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Are -functions able to solve Diophantine equations? An - - PowerPoint PPT Presentation

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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

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Are ζ-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 ζ-functions able to solve Diophantine equations?

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university-logo ζ-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 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + · · · = π 4 (already known to GREGORY and MADHAVA)

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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 ζ-functions able to solve Diophantine equations?

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university-logo ζ-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) : χ : (Z/NZ)× → C× extends to N χ(n) := χ(n mod N), (n, N) = 1; 0,

  • therwise.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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. χ : L(s, χ) =

∞

  • n=1

χ(n) ns , s ǫ C, ℜ(s) > 1. satisfies:

  • Euler product

L(s, χ) =

  • p

1 1 − χ(p)p−s ,

  • meromorphic continuation to C,
  • functional equation.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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

χ ≡ 1 : Riemann ζ-function ζ(s) =

∞

  • n=1

1 ns =

  • p

1 1 − p−s , χ1 : (Z/4Z)× = {1, 3} → C×, χ1(1) = 1, χ1(3) = −1 L(1, χ1) = 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + · · · = π 4

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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

χ ≡ 1 : Riemann ζ-function ζ(s) =

∞

  • n=1

1 ns =

  • p

1 1 − p−s , χ1 : (Z/4Z)× = {1, 3} → C×, χ1(1) = 1, χ1(3) = −1 L(1, χ1) = 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + · · · = π 4

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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 ζ-functions able to solve Diophantine equations?

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university-logo ζ-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(MIH ˇ

AILESCU, 2002):

xp − yq = 1, has unique solution 32 − 23 = 1 in 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 ζ-functions able to solve Diophantine equations?

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university-logo ζ-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(MIH ˇ

AILESCU, 2002):

xp − yq = 1, has unique solution 32 − 23 = 1 in 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 ζ-functions able to solve Diophantine equations?

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university-logo ζ-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

ζm primitive mth root of unity Z[ζm] the ring of integers of Q(ζm), e.g. for m = 4 with i2 = −1 we have in Z[i] = {a + bi|a, b ǫ Z} : x3 − y2 = 1 ⇔ x3 = (y + i)(y − i) and for m = pn we have in Z[ζpn] : xpn+ypn = (x + y)(x + ζpny)(x + ζ2

pny) · . . . · (x + ζpn−1 pn

y) = zpn.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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 factorisation to conclude a contradiction from the assumption that the Catalan or Fermat equation has a non-trivial solution. Problem: In general, Z[ζm] is not a unique factorisation domain (UFD), e.g. Z[ζ23]!

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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 factorisation to conclude a contradiction from the assumption that the Catalan or Fermat equation has a non-trivial solution. Problem: In general, Z[ζm] is not a unique factorisation domain (UFD), e.g. Z[ζ23]!

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations 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[ζm]-submodules) 0 = a ⊆ Z[ζm] we have unique factorisation into prime ideals Pi = 0 : a =

n

  • i=1

Pni

i

Principal ideals: (a) = Z[ζm]a

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations 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(ζm)) : = { ideals of Z[ζm]}/{ principal ideals of Z[ζm]} ∼ = Pic(Z[ζm]) Fundamental Theorem of algebraic number theory: #Cl(Q(ζm)) < ∞ and hQ(ζm) := #Cl(Q(ζm)) = 1 ⇔ Z[ζm] is a UFD Nevertheless, Cl(Q(ζm)) is difficult to determine, too many relations!

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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 L-function solves the problem

How can we compute hQ(i)? It is a mystery that L(s, χ1) knows the answer!

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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 L-function solves the problem

How can we compute hQ(i)? It is a mystery that L(s, χ1) knows the answer!

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-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 cyclotomic character

Gauß: G(Q(ζN)/Q)

κN ∼ =

(Z/NZ)×

with g(ζN) = ζκN(g)

N

for all g ǫ G(Q(ζN)/Q) N = 4 : ⇒ χ1 is character of Galois group G(Q(i)/Q) ⇒ L(s, χ1) (analytic) invariant of Q(i).

