Diophantine Approximation and Nevanlinna Theory

122 Pages·2010·0.57 MB·English

Diophantine Approximation and Nevanlinna
Theory
PaulVojta
AbstractAswasoriginallyobservedbyC.F.Osgoodandfurtherdevelopedbythe
author, there is a formal analogy between Nevanlinna theory in complex analysis
andcertainresultsindiophantineapproximation.Thesenotesdescribethisanalogy,
after briefly introducing the theory of heights and Weil function in number theory
andthemainconceptsofNevanlinnatheory.Parallelconjecturesarethenpresented,
inNevanlinnatheory(“Griffiths’conjecture”)andtheauthor’sconjectureonratio-
nalpoints.Followingthis,recentworkisdescribed,highlightingworkofCorvaja,
Zannier,Evertse,andFerrettioncombiningSchmidt’sSubspaceTheoremwithge-
ometricalconstructionstoobtainpartialresultsonthisconjecture.Counterpartsof
theseresultsinNevanlinnatheoryarealsogiven(duetoRu).Thesenotesalsode-
scribe parallel extensions of the conjectures in Nevanlinna theory and diophantine
approximation, to involve finite ramified coverings and algebraic points, respec-
tively.Variantsoftheseconjecturesinvolvingtruncatedcountingfunctionsarealso
introduced,andtherelationsofthesevariousconjectureswiththeabcconjectureof
MasserandOesterle´ arealsodescribed.
0 Introduction
BeginningwiththeworkofC.F.Osgood[1981],ithasbeenknownthatthebranch
ofcomplexanalysisknownasNevanlinnatheory(alsocalledvaluedistributionthe-
ory)hasmanysimilaritieswithRoth’stheoremondiophantineapproximation.This
wasextendedbytheauthor[Vojta,1987]toincludeanexplicitdictionaryandtoin-
cludegeometricresultsaswell,suchasPicard’stheoremandMordell’sconjecture
(Faltings’theorem).ThelatteranalogytiesinwithLang’sconjecturethataprojec-
PaulVojta
DepartmentofMathematics,UniversityofCalifornia,970EvansHall#3840,Berkeley,CA 94720-
3840e-mail:[email protected]
PartiallysupportedbyNSFgrantDMS-0500512
113
114 PaulVojta
tive variety should have only finitely many rational points over any given number
field(i.e.,isMordellic)ifandonlyifitisKobayashihyperbolic.
Thiscircleofideashasdevelopedfurtherinthelast20years:Lang’sconjecture
onsharpeningtheerrorterminRoth’stheoremwascarriedovertoaconjecturein
Nevanlinnatheorywhichwasprovedinmanycases.Intheotherdirection,Bloch’s
conjectures on holomorphic curves in abelian varieties (later proved; see Section
14 for details) led to proofs of the corresponding results in number theory (again,
see Section 14). More recently, work in number theory using Schmidt’s Subspace
TheoremhasledtocorrespondingresultsinNevanlinnatheory.
ThisrelationwithNevanlinnatheoryisinsomesensesimilartothe(mucholder)
relationwithfunctionfields,inthatoneoftenlookstofunctionfieldsorNevanlinna
theory for ideas that might translate over to the number field case, and that work
overfunctionfieldsorinNevanlinnatheoryisofteneasierthanworkinthenumber
fieldcase.Ontheotherhand,bothfunctionfieldsandNevanlinnatheoryhavecon-
cepts that (so far) have no counterpart in the number field case. This is especially
true of derivatives, which exist in both the function field case and in Nevanlinna
theory.Inthenumberfieldcase,however,onewouldwantthe“derivativewithre-
spectto p,”whichremainsasamajorstumblingblock,although(separate)workof
BuiumandofMinhyongKimmayultimatelyprovidesomeanswers.Thesearchfor
suchaderivativeisalsoaddressedinthesenotes,usingapotentialapproachusing
successiveminima.
Itisimportanttonote,however,thattherelationwithNevanlinnatheorydoesnot
“gothrough”thefunctionfieldcase.Althoughitispossibletolookatthefunction
field case over C and apply Nevanlinna theory to the functions representing the
rationalpoints,thisisnottheanalogybeingdescribedhere.Instead,intheanalogy
presentedhere,oneholomorphicfunctioncorrespondstoinfinitelymanyrationalor
algebraic points (whether over a number field or over a function field). Thus, the
analogy with Nevanlinna theory is less concrete, and may be regarded as a more
distantanalogythanfunctionfields.
