's theorem [5℄, where V is a urve and the
rank of G is stritly less than the dimension of A, it is elementary non-Arhimedean analysis, more
speially p-adi abelian integrals, that ompletes the proof. Lang makes lear in several dierent
plaes ([28℄, [32℄, notes to hapter 8, [36℄, I.6) that Chabauty's theorem was a denite fator in the
formulation of his onjeture. This then invites a return to our main theme, as we remind ourselves
that non-Arhimedean analysis has ome to be viewed protably over the last several deades as a
projetion of analysis on Galois groups, a perspetive of whih Lang was well aware ([33℄, hapter
4). As suh, it has something quite substantial to say about non-torsion points, at least on ellipti
urves ([18℄, for example). Hodge theory is again a key ingredient, this time as the medium in whih
to realize suh a projetion [43℄.
It is by now known even to the general publi that a areful study of Galois ations underlies
the theorem of Wiles [49℄, and roughly one-half of the diÆulties in the theorem of Faltings [8℄.
There, Galois representations must be studied in onjuntion with an array of intriate auxiliary
onstrutions. However, the most basi step in the Galois-theoreti desription of non-torsion points,
remarkable in its simpliity, goes through the Kummer exat sequene
Æ
0!A[m℄(F )!A(F )!A(F ) !
H 1 ( ; A[m℄)!H 1 ( ; A)!:
In this ase, an easy study of speialization allows us to loate the image of Æ inside a subgroup
H 1 ( S ; A[m℄) of ohomology lasses with restrited ramiation, whih then form a nite group.
We dedue thereby the niteness of A(F )=mA(F ), the weak Mordell-Weil theorem. Apparently, a
streamlined presentation of this proof, systematially emphasizing the role of Galois ohomology, rst
appears in Lang's paper with Tate [38℄. There, they also emphasize the interpretation of Galois
ohomology groups as lassifying spaes for torsors, in this ase, for A and A[m℄. (We reall that
a torsor for a group U in some ategory is an objet orresponding to a set with simply transitive
U -ation, where the extra struture of the ategory, suh as Galois ations, prevent them from being
trivial. See, for example, [40℄, III.4.) This onstrution has been generalized in one diretion to study
non-torsion algebrai yles by assoiating to them extensions of motives [7℄. More pertinent to the
present disussion, however, is a version of the Kummer map that avoids any attempt to abelianize,
taking values, in fat, in non-abelian ohomology lasses.
In the ourse of preparing this artile, I looked into Lang's magnum opus [32℄ for the rst time
in many years and was a bit surprised to nd a setion entitled `non-abelian Kummer theory.' What
is non-abelian there is the Galois group that needs to be onsidered if one does not assume a priori
that the torsion points of the group variety are rational over the ground eld. The eld of m-division
points of the rational points will then have a Galois group H of the form
0!A[m℄!H !M !0
2
where M GL2g (Z=m). Thus, `non-abelian' in this ontext is used in the same sense as in the
referene to non-abelian Iwasawa theory. But what is neessary for hyperboli urves is yet another
layer of non-ommutativity, this time in the oeÆients of the ation. Given a variety X with a
b) lassifying nite etale
rational point b, we an ertainly onsider the etale fundamental group ^1 (X;
overs of X . But the same ategory assoiates to any other point x 2 X (F ) the set of etale paths
^1 (X ; b; x)
b). All these live inside the ategory of pro-nite
from b to x whih is naturally a torsor for ^1 (X;
b)) that
sets with Galois ation. There is then a non-abelian ontinuous ohomology set H 1 ( ; ^1 (X;
lassies torsors, and a non-abelian Kummer map
!H
Æ na : X (F )
1
b))
( ; ^1 (X;
sending a point x to the lass of the torsor ^1 (X ; b; x). This is obviously a basi onstrution whose
importane, however, has begun to emerge only in the last twenty or so years. It relies very muh on the
exible use of varying base-points in Grothendiek's theory of the fundamental group, and it appears
to have taken some time after the ineption of the arithmeti 1 theory [45℄ for the importane of suh
a variation to be properly appreiated [12, 6, 16℄. In fat, the impetus for taking it seriously ame
also for the most part from Hodge theory [13, 14℄. As far as Diophantine problems are onerned,
in a letter to Faltings [12℄ written shortly after the proof of the Mordell onjeture, Grothendiek
proposed the remarkable onjeture that Æna should be a bijetion for ompat hyperboli urves. He
expeted suh a statement to be diretly relevant to the Mordell problem and probably its variants
like onjeture (2). This expetation appears still to be rather reasonable. For one thing, it is evident
that the onjeture is a hyperboli analogue of the niteness onjeture for Tate-Shafarevih groups.
