Park City lectures on elliptic curves over function fields

0.65 MB·English

by  Douglas Ulmer

1
Contents
1
0
2

n Elliptic curves over function fields
a Douglas Ulmer 1
J

0 Elliptic curves over function fields 3
1
Introduction 3

]
T Lecture 0. Background on curves and function fields 5
N 1. Terminology 5
2. Function fields and curves 5
.
h
3. Zeta functions 6
t
a 4. Cohomology 7
m 5. Jacobians 7
[ 6. Tate’s theorem on homomorphisms of abelian varieties 9

1
Lecture 1. Elliptic curves over function fields 11
v
1. Elliptic curves 11
9
3 2. Frobenius 12
9 3. The Hasse invariant 13
1 4. Endomorphisms 14
.
1 5. The Mordell-Weil-Lang-N´erontheorem 14
0 6. The constant case 15
1
7. Torsion 16
1
8. Local invariants 18
:
v 9. The L-function 19
i
X 10. The basic BSD conjecture 20
11. The Tate-Shafarevich group 20
r
a 12. Statements of the main results 21
13. The rest of the course 22
Lecture 2. Surfaces and the Tate conjecture 25
1. Motivation 25
2. Surfaces 25
3. Divisors and the N´eron-Severigroup 26
4. The Picard scheme 27
5. Intersection numbers and numerical equivalence 27
6. Cycle classes and homologicalequivalence 28
7. Comparison of equivalence relations on divisors 29
8. Examples 29
9. Tate’s conjectures T and T 31
1 2
10. T and the Brauer group 32
1
i
ii
11. The descent property of T 34
1
12. Tate’s theorem on products 34
13. Products of curves and DPC 35
Lecture 3. Elliptic curves and elliptic surfaces 37
1. Curves and surfaces 37
2. The bundle ω and the height of 40
E
3. Examples 40
4. and the classification of surfaces 42
E
5. Points and divisors, Shioda-Tate 43
6. L-functions and Zeta-functions 44
7. The Tate-Shafarevich and Brauer groups 45
8. The main classical results 46
9. Domination by a product of curves 47
10. Four monomials 47
11. Berger’s construction 48
Lecture 4. Unbounded ranks in towers 51
1. Grothendieck’s analysis of L-functions 51
2. The case of an elliptic curve 54
3. Large analytic ranks in towers 55
4. Large algebraic ranks 58
Lecture 5. More applications of products of curves 61
1. More on Berger’s construction 61
2. A rank formula 62
3. First examples 63
4. Explicit points 64
5. Another example 65
6. Further developments 65
Bibliography 67
Elliptic curves over function fields
Douglas Ulmer

IAS/ParkCityMathematicsSeries
VolumeXX,XXXX
Elliptic curves over function fields
Douglas Ulmer
Introduction
These are the notes from a course of five lectures at the 2009 Park City Math
Institute. The focus is on elliptic curves over function fields over finite fields. In
the first three lectures, we explain the main classical results (mainly due to Tate)
on the Birchand Swinnerton-Dyer conjecture in this context and its connection to
the Tate conjecture about divisors on surfaces. This is preceded by a “Lecture 0”
on background material. In the remaining two lectures, we discuss more recent
developmentsonelliptic curvesoflargerankandconstructionsofexplicitpoints in
high rank situations.
A greatdealof this materialgeneralizes naturally to the context of curves and
Jacobiansofany genus overfunctionfieldsoverarbitrary ground fields. Thesegen-
eralizations were discussed in a course of 12 lectures at the CRM in Barcelona
in February, 2010, and will be written up as a companion to these notes, see
[Ulm11]. Unfortunately, theorems on unbounded ranks over function fields are
currently known only in the context of finite ground fields.
Finally, we mention here that very interesting theorems of Gross-Zagier type
existalsointhefunctionfieldcontext. Thesewouldbethesubjectofanotherseries
of lectures and we will not say anything more about them in these notes.
It is a pleasure to thank the organizersof the 2009 PCMI for the invitation to
speak, the students for their interest, enthusiasm, and stimulating questions, and
the “elder statesmen”—Bryan Birch, Dick Gross, John Tate, and Yuri Zarhin—
for their remarks and encouragement. Thanks also to Keith Conrad for bringing
the fascinating historical articles of Roquette [Roq06] to my attention. Last but
not least, thanks are due as well to Lisa Berger, Tommy Occhipinti, Karl Rubin,
Alice Silverberg, Yuri Zarhin, and an anonymous referee for their suggestions and
TEXnical advice.
SchoolofMathematics, GeorgiaInstitute ofTechnology, Atlanta,GA30332
E-mail address: [email protected]
(cid:13)c2011AmericanMathematicalSociety
3