Thesenotesdescribesomeoftheworkinthisarea,includingmuchofthenec-
essarybackgroundindiophantinegeometry.Specifically,Sections1–3recallbasic
definitionsofnumbertheoryandthetheoryofheightsofelementsofnumberfields,
culminatinginthestatementofRoth’stheoremandsomeequivalentformulationsof
thattheorem.Thispartassumesthatthereaderknowsthebasicsofalgebraicnumber
theory and algebraic geometry at the level of Lang [1970] and Hartshorne [1977],
respectively.Someproofsareomitted,however;forthosetheinterestedreadermay
refertoLang[1983].
Sections4–6brieflyintroduceNevanlinnatheoryandtheanalogybetweenRoth’s
theorem and the classical work of Nevanlinna. Again, many proofs are omitted;
referencesincludeShabat[1985],Nevanlinna[1970],andGoldbergandOstrovskii
[2008]forpureNevanlinnatheory,andVojta[1987]andRu[2001]fortheanalogy.
Sections7–15generalizethecontentoftheearliersections,inthemoregeometric
contextofvarietiesovertheappropriatefields(numberfields,functionfields,orC).
Again,proofsareoftenomitted;mostmaybefoundinthereferencesgivenabove.
DiophantineApproximationandNevanlinnaTheory 115
Section 14 in particular introduces the main conjectures being discussed here:
Conjecture 14.2 in Nevanlinna theory (“Griffiths’ conjecture”) and its counterpart
innumbertheory,theauthor’sConjecture14.6onrationalpointsonvarieties.
Sections16and17roundoutthefirstpartofthesenotes,bydiscussingthefunc-
tion field case and the subject of the exceptional sets that come up in the study of
higherdimensionalvarieties.
In both Nevanlinna theory and number theory, these conjectures have been
provedonlyinveryspecialcases,mostlyinvolvingsubvarietiesofsemiabelianva-
rieties.Thisincludesthecaseofprojectivespaceminusacollectionofhyperplanes
in general position (Cartan’s theorem and Schmidt’s Subspace Theorem). Recent
workofCorvaja,Zannier,Evertse,Ferretti,andRuhasshown,however,thatusing
geometricconstructionsonecanuseSchmidt’sSubspaceTheoremandCartan’sthe-
oremtoderiveotherweakspecialcasesoftheconjecturesmentionedabove.Thisis
thesubjectofSections18–22.
Sections23–27presentgeneralizationsofConjectures14.2and14.6.Conjecture
14.2,inNevanlinnatheory,canbegeneralizedtoinvolvetruncatedcountingfunc-
tions(aswasdonebyNevanlinnainthe1-dimensionalcase),andcanalsobeposed
in the context of finite ramified coverings. In number theory, Conjecture 14.6 can
alsobegeneralizedtoinvolvetruncatedcountingfunctions.Thesimplestnontrivial
caseofthisconjecture,involvingthedivisor[0]+[1]+[∞]onP1,isthecelebrated
“abc conjecture” of Masser and Oesterle´. Thus, Conjecture 22.5 can be regarded
as a generalization of the abc conjecture as well as of Conjecture 14.6. One can
also generalize Conjecture 14.6 to treat algebraic points; this corresponds to finite
ramifiedcoveringsinNevanlinnatheory.ThisisConjecture24.1,whichcanalsobe
posedusingtruncatedcountingfunctions(Conjecture24.3).
Sections28and29brieflydiscussthequestionofderivativesinNevanlinnathe-
ory, and Nevanlinna’s “Lemma on the Logarithmic Derivative.” A geometric form
ofthislemma,duetoR.Kobayashi,M.McQuillan,andP.-M.Wong,isgiven,andit
isshownhowthisformleadstoaninequalityinNevanlinnatheory,duetoMcQuil-
lan,calledthe“tautologicalinequality.”Thisinequalitythenleadstoaconjecturein
numbertheory(Conjecture29.1),whichofcourseshouldthenbecalledthe“tauto-
logicalconjecture.”Thisconjectureisdiscussedbriefly;itisofinterestsinceitmay
shedsomelightonhowonemighttake“derivatives”innumbertheory.