And then, profound progress is represented by the work of Nakamura, Tamagawa, and Mohizuki
[42, 46, 41℄, where a statement of this sort is proved when points in the number eld are replaed by
dominant maps from other varieties. Some marginal insight might also be gleaned from [22℄ and [23℄
where a unipotent analogue of the Kummer map is related to Diophantine niteness theorems. There,
the ambient spae inside whih the analysis takes plae is a lassifying variety Hf1 ( v ; Unet ) of torsors for
the loal unipotent etale fundamental group (rather than the Jaobian), while the nitely-generated
group G is replaed by the image of a map
! Hf (
Hf1 ( S ; Unet )
1
et
v ; Un
)
oming from a spae of global torsors. Thereby, one obtains a new manifestation of the analyti strategy proving X \ Im[Hf1 ( S ; Unet )℄ to be nite in some very speial irumstanes, and in general for a
hyperboli urve over Q if one admits standard onjetures from the theory of mixed motives (for example, the Fontaine-Mazur onjeture on geometri Galois representations). Fortunately, Chabauty's
original method ts naturally into this setting as the tehnial foundation of the analyti part now
beomes non-abelian p-adi Hodge theory and iterated integrals. Inidentally, some sense of the Diophantine ontent of these ideas an already be gained by deriving the injetivity of Æna from the
Mordell-Weil theorem.
It should be lear at this point that the Galois theory of the title refers in general to the theory
of the fundamental group. Serge Lang was profoundly onerned with the fundamental group for a
good part of his mathematial life. A rather haphazard list of evidene might be omprised of:
-his foundational work on unramied lass eld theory for varieties over nite elds, where he
proves the surjetivity of the reiproity map among many other things [25, 26℄;
-his study of the ubiquitous `Lang torsor' [27℄;
-his work with Serre on fundamental groups of proper varieties in arbitrary harateristi [37℄;
-his extensive study with Kubert of the modular funtion eld [24℄;
-his work with Katz [21℄ on niteness theorems for relative 1 's that made possible the subsequent
proof by Bloh [1℄, and then Kato and Saito [19, 20℄ of the niteness of CH0 for arithmeti shemes.
3
Besides these inuential papers, the reader is referred to his beautiful AMS olloquium letures
[31℄ for a global perspetive on the role of overing spaes in arithmeti.
Even towards the end of his life when his published work went in an inreasingly analyti diretion,
he had a keen interest both in fundamental groups and in the analogy between hyperboli manifolds
and number elds wherein fundamental groups play a entral role. In my last year of graduate shool,
he urged me strongly to study the work of Kato and Saito (and apply it to Arakelov theory!) even
though it had been years sine he had himself been involved with suh questions. From the Spring
of 2004, I reall a harateristially animated exhange in the ourse of whih he explained to me a
theorem of Geyer [11℄ stating that abelian subgroups of Gal(Q =Q ) are pro-yli. It was lear that
he pereived this fat to t niely into his vivid ideas about the heat kernel [17℄, but in a manner
that I failed (and still fail) to omprehend properly. (He was unfortunately seretive with his deeper
reetions on the arithmeti signiane of his later work, allowing only informal glimpses here and
there. It is tempting but probably premature to speulate about a Galois theory that enompasses
even Arhimedean analysis.) The preoupation with hyperboli geometry that was evident even from
the 70's ([30, 34, 35℄ and [36℄, hapters 8 and 9) ould rather generally be onstrued as reeting a
persistent intuition about the relevane of fundamental groups to Diophantine problems. (An intuition
that was shared by Grothendiek [12℄ and even Weil [48℄.)
As for the diret appliation of non-abelian fundamental groups to Diophantine geometry that we
have outlined here, one an onviningly plae it into the general framework of Lang's inquiries. He
is disussing the theorem of Siegel in the following paragraph from the notes to hapter 8 of [32℄:
The general version used here was presented in [28℄ following Siegel's (and Mahler's)
method. The Jaobian replaes the theta funtion, as usual, and the mehanism of the
overing already used by Siegel appears here in its full formal larity. It is striking to
observe that in [25℄, I used the Jaobian in a formally analogous way to deal with the lass
eld theory in funtion elds. In that ase, Artin's reiproity law was redued to a formal
omputation in the isogeny u 7! u(q) u of the Jaobian. In the present ase, the heart of
the proof is redued to a formal omputation of heights in the isogeny u 7! mu + a.
We have emphasized above the importane of the Kummer map
x
7! [^ (X ; b; x)℄ 2 H
1
1
( ; ^1 (X ; b)):
^ ), the
b) by its abelian quotient H1 (X;
Z
When X is dened over a nite eld Fq and we replae ^1 (X;
map takes values in
Z
^ ))
H 1 (Gal( q = q ); H1 (X;
F F
Fr
Z
^ )=[(F r
= H1 (X;
Z
^ )℄;
1)H1 (X;
2 Gal(F q =Fq ) being the Frobenius element. But this last group is nothing but the kernel
^1ab (X )0
of the struture map
!^ (Spe(Fq )):
^1ab (X )
1
Thus the abelian quotient of the Kummer map beomes identied with the reiproity map [19℄
CH0 (X )0
!^ab (X )
1
0
of unramied lass eld theory evaluated on the yle (x) (b). In other words, the reiproity map
is merely an `abelianized' Kummer map in this situation. There is no hoie but to interprete the
reiproity law [19, 20℄ as an `abelianized Grothendiek onjeture' over nite elds.
Of ourse it is hard to imagine exatly what Lang himself found striking in the analogy when he
wrote the lines quoted above. What is not hard to imagine is that he would have been very muh at
home with the ideas surrounding Grothendiek's onjeture and the non-abelian Kummer map.
4
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