1

Audrey TerrasU.C.S.D.

2006

more details can be found in

my webpage: www.math.ucsd.edu/~aterras/newbook.pdf

2

The Riemann zeta function for Re(s) > 1

( )ζ∞

∑ ∏-1-s

sn=1 p=prime

1(s) = = 1 - p .n

z Riemann (1859) extended to all complex swith pole at s=1

z Functional equation relates value at s and 1-s

z Riemann hypothesis (non real zeros ζ(s)=0 are on the line Re(s)=1/2). This now checked for 1013 billion zeros. (work of X. Gourdon and P. Demichel). See Ed Pegg Jr.’s website.

π ζΛ Γ Λ-s/2(s) = (s/2) (s) = (1-s)

3

Graph of z=|ζ(x+iy) | showing the pole at x+iy=1 and the first 6 zeros which are on the line x=1/2, of course. The picture was made by D. Asimov and S. Wagon to accompany their article on the evidence for the Riemann hypothesis as of 1986.

z=»zeta Hx+iyL»

-1

-0.5

0

0.5

1

x

-20020

y

0

5

10

z

0

5

10

z

4

Pulchritudinous Primes Webpage

z duality between primes & complex zeros of zeta using Hadamard product over zeros

z prime number theorem

proved by Hadamard and de la Vallée Poussin (1896-1900)Their proof requires complex analysis

z statistics of Riemann zero spacings studied by Odlyzko (GUE)www.dtc.umn.edu/~odlyzko/doc/zeta.htm

{ }≤ ∼ → ∞x# p = prime p x , as x

logx

5

Odlyzko’sComparison of Spacings of

Zeros of Zeta and

Eigenvalues of Random

HermitianMatrix.

Many Kinds of ZetaDedekind zeta of an algebraic number field F, where primes become prime

ideals p and infinite product of terms(1-Np-s)-1, where Np = norm of p = #(O/p), O=ring of integers in F

( )( ) ( )

[ ] 0

( ) 1 s j C

C j

Z s e ν− +

≥

= −∏ ∏Duality between spectrum ∆ on M &

lengths closed geodesics in MZ(s+1)/Z(s) is more like Riemann zeta

Selberg zeta associated to a compact Riemannian manifold M=G\H, H = upper half plane with

ds2=(dx2+dy2)y-2

G=discrete subgroup of group of real fractional linear transformationsprimes = primitive closed geodesics C in M of length ν(C), (primitive means only go around once)

6

What the Dedekind Zeta says about the Number Field

7

Zeta Functions of Graphs

We will see they have similar properties and applications to those of number theory

But first we need to figure out what primes in graphs are

This requires us to label the edges

X = finite connected (not-necessarily regular graph)

Orient the edges. Label them as follows. Here the inverse edge has opposite orientation.

2|E|

1 2 |E|

-1 -11|E|+1 |E|

e ,e ,...,e ,

e = e ,...,e = ee1

e7

8

prime [C] in X = equivalence class of closedbacktrackless, tailless, primitive paths (walks) in X. C = a1a2…as, ai = oriented edge

length ν(C)=sbacktrackless means ≠ -1

i+1 ia a

≠ -1s 1a aTailless means

primitive means C≠Dm

[C] = {a1a2…as, a2…as a1, … , asa1a2…as-1}

EXAMPLES of Primes in Graph

[C] =[ e1e2e3]

[C’]=[e7e10e12e8]

e1

e3 e2

e7

e10

e12

e8

9

Ihara Zeta Function

Ihara’s Theorem (Bass, Hashimoto, etc.)A = adjacency matrix of XQ = diagonal matrix jth diagonal entry = degree jth vertex -1;

r = rank fundamental group = |E|-|V|+1

ζ -1 2 r-1 2V(u,X) =(1-u ) det(I-Au+Qu )

Here V is for vertex

( )νζ ∏-1(c)

V[C]

(u,X) = 1-uprimes in X

Example

Remove an edge from K4

Call the new graph X

( )ζ-1

V 4

2 2 2 3

u,K =

(1 - u ) (1 - u)(1 - 2u)(1 + u + 2u )

( )ζ -1V

2 2 2 2 3

u, X =(1 - u )(1 - u)(1 + u )(1 + u + 2u )(1 - u - 2 u )

10

z=»1êzeta_K4 »0

1

2

x

-10 -5 0 5 10

y

0

500

1000

1500

2000

z

0

500

1000

1500

2000

z

Graph of z=1/|ζK4(2-(x+iy))|

Drawn by Mathematica

Comparison of 2 zetas ζ(u,K4)-1 and ζ(u,K4-edge)-1

z=»1êzeta_K4 »

-1

-0.5

0

0.5

1

x

-1

-0.5

0

0.5

1

y

0

1000

2000

3000

4000

z

-1

-0.5

0

0.5

1

x

z=»1êzeta_K4 −edge »

-1

-0.5

0

0.5

1

x

-1

-0.5

0

0.5

1

y

0

200

400

z

-1

-0.5

0

0.5

1

x

5 poles

9 poles

11

Some History 1960-1990

Ihara defined the zeta as a product over p-adic group elements.