Theabcconjectureinfusesmuchofthistheory,notonlybecauseaNevanlinna-
like conjecture with truncated counting functions contains the abc conjecture as a
specialcase,butalsobecauseotherconjecturesalsoimplytheabcconjecture,and
therefore are “at least as hard” as abc. Specifically, Conjecture 24.1, on algebraic
points, implies the abc conjecture, even if known only in dimension 1, and Con-
jecture14.6,onrationalpoints,alsoimpliesabcifknowninhighdimensions.This
latterimplicationisthesubjectofSection30.Finally,implicationsintheotherdi-
rectionareexploredinSection31.
116 PaulVojta
1 NotationandBasicResults:NumberTheory
We assume that the reader has an understanding of the fundamental basic facts
of number theory (and algebraic geometry), up through the definitions of (Weil)
heights. References for these topics include [Lang, 1983] and [Vojta, 1987]. We
do,however,recallsomeofthebasicconventionsheresincetheyoftendifferfrom
authortoauthor.
Throughoutthesenotes,kwillusuallydenoteanumberfield;ifso,thenO will
k
denote its ring of integers and M its set of places. This latter set is in one-to-one
k
correspondence with the disjoint union of the set of nonzero prime ideals of O ,
k
thesetofrealembeddingsσ: k(cid:44)→R,andthesetofunorderedcomplexconjugate
pairs (σ,σ) of complex embeddings σ: k (cid:44)→C with image not contained in R.
SuchelementsofM arecallednon-archimedeanorfiniteplaces,realplaces,and
k
complexplaces,respectively.
Therealandcomplexplacesarecollectivelyreferredtoasarchimedeanorinfi-
niteplaces.ThesetoftheseplacesisdenotedS .Itisafiniteset.
∞
Toeachplacev∈M weassociateanorm(cid:107)·(cid:107) ,definedforx∈kby(cid:107)x(cid:107) =0if
k v v
x=0and
(1.1)

(Ok:p)ordp(x) ifvcorrespondstop⊆Ok;
(cid:107)x(cid:107)v= |σ(x)| ifvcorrespondstoσ: k(cid:44)→R;and
|σ(x)|2 ifvisacomplexplace,correspondingtoσ: k(cid:44)→C
ifx(cid:54)=0.Hereord (x)meanstheexponentofpinthefactorizationofthefractional
p
ideal(x).Ifweusetheconventionthatord (0)=∞,then(1.1)isalsovalidwhen
p
x=0.
Wereferto(cid:107)·(cid:107) asanorminsteadofanabsolutevalue,because(cid:107)·(cid:107) doesnot
v v
satisfythetriangleinequalitywhenvisacomplexplace.However,let

0 ifvisnon-archimedean;

(1.2) Nv= 1 ifvisreal;and
2 ifviscomplex.
Thenthenormassociatedtoaplacevofksatisfiestheaxioms
(1.3.1) (cid:107)x(cid:107) ≥0,withequalityifandonlyifx=0;
v
(1.3.2) (cid:107)xy(cid:107) =(cid:107)x(cid:107) (cid:107)y(cid:107) forallx,y∈k;and
v v v
(1.3.3) (cid:107)x1+···+xn(cid:107)v≤nNvmax{(cid:107)x1(cid:107)v,...,(cid:107)xn(cid:107)v}forallx1,...,xn∈k,n∈N.
(Inthesenotes,N={0,1,2,...}.)
Someauthorstreatcomplexconjugateembeddingsasdistinctplaces.Wedonot
dosohere,becausetheygiverisetothesamenorms.
Notethat,ifx∈k,thenx liesintheringofintegersifandonlyif(cid:107)x(cid:107) ≤1for
v
all non-archimedean places v. Indeed, if x(cid:54)=0 then both conditions are equivalent
tothefractionalideal(x)beingagenuineideal.
DiophantineApproximationandNevanlinnaTheory 117
LetLbeafiniteextensionofanumberfieldk,andletwbeaplaceofL.Ifwis
non-archimedean,correspondingtoanonzeroprimeidealq⊆O ,thenp:=q∩O
L k
is a nonzero prime of O , and gives rise to a non-archimedean place v∈M . If v
k k
arisesfromwinthisway,thenwesaythatwliesoverv,andwritew|v.Likewise,if
wisarchimedean,thenitcorrespondstoanembeddingτ: L(cid:44)→C,anditsrestriction
τ(cid:12)(cid:12)k: k(cid:44)→Cgivesrisetoauniquearchimedeanplacev∈Mk,andagainwesaythat
wliesovervandwritew|v.