LECTURE 0
Background on curves and function fields
This “Lecture 0” covers definitions and notations that are probably familiar
to many readers and that were reviewed very quickly during the PCMI lectures.
Readers are invited to skip it and refer back as necessary.
1. Terminology
Throughout,weusethelanguageofschemes. Thisisnecessarytobeonfirmground
when dealing with some of the more subtle aspects involving non-perfect ground
fields and possibly non-reduced group schemes. However, the instances where we
use any hard results from this theory are isolated and students should be able to
followreadilythemainlinesofdiscussion,perhapswiththe assistanceofafriendly
algebraic geometer.
Throughout, a variety over a field F is a separated, reduced scheme of finite
type over SpecF. A curve is a variety purely of dimension 1 and a surface is a
variety purely of dimension 2.
2. Function fields and curves
Throughout,p will be aprime number andF will denote the fieldwith q elements
q
withq apowerofp. Wewrite forasmooth,projective,andabsolutelyirreducible
C
curve of genus g over F and we write K = F ( ) for the function field of over
q q
C C
F . Themostimportantexampleis when =P1,the projectiveline,inwhichcase
q
C
K =F ( )=F (t) is the field of rational functions in a variable t over F .
q q q
C
We write v for a closed point of , or equivalently for an equivalence class of
C
valuations of K. For each such v we write for the local ring at v (the ring
(v)
of rational functions on regular at v), m O for the maximal ideal (those
v (v)
functionsvanishingatv),Candκ = /m fo⊂rthOeresiduefieldatv. Theextension
v (v) v
O
κ /F is finite and we set deg(v)=[κ :F ] and q =qdeg(v) so that κ =F .
v q v q v v ∼ qv
For example, in the case where = P1, the “finite” places of correspond
C C
bijectively to monic irreducible polynomials f F [t]. If v corresponds to f, then
q
∈
isthesetofratiosg/hwhereg,h F [t]andf doesnotdivideh. Themaximal
(v) q
Oideal m consists of ratios g/h where∈f does divide g, and the degree of v is the
v
degreeoff as apolynomialint. Thereis onemore place ofK,the “infinite” place
v = . The localring consists of ratios g/h with g,h F [t] and deg(g) deg(h).
q
∞ ∈ ≤
The maximal ideal consists of ratios g/h where deg(g)<deg(h) and the degree of
v = is 1. The finite and infinite places of P1 give all closed points of P1.
∞
We write Ksep for a separable closure of K and let G = Gal(Ksep/K). We
K
write F for the algebraicclosure of F in Ksep. For eachplace v of K we have the
q q
decomposition group D (defined only up to conjugacy), its normal subgroup the
v
inertiagroupI D ,andFr the(geometric)Frobeniusatv,acanonicalgenerator
v v v
⊂
5
6 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS
of the quotient Dv/Iv ∼= Gal(Fq/Fq) that acts as x 7→ xqv−1 on the residue field at
a place w dividing v in a finite extension F Ksep unramified over v.
⊂
Generalreferencesforthissectionandthenextare[Gol03],[Ros02],and[Sti09].
3. Zeta functions
Let be a variety over the finite field F . Extending the notation of the previous
q
X
section, if x is a closed point of , we write κ for the residue field at x, q for its
x x
X
cardinality, and deg(x) for [κ :F ].
x q
We define the Z and ζ functions of via Euler products:
X
−1
Z( ,T)= 1 Tdeg(x)
X −
Yx (cid:16) (cid:17)
and
ζ( ,s)=Z( ,q−s)= 1 q−s −1
X X − x
x
Y(cid:0) (cid:1)
where the products are over the closed points of . It is a standard exercise to
X
show that
Tn
Z( ,T)=exp N
n
X  n 
n≥1
X
whereN isthenumberofF -valuedpointsof . Itfollowsfromacrudeestimate
n qn
X
forthenumberofF pointsof thattheEulerproductdefiningζ( ,s)converges