Serre saw the graph theory interpretation. Hashimoto and Bass extended the theory.

Remarks

• Tomorrow we outline Bass’s proofBass’s proof of Ihara’s theorem. It will involve defining an edge zeta function with more variables

•• Another proofAnother proof of the Ihara theorem for regular graphs uses the SelbergSelberg trace formulatrace formula on the tree. For the trivial representation, see A.T., Fourier Analysis on Finite Groups & Applics; for general case, see and Venkov & Nikitin, St. Petersberg Math. J., 5 (1994)

12

Ì RH is true for “most” graphs (Alon Conjecture) but it can be false

Ì Hashimoto [Adv. Stud. Pure Math., 15 (1989)] provesIhara ζ for certain graphs is essentially the ζ function of a Shimura curve over a finite field

Ì κ(X)=the number of spanning trees of X, thecomplexity

analog of value of Dedekind zeta at 0Ì If graph is (q+1)-regular, the Riemann Hypothesis,

for (poles) and variable s with u=q-s, means graph isRamanujan i.e., non-trivial spectrum of adjacency matrix is contained in the spectrum for the universalcovering tree which is the interval (-2√q, 2√q) [ Lubotzky, Phillips & Sarnak, Combinatorica, 8 (1988)]

Ì If graph is (q+1)-regular, Ihara zeta has functional equations relating value at u and 1/(qu), which is s and 1-s

κζ

(r)r+1 r

X

1 (0) = (-1) 2 (r - 1) (X)

The Edge Matrix W1

Define W1 to be the 2m×2m matrix with i j entry 1 if edge i feeds into edge j,provided that ei ≠ the inverse of ej, otherwise the i j entry is 0.

We will prove the following theorem next time.

Theorem. ζ(u,X)-1=det(I-W1u).

Corollary.

The poles of Ihara zeta are the reciprocals of the eigenvalues of W1.

13

Theorem. (Graph prime number theorem) If the graph is connected and m divides ∆,

π(m) = #{prime paths of length m} ~ ∆R-m/m, as m → ∞

Proof.

Define R as the radius of convergence of the Ihara zeta. It is also the closest pole to 0. For a (q+1)-regular graph, R=1/q.

Define ∆ as the g.c.d. of the prime lengths.

If Nm=# {closed paths C,length m,no backtrack,no tails}

we have

Then ζ(u,X)-1=det(I-W1u) implies

ζ ∞

∑ mm

m=1

dlog (u,X)u = N udu

( )λ

λ λ

ζλ

λ λ

≥ ∈

∈ ≥ ≥ ∈

= − −

= =

∑ ∑

∑ ∑ ∑ ∑

1

1 1

mn 1 ( )

( ) 1 1 ( )

log (u) = N log(1 u)

u

m

Spec W

m mm

Spec W m m Spec W

du udu

um m

The dominant terms in this sum are those coming from the eigenvalues λ of W1 with |λ|=1/R.λ

λ

∈∑

1

m

mSpec(W )

N =m

14

By this theorem,

π

λλ

λ ∆∆

∈∑ ∑∼

1

amm -m 2 i

mspec(W ) a=1

| | maximal

|R|N = em m

The sum is 0 unless m divides ∆, when it is ∆R-m/m

Theorem. (Kotani and Sunada)The poles of ζ on |u|=R have the form Re2pia/∆, where a=1,2,....,∆.

λ

λ

∈∑

1

m

mSpec(W )

N =m

Möbius inversion implies|

1( ) ( ) d

d m

mm N

m dπ µ= ∑

( ) πζ

≥∏

- (n)n

n 1(u,X) = 1-u

Next we need a formula to relate N and π

ζπ

≥∑∑ m

m 1 d|m

dlog (u,X)u = d (d)udu

This implies

π∑md|m

N = d (d)

Note that π(1)=0 so that when m is prime p, we have Np=π(p).Thus the prime number theorem follows.

Ì

15

Tetrahedron example

= 24x3 + 24x4 + 96x6 + 168x7 + 168x8 + 528x9 + 1200x10 + 1848x11 + 3960x12 + 8736x13 + 16128x14 + 31944x15 + 66888x16

⇒ 8 prime paths of length 3 on the tetrahedron. Check it! We count 4 plus their inverses to get 8. ⇒ 6 paths of length 4. We agree.⇒ 0 paths of length 5 ⇒ 16 paths of length 6. That is harder to check.

Question: 528 is not divisible by 9. Shouldn't it be?Answer: You only know m divides Nm when m is prime.

ζ≥

∑ mX m

m 1

dx log (x) = N xdx

|

1( ) ( ) d

d m

mm N

m dπ µ= ∑

All poles except -1 of ζX(u) for a random graph with 80 vertices are denoted by little boxes.