Foreachv∈M ,thesetofw∈M lyingoveritisnonemptyandfinite.Ifw|v
k L
thenwealsosaythatvliesunderw.
IfSisasubsetofM ,thenwesayw|SifwliesoversomeplaceinS;otherwise
k
wewritew(cid:45)S.
Ifw|v,thenwehave
(1.4) (cid:107)x(cid:107) =(cid:107)x(cid:107)[Lw:kv] forallx∈k,
w v
whereL andk denotethecompletionsofLandkatwandv,respectively.Wealso
w v
have
(1.5) ∏ (cid:107)y(cid:107) =(cid:107)NLy(cid:107) forallv∈M andally∈L.
w k v k
w∈ML
w|v
This is proved by using the isomorphism L⊗kkv ∼= ∏w|vLw; see for example
[Neukirch,1999,Ch.II,Cor.8.4].
Let L/K/k be a tower of number fields, and let w(cid:48) and v be places of L and k,
respectively.Thenw(cid:48)|vifandonlyifthereisaplacewofK satisfyingw(cid:48)|wand
w|v.
Thefieldk=Qhasnocomplexplaces,onerealplacecorrespondingtotheinclu-
sion Q⊆R, and infinitely many non-archimedean places, corresponding to prime
rationalintegers.Thus,wewrite
MQ={∞,2,3,5,...}.
PlacesofanumberfieldsatisfyaProductFormula
(1.6) ∏ (cid:107)x(cid:107) =1 forallx∈k∗.
v
v∈Mk
Thisformulaplaysakeyroleinnumbertheory:itisusedtoshowthatcertainex-
pressionsfortheheightarewelldefined,anditalsoimpliesthatif∏v(cid:107)x(cid:107)v<1then
x=0.
The Product Formula is proved first by showing that it is true when k=Q (by
directverification)andthenusing(1.5)topasstoanarbitrarynumberfield.
118 PaulVojta
2 Heights
Theheightofanumber,orofapointonavariety,isameasureofthecomplexityof
thatnumber.Forexample,100/201and1/2areveryclosetoeachother(usingthe
normattheinfiniteplace,atleast),butthelatterisamuch“simpler”numbersince
itcanbewrittendownusingfewerdigits.
We define the height (also called the Weil height) of an element x∈k by the
formula
(2.1) H (x)= ∏ max{(cid:107)x(cid:107) ,1}.
k v
v∈Mk
Asanexample,considerthespecialcaseinwhichk=Q.Writex=a/bwitha,b∈Z
relatively prime. For all (finite) rational primes p, if pi is the largest power of p
dividinga,and pj isthelargestpowerdividingb,then(cid:107)a(cid:107) =p−iand(cid:107)b(cid:107) =p−j,
p p
and therefore max{(cid:107)x(cid:107) ,1}= pb. Therefore the product of all terms in (2.1) over
p
allfiniteplacesvisjust|b|.Attheinfiniteplace,wehave(cid:107)x(cid:107) =|a/b|,sothisgives
∞
(2.2) HQ(x)=max{|a|,|b|}.
Similarly,ifP∈Pn(k)forsomen∈N,wedefinetheWeilheighth (P)asfol-
k
lows. Let [x :···:x ] be homogeneous coordinates for P (with the x always as-
0 n i
sumedtolieink).Thenwedefine
(2.3) H (P)= ∏ max{(cid:107)x (cid:107) ,...,(cid:107)x (cid:107) }.
k 0 v n v
v∈Mk
BytheProductFormula(1.6),thisquantityisindependentofthechoiceofhomo-
geneouscoordinates.
IfweidentifykwithA1(k)andidentifythelatterwithasubsetofP1(k)viathe
standard injection i: A1 (cid:44)→P1, then we note that H (x)=H (i(x)) for all x∈k.
k k
Similarly,wecanidentifyknwithAn(k),andthestandardembeddingofAnintoPn
givesusaheight
H (x ,...,x )= ∏ max{(cid:107)x (cid:107) ,...,(cid:107)x (cid:107) ,1}.
k 1 n 1 v n v
v∈Mk
Theheightfunctionsdefinedsofar,allusingcapital‘H,’arecalledmultiplicative
heights.Usuallyitismoreconvenienttotaketheirlogarithmsanddefinelogarith-
micheights:
(2.4) h (x)=logH (x)= ∑ log+(cid:107)x(cid:107)
k k v
v∈Mk
and
h ([x :···:x ])=logH ([x :···:x ])= ∑ logmax{(cid:107)x (cid:107) ,...,(cid:107)x (cid:107) }.
k 0 n k 0 n 0 v n v
v∈Mk
DiophantineApproximationandNevanlinnaTheory 119
Here
log+(x)=max{logx,0}.
