are l functions able to solve diophantine equations

7/27/2019 Are l Functions Able to Solve Diophantine Equations

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-functions and Diophantine equations

The function field case

Classical Iwasawa theory

Non-commutative Iwasawa Theory

Are -functions able to solve Diophantine

equations?An introduction to (non-commutative) Iwasawa theory

Otmar Venjakob

Mathematical Institute

University of Heidelberg

CMS Winter 2007 Meeting

Otmar Venjakob Are -functions able to solve Diophantine equations?


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L-functions

Diophantine Equations

The analytic class number formula

Leibniz (1673)

1 13

+ 15

17

+ 19

111

+ = 4

(already known to GREGORY and MADHAVA)



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L-functions



Special values of L-functions



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L-functions



N 1, (Z/NZ) units of ring Z/NZ.

Dirichlet Character (modulo N) :

: (Z/NZ) C

extends to N

(n

) := (nmod N), (n, N) = 1;

0, otherwise.



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L-functions



Dirichlet L-function w.r.t. :

L(s, ) =

n=1

(n)

ns, s C, (s) > 1.

satisfies:- Euler product

L(s, ) =

p

1

1

(p)ps

,

- meromorphic continuation to C,- functional equation.


f i d Di h i i


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L-functions



Examples

1 : Riemann -function

(s) =

n=1

1

ns=

p

1

1 ps,

1 : (Z/4Z) = {1, 3} C, 1(1) = 1, 1(3) = 1

L(1, 1) = 1 13

+ 15

17

+ 19

111

+ = 4


f ti d Di h ti ti


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L-functions



Examples

1 : Riemann -function

(s) =

n=1

1

ns=

p

1

1 ps,

1 : (Z/4Z) = {1, 3} C, 1(1) = 1, 1(3) = 1

L(1, 1) = 1 13

+ 15

17

+ 19

111

+ = 4


functions and Diophantine equations


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L-functions







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L-functions



Conjectures of Catalan and Fermat

p, q prime numbers

Catalan(1844), Theorem(MIHAILESCU, 2002):

xp

yq = 1,

has unique solution

32 23 = 1in Z with x, y > 0.

Fermat(1665), Theorem(WILES et al., 1994):

xp + yp = zp, p > 2,

has no solution in Z with xyz = 0.Otmar Venjakob Are -functions able to solve Diophantine equations?



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L-functions



Conjectures of Catalan and Fermat

p, q prime numbers

Catalan(1844), Theorem(MIHAILESCU, 2002):

xp

yq = 1,

has unique solution

32 23 = 1in Z with x, y > 0.

Fermat(1665), Theorem(WILES et al., 1994):

xp + yp = zp, p > 2,

has no solution in Z with xyz = 0.Otmar Venjakob Are -functions able to solve Diophantine equations?



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L-functions



Factorisation over larger ring of integers

m primitive mth root of unity

Z[m] the ring of integers of Q(m),

e.g. for m = 4 with i2 = 1 we have in Z[i] = {a+ bi|a, b Z} :x3 y2 = 1 x3 = (y + i)(y i)

and for m = pn

we have in Z[pn] :

xpn

+ypn

= (x + y)(x + pny)(x + 2pny) . . . (x + p

n1pn y) = z

pn.


-functions and Diophantine equationsL f ti


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L-functions



The strategy

Hope: Use unique prime factorisationto conclude a

contradiction from the assumption that the Catalan or Fermatequation has a non-trivial solution.

Problem: In general, Z[m] is not a unique factorisation domain(UFD), e.g. Z[23]!


-functions and Diophantine equationsL f ti


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p q




L-functions



The strategy

Hope: Use unique prime factorisationto conclude a

contradiction from the assumption that the Catalan or Fermatequation has a non-trivial solution.

Problem: In general, Z[m] is not a unique factorisation domain(UFD), e.g. Z[23]!


-functions and Diophantine equationsL functions


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p q




L-functions



Ideals

Kummer: Replace numbers by ideal numbers:

For ideals(=Z[m]-submodules) 0= a

Z[m] we have unique

factorisation into prime ideals Pi = 0 :

a =

ni=1

Pnii

Principal ideals: (a) = Z[m]a


-functions and Diophantine equationsL functions


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L-functions



Ideal class group

Cl(Q(m)) : = { ideals of Z[m]}/{ principal ideals of Z[m]}= Pic(Z[m])

Fundamental Theorem of algebraic number theory:

#Cl(Q(m)) < and

hQ(m) := #Cl(Q(m)) = 1 Z[m] is a UFD

Nevertheless, Cl(Q(m)) is difficult to determine, too many

relations!Otmar Venjakob Are -functions able to solve Diophantine equations?