The 5 circles are centered at the origin and have radii

R, q-1/2, R1/2, (pq)-1/4, p-1/2

q+1=max degree,

p+1=min degree

R=radius of convergence

of Euler product for ζX(u)

Experiments on Locations of Poles of Ihara Zeta of Irregular

Graphs- joint work with Matthew Horton

Kotani & Sunada, J. Math. Soc. U. Tokyo, 7 (2000) show imaginary poles lie between pink and outside circles; all poles between inner circle and circle of radius 1

Ramanujan graph in regular case would have only 2 circles; inner and rest are same

All poles but ±q on green circle; radius √q

16

Derek Newland’s

Experiments

Mathematica experiment with random 53-regular graph

( 2000 vertices)

Top row = distributions for eigenvalues of A on left and

Imaginary parts of the zeta poles on right.

Bottom row contains their respective level spacings.

Red line on bottom left: Wigner surmise, y = (πx/2)exp(-πx2/4).

Exercises1. List all the zeta functions you can and what they are good for.

There is a website that claims to list lots of them: www.maths.ex.ac.uk/~mwatkins

2. Compute Ihara zeta functions for the cube, dodecahedron, buckyball, your favorite graph. Mathematica should help.

3. Do the basic graph theory exercise 14 on page 24 of the manuscript on my website:

www.math.ucsd.edu/~aterras/newbook.pdf4. Show that the radius of convergence of the Ihara zeta of a

(q+1)-regular graph is R=1/q. Explain why the closest pole of zeta to the origin is at R.

5. Prove the functional equations of Ihara zeta for a regular graph. See p. 25 of my manuscript.

6. Look up the paper of Kotani and Sunada and figure out their proof. You need the Perron – Frobenius theorem from linear algebra. [Zeta functions of graphs, J. Math. Soc. Univ. Tokyo, 7(2000)].

17

6. Fill in the details in the proof of the graph theory prime number theorem.

7. Prove the prime number theorem for a (q+1)-regular graph using Ihara’s theorem with its 3-term determinant rather than the 1/det(I-uW1) formula.

8. Exercise 16 on page 28 of my manuscript. This is a Mathematica exercise to plot poles of Ihara zetas.

1

Audrey TerrasU.C.S.D.

2006

Basic Assumptions

graphs are

connected,

with r=rank fundamental group > 1,

no degree 1 vertices (called leaf vertex, hair, danglers, ...)

2

Remarks

Here we outline Bass’s proof of Ihara’s theorem. It will involve defining an edge zeta function with more variablesAnother proof of the Ihara theorem for regular graphs uses the Selbergtrace formula on the tree. For the trivial representation, see A.T., Fourier Analysis on Finite Groups & Applics; for general case, see and Venkov & Nikitin, St. PetersbergMath. J., 5 (1994)

Edge ZetasOrient the edges of the graph. Multiedge matrix W has ab entry wab in C, w(a,b)=wab if the edges a and b look like

a b

1 1 2 2 3 1( ) ( , ) ( , ) ( , ) ( , )E s s sN C w a a w a a w a a w a a−= L

Otherwise set wab=0.

( ) 1

[ ]

( , ) 1 ( )E EC

W X N Cζ −= −∏

For a prime C = a1a2…as, define the edge norm

Define the edge zeta as

and a is not the inverse

of b

3

Properties of Edge Zeta

v Set all non-0 variables wab=u in the edge zeta & get Ihara zeta

v Edge zeta is the reciprocal of a polynomial given by a much simpler determinant formula than the Ihara zeta

v Even better, the proof is simpler

( ) 1( , ) detE W X I Wζ −= −

Determinant Formula

For Edge Zeta

From this Bass gives an ingenious proof of Ihara’s theorem.

Reference:Stark and T., Adv. in Math., Vol.

121 and 154 and ? (1996 and 2000 and 2006)

4

Example

D=Dumbbell Graph

11 12

23 26

33 351

42 44

51 54

65 66

1 0 0 0 00 1 0 00 0 1 0 0

( , ) det0 0 1 0 0

0 0 1 00 0 0 0 1

E

w ww w

w wW D

w ww w

w w

ζ −

− − −

= −

− −

e2 and e5 are the vertical edges.

Specialize all variables with 2 and 5 to be 0 and get zeta function of subgraph with vertical edge removed. Fission

Diagonalizes the matrix.

e2 e5

e1 e4

e3 e6

Proof of the Determinant Formula

1

[ ] 1

( )log

jE

EC j

N Cj

ζ −

≥

= ∑ ∑,

iji j ij

L ww∂

=∂∑

( )1

1 1( )

1 1log ( ) ,

C primitive, no backtrack, no tails

kE E

m C kC m

L L N Cm k

ν

ζ −

≥ ≥=

= ∑ ∑ ∑( ) ( )k k

E ELN C kmN C=

1

1

log ( ) ( )

Here C need not be primitive, still no backtrack, no tails though.

mE E

C m

L N C Tr Wζ −

≥

= =∑ ∑

5

( ) 1

1

logdet ( )m

m

L I W Tr W−

≥

− = ∑This proves L(log( determinant formula)).

So we get the formula

up to a multiplicative constant. The proof ends by

noting that both sides are 1 when all the wij are 0.