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory L-functions Diophantine Equations The analytic class number formula

Analytic class number formula for imaginary quadratic number fields: hQ(i) = #µ(Q(i)) √ N 2π L(1, χ1) = 4 · 2 2π L(1, χ1) = 4 πL(1, χ1) = 1 (by Leibniz’ formula) ⇒ Z[i] is a UFD.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory L-functions Diophantine Equations The analytic class number formula

Analytic class number formula for imaginary quadratic number fields: hQ(i) = #µ(Q(i)) √ N 2π L(1, χ1) = 4 · 2 2π L(1, χ1) = 4 πL(1, χ1) = 1 (by Leibniz’ formula) ⇒ Z[i] is a UFD.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory L-functions Diophantine Equations The analytic class number formula

Analytic class number formula for imaginary quadratic number fields: hQ(i) = #µ(Q(i)) √ N 2π L(1, χ1) = 4 · 2 2π L(1, χ1) = 4 πL(1, χ1) = 1 (by Leibniz’ formula) ⇒ Z[i] is a UFD.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory L-functions Diophantine Equations The analytic class number formula

A special case of the Catalan equation

Since ‘L(s, χ1) knows the arithmetic’ of Q(i), it is able to solve

  • ur 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 ǫ Z taking Re(−) and Im(−) gives: y = 0, contradiction!

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory L-functions Diophantine Equations The analytic class number formula

Regular primes

Similarly ζ(s) ‘knows’ for which p Cl(Q(ζp))(p) = 1 holds! Then the Fermat equation does not have any non-trivial

  • solution. But 37|hCl(Q(ζ37))!

Iwasawa: Cl(Q(ζpn))(p) =? for n ≥ 1.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

The function field case

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Number fields versus function fields

Q ← → Fl(X) = K(P1

Fl)

K/Q number field ← → K(C)/Fl(X) function field C ⊆ Pn

Fl

smooth, projective curve, i.e. K(C)/Fl(X) finite extension

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Counting points on C

Nr := #C(Flr ) cardinality of Flr -rational points φ : C → C Frobenius-automorphism xi → xl

i

Lefschetz-Trace-Formula Nr = #{Fix points of C(Fl) under φr} =

2

  • n=0

(−1)nTr

  • φr|Hn(C)
  • Otmar Venjakob

Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

ζ-function of C, WEIL (1948)

ζC(s) : =

  • x ǫ |C|

1 1 − (#k(x))−s , s ǫ C, ℜ(s) > 1, = exp ∞

  • r=1

Nr tr r

  • ,

t = l−s =

2

  • n=0

det(1 − φt|Hn(C))(−1)n+1 = det(1 − φt|“Pic0(C)”) (1 − t)(1 − lt) ǫ Q(t)

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Riemann hypothesis for C

ζC is a rational function in t and has poles at: s = 0, s = 1 zeroes at certain s = α satisfying ℜ(α) = 1

2.

Can the Riemann ζ-function also be expressed as rational function?

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory p-adic ζ-functions Main Conjecture

Classical Iwasawa theory

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory p-adic ζ-functions Main Conjecture

Tower of number fields

Studying the class number formula in a whole tower of number fields simultaneously: Q ⊆ F1 ⊆ . . . ⊆ Fn ⊆ Fn+1 ⊆ . . . ⊆ F∞ :=

  • n≥0

Fn. with Fn := Q(ζpn), 1 ≤ n ≤ ∞, Zp = lim ← −

n

Z/pnZ ⊆ Qp := Quot(Zp), κ : G := G(F∞/Q)

∼ =

Z×

p ,

g(ζpn) = ζκ(g)

pn

for all g ǫ G, n ≥ 0 F∞ Fn Q

G

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory p-adic ζ-functions Main Conjecture

Z×

p ∼

= Z/(p − 1)Z × Zp, i.e. G = ∆ × Γ with ∆ = G(F1/Q) ∼ = Z/(p − 1)Z and Γ = G(F∞/F1) = < γ > ∼ = Zp. Iwasawa-Algebra Λ(G) := lim ← −

G′G open

Zp[G/G′] ∼ = Zp[∆][[T]] with T := γ − 1.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory p-adic ζ-functions Main Conjecture

p-adic functions

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

p−1

  • i=1

Q(Zp[[T]]). Z = (Z1(T), . . . , Zp−1(T)) ǫ Q(G) are functions on Zp : for n ǫ N Z(n) := Zi(n)(κ(γ)n − 1) ǫ Qp ∪ {∞}, i(n) ≡ n mod (p − 1)