The equation (1.5) tells us how heights change when the number field k is ex-
tendedtoalargernumberfieldL:
(2.5) h (x)=[L:k]h (x)
L k
and
(2.6) h ([x :···:x ])=[L:k]h ([x :···:x ])
L 0 n k 0 n
forallx∈kandall[x ,...,x ]∈Pn(k),respectively.
0 n
Then,givenx∈Q,wedefine
1
h (x)= h (x)
k L
[L:k]
for any number field L⊇k(x), and similarly given any [x :···:x ]∈Pn(Q), we
0 n
define
1
h ([x :···:x ])= h ([x :···:x ])
k 0 n L 0 n
[L:k]
for any number field L⊇k(x ,...,x ). These expressions are independent of the
0 n
choiceofLby(2.5)and(2.6),respectively.
FollowingEGA,ifxisapointonPn,thenκ(x)willdenotetheresiduefieldof
k
thelocalringat x.If x isaclosedpointthen thehomogeneouscoordinatescanbe
chosensuchthatk(x ,...,x )=κ(x).
0 n
Withthesedefinitions,(2.5)and(2.6)remainvalidwithouttheconditionsx∈k
and[x :···:x ]∈Pn(k),respectively.
0 n
Itiscommontoassumek=Qandomitthesubscriptk.Theresultingheightsare
calledabsoluteheights.
It is obvious from (2.1) that h (x)≥0 for all x∈k, and that equality holds if
k
x=0orifxisarootofunity.Conversely,h (x)=0implies(cid:107)x(cid:107) ≤1forallv;if
k v
x(cid:54)=0 then the Product Formula implies (cid:107)x(cid:107) =1 for all v. Thus x must be a unit,
v
andtheknownstructureoftheunitgroupthenleadstothefactthatxmustbearoot
ofunity.
Therefore,thereareinfinitelymanyelementsofQwithheight0.Ifonebounds
thedegreeofsuchelementsoverQ,thenthereareonlyfinitelymany;moregener-
ally,wehave:
Theorem2.7.(Northcott’s finiteness theorem) For any r ∈Z and any C ∈R,
>0
thereareonlyfinitelymanyx∈Qsuchthat[Q(x):Q]≤randh(x)≤C.Moreover,
givenanyn∈Nthereareonlyfinitelymanyx∈Pn(Q)suchthat[κ(x):Q]≤rand
h(x)≤C.
Thefirstassertionisprovedusingthefactthat,foranyx∈Q,ifoneletsk=Q(x),
thenH (x)iswithinaconstantfactorofthelargestabsolutevalueofthelargestcoef-
k
ficientoftheirreduciblepolynomialofxoverQ,whenthatpolynomialismultiplied
120 PaulVojta
by a rational number so that its coefficients are relatively prime integers. The sec-
ondassertionthenfollowsasaconsequenceofthefirst.Fordetails,see[Lang,1991,
Ch.II,Thm.2.2].
Thisresultplaysacentralroleinnumbertheory,since(forexample)provingan
upperboundontheheightsofrationalpointsisequivalenttoprovingfiniteness.
3 Roth’sTheorem
K.F.Roth[1955]provedakeyandmuch-anticipatedtheoremonhowwellanalge-
braicnumbercanbeapproximatedbyrationalnumbers.Ofcourserationalnumbers
aredenseinthereals,butifonelimitsthesizeofthedenominatorthenonecanask
meaningfulandnontrivialquestions.
Theorem3.1.(Roth)Fixα∈Q,ε>0,andC>0.Thenthereareonlyfinitelymany
a/b∈Q,whereaandbarerelativelyprimeintegers,suchthat
(cid:12)a (cid:12) C
(3.1.1) (cid:12) −α(cid:12)≤ .
(cid:12)b (cid:12) |b|2+ε
Example3.2.AsadiophantineapplicationofRoth’stheorem,considerthediophan-
tineequation
(3.2.1) x3−2y3=11, x,y∈Z.