{

An exercise in matrix calculus gives

( ) 1( , ) detE W X I Wζ −= −

1 2 1 2( , ) (1 ) det( )rV u X u I Au Quζ − −= − − +

( ) 1( , ) detE W X I Wζ −= −

6

1, if v is starting vertex of edge e0, otherwiseves

=

1, if v is the terminal vertex of edge e0, otherwisevet

=

Define Define starting matrix S and terminal matrix T

| |

| |

00E

E

IJ

I

=

j j+|E|

, TJ=Ssince start (end) of e is end (start) of e

SJ T=

t|V| T , Q+I t tA S SS TT= = =

Note: matrix A counts number of undirected edges connecting 2 distinct vertices and twice # of loops at each vertex.

Then we see that

Part 1 of Bass Proof

1 ijW matrix obtained from W by setting all non-zero w equal to 1t

1

j j±|E|

W + J = T , where J compensates for not allowing edge e to feed into e

S

2| | | |

2| | 2| | 1

2| || |

2| |2| |

0 (1 )0

0

0

V Vt

E E

VVt t

EE

I I u SuT I I W u

II Au Qu SuT S u II Ju

− −

− + = −+

Below all matrices are (|V|+2|E|) x (|V|+2|E|)

with |V| x |V| first block

Then take determinants of both sides to see2 | | 2

2| | 1 | | 2| |(1 ) det( ) det( ) det( )VE V Eu I W u I Au Qu I Ju− − = − + +

Part 2 of Bass Proof

7

2 | | 22| | 1 | | 2| |(1 ) det( ) det( )det( )V

E V Eu I W u I Au Qu I Ju− − = − + +

2

I 0 implies ( )

-Iu I 0 (1 )I Iu I Iu

I Ju I JuIu I I u

+ = + = −

So det(I+Ju)=(1-u2)|E|

Since r-1=|E|-|V|, for a connected graph, the Ihara formula for the vertex zeta function follows from the edge zeta determinant formula.

End of Bass Proof

Next we define a zeta function invented by Stark which has several advantages over the edge zeta.

It can be used to compute the edge zeta using smaller determinants.

It gives the edge zeta for a graph in which an edge has been fused.

8

Fundamental Group of X can be identified with group generated by edges left out of a spanning tree

Imitate the definition of the edge zeta function. Define for a prime path

1 11 1, { , , }s j rC a a where a e e± ±= ∈L …

the path norm

2r ×2r multipath matrix Z has ij entry zij in C if 1

j ie e−≠ and zij =0, otherwise.

1

1 11

( ) ( , ) ( , )s

P s i ii

N C z a a z a a−

+=

= ∏

( ) 1

[ ]

( , ) 1 ( )P PC

Z X N Cζ −= −∏

1 11 1,... , ,...,r re e e e− −

Define the path zeta function

edges left out of a spanning tree T of X are inverse edges are

edges of the spanning tree T are with inverse edges

If , write the part of the path between ei and ej as the (unique) product

A prime cycle C is first written as a product of the generators of the fundamental group ej and then a product of actual edges ej and tk.

Now specialize the multipath matrix Z to Z(W) with entries

Then

1,... re e1 1

1 1 2,...,r r re e e e− −+ = =

1 | | 1,..., Xt t −

| | 2| | 2,...X Xt t −

1i je e−≠

1 nk kt tL

1 1

1

1

( , ) ( , ) ( , )n

n

ij i k k j k kz w e t w t e w t tν ν

ν+

−

=

= ∏

( ( ), ) ( , )P EZ W X W Xζ ζ=

9

spanning trees

A tree is a graph without cycles.A spanning tree for a graph X is a subgraph which is

a tree and which contains all the vertices of X.

the red graph is a spanning tree for K4

Recall that the edge zeta involved a 6x6 determinant.

The path zeta is only 4x4.

Maple computes it much faster than the 6x6.

1 01 0

0 10 1

aa ab bc ab bf

ce ea cc ce ed

db bc dd db bf

fe ea fe ed ff

w w w w ww w w w w

w w w w ww w w w w

− − − −

b e

a d

c f

Fusion: shrink edge b to a point. The edge zeta of the new graph obtained by setting wxbwby=wxy in specialized path zeta & same for e instead of b.

10

ExercisesExercises1) Show that any non-cycle graph has an infinite number of

primes. Show that the primes do not in general commute and do not in general give rise to unique factorization.

2) Show that any non-bipartite k-regular graph, with k≥3, has the constant ∆ equal to 1. A bipartite graph has the property that the vertices are a union of 2 disjoint sets A and B such that no 2 vertices in A are adjacent and no 2 vertices in B are adjacent.

3) Compute the edge zeta function for your favorite graph; e.g. K4 or K4-edge.

4) Fill in the details of the proof that 1/ζE(W,X)=det(I-W).5) Fill in the details of the proof that the formula in exercise 4

implies Ihara’s 3-term determinant formula for the vertex zeta.

6) Write a Mathematica (or whatever) program to do the process that specializes the path zeta to get the edge zeta.