qn
X X
in the half plane Re(s)>dim .
X
If is smooth and projective, then it is known that Z( ,T) is a rational
X X
function of the form
dimX−1P (T)
i=0 2i+1
dimX P (T)
Q i=0 2i
where P (T) = (1 T), P (T) = (1 qdimXT), and for all 0 i 2dim
0 − 2dimX Q − ≤ ≤ X
P (T)is a polynomialwith integercoefficientsandconstantterm1. We denote the
i
inverse roots of P by α so that
i ij
P (T)= (1 α T)
i ij
−
j
Y
The inverse roots α of P (T) are algebraic integers that have absolute value
ij i
qi/2 in every complex embedding. (We say that they are Weil numbers of size
qi/2.) It follows that ζ( ,s) has a meromorphic continuation to the whole s
X
plane, with poles on the lines Res 0,...,dim and zeroes on the lines Res
∈{ X} ∈
1/2,...,dim 1/2 . ThisistheanalogueoftheRiemannhypothesisforζ( ,s).
{ X− } X
It is also known that the set of inverse roots of P (T) (with multiplicities) is
i
stable under α q/α . Thus ζ( ,s) satisfies a functional equation when s is
ij ij
7→ X
replaced by dim s.
X −
In the case where is a curve, P (T) has degree 2g (g = the genus of ) and
1
X C
has the form
2g
P (T)=1+ +qgT2g = (1 α T).
1 1j
··· −
j=1
Y
Thus ζ( ,s) has simple poles for s 2πi Z and s 1+ 2πi Z and its zeroes lie on
C ∈ logq ∈ logq
the line Res=1/2.
LECTURE0. BACKGROUND ON CURVES AND FUNCTION FIELDS 7
For a fascinating history of the early work on zeta functions and the Riemann
hypothesis for curvesover finite fields, see [Roq06] and parts I and II of that work.
4. Cohomology
Assumethat isasmoothprojectivevarietyoverk =F . Wewrite for F .
X q X X×Fq q
Note that G =Gal(F /F ) acts on via the factor F .
k q q q
X
Choose a prime ℓ = p. We have ℓ-adic cohomology groups Hi( ,Q ) which
ℓ
6 X
are finite-dimensional Q -vector spaces and which vanish unless 0 i 2dim .
ℓ
≤ ≤ X
Functorialityin givesacontinuousactionofGal(F /F ). Sincethegeometric
q q
Frobenius (Fr (a)=Xaq−1) is a topological generator of Gal(F /F ), the character-
q q q
istic polynomial of Fr on Hi( ,Q ) determines the eigenvalues of the action of
q ℓ
X
Gal(F /F ); in fancierlanguage,itdetermines the actionup to semi-simplification.
q q
An important result (inspired by [Wei49] and proven in great generality in
[SGA5])saysthatthe factorsP ofZ( ,t)arecharacteristicpolynomialsofFrobe-
i
X
nius:
(4.1) P (T)=det(1 T Fr Hi( ,Q )).
i q ℓ
− | X
From this point of view, the functional equation and Riemann hypothesis for
Z( ,T) are statements about duality and purity.
X
To discuss the connections, we need more notation. Let Z (1) = lim µ (F )
ℓ ℓn q
n
and Q (1) = Z (1) Q , so that Q (1) is a one-dimensional Q -vec←to−r space on
ℓ ℓ ⊗Zℓ ℓ ℓ ℓ
which Gal(F /F ) acts via the ℓ-adic cyclotomic character. More generally, for
q q
n > 0 set Q (n) = Q (1)⊗n (n-th tensor power) and Q ( n) = Hom(Q (n),Q ),
ℓ ℓ ℓ ℓ ℓ
−
so that for all n, Q (n) is a one-dimensional Q -vector space on which Gal(F /F )
ℓ ℓ q q
acts via the nth power of the ℓ-adic cyclotomic character.
We have H0( ,Q ) = Q (with trivial Galois action) and H2dimX( ,Q ) =
X ℓ ∼ ℓ X ℓ ∼
Q (dim ). The functionalequationfollowsfromthe factthatwehavea canonical
ℓ
X
non-degenerate,Galois equivariant pairing
Hi( ,Q ) H2dimX−i( ,Q ) H2dimX( ,Q )=Q (dim ).
X ℓ × X ℓ → X ℓ ∼ ℓ X
Indeed, the non-degeneracy of this pairing implies that if α is an eigenvalue of Fr
q
on Hi( ,Q ), then qdimX/α is an eigenvalue of Fr on H2dimX−i( ,Q ).
ℓ q ℓ
X X
The Riemann hypothesis in this context is the statement that the eigenvalues
ofFr onHi( ,Q )arealgebraicintegerswithabsolutevalueqi/2 ineverycomplex
q ℓ
X
embedding.
See[SGA41]or[Mil80]foranoverviewof´etalecohomologyanditsconnections
