arithmetic geometry for physicists€¦ · arithmetic geometry for physicists minhyongkim...

Arithmetic Geometry for Physicists

Minhyong Kim

Seoul, August, 2017

I. Splitting primes

Splitting primesGiven a polynomial f ∈ Z[x ], the set of splitting primes

Sp(f )

of f consists of the primes that

–do not divide the discriminant ∆(f ) of f ;–divide some value f (n), n = 1, 2, 3, . . ..

Recall that

∆(ax2 + bx + c) = b2 − 4ac ; ∆(x3 + bx + c) = −4b3 − 27c2,

etc.

In general ∆(f ) = an∏

i<j(αi − αj)2, where an is the leading

coefficient and αi are the roots of f .

For p - ∆(f ), p ∈ Sp(f ) if and only if f (x) = 0 has a solution inFp = Z/p.

Splitting primes

Example A:

g(x) = x2 − 5.∆(g) = 20.

n = 1 2 3 4 5 6 7 8 9 10g(n) = −4 −1 4 11 20 31 44 59 76 95

The primes that occur are

Sp(g) = 11, 31, 59, 19, . . .

These are p ≡ ±1 mod 5.

Splitting primes

Example B:

h(x) = x3 + x2 − 2x − 1.

∆(h) = 49.We compute

n = 1 2 3 4 5h(n) = −1 7 29 71 139

n = 6 7 8 9h(n) = 239 13 · 43 13 · 43 7 · 113

So we see that

Sp(h) = 13, 29, 43, 71, 113, 239, . . .

These are p ≡ ±1 mod 7.

Splitting primes

Example C:

f (x) = 13249 + 140x + 1588x2 − 2x3 + 67x4 + x6.

Values f (n), where n = 1, 2, 3, . . . , 10:

15043 = 72 · 307, 21001, 34063 = 23× 1481,

60337, 110899, 204313 = 173 · 1181,

369871 = 59 · 6269, 651553 = 72 · 13297, 1112707,

1841449 = 232 · 592, . . .

Primes dividing the discriminant: 2,3,7,23, 191.

What is Sp(f )?

Splitting primes

TheoremThe splitting primes of f are exactly p such that the coefficient ofqp is 2 in the series

F (q) = (1/2)∑

n,m∈Z[qm

2+mn+6n2 − q2m2+mn+3n2].

Fact:

F (q) is a modular cusp form of weight 1 for the group Γ0(23). withcharacter

( ·23

).

Splitting primesRecall that an algebraic number field K is a finite field extension ofQ.

This means thatK ' Qd

with a multiplication extending that of Q.

For example,

Q[i ] ' Q2, (a, b)(c, d) = (ac − bd , ad + bc);

Q[√−23] ' Q2, (a, b)(c , d) = (ac − 23bd , ad + bc).

Q[ζ5] ' Q4;

where ζ5 = exp(2πi/5).

The dimension d is also called the degree of the field K anddenoted [K : Q].

We say K is Galois if |Aut(K )| = d .

Splitting primesAn algebraic number field K admits a subring OK referred to as itsring of algebraic integers. It is the maximal subring of K isomorphicto Zd .

For example,OQ(i) = Z[i ];

OQ[ζ5] = Z[ζ5];

OQ[√−23] = Z[(1 +

√−23)/2].

Among other things, an algebraic number field provides us withnew primes. That is, given a usual prime p, we can write

(p) = P1P2 · · ·Ps

for prime ideals Pi in OK .

For example, in Z[√−23]

(3) = (3,√−23 + 1)(3,

√−23− 1); (23) = (

√−23)2; (5) = (5)

Splitting primes

We say p is unramified in K if the Pi are all distinct. Otherwise pis ramified in K .

For example, 3 is unramified and 23 is ramified in Q[√−23].

Each k(Pi ) := OK/Pi is a finite field extension of Fp = Z/p, andwe have the equality∑

i

[k(Pi ) : Fp] = d = [K : Q].