7) Prove that if

( ) 1

1

log det ( )m

m

L I W Tr W−

≥

− = ∑,

iji j ij

L ww∂

=∂∑

Hint: Use the fact that you can write the matrix W (which is not symmetric) as a product W=U-1TU, where U is orthogonal and T is upper triangular by Gram-Schmidt.

1

a stroll through the graph zeta garden, III

Audrey Terraswww.math.ucsd.edu/~aterras/

newbook.pdf

Artin L-Functions

§ 1st number field case§ 2nd graph theory version

2

K ⊃ F number fields with K/F Galois

OK ⊃ OF rings of integers

P ⊃ p prime ideals (p unramified , i.e., pÀ P2)

Frobenius Automorphism when p is unramified.

σP generates finite Galois group, Gal((OK /P)/(OF/p ))determined by p up to conjugation if P/p unramifiedf (P/p) = order of σP = [OK /P: OF/p] g (P/p)=number of primes of K dividing p

Artin L-Function for s∈C, π a representation of Gal(K/F). Give only the formula for unramified primes p of F. Pick P a prime in OK dividing p.

( ),//Gal FKFK

σ

= ∈

PP( ) (mod ), K

Nx x x Oσ ≡ ∈Pp P

1

( , )" " det 1/ sL s NFK

π π

−

− = −

∏

pp

P

Artin L-Functions of Number Fields

Np=|OK/p|

z Factorization

z Tchebotarev Density Theorem ⇒ ∀ σ in Gal(K/F), ∃ ∞ -ly many prime ideals p of OF

such that ∃ P in OK dividing p with Frobenius

z Artin Conjecture: L(s,π) entire, for non-trivialirreducible rep π (proved in function field casenot number field case)

References: Stark's paper in From Number Theory to Physics,edited by Waldschmidt et al

Lang or Neukirch, Algebraic Number TheoryRosen, Number Theory in Function Fields

d

irreducibledegree d

( ) L(s, )K s π

π

π

ζ π= ∏

/{ }

K Fσ

=

P

3

field ring prime ideal finite field

K= Q(√m) OK= Z[√m] p ⊃ pOK OK/p

F= Q OF=Z pZ Z/pZf=degree of OK/p over OF/pOFg=# of such p,

efg=2

Assume, m is a square-free integer congruent to2 or 3 (mod 4).

3 CASES

1) p inert: f=2. pOK = prime ideal in K, m T x2 (mod p)

2) p splits: g=2. pOK = p p', p ≠ p', m ≡ x2 (mod p)

3) p ramifies: e=2. pOK = p2, p divides 4m

Gal(K/F)={1,-1}

Frobenius automorphism= Legendre Symbol =

p F 4m implies p has 50% chance of being in Case 1(and 50% chance of being in case 2)

Assume, m is a square-free integer ≡ 2 or 3 (mod 4).

K=F(√m)/F, F=Q

Decomposition of Primes in Quadratic Extensions

1, in case 14 1, in case 2

0, in case 3

mp

−=

4

Theorem. Set C(p)=the conjugacy class of the Frobeniusautomorphism of prime ideals P of K above p. Then, for every conjugacy class C in G=Gal(K/F),

For a set S of primes of F, define the analytic density of S.

( )1( ) lim

log 1p S

s

spS

sδ ∈

→ +

− = − −

∑

The proof requires the facts that L(s,π) continues to s=1 with no pole or zero if π≠1, while L(s,1) = ζF(s) has a simple pole at s=1.

Tchebotarev Density Theorem

{ } | |( ) .

| |C

p C p CG

δ = =

Proof. Sum the logs of the Artin L-functions ×conjugates of characters χπ over all irreducible reps π

of G. As s → 1+,

( )

( )

1log log , ( )

1

( ( )) ( )

| || |

s

p

s

pC p C

L s Cs

C p p C

pGC

ππ

π ππ

π χ

χ χ−

−

=

− ∑

∑∑

∑

∼

∼

∼

Taking logs of Euler product for L(s,π) - only 1st term of Taylerexpansion contributes. Then use the orthogonality relations of the characters of the irreducible reps π of G.

5

Graph Galois Theory for unramified coverings

Graph Y an unramified covering of Graph X means (assuming no loops or multiple edges)

π:Y→X is an onto graph map such that for every x∈X & for every y ∈ π-1(x),

π maps the points z ∈ Y adjacent to y 1-1, onto the points w ∈ X adjacent to x.

Normal d-sheeted Covering means: ∃ d graph isomorphisms σ1 ,..., σdmapping Y → Y such that π σj (y) = π (y) ∀ y ∈ YThe Galois group G(Y/X) = { σ1 , ... , σd }.

Note: We do not assume graphs are regular! We do assume that they are connected, without “danglers” (degree 1 vertices)

Gives generalization of Cayley & Schreiergraphs

(x,σ)

Y

πx

X

First pick a spanning tree T in X (no cycles, connected, includes all vertices of X).

Second make n=|G| copies of thetree T in X. These are the sheets of Y.

elements of Y: (x,τ), x∈X and τ∈G

Label the sheets with τ∈G. Group action of G on sheets.