-functions and Diophantine equationsL-functions


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L functions



The L-function solves the problem

How can we compute

hQ(i)?

It is a mystery that

L(s, 1)

knows the answer!



Th f i fi ldL-functions


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L functions



The L-function solves the problem

How can we compute

hQ(i)?

It is a mystery that

L(s, 1)

knows the answer!



Th f ti fi ldL-functions


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The cyclotomic character

Gau:

G(Q(N)/Q)N=

// (Z/NZ)

with g(N) = N(g)

N for all g G(Q(N)/Q)

N = 4 :

1 is character of Galois group G(Q(i)/Q)

L(s, 1) (analytic) invariant of Q(i).



The function field caseL-functions


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Analytic class number formula for imaginary quadratic numberfields:

hQ(i) = #(Q(i))N

2L(1, 1)

=4 22

L(1, 1)

=

4

L(1, 1) = 1 (by Leibniz formula)

Z[i] is a UFD.



The function field caseL-functions


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hQ(i) = #(Q(i))N2

L(1, 1)

=4 22

L(1, 1)

=

4


Z[i] is a UFD.


-functions and Diophantine equationsThe function field case

L-functions


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hQ(i) = #(Q(i))N2

L(1, 1)

=4 22

L(1, 1)

=

4


Z[i] is a UFD.



L-functions


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A special case of the Catalan equation

Since L(s, 1) knows the arithmetic of Q(i), it is able to solveour problem:

Claim: x3

y2

= 1 has no solution in Z.

In the decomposition x3 = (y + i)(y i) the factors (y + i) and(y i) are coprime (easy!)

y + i = (a+ bi)3 for some a, b Z

taking Re() and Im() gives: y = 0, contradiction!



L-functions

Di h i E i


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Regular primes

Similarly (s) knows for which p

Cl(Q(p))(p) = 1

holds! Then the Fermat equation does not have any non-trivial

solution. But 37|hCl(Q(37))!

Iwasawa:

Cl(Q(pn))(p) =? for n 1.




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Number fields versus function fields

Q Fl(X) = K(P1Fl)

K/Q number field K(C)/Fl(X) function field

C PnFl smooth, projective curve, i.e.

K(C)/Fl(X) finite extension




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Counting points on C

Nr := #C(Flr) cardinality of Flr-rational points

: C C Frobenius-automorphismxi

xli

Lefschetz-Trace-Formula

Nr = #{Fix points of C(Fl) under r}

=2

n=0

(1)nTrr|Hn(C)




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-function of C, WEIL (1948)

C(s) : =

x |C|

1

1 (#k(x))s, s C, (s) > 1,

= exp

r=1

Nrtr

r

, t = ls

=2

n=0

det(1

t

|Hn(C))(1)

n+1

=det(1 t|Pic0(C))

(1 t)(1 lt) Q(t)



Cl i l I h


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Riemann hypothesis for C

C is a rational function in t and has

poles at: s = 0, s = 1

zeroes at certain s = satisfying () = 12 .

Can the Riemann -function also be expressed as rationalfunction?



Cl i l I th

p-adic -functions

M i C j t


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Main Conjecture





p-adic -functions

Main Conjecture


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Main Conjecture

Tower of number fields

Studying the class number formula in a whole tower of numberfields simultaneously:

Q F1 . . . Fn Fn+1 . . . F :=

n0

Fn.

with Fn := Q(pn), 1 n ,

Zp = limn

Z/pnZ Qp := Quot(Zp),

: G := G(F/Q)=

// Zp ,

g(pn) = (g)pn for all g G, n 0

F

Fn

Q

G




p-adic -functions

Main Conjecture


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Main Conjecture

Zp = Z/(p 1)Z Zp, i.e. G = with

= G(F1/Q) = Z/(p 1)Z and = G(F/F1) = < > = Zp.

Iwasawa-Algebra

(G) := limGG open

Zp[G/G] = Zp[][[T]]

with T := 1.




p-adic -functions

Main Conjecture


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Main Conjecture

p-adic functions

Maximal ring of quotients of (G) : Q(G) =

p1

i=1 Q(

Zp[[T]]).

Z = (Z1(T), . . . , Zp1(T)) Q(G) are functions on Zp : for n N

Z(n) := Zi(n)(()n 1) Qp {}, i(n) nmod (p 1)




p-adic -functions

Main Conjecture


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Main Conjecture

Analytic p-adic -function

KUBOTA, LEOPOLDT and IWASAWA:

p Q(G) such that for k < 0

p(k) = (1 pk)(k),

i.e. p interpolates - up to the Euler-factor at p - the Riemann

-function p-adically.




p-adic -functions

Main Conjecture


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Main Conjecture

Ideal class group over F

IWASAWA: #Cl(Fn)(p) = pnrkZp(X)+const where

X := X(F) = limn

Cl(Fn)(p) with G-action,

Zp(1) := limn

pn with G-action,

X Zp Qp, Qp(1) := Zp(1) Zp Qp

finite-dimensional Qp-vector spaces with operation by .




p-adic -functions

Main Conjecture


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y


j

Iwasawa Main Conjecture

MAZUR and WILES (1986):

p det(1

T

|X

Zp Qp)

det(1 T|Qp(1)) mod (G)

i

det(1 T|Hi)(1)i+1 mod (G)

analytic algebraic p-adic -function

Trace formula




p-adic -functions

Main Conjecture


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y


j

The analogy

function field number field

Fl = Fl() F = Q((p)) C GmC p

Pic0(C) X= Pic(F)





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Non-commutative

Iwasawa Theory





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From Gm to arbitrary representations

up to now:

coefficients in cohomology: Zp(1)

Gm, (p)

33fffffffffffffffff

,,XXXXXXXXX

XXXXXXXX

tower of number fields: F = Q((p))





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Generalisations

: GQ GL(V)(continuous) representation with V = Qnpand Galois-stable lattice T = Znp.

coefficients in cohomology: T

33hhhhhhhhhhhhhhh

**VVVVVVV

VVVVVVV

tower of fields: K = Qker()

Example: E elliptic curve over Q,T = TpE = lim

n

E[pn] = Z2p, V := T Zp Qp.




N t ti I Th


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p-adic Lie extensions

F K such that

G:= G(K/Q)

GLn(Zp)

is a p-adic Lie group

with subgroup H such that

:= G/H = Zp

KH

zzzzzzzz

F

Q

G




Non commutative Iwasawa Theory


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Philosophy

Attach to (, V)

analytic p-adic L-function L(V, K) with interpolationproperty

L(V, K) L(1, V )for : G GLn(Zp) with finite image.algebraic p-adic L-function F(V, K).

Problem: (G) in general not commutative! Non-commutativedeterminants?






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Non-commutative Iwasawa Main Conjecture

COATES, FUKAYA, KATO, SUJATHA, V.:

There exists a canonical localisation (G)S of (G), such thatF(V, K) exists in

K1((G)S).

Also L(V, K) should live in this K-group.

Main Conjecture:

L(V, K) F(V, K) mod K1((G)).






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Non-commutative characteristic polynomials

M (G)-module, which is finitely generated (H)-moduleBURNS, SCHNEIDER, V.:

(

G)S

(H) M

1

=

// (

G)S

(H) M

induces det(1 T|M) K1((G)S).Main Conjecture over K :

Trace formula" in K1((G)S) mod K1((G)):L(K,Zp(1)) det(1 T|H(K,Zp(1)))






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New Congruences

If G = Zp Zp and coefficients: Zp(1)

KATO: K1((G))

//

iOi[[T]]

K1((G)S) // i Quot(Oi[[T]])

L(K/Q) // (Lp(i, F))i

Existence of L(K/Q) congruences between Lp(i, F)Main Conjecture /K Main Conjecture /F for all i

Similar results by RITTER, WEISS for finite H.Otmar Venjakob Are -functions able to solve Diophantine equations?





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A theorem for totally real fields

F totally real, Fcyc K totally real,G = Zp Zp

MAHESH KAKDE, a student of Coates, recently announced:

Theorem (Kakde)

Assume Leopoldts conjecture for F. Then the non-commutative

Main Conjecture for the Tate motive (i.e. for = Gm) holds overK/F.


are l functions able to solve diophantine equations

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