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

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 university-logo Mathematical Institute University of Heidelberg CMS Winter 2007 Meeting Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Leibniz (1673) 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + · · · = π 4 university-logo (already known to G REGORY and M ADHAVA ) Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Special values of L -functions university-logo Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory ( Z / N Z ) × N ≥ 1 , Z / N Z . units of ring Dirichlet Character (modulo N ) : χ : ( Z / N Z ) × → C × extends to N � χ ( n mod N ) , ( n , N ) = 1; university-logo χ ( n ) := 0 , otherwise. Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Dirichlet L -function w.r.t. χ : ∞ χ ( n ) � L ( s , χ ) = n s , s ǫ C , ℜ ( s ) > 1 . n = 1 satisfies: - Euler product 1 � L ( s , χ ) = 1 − χ ( p ) p − s , university-logo p - meromorphic continuation to C , - functional equation. Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Examples χ ≡ 1 : Riemann ζ -function ∞ 1 1 � � ζ ( s ) = n s = 1 − p − s , p n = 1 χ 1 : ( Z / 4 Z ) × = { 1 , 3 } → C × , χ 1 ( 1 ) = 1 , χ 1 ( 3 ) = − 1 university-logo L ( 1 , χ 1 ) = 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + · · · = π 4 Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Examples χ ≡ 1 : Riemann ζ -function ∞ 1 1 � � ζ ( s ) = n s = 1 − p − s , p n = 1 χ 1 : ( Z / 4 Z ) × = { 1 , 3 } → C × , χ 1 ( 1 ) = 1 , χ 1 ( 3 ) = − 1 university-logo L ( 1 , χ 1 ) = 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + · · · = π 4 Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Diophantine Equations university-logo Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Conjectures of Catalan and Fermat p , q prime numbers Catalan (1844), Theorem(M IH ˇ AILESCU , 2002): x p − y q = 1 , has unique solution 3 2 − 2 3 = 1 in Z with x , y > 0 . university-logo Fermat (1665), Theorem(W ILES et al., 1994): x p + y p = z p , p > 2 , has no solution in Z with xyz � = 0 . Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Conjectures of Catalan and Fermat p , q prime numbers Catalan (1844), Theorem(M IH ˇ AILESCU , 2002): x p − y q = 1 , has unique solution 3 2 − 2 3 = 1 in Z with x , y > 0 . university-logo Fermat (1665), Theorem(W ILES et al., 1994): x p + y p = z p , p > 2 , has no solution in Z with xyz � = 0 . Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Factorisation over larger ring of integers ζ m primitive m th root of unity Z [ ζ m ] the ring of integers of Q ( ζ m ) , e.g. for m = 4 with i 2 = − 1 we have in Z [ i ] = { a + bi | a , b ǫ Z } : x 3 − y 2 = 1 ⇔ x 3 = ( y + i )( y − i ) university-logo and for m = p n we have in Z [ ζ p n ] : x p n + y p n = ( x + y )( x + ζ p n y )( x + ζ 2 p n y ) · . . . · ( x + ζ p n − 1 y ) = z p n . p n Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory 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 [ ζ m ] is not a unique factorisation domain university-logo (UFD), e.g. Z [ ζ 23 ] ! Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory 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 [ ζ m ] is not a unique factorisation domain university-logo (UFD), e.g. Z [ ζ 23 ] ! Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Ideals Kummer: Replace numbers by ‘ideal numbers’: For ideals(= Z [ ζ m ] -submodules) 0 � = a ⊆ Z [ ζ m ] we have unique factorisation into prime ideals P i � = 0 : n � P n i a = i university-logo i = 1 Principal ideals: ( a ) = Z [ ζ m ] a Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Ideal class group Cl ( Q ( ζ m )) : = { ideals of Z [ ζ m ] } / { principal ideals of Z [ ζ m ] } ∼ = Pic ( Z [ ζ m ]) Fundamental Theorem of algebraic number theory: # Cl ( Q ( ζ m )) < ∞ university-logo and h Q ( ζ m ) := # Cl ( Q ( ζ m )) = 1 ⇔ Z [ ζ m ] is a UFD Nevertheless, Cl ( Q ( ζ m )) is difficult to determine, too many relations! Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory The L -function solves the problem How can we compute h Q ( i ) ? It is a mystery that L ( s , χ 1 ) university-logo knows the answer! Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory The L -function solves the problem How can we compute h Q ( i ) ? It is a