σ(sheet τ) = sheet(στ)σ(x,τ)=(x, στ)

How to Label the Sheets of a GaloisCovering

Given G, get examples Y by giving permutation representation of generators of G to lift edges of X left out of T.

6

Spanning Tree in X is red. Corresponding sheets of Y are also red

Cube covers Tetrahedron

Example 1. Quadratic Cover

Non-normal cubic cover of K4

Exercise. Explain why the cover is not normal.

d

b

7

¯ L(u, ρ,Y/X)-1 = (1-u2) (1+u) (1+2u) (1-u+2u2)3

¯ ζ(u,Y)-1 = L(u,ρ,Y/X)-1 ζ(u,X)-1

¯ ζ(u,X)-1 = (1-u2)2(1-u)(1-2u) (1+u+2u2)3

Exercise. Show that for any unramified possibly irregular, non-normal cover Y of X ,

ζ(u,X)-1 divides ζ(u,Y)-1.

Exercise. Factor the zeta function of the non-normal cubic cover of K4 on the preceding page. Show that

ζ(u,X)-1 divides ζ(u,Y)-1.

Definition. A graph Z is an intermediate covering to the covering Y/X with projection map π:Y → X means there are projection maps π1:Z→X and π2:Y →Z so that π = π1 ° π2.

Evil Exercise. State and prove the fundamental theorems of graph Galois theory. See my book for some answers.

www.math.ucsd.edu/~aterras/newbook.pdfNote: First you need to define what you mean by intermediate coverings and when we should say such coverings are equal.

Definitions. 2 intermediate covers Z and Z’ to Y/X are covering isomorphic(conjugate) if there is an graph isomorphism i:Z→Z’such that π’1 ° i = π1 .

For us to say Z=Z’ we needalso i ° π2 = π’2 .

8

Fundamental Theorems of Graph Galois Theory

Theorem. Suppose Y/X is an unramified normal covering with Galois group G=G(Y/X).

1) We have a 1-1 correspondence between subgroups H of G and graphs Z(H) intermediate to Y/X. Each intermediate graph Z to Y/X corresponds to some subgroup H(Z) of G. Write Z ↔ H when intermediate graph Z corresponds to

subgroup H.2) If Zi ↔ Hi, for i=1,2, then Z1 is intermediate to Y/Z2 iff

H1 ⊂ H2.3) 2 intermediate graphs to Y/X correspond to conjugate

subgroups of G iff they are covering isomorphic. 4) If Z is an intermediate covering to Y/X, then Z is itself a

normal covering of X iff the corresponding subgroup H of G is a normal subgroup. Then Gal(Z/X) ≅ G/H.

A bit of the proof1) Let H be a subgroup of G. The points of Y have the form

(x,σ), with x∈X and σ∈G. Define the vertices of Z to be points (x,Hσ) with x∈X and σ∈G.

Define (a,Hσ) and (b,Hτ) to be adjacent in Z iff there are elements h and h’ of H so that (a,hσ) and (b,h’τ) have an edge joining them in Y.

Recall Definition. A graph Z is an intermediate covering to the covering Y/X with projection map π:Y → X means there are projection maps π1:Z→X and π2:Y →Z so that π = π1 ° π2.

2) The Galois group G(Y/Z) should be H. Fix v in X.

H={ σ∈G | σ(v,1) ∈ π2-1(π2(v,1))}.

Check this defines a subgroup of G. See my

manuscript p. 47.

9

a'' c'

d''b''

b'd'

c" a′

Example of Splitting of Primes in Quadratic Cover

f=2

Picture of Splitting of Prime which is inert; i.e., f=2, g=1, e=1

1 prime cycle D above, & D is lift of C2.

a d

c

b

d''

c"

b''

a'

b'd'

a'' c'

c

b

a d

Example of Splitting of Primes in Quadratic Cover

g=2

Picture of Splitting of Prime which splits completely; i.e., f=1, g=2, e=1

2 primes cycles above

10

X x C Exercise: Compute Frob(D) on for

examples in cube over K4, G={1,σ}.

Frob(D) = = τσ-1 ∈ G=Gal(Y/X)

where τσ-1 maps sheet σ to sheet τ

Y/XD

D a prime above C

not necessarily closed

( D a prime above C isclosed and is obtained

by f liftings like )

°( ) ( )length C length C=

°C

°CY

(x,σ)

π

= the unique lift of C in Y starting at (x,σ) ending at (x,τ)

°C

(x,τ)

Frobenius Automorphism

1

a stroll through the graph zeta garden, IV

Audrey Terras

§more details can be found inwww.math.ucsd.edu/~aterras/newbook.pdf

§This is a very rough draft of a book I am trying to write. Lists of typos will be appreciated

Applications of Artin L-Functions on Graphs

• 1st number field case• 2nd graph theory version

2

OF p OF/p

3

Q Z pZ Z/pZ

More details are in Stark’s article in From Number Theory to Physics, edited by Waldschmidt et al

( )3 2F =Q

Example 2. Galois Extension of Non-normal Cubicfield ring prime ideal finite fieldK=F(e2πi/3) OK P OK/P

2

Splitting of Rational Primes in OF - Type 1. Primes that Split Completely: pOF= p1 p2 p3 , with distinct pi of degree 1 (p=31 is 1st example),

Frobenius of prime P above pi has order 1density 1/6 by Tchebotarev

There are also 2 other types of unramified primes.

Exercise. Do a function field analog of the preceding example in the spirit of E. Artin’s thesis. Artin’s thesis did quadratic extensions of the field of rational functions of 1 indeterminate with coefficients in a finite field.

3

G=S3, H={(1),(23)} fixes Y3.

a(1)=(a,(1)), a(2)=(a,(13)), a(3)=(a,(132), a(4)=(a,(23)),a(5)=(a,(123)),a(6)=(a,(12)).

Here we use standard cycle notation for elements of the symmetric group.

a(x)

a

1

2

4

2’

1’4’

3’3’’’

4’’’1’’’

2’’’

4’’

2’’

1’’

3’’

3(6)

1(6)2(6)

4(6)

4(5)

2(5)

1(5)

3(5)

3(4)

1(4)

4(4)

2(4)

4(1)

1(1)

2(1)

3(1)

3(2)

1(2)

4(2)2(2)

4(3)1(3)2(3)

3(3)

Y6

Y3

X

x=1,2,3

a(x),a(x+3)

a(x)

3

Prime Splitting Completelypath in X (list vertices) 14312412431

f=1, g=3 3 lifts to Y31’4’3’’’1’’’2’’’4’’1’’2’’4’’’3’1’1’’4’’3’’1’’2’’4’’’1’’’2’’’4’’3’’1’’1’’’4’’’3’1’2’4’1’2’4’3’’’1’’’

Frobenius trivial ⇒ density 1/6

1

23

4

page 60 of my manuscript on my website gives examples of primes with other splitting behavior in the non-Galois cubic extension

4

Ì ζ(u,X)-1 = (1-u2)(1-u)(1+u2)(1+u+2u2)(1-u2-2u3)Ì ζ(u,Y3)

-1 = ζ(u,X)-1 (1-u2)2(1-u-u3+2u4)×(1-u+2u2-u3+2u4)(1+u+u3+2u4)

×(1+u+2u2+u3+2u4)Ì ζ(u,Y6)

-1 = ζ(u,Y3)-1 (1-u2)3(1+u)(1+u2)(1-u+2u2)

×(1-u2+2u3) (1-u-u3+2u4) (1-u+2u2-u3+2u4) ×(1+u+u3+2u4)(1+u+2u2+u3+2u4)

It follows that, as in number theory ζ(u,X)2 ζ(u,Y6) = ζ(u,Y2) ζ(u,Y3)2

Y2 is an intermediate quadratic extension between Y6 and X. See Stark & Terras, Adv. in Math., 154 (2000), Fig. 13, for more

Frob(D) = = τσ-1 ∈ G=Gal(Y/X)

where τσ-1 maps sheet σ to sheet τ

Y/XD

Frobenius Automorphism

D a prime in Y above C a prime of X consists of f lifts of C.

Suppose Y/X a normal cover. Given a path p of X, there is a unique lift of pto a path p on Y starting on sheet 1 and having the same length as p.

Here you may take p to be an edge if you like.

X x C

Y

(x,σ)

π

(x,τ)

C

5

Artin L-functions involve (irreducible) representations πof the Galois group. If you do not know about this subject, assume the Galois group G is abelian, or even better cyclic. In that case a representation is a group homomorphism χ:G→TT=group under multiplication of complex numbers of absolute value 1.

Moreover χ is given by complex exponentials:

G=Z/nZ under + and χ(x mod n) = exp(2π iax/n)

Here a is a fixed element of G. As you run through all n elements of the cyclic group G you get all the representations of G.

A reference for representations of finite groups is my book, Fourier Analysis on Finite Groups and Applications.

1) Replace D with different prime above C and see thatConjugacy class of Frob(D) ∈ Gal(Y/X) unchanged.

2) Varying x = start of C does not change Frob(D).3) Frob(D)j = Frob(Dj) .

ρ = representation of G=Gal(Y/X), u∈C, |u| small

[C]=primes of Xν(C)=length C, D a prime in Y over C

1( )

[ ]

/( , , / ) det 1 C

C

Y XL u Y X u

Dνρ ρ

− = −

∏

Artin L-Function

Properties of Frobenius Exercise: Prove these properties

6

Copy from Lang, Algebraic Number Theory

1)L(u,1,Y/X) = ζ(u,X) = Ihara zeta function of X (our analogue of the Dedekind zeta function, also Selberg zeta)

2)

product over all irreducible reps of G, dρ=degree ρ

µ( , ) ( , , / )d

G

u Y L u Y X ρ

ρ

ζ ρ∈

= ∏

Properties of Artin L-Functions

Proofs of 1) and 2) require basic facts about reps of finite groups. See A. T., Fourier Analysis on Finite Groups and Applications.

2

1

( 1)2

( , , / )

(1 ) de t ( ' ' ' )r d

L u Y X

u I A u Q uρρ

ρχ −

−= − − +r = rank fundamental group of X = |E|-|V|+1

ρ = representation of G = Gal(Y/X), d = dρ = degree ρ

Definitions. nd×nd matrices A′, Q′, I′, n=|X|nxn matrix A(g), g ∈ Gal(Y/X), has entry for α,β∈X given by

(A(g))α,β = # { edges in Y from (α,e) to (β,g) }, e=identity ∈ G.

Q = diagonal matrix, jth diagonal entry = qj = (degree of jth vertex in X)-1,

Q′ = Q⊗Id , I′ = Ind = identity matrix.

' ( ) ( )g G

A A g gρ ρ∈

= ⊗∑

Ihara Theorem for L-Functions

7

Y=cube, X=tetrahedron: G = {e,g}representations of G are 1 and ρ: ρ(e) = 1, ρ(g) = -1A(e)u,v = #{ length 1 paths u′ to v′ in Y}A(g)u,v = #{ length 1 paths u′ to v′′ in Y}

0 1 0 0 0 0 1 11 0 1 1 0 0 0 0

( ) ( )0 1 0 0 1 0 0 10 1 0 0 1 0 1 0

A e A g

= =

0 1 1 11 0 1 1

' ( ) ( )1 1 0 11 1 1 0

A A e A gρ

− − = − = − − − −

A′1 = A = adjacency matrix of X = A(e)+A(g)

a d

c

b

a'' c'

d''b''

b'd'

c" a′

(u,e)=u'(u,g)=u"

EXAMPLE

You can also consider edge and path Artin L-functions. See my proto-book.

www.math.ucsd.edu/~aterras/newbook.pdf

Generalizations of Cayley and Schreier graphs are also described in my manuscript above. This leads to the construction of the graphs on the following slide.

8

2 connected regular graphs (without loops & multiple edges) which are isospectral but not isomorphic

Application of Galois Theory of Graph Coverings. You can’t hear the shape of a graph.

z See A.T. & Stark in Adv. in Math., Vol. 154 (2000) for the details or my manuscript.

z Method goes back to algebraic number theorists who found number fields Ki which are non isomorphic buthave the same Dedekind zeta. SeePerlis, J. Number Theory, 9 (1977).

z Galois group is GL(3,F2), order 168z appears in Buser,Geometry & Spectra of

Compact Riemann Surfaces also Gordon, Webb,& Wolpert (isospectral non-isomorphic planardrums), Inventiones Math.(1992)

Audrey

Harold

z Robert Perlis and Aubi Melleinhave used the same methods tofind many examples of isospectralnon isomorphic graphs with multipleedges and components. 2 suchare on the right.

9

There are nice constructions of isospectal non-isomorphic graphs in paper of Lubotzky, Samuels and Vishne

Exercises1) Consider more examples of Galois covers and

their zeta and L-functions. There are lots in my manuscript

www.math.ucsd.edu/~aterras/newbook.pdfon my website or the joint papers with H. Stark in Advances in Math.

2) Write up a proof of the Tchebotarev densitytheorem for Artin L-functions of graphs.

3) Look at Avi Wigderson’s website and find out what zig-zag products of graphs are. Compute their zeta functions. Are their any divisibility properties?

10

1) Find the meaning of the Riemann hypothesis for irregular graphs. Are there functional equations? Paper of Joel Friedman, Shlomo Hoory & Angel is relevant. They consider n-covers of K4-edge as n goes to infinity and find a region in the complex plane approached by the eigenvalues of the W1 matrix. H.M.S. & A.T. discuss the RH for irregular graphs in our Part III, to appear soon in Advances in Math.

2) Are there analogs of Artin L-functions for higher dimensional things – hypergraphs or buildings ? Papers of Winnie Li (Ramanujan buildings) and Anton Deitmar (Selbergzetas for buildings) and various other people should be consulted.

3) Connect the zeta polynomials of graphs to other polynomialsassociated to graphs and knots (Tutte, Alexander, andJones polynomials). Papers exist. But connection ismysterious to me.

Research Problems

4) Is there a graph analog of regulator, Stark Conjectures, classfield theory for abelian graph coverings? Or more simply aquadratic reciprocity law, fundamental units? The ideal classgroup is the Jacobian of a graph and has order = number ofspanning trees (paper of Roland Bacher, Pierre de la Harpeand Tatiana Nagnibeda). There is an analog of Brauer-Siegeltheory (see H.S. and A.T. , Part III).

5) Can you prove an analog of the theorem of Katz and Sarnak for function field zeta functions? This says for almost all curvesover a finite field as the genus and order of the field go to infinity, the imaginary parts of the zeros of zetas have GUElevel spacing (meaning the spacings look like those Odlyzkofound for the high zeros of Riemann zeta as in the picture I showed in Lecture 1). See the very difficult book of Katz andSarnak. A great more elementary reference giving background on this subject is the new book by Steven J. Miller and RaminTakloo-Bighash, An Invitation to Modern Number Theory. Seetheir bibliography on the web too! See also Derek Newland’sPh.D. thesis from U.C.S.D. 2005.

11