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory p-adic ζ-functions Main Conjecture

Analytic p-adic ζ-function

KUBOTA, LEOPOLDT and IWASAWA: ζp ǫ Q(G) such that for k < 0 ζp(k) = (1 − p−k)ζ(k), i.e. ζp interpolates - up to the Euler-factor at p - the Riemann ζ-function p-adically.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory p-adic ζ-functions Main Conjecture

Ideal class group over F∞

IWASAWA: #Cl(Fn)(p) = pnrkZp(X)+const where X := X(F∞) = lim ← −

n

Cl(Fn)(p) with G-action, Zp(1) := lim ← −

n

µpn with G-action, X − ⊗Zp Qp, Qp(1) := Zp(1) ⊗Zp Qp finite-dimensional Qp-vector spaces with operation by γ.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory p-adic ζ-functions Main Conjecture

Iwasawa Main Conjecture

MAZUR and WILES (1986): ζp ≡ det(1 − γT|X − ⊗Zp Qp) det(1 − γT|Qp(1)) mod Λ(G)× ≡

  • i

det(1 − γT|Hi)(−1)i+1 mod Λ(G)× analytic algebraic p-adic ζ-function ‘Trace formula’

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory p-adic ζ-functions Main Conjecture

The analogy

function field number field Fl = Fl(µ) F∞ = Q(µ(p)) φ γ C Gm ζC ζp Pic0(C) X‘=’ Pic(F∞)

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Non-commutative Iwasawa Theory

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

From Gm to arbitrary representations

up to now: coefficients in cohomology: Zp(1) Gm, µ(p)

  • tower of number fields: F∞ = Q(µ(p))

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Generalisations

ρ : GQ → GL(V) (continuous) representation with V ∼ = Qn

p

and Galois-stable lattice T ∼ = Zn

p.

coefficients in cohomology: T ρ

  • tower of fields: K∞ = Q

ker(ρ)

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

n

E[pn] ∼ = Z2

p, V := T ⊗Zp Qp.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory 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 Γ := G/H ∼ = Zp K∞

H

  • F∞

Γ

Q

G

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Philosophy

Attach to (ρ, V) analytic p-adic L-function L(V, K∞) with interpolation property L(V, K∞) ∼ L(1, V ⊗ χ) for χ : G → GLn(Zp) with finite image. algebraic p-adic L-function F(V, K∞). Problem: Λ(G) in general not commutative! Non-commutative determinants?

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Non-commutative Iwasawa Main Conjecture

COATES, FUKAYA, KATO, SUJATHA, V.: There exists a canonical localisation Λ(G)S of Λ(G), such that F(V, K∞) exists in K1(Λ(G)S). Also L(V, K∞) should live in this K-group. Main Conjecture: L(V, K∞) ≡ F(V, K∞) mod K1(Λ(G)).

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

Non-commutative characteristic polynomials

M Λ(G)-module, which is finitely generated Λ(H)-module BURNS, SCHNEIDER, V.: Λ(G)S ⊗Λ(H) M

“1−γ” ∼ =

Λ(G)S ⊗Λ(H) M

induces “ det(1 − γT|M)” ǫ K1(Λ(G)S). Main Conjecture over K∞ : “Trace formula" in K1(Λ(G)S) mod K1(Λ(G)): L(K∞, Zp(1)) ≡ det(1 − γT|H•(K∞, Zp(1)))

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory Non-commutative Iwasawa Theory

New Congruences

If G ∼ = Zp ⋊ Z×

p and coefficients: Zp(1)

KATO: K1(Λ(G))

  • χi Oi[[T]]×
  • K1(Λ(G)S)

χi Quot(Oi[[T]])×

L(K∞/Q)

(Lp(χi, F∞))i

Existence of L(K∞/Q) ⇐ ⇒ congruences between Lp(χi, F∞) Main Conjecture /K∞ ⇐ ⇒ Main Conjecture /F∞ for all χi Similar results by RITTER, WEISS for finite H.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?

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university-logo ζ-functions and Diophantine equations The function field case Classical Iwasawa theory 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 Leopoldt’s conjecture for F. Then the non-commutative Main Conjecture for the Tate motive (i.e. for = Gm) holds over K∞/F.

Otmar Venjakob Are ζ-functions able to solve Diophantine equations?