√
If (x,y) is a solution, then x/y must be close to 32 (assuming |x| or |y| is large,
whichwouldimplybotharelarge):
(cid:12)(cid:12)(cid:12)x−√32(cid:12)(cid:12)(cid:12)=(cid:12)(cid:12)(cid:12) √11 √ (cid:12)(cid:12)(cid:12)(cid:28) 1 .
(cid:12)y (cid:12) (cid:12)y(x2+xy32+y2 34)(cid:12) |y|3
ThusRoth’stheoremimpliesthat(3.2.1)hasonlyfinitelymanysolutions.
Moregenerally,if f ∈Z[x,y]ishomogeneousofdegree≥3andhasnorepeated
factors, then for any a∈Z f(x,y)=a has only finitely many integral solutions.
ThisiscalledtheThueequationandhistoricallywasthedrivingforcebehindthe
development of Roth’s theorem (which is sometimes called the Thue-Siegel-Roth
theorem,sometimesalsomentioningSchneider,Dyson,andMahler).
The inequality (3.1.1) is best possible, in the sense that the 2 in the exponent
ontheright-handsidecannotbereplacedbyasmallernumber.Thiscanbeshown
usingcontinuedfractions.Ofcourseonecanconjectureasharpererrorterm[Lang
andCherry,1990,Intro.toCh.I].
Ifa/bisclosetoα,thenafteradjustingC onecanreplace|b|intheright-hand
sideof(3.1.1)withHQ(a/b)(see(2.2)).Moreover,thetheoremhasbeengeneral-
ized to allow a finite set of places (possibly non-archimedean) and to work over a
numberfield:
DiophantineApproximationandNevanlinnaTheory 121
Theorem3.3.Letkbeanumberfield,letSbeafinitesetofplacesofkcontaining
all archimedean places, fix α ∈Q for each v∈S, let ε >0, and letC>0. Then
v
onlyfinitelymanyx∈ksatisfytheinequality
C
(3.3.1) ∏min{1,(cid:107)x−α (cid:107) }≤ .
v v H (x)2+ε
v∈S k
(Strictlyspeaking,Scanbeanyfinitesetofplacesatthispoint,butrequiringSto
contain all archimedean places does not weaken the theorem, and this assumption
willbenecessaryinSection5.See,forexample,(5.3).)
Taking−logofbothsidesof(3.3.1),dividingby[k:Q],andrephrasingthelogic,
theabovetheoremisequivalenttotheassertionthatforallc∈Rtheinequality
(cid:13) (cid:13)
(3.4) 1 ∑log+(cid:13)(cid:13) 1 (cid:13)(cid:13) ≤(2+ε)h(x)+c
[k:Q]v∈S (cid:13)x−αv(cid:13)v
holdsforallbutfinitelymanyx∈k.
Inwriting(3.3.1),weassumethatonehaschosenanembeddingi : k¯(cid:44)→k over
v v
kforeachv∈S.Otherwisetheexpression(cid:107)x−α (cid:107) maynotmakesense.
v v
This is mostly a moot point, however, since we may restrict to α ∈ k for
v
all v. Clearly this restricted theorem is implied by the theorem without the ad-
ditional restriction, but in fact it also implies the original theorem. To see this,
suppose k, S, ε, and c are as above, and assume that α ∈ Q are given for all
v
v∈S. Let L be the Galois closure over k of k(α :v∈S), and let T be the set
v
of all places of L lying over places in S. We assume that L is a subfield of k¯,
(cid:12)
so that αv ∈ L for all v ∈ S. Then (iv)(cid:12)L: L → kv induces a place w0 of L over
v, and all other places w of L over v are conjugates by elements σ ∈Gal(L/k):
w
(cid:107)x(cid:107) =(cid:107)σ−1(x)(cid:107) forallx∈L.Lettingα =σ (α )forallw|v,wethenhave
w w w0 w w v
(cid:107)x−α (cid:107) =(cid:107)σ−1(x−α )(cid:107) =(cid:107)x−α (cid:107)[Lw0:kv]forallx∈kby(1.4),andtherefore
w w w w w0 v v
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
w∑|vlog+(cid:13)(cid:13)(cid:13)x−1αw(cid:13)(cid:13)(cid:13)w=w∑|v[Lw0 :kv]log+(cid:13)(cid:13)(cid:13)x−1αv(cid:13)(cid:13)(cid:13)v=[L:k]log+(cid:13)(cid:13)(cid:13)x−1αv(cid:13)(cid:13)(cid:13)v
sinceL/kisGalois.Thus
(cid:13) (cid:13) (cid:13) (cid:13)
1 ∑log+(cid:13)(cid:13) 1 (cid:13)(cid:13) = 1 ∑ log+(cid:13)(cid:13) 1 (cid:13)(cid:13)
[k:Q]v∈S (cid:13)x−αv(cid:13)v [L:Q]w∈T (cid:13)x−αw(cid:13)w
for all x∈k. Applying Roth’s theorem over the field L (where now α ∈L for all
w
w∈T)thengives(3.4)foralmostallx∈k.
Finally,wenotethatRoth’stheorem(asnowrephrased)isequivalenttothefol-
lowingstatement.
Theorem3.5.Let k be a number field, let S⊇S be a finite set of places of k, fix
∞
distinctα ,...,α ∈k,letε >0,andletc∈R.Thentheinequality
1 q
122 PaulVojta
(3.5.1) 1 ∑∑q log+(cid:13)(cid:13)(cid:13) 1 (cid:13)(cid:13)(cid:13) ≤(2+ε)h(x)+c
[k:Q]v∈Si=1 (cid:13)x−αi(cid:13)v
holdsforalmostallx∈k.
Indeed, given α ∈k for all v∈S, let α ,...,α be the distinct elements of the
v 1 q
set{α :v∈S}.Then
v
1 ∑log+(cid:13)(cid:13)(cid:13) 1 (cid:13)(cid:13)(cid:13) ≤ 1 ∑∑q log+(cid:13)(cid:13)(cid:13) 1 (cid:13)(cid:13)(cid:13) ,
[k:Q]v∈S (cid:13)x−αv(cid:13)v [k:Q]v∈Si=1 (cid:13)x−αi(cid:13)v
soTheorem3.5impliestheearlierformofRoth’stheorem(asmodified).
Conversely, given distinct α ,...,α ∈k, we note that any given x∈k can be
1 q
closetoonlyoneoftheα ateachplacev(wherethevalueofimaydependonv).
i
Therefore,foreachv,
∑q log+(cid:13)(cid:13)(cid:13) 1 (cid:13)(cid:13)(cid:13) ≤log+(cid:13)(cid:13)(cid:13) 1 (cid:13)(cid:13)(cid:13) +cv
i=1 (cid:13)x−αi(cid:13)v (cid:13)x−αv(cid:13)v
forsomeconstantc independentofxandsomeα ∈{α ,...,α }dependingonx
v v 1 q
andv.Thus,foreachx∈k,thereisachoiceofα foreachv∈Ssuchthat
v
1 ∑∑q log+(cid:13)(cid:13)(cid:13) 1 (cid:13)(cid:13)(cid:13) ≤ 1 ∑log+(cid:13)(cid:13)(cid:13) 1 (cid:13)(cid:13)(cid:13) +c(cid:48),
[k:Q]v∈Si=1 (cid:13)x−αi(cid:13)v [k:Q]v∈S (cid:13)x−αv(cid:13)v
wherec(cid:48)isindependentofx.Sincethereareonlyfinitelymanychoicesofthesystem
{α :v∈S}, finitely many applications of the earlier version of Roth’s theorem
v
sufficetoimplyTheorem3.5.
4 BasicsofNevanlinnaTheory
Nevanlinna theory, developed by R. and F. Nevanlinna in the 1920s, concerns the
distributionofvaluesofholomorphicandmeromorphicfunctions,inmuchthesame
waythatRoth’stheoremconcernsapproximationofelementsofanumberfield.
OnecanthinkofitasageneralizationofatheoremofPicard,whichsaysthata
nonconstantholomorphicfunctionfromCtoP1 canomitatmosttwopoints.This,
inturn,generalizesLiouville’stheorem.
An example relevant to Picard’s theorem is the exponential function ez, which
omits the values 0 and ∞. When r is large, the circle |z|=r is mapped to many
values close to ∞ (when Rez is large) and many values close to 0 (when Rez is
highlynegative).
So even though ez omits these two values, it spends a lot of time very close to
them. This observation can be made precise, in what is called Nevanlinna’s First
MainTheorem.Inordertostatethistheorem,weneedsomedefinitions.