2
with the Weil conjectures.
5. Jacobians
5.1. Picard and Albanese properties
We briefly review two (dual) universal properties of the Jacobian of a curve that
we will need. See [Mil86b] for more details.
We assume throughout that the curve has an F -rational point x, i.e., a
q
C
closed point with residue field F . If T is another connected variety over F with
q q
an F -rational point t, a divisorial correspondence between (C,x) and (T,t) is an
q
invertiblesheaf onC T suchthat and aretrivial. Twodivisorial
L ×Fq L|C×t L|x×T
correspondences are equal when they are isomorphic as invertible sheaves. Note
8 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS
that the set of divisorial correspondences between ( ,x) and (T,t) forms a group
C
under tensor product and is thus a subgroup of Pic( T). We write
C×
DivCorr(( ,x),(T,t)) Pic( T)
C ⊂ C×
for this subgroup. One may think of a divisorialcorrespondence as giving a family
of invertible sheaves on C: s .
C×s
7→L|
Let J = J be the Jacobian of and write 0 for its identity element. Then
C
C
J is a g-dimensional abelian variety over F and it carries the “universal divisorial
q
correspondence with C.” More precisely, there is a divisorial correspondence
M
between (C,x) and (J,0) such that if S is another connected variety over F with
q
F -rational point s and is a divisorial correspondence between (C,x) and (S,s),
q
L
then there is a unique morphism φ : S J sending s to 0 such that = φ∗ .
→ L M
(Of course depends on the choice of base point x, but we omit this from the
M
notation.)
It follows that there is a canonicalmorphism, the Abel-Jacobimorphism, AJ :
J sending x to 0. Intuitively, this corresponds to the family of invertible
C →
sheaves parameterized by that sends y to (y x). More precisely, let
C
C ∈ C O −
∆ C be the diagonal, let
⊂C×
D =∆ x x,
− ×C−C×
andlet = (D)whichisadivisorialcorrespondencebetween(C,x)anditself.
C×C
L O
The universal property above then yields the morphism AJ : J. It is known
C →
thatAJ isaclosedimmersionandthatitsimagegeneratesJ asanalgebraicgroup.
The second universalproperty enjoyed by J (or rather by AJ) is the Albanese
property: it is universal for maps to abelian varieties. More precisely, if A is an
abelianvariety andφ: A is a morphismsending x to 0, then there is a unique
C →
homomorphism of abelian varieties ψ :J A such that φ=ψ AJ.
→ ◦
Combiningthetwouniversalpropertiesgivesausefulconnectionbetweencorre-
spondencesandhomomorphisms: Suppose and arecurvesoverF withrational
q
C D
points x and y . Then we have an isomorphism
∈C ∈D
(5.1.1) DivCorr((C,x),(D,y))∼=Hom(JC,JD).
Intuitively,givenadivisorialcorrespondenceon , wegetafamilyofinvertible
C×D
sheaves on parameterized by and thus a morphism J . The Albanese
D
D C C →
property then gives a homomorphismJ J . We leave the precise versionas an
C D
→
exercise,or see [Mil86b, 6.3]. We willuse this isomorphismlater to understandthe
N´eron-Severigroup of a product of curves.
5.2. The Tate module
Let A be an abelian variety of dimension g over F , for example the Jacobian of
q
a curve of genus g. (See [Mil86a] for a brief introduction to abelian varieties and
[Mum08]foramuchmorecompletetreatment.) Chooseaprimeℓ=p. LetA[ℓn]be
6
thesetofF pointsofAoforderdividingℓn. Itisagroupisomorphicto(Z/ℓnZ)2g
q
with a linear action of Gal(F /F ). We form the inverse limit
q q
T A=limA[ℓn]
ℓ
←n−
wherethe transitionmapsaregivenbymultiplicationby ℓ. LetV A=T A Q ,
ℓ ℓ ⊗Zℓ ℓ
a 2g-dimensional Q -vector space with a linear action of Gal(F /F ). It is often
ℓ q q
called the Tate module of A.