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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-functions and Diophantine equations
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
Leibniz (1673)
1 13
+ 15
17
+ 19
111
+ = 4
(already known to GREGORY and MADHAVA)
Otmar Venjakob Are -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
L-functions
Diophantine Equations
The analytic class number formula
Special values of L-functions
Otmar Venjakob Are -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
L-functions
Diophantine Equations
The analytic class number formula
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.
Otmar Venjakob Are -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
L-functions
Diophantine Equations
The analytic class number formula
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.
Otmar Venjakob Are -functions able to solve Diophantine equations?
f i d Di h i i
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-functions and Diophantine equations
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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
Otmar Venjakob Are -functions able to solve Diophantine equations?
f ti d Di h ti ti
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-functions and Diophantine equations
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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
Otmar Venjakob Are -functions able to solve Diophantine equations?
functions and Diophantine equations
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-functions and Diophantine equations
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
Diophantine Equations
Otmar Venjakob Are -functions able to solve Diophantine equations?
functions and Diophantine equations
-
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-functions and Diophantine equations
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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?
-functions and Diophantine equations
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-functions and Diophantine equations
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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?
-functions and Diophantine equations
-
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functions and Diophantine equations
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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.
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsL f ti
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functions and Diophantine equations
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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]!
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsL f ti
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p q
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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]!
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsL functions
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p q
The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsL functions
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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L-functions
Diophantine Equations
The analytic class number formula
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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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L functions
Diophantine Equations
The analytic class number formula
The L-function solves the problem
How can we compute
hQ(i)?
It is a mystery that
L(s, 1)
knows the answer!
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equations
Th f i fi ldL-functions
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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
L functions
Diophantine Equations
The analytic class number formula
The L-function solves the problem
How can we compute
hQ(i)?
It is a mystery that
L(s, 1)
knows the answer!
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equations
Th f ti fi ldL-functions
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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
Diophantine Equations
The analytic class number formula
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).
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equations
The function field caseL-functions
-
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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
Diophantine Equations
The analytic class number formula
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.
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equations
The function field caseL-functions
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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
Diophantine Equations
The analytic class number formula
Analytic class number formula for imaginary quadratic numberfields:
hQ(i) = #(Q(i))N2
L(1, 1)
=4 22
L(1, 1)
=
4
L(1, 1) = 1 (by Leibniz formula)
Z[i] is a UFD.
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
L-functions
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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
Diophantine Equations
The analytic class number formula
Analytic class number formula for imaginary quadratic numberfields:
hQ(i) = #(Q(i))N2
L(1, 1)
=4 22
L(1, 1)
=
4
L(1, 1) = 1 (by Leibniz formula)
Z[i] is a UFD.
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
L-functions
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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
Diophantine Equations
The analytic class number formula
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!
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
L-functions
Di h i E i
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The function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
Diophantine Equations
The analytic class number formula
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.
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
-
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
The function field case
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
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
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
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)
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
-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)
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Cl i l I h
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
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?
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Cl i l I th
p-adic -functions
M i C j t
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
Main Conjecture
Classical Iwasawa theory
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
p-adic -functions
Main Conjecture
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
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
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
p-adic -functions
Main Conjecture
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
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.
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
p-adic -functions
Main Conjecture
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
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)
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
p-adic -functions
Main Conjecture
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
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.
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
p-adic -functions
Main Conjecture
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Classical Iwasawa theory
Non-commutative Iwasawa Theory
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 .
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
p-adic -functions
Main Conjecture
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y
Non-commutative Iwasawa Theory
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
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
p-adic -functions
Main Conjecture
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y
Non-commutative Iwasawa Theory
j
The analogy
function field number field
Fl = Fl() F = Q((p)) C GmC p
Pic0(C) X= Pic(F)
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
-
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Non-commutative Iwasawa Theory
Non-commutative
Iwasawa Theory
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
-
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Non-commutative Iwasawa Theory
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))
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
-
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Non-commutative Iwasawa Theory
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.
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
N t ti I Th
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Non-commutative Iwasawa Theory
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
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
Non commutative Iwasawa Theory
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Non-commutative Iwasawa Theory
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?
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
Non commutative Iwasawa Theory
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Non-commutative Iwasawa Theory
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)).
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
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Non-commutative Iwasawa Theory
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)))
Otmar Venjakob Are -functions able to solve Diophantine equations?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
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Non commutative Iwasawa Theory
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?
-functions and Diophantine equationsThe function field case
Classical Iwasawa theory
Non-commutative Iwasawa Theory
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Non commutative Iwasawa Theory
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.
Otmar Venjakob Are -functions able to solve Diophantine equations?
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