This is (essentially) because

Fdp ' Zd/pZd ' OK/pOK = OK/

∏i

Pi '∏i

OK/Pi .

In particular, there are at most d such Pi .

Splitting primes

Say p splits (or splits completely) in K if p is unramified and

(p) = P1P2 · · ·Pd

for d = dimQK . (⇔ [k(Pi ) : Fp] = 1 for all i .)

Denote by Sp(K ) the set of primes that split in K .

Examples:

In Q[√−23], 3 splits, but 5 does not.

In Q[ζn], p - n splits if and only if p ≡ 1 mod n.

Splitting primes

Any algebraic number field can be written K = Q[α] for a root α ofan irreducible monic polynomial f (x) ∈ Z[x ].

In this form,

– [K : Q] = deg(f );

– If p - ∆(f )⇒ p is unramified;

– When K is Galois, an unramified p is split if and only if

f (x) ≡ 0 mod p

has a root, i.e. if and only if p|f (n) for some n. That is,

Sp(K ) ≈ Sp(f ),

where the ≈ means that Sp(K ) \ Sp(f ) and Sp(f ) \ Sp(K ) areboth finite.

Splitting primes

From here, assume K is Galois.

For every prime p ∈ Z, considering

(p) = P1P2 · · ·Ps

get a collection of subgroups

D(Pi ) := g ∈ Gal(K/Q) | gPi = Pi ⊂ Gal(K/Q),

the decomposition groups of the Pi .

These are all conjugate inside Gal(K/Q): that is, for all i , j , thereis an aij ∈ Gal(K/Q) such that

D(Pi ) = aijD(Pi )a−1ij .

Splitting primes

Thus, given a Galois field K , each prime p determines a conjugacyclass of subgroups

D(P)P|pin Gal(K/Q).

The action of D(Pi ) on k(Pi ) = OK/Pi induces a homomorphism

D(Pi ) - Gal(k(Pi )/Fp) = 〈σPi〉.

where σPiis the p-power map x 7→ xp.

When p is unramified, that is, for most p, this map is anisomorphism:

D(Pi ) ' Gal(k(Pi )/Fp).

Splitting primes

Thus, we get an element

FrKPi∈ D(Pi ) ⊂ Gal(K/Q),

corresponding to σ−1Pi

characterised by the condition

FrKPi(a) = a1/p mod Pi

for all a ∈ OK .

Can put this together for all Pi to get a conjugacy class of elements

FrKp = FrKPi ⊂ Gal(K/Q),

the Frobenius class of p.

Note that when Gal(K/Q) is abelian, e.g., K = Q(ζn), this is asingle element.

Splitting Primes

Q ⊂ K ⊃ OK ⊃ Z

(p) = P1P2 · · ·Ps

k(Pi ) = OK/Pi ⊃ Fp

D(Pi ) = g ∈ Gal(K/Q) : gPi = Pi

D(Pi ) ⊂ Gal(K/Q)

← (if unramified)

FrKPi∈Gal(k(P))

'

?

FrKPii ⊂ Gal(K/Q)

Splitting primesGalois theory and splitting primes:

I. p is split in K if and only if FrKp = 1.

This is because p split is equivalent to [k(Pi ) : Fp] = 1 for all i ,which is exactly when all the extensions k(Pi ) of Fp are trivial.

II. Chebotarev Density Theorem:

Let C ⊂ Gal(K/Q) be a conjugacy class. Then there are infinitelymany primes p such that

FrKp = C .

In fact, the density of this set of primes is

|C |/|G |.

In particular, the density of Sp(K ) is exactly 1/[K : Q].

Splitting primes

Remark on classification of algebraic number fields:

Let K and K ′ be two finite Galois extensions of Q. Then

K ' K ′

if and only ifSp(K ) ≈ Sp(K ′).

So, in some sense, the classification of K is the same as theclassification of the possible Sp(K ).

Splitting primesSome infinite Galois groups:

Gal(Q/Q) = lim←−F

Gal(F/Q),

where F runs over Galois number fields.

For a number field F ,

Gal(F/F ) = Gal(Q/F ) = g ∈ Gal(Q/Q) | g |F = Id

Let S be a finite set of primes. Then

Gal(QS/Q) = lim←−F

Gal(F/Q),

where F runs over the Galois extensions of Q that are unramifiedoutside S .

Similarly,

Gal(FS/F )

where S is a finite set of primes in OF .

Splitting primes

Galois groups like

Gal(QS/Q) = lim←−F

Gal(F/Q),

are very important in practice, because these are the ones that arisein nature, and which one can tools to understand.

For a prime p /∈ S , can define conjugacy class Frp ⊂ Gal(QS/Q).

Splitting primes

Return to the modular form

F (q) = (1/2)∑

n,m∈Z[qm

2+mn+6n2 − q2m2+mn+3n2]

=∞∑n=1

anqn

of weight 1, level 23 and character( ·23

).

This means that for

γ =

(a bc d

)∈ Γ0(23),

we have

F (γτ) =

(d

23

)(cτ + d)F (τ).

Splitting primes

It has an associated L-series

L(F , s) =∞∑n=1

ann−s

=∏p

1(1− app−s +

( p23

)p−2s)

.

This product decomposition is a manifestation of the fact that F isan eigenform for a Hecke algebra.

Splitting primesDeligne and Serre associated to F a continuous representation

ρF : Gal(Q/Q) - GL2(C)

with the following properties:

–G = Im(ρF ) ⊂ GL2(C) is finite, so that

K = [Q]Ker(ρF ),

is an algebraic number field such that Gal(K/Q) ' G ;

–The primes that ramify in OK are exactly the primes dividing thelevel of F , i.e., p = 23;

–If p 6= 23, then the conjugacy class Frp in G ⊂ GL2(C), satisfies

Tr(Frp) = ap

and det(Frp) =( p23

).

This property is essentially equivalent to an equality

L(F , s) = L(ρF , s).

Splitting primes

The L-function of the Galois representation is defined as

L(ρF , s) =∏p

1det([I − p−sFrp]|(C2)I (p))

.

Here, strictly speaking, I should pick a prime P|p in K . Then thereis an exact sequence

0 - I (P) - D(P) - Gal(k(P)/Fp) - 0,

defining the subgroup I (P), called the inertia subgroup.

The factors corresponding to p on the right are actually

1det([I − p−sFrP]|(C2)I (P))

.

Splitting primesThe representation and field K has been constructed in such a waythat ap = 2 if and only if Tr(Frp) = 2.

However, the eigenvalues of Frp are roots of 1. Hence

Tr(Frp) = 2

if and only if both eigenvalues are 1, i.e. Frp = Id .

Hencep | p - 23, ap = 2 = Sp(K ).

But also, K = Q(α) for α a root of

f (x) = 13249 + 140x + 1588x2 − 2x3 + 67x4 + x6.

Thus, for p - ∆(f ), p ∈ Sp(f ) ⊂ Sp(K ) if and only if ap = 2.

Splitting primes

To sum:

Cuspidal Hecke eigenform F

- K = (Q)Ker(ρF ) = Q(α)

where α is a root of f (x).

Field was constructed so that we know p such that Frp = 1 inGal(K/Q) = Im(ρF ).

Splitting primes

Langlands (refined by many other people):

Given any continuous irreducible representation

ρ : Gal(Q/Q) - GLn(C),

we should haveL(ρ, s) = L(F , s)

for some ‘automorphic form’ F .

Actually, F must be replaced in general by an automorphicrepresentation π of GLn(AQ) and there should be a way toassociate ρπ to π.

Similar statement for irreducible continuous motivic representation

ρ : Gal(Q/Q) - GLn(Qp)

Splitting primes

Even more generally, given a motivic continuous representation

ρ : Gal(Q/Q) - G∨(Qp),

we should haveρ ' ρπ

for an automorphic representation π of

G (AQ),

where G and G∨ are Langlands dual pairs of reductive groups.

General idea is that homomorphisms like ρ should parametriseautomorphic representations of G (AQ).

Splitting primes

Remark 1:

Galois representations are hard to construct: They are essentiallyalways constructed as

H iet(X ,Qp)

where X is a variety defined over Q. This admits an action ofGal(Q/Q) that factors through some

ΓS = Gal(QS/Q)

for a finite set S of primes. Here, QS is the field obtained as acompositum of all finite algebraic extensions F that are unramifiedoutside the finite set S of primes.

Splitting primes

Remark 2:

Galois representations are arithmetic gauge fields.

In geometry there is a correspondence

Bundle with flat connection on X

←→ Locally constant sheaf on X

←→ Representation of π1(X )

Splitting primes

WhenX = Spec(Z[1/N]),

for N =∏

p∈S p, then

π1(X ) = Gal(QS/Q).

Thus a representation of Gal(QS/Q) is interpreted as an arithmeticflat connection on Spec(Z[1/N)).

Splitting PrimesRemark 3:

Galois representations are motives.

Given K Galois with Galois group G ,

H0(Spec(K ),C) ' C[G ]

'∏ρ

dim(ρ)ρ,

whereρ : Gal(Q/Q) - GLn(C)

run over irreducible representations that occur in H0(Spec(K ),C).

That is, the ρ are constituent motives of Spec(K ), manifested as

ζK (s) =∏ρ

L(ρ, s)dim(ρ).

Splitting primes

Remark 4:

The group ΓS is essentially a topologically finitely-generated group.Its representations lie in moduli spaces

M(ΓS ,G )

of representationsρ : ΓS

- G

where G is a p-adic Lie group.The structure of this moduli space is very similar to a space

M(M,G )

of G -connections on a three manifold M studied by physicists.

Splitting primes

Remark 5:

In physics:

strings

gauge fields

automorphic forms

-

Interlude: What is arithmetic geometry?


Arithmetic geometry is the study of arithmetic geometries.

These are schemes of finite type, i.e., schemes that are finite unionsof those of the form

Spec(Z[x1, x2, . . . , xn]/(f1, f2, . . . , fm)),

and maps between them, for example,

Spec(Z[x1, x2, . . . , xn]/(f1, f2, . . . , fm)) - Spec(Z)

and

Spec(Z) - Spec(Z[x1, x2, . . . , xn]/(f1, f2, . . . , fm)).

Interlude: What is arithmetic geometry?Recall that scheme theory associates to a commutative ring A withunity, a (ringed) space

Spec(A),

whose underlying set consists of the prime ideals in A.

For example,Spec(C[x ]) = (0) ∪ C.

More generally,

Spec(C[x1, x2, . . . , xn]/(f1, f2, . . . , fm))

is the variety given as the zero set of the fi , enriched by adding apoint for each subvariety.

Spec(Z) = (0) ∪ 2, 3, 5, 7, 11, . . . , 37, . . . , 691, . . . , 1112707, . . .

Spec(Q[x ]) = (0) ∪ irreducible polynomials/Q×


The basic open subsets in Y = Spec(A) are of the form

Ua := Spec(A[1/a]) ⊂ Y .

There is a sheaf of rings OY on Y called the structure sheafdetermined by the values

OY (Ua) = A[1/a].

A general scheme is a topological space X equipped with a sheaf ofrings OX such that the pair (X ,OX ) is locally isomorphic to(Spec(A),OSpec(A)).


The basic maps of schemes are of the form

Spec(B) - Spec(A)

determined by a ring homomorphism A - B . At the level ofsets, one takes the inverse image of prime ideals.

General maps between schemes are locally of this form.

Suppose x is a point in X . Then x ∈ Spec(A) ⊂ X for some A, sothat x corresponds to a prime ideal P in A.

Thus, we get the map

Spec(A/P) - Spec(A) ⊂ X ,

which encodes the information of x .


The fieldk(x) := Frac(A/P)

is called the residue field at x , and we have the map

Spec(k(x)) - Spec(A/P) - Spec(A) - X .

This map is often identified with the point x .

Example:

X = Spec(Z[t]).

x = (t, p)

Spec(Fp) ⊂ - X .


X = Spec(Z[t]).

x = (t)

Spec(Q) ⊂ - Spec(Z) ⊂ - X .

x = (p)

Spec(Fp(t)) ⊂ - Spec(Fp[t]) ⊂ - X .


X = Spec(Z[t]).

x = (f (t)), where f ∈ Z[t] is an irreducible polynomial.

Spec(F ) ⊂ - Spec(Z[t]/(f (t))) ⊂ - X ,

where F = Q[t]/(f (t)) is an algebraic number field.

x = (p, f (t)), where f (t) is degree d and irreducible mod p.

Spec(Fpd ) = Spec(Fp[t]/(f (t))) ⊂ - X .


X = Spec(Z[t]).

x = (0)

Spec(Q(t)) ⊂ - Spec(Z[t]) = X .

Algebra or Geometry?

Figure: Mumford’s picture of Spec(Z[x ])

Interlude: What is arithmetic geometry?Another point that arises from x is

Spec(k(x)) - Spec(k(x)) - X ,

where k(x) is the separable closure of k(x). This map is sometimescalled the geometric point with image x .

For example, Z - Z/p = Fp⊂ - Fp gives rise to the geometric

pointSpec(Fp) - Spec(Fp) ⊂ - Spec(Z).

Also,Spec(Q) - Spec(Q) ⊂ - Spec(Z).

Spec(Q(t)) - Spec(Q(t)) ⊂ - Spec(Z[t]).

Interlude: What is arithmetic geometry?The arrow

Spec(Z[x1, x2, . . . , xn]/(f1, f2, . . . , fm)) - Spec(Z)

is opposite to the inclusion

Z ⊂ - Z[x1, x2, . . . , xn]/(f1, f2, . . . , fm)

but the geometric perspective encourages us to view the scheme asa fibre-bundle over Spec(Z).

Reducing modulo a prime p gives us an inclusion

Fp⊂ - Fp[x1, x2, . . . , xn]/(f1, f2, . . . , fm),

which in geometric form

Spec(Fp[x1, x2, . . . , xn]/(f1, f2, . . . , fm)) - Spec(Fp)

is the fibre of the previous map over the point p.


Other important basic morphisms include:Inclusion of a point:

Spec(Fp) ⊂ - Spec(Z);

Inclusion of a point through a tubular neighbourhood:

Spec(Fp) ⊂ - Spec(Zp) - Spec(Z);

An open embedding:

Spec(Z[1/N)) ⊂ - Spec(Z);

Refinements of open embeddings:

Spec(Z) ⊃ Spec(Z[1/2]) ⊃ Spec(Z[1/6]) ⊃ Spec(Z[1/30]) · · · .


Note that if N =∏k

i=1 peii , then

Spec(Z[1/N]) = Spec(Z[1/p1, 1/p2, . . . , 1/pk ])

= Spec(Z) \ p1, p2, . . . , pk.

Sometimes consider limits like

Spec(Q) = ∩NSpec(Z[1/N]).

A variety over a finite field looks like

Spec(Fp[x1, x2, . . . , xn]/(f1, f2, . . . , fm)) - Spec(Fp) ⊂ - Spec(Z),

which is fibred over Spec(Z) with only one fibre.


The arrow

Spec(Z) - Spec(Z[x1, x2, . . . , xn]/(f1, f2, . . . , fm)).

is opposite to a ring homomorphism

Z[x1, x2, . . . , xn]/(f1, f2, . . . , fm)) - Z;

Hence, it corresponds to an integral solution of the system ofequations

f1 = 0, f2 = 0, . . . , fm = 0.

Similarly, an arrow

Spec(A) - Spec(Z[x1, x2, . . . , xn]/(f1, f2, . . . , fm))

corresponds to solutions with entries in A (e.g. Zp, Fp, Q, R, ...).


Equationy2 = x3 − 2

has solution(234092288158675600

,113259286337279449455096000

)


Picture 1:y2 = x3 − 2

×P

View of classical algebraic geometry, say à là Weil.


Picture 2:X

B

P

]]

B = Spec(Q)

X = Spec(Q[x , y ]/(y2 − x3 + 2))

This view emphasised by A. Grothendieck.


Picture 3:XN

BN

P

]]

BN = Spec(Z[1/N])

XN = Spec(Z[1/N][x , y ]/(y2 − x3 + 2))

where, N = 3830.


We have449455096000 = 26 · 53 · 3833

and58675600 = 24 · 53 · 3832

So solution(234092288158675600

,113259286337279449455096000

)

has coordinates in ring

Z[1

2 · 5 · 383] = Z[1/3830].

Should think of it as having ‘poles’ at (2), (5), (383).

Interlude: What is arithmetic geometry?Can also consider the homogeneous equation

zy2 − x3 + 2z3 = 0

or the scheme

X = Proj(Z[x , y , z ]/(zy2 − x3 + 2z3).

This is a union of three pieces

Spec(Z[x , y ]/(y2 − x3 + 2);

Spec(Z[x , z ]/(z − x3 + 2z3);

Spec(Z[y , z ]/(zy2 − 1 + 2z3).

Then the point

(89426414521128940 : 2456480661369244231 :

65930118577144000)

is a mapSpec(Z) - X.


Picture 4:

Spec(Z)

X7 X13 · · ·

· · ·

X · · ·

2 3 5 7 11 13 379 383 389

P


Topology:

Basic open sets in the Zariski topology on Spec(A) are of the formSpec(A[1/a]). Grothendieck topologies regard suitable maps

Spec(B) - Spec(A)

as open sets.

Intersections of open sets are replaced by fibre products

Spec(B)×Spec(A) Spec(B) -- Spec(B) - Spec(A),

whereSpec(B)×Spec(A) Spec(B) = Spec(B ⊗A B).


The most important one (so far) is the étale topology.

A mapSpec(B) - Spec(A)

is called étale essentially if B is locally of the form

A[x ]/(f (x))

where f ∈ A[x ] is a monic polynomial such that ∆(f ) ∈ A×.

The key point is that all fibers

Spec(B ⊗A (A/m)) = Spec((A/m)[x ]/(f )) - Spec(A/m)

have the same number of geometric points.


For the map

Spec(C[x ][y ]/(yn − x)) - Spec(C[x ]),

we have∆ = (−1)n−1nnxn−1.

This is *not* a unit in C[x ], so the map is not étale.

However,

Spec(C[x , x−1][y ]/(yn − x)) - Spec(C[x , x−1])

is étale.


Similarly,

Spec(Fp[x , x−1][y ]/(yn − x)) - Spec(Fp[x , x−1])

is étale if p - n.

The mapSpec(Z[x ]/(x2 + 23)) - Spec(Z)

is not étale,

ButSpec(Z[1/46][x ]/(x2 + 23)) - Spec(Z[1/46])

is étale.


Together with the notion of an étale open map

Y - X ,

have the notion of an étale open covering

(Ui- X )i∈I .

Can use this to form a complex C ((Ui ),Z/n):∏i

C (Ui ,Z/n) -∏i<j

C (Ui ×X Uj ,Z/n)

-∏

i<j<k

C (Ui ×X Uj ×X Uk ,Z/n) - · · ·


Étale cohomology with coefficients in Z/n is then defined as

H i (X ,Z/n) := H i (lim−→(Ui )

C ∗((Ui ),Z/n)).

[This is not always right, but good enough for many purposes.]

Also,H i (X ,Zp) := lim←−H i (X ,Z/pn),

andH i (X ,Qp) ' H i (X ,Zp)⊗Qp.


There is a collection of techniques for computing étale cohomology,for example,

(1) Comparison with complex cohomology;(2) Comparison with (pro-finite) group cohomology;(3) Long exact sequences depending on ‘breaking up’ X .(4) Spectral sequences depending on maps between spaces;(5) Generalisation to sheaves and ‘breaking up sheaves’.(6) Operations on sheaves;(7) Various base-change theorems;(8) Lefschetz trace formula.(9) etc.


For example, if X is a variety over Q, then

H i (X ,Z/n) ' H i (X (C),Z/n)

and the same for all the other coefficients.

On the other hand, suppose X is defined over Q and X is the samevariety regarded as being over Q.

The difference is between

Spec(Q[x1, . . . , xn]/(f1, . . . , fm))

andSpec(Q[x1, . . . , xn]/(f1, . . . , fm)).


ThenH i (X ,Z/n) ' H i (X (C),Z/n).

But the left hand side has a natural action of

Gal(Q/Q).

Thus, we end up with ‘arithmetic symmetry’ acting on complexcohomology.

Interlude: What is arithmetic geometry?Simple application:

Suppose X is a complex variety and σ ∈ Aut(C).

Let X σ be obtained from X by changing all the coefficients of thedefining equation for X using σ.

For example, suppose the equations defining X has all coefficientsalgebraic except for some occurrences of π. There is a σcorresponding to substitution of all occurrences of π by e.

ThenH i (X ,Z) ' H i (X σ,Z).

Proof: The map σ : X - X σ is a continuous isomorphism forthe étale topology and induces

H i (X ,Z/n) ' H i (X σ,Z/n)

for all n.


But more serious application is when X is defined over Q, so thatone ends up with a Galois representation

ρ : Gal(Q/Q) - Aut(H i (X ,Qp)).

These are the main examples of Galois representations studied inarithmetic geometry.


Example: X = Gm = Spec(Q[x , x−1]). Then

H1(X ,Zp) ' Zp(−1) := Hom(Zp(1),Zp),

as a representation of Gal(Q/Q), where

Zp(1) := lim←−µpn ,

and µi is the group of i−th root of unity. Thus,

Zp(1) ' Zp

as a group, but the Galois action is determined by the quotients

µpn ' Z/pn.


Example: When E is an elliptic curve,

H1(X ,Zp) ' TpE ⊗Zp Zp(−1).

Here,TpX = lim←−

n

E [pn].

This is still quite difficult when E doesn’t have complexmultiplication. Most of the time

Im(Gal(Q/Q)) ' Aut(H1(E ,Zp)).

When E has complex multiplication the action can be describedrelatively explicitly in terms of class field theory and the maintheorem of complex multiplication.


Most other cases are difficult. Among the best understood areGalois representations associated to modular forms. which areobtained as pieces of the étale cohomology of Kuga-Sato varieties.

One proves identities like

ap = Tr(Frp)

by relating the action of Frp to the action of a Heckecorrespondence Tp, and using the fact that ap is eigenvalue of Tp

on the modular form F .


This kind of phenomena persists for the cohomology of Shimuravarieties, where the étale cohomology tends to have actions of bothGalois groups and Hecke algebra. This often allows associations like

π 7→ ρπ

from automorphic representations to Galois representations.

For automorphic representations of groups that do not haveassociated Shimura varieties, one uses versions of Langlandsfunctoriality to move the computations to Shimura varieties.


However, one quite general fact is that the homomorphism

Gal(Q/Q) - Aut(H i (X ,Zp))

factors through a quotient

Gal(QS/Q) ' π1(Spec(Z[1/S))

and defines a point in a moduli space

M(Spec(Z[1/S ]),GLn(Zp)).


Reason:

Can ‘spread out’ X into

X ⊂ - XS

Spec(Q)?

⊂- Spec(Z[1/S ]?

where the right hand side is a fibre bundle.


General theme:

Arithmetic geometries are quite different from the manifolds thatcome up in physics. However, moduli spaces of G -bundles on suchspaces are quite close to moduli spaces of physics.

In particular, whenX = Spec(Z[1/N]),

the moduli spaceM(X ,GLn(Zp))

of principal GLn(Zp)-bundles on X parametrises Galoisrepresentations of the sort standardly studied in number theory, andone might attempt to look at them from the point of view ofphysics, especially TQFT.


M(X ,GLn(Zp))≤wMot⊂loc- [

∏′p WD(Xv ,GLn(Zp))C]≤w

M(X ,GLn(Zp))⊃M(X ,GLn(Zp))Mot

?

∩

⊂loc-∏′

p WD(Xv ,GLn(Zp))C

?

∩


The elements

((φp,Np))p ∈′∏p

WD(Xv ,GLn(Zp))C

consist of the following data:

– φp ∈ GLn(C), semi-simple;– Np ∈ Mn(C) nilpotent, such that

φ−1p Npφp = pNp,

and Np = 0 for most p;– There is a w such that

‖φp‖ ≤ pw

for all p.


L(((φp,Np))p)) =∏p

1det([I − φp]|(Cn)N=0 .

Will not converge in general. Get a function

L : [′∏p

WD(Xv ,GLn(Zp))C]≤w - C

for w = w0 sufficiently small, say w0 = −2.


Change

((φp,Np))p - ((φp,Np))p(s) := ((p−sφp,Np))p.

Then

((p−sφp,Np))p(s) ∈ [′∏p

WD(Xv ,GLn(Zp))C]≤w0

for Re(s) >> 0.

When ((φp,Np))p = loc(ρ), then write

L(ρ) = L(loc(ρ))

andL(ρ(s)) = L(loc(ρ)(s)).


Within region of convergence, if n ∈ Z,

L(ρ(n)) = L(ρ(n)).

Here, left hand side refers to ρ⊗ Zp(n).


(Part of) Hasse-Weil conjecture:

Ifρ ∈M(X ,GLn(Zp))≤wMot ,

thenL(ρ(s))

extends to analytic function of s.

As a consequence, one can define

L :M(X ,GLn(Zp))Mot- C

byL(ρ) = L([ρ(−n)](n))

for n-sufficiently large.


Conjectures of Beilinson and Deligne:

L(ρ)/Ωρ ∈ Q,

where Ωρ is a period.


(Part of the) Main conjecture of Iwasawa theory:

There exists a p-adic analytic extension given by the vertical arrow:

M(X ,GLn(Zp)) ⊃M(X ,GLn(Zp))Mot

Cp

Lp

?

L/Ω

Lp is called the p-adic L-function.


The moduli spaceM(X ,GLn(Zp)) carries a a determinant linebundle

DET - M(X ,GLn(Zp))

whose fibre over [ρ] is

DET[ρ] = detH∗c (X , ρ).

Kato has conjectured the existence of a canonical trivialisation

Z ∈ Γ(M(X ,GLn(Zp)),DET )

more or less equivalent to the main conjecture.

Can one give ‘physics constructions’ of such trivialisations?


L-functions are believed to be a powerful unified approach toarithmetic geometry.

For example, if ρE = TpE is the representation given by the torsionpoints of an elliptic curve E/Q, then it is conjectured that

ords=0L(ρ(s)) = rankE (Q).

However, even in the best of all possible worlds, L-functions carrylinearised information about arithmetic schemes.

A wild hope is that a more dynamic quantum theory of spaces likeM(X ,G ) will help with the study of non-linear invariants as well.

arithmetic geometry for physicists€¦ · arithmetic geometry for physicists minhyongkim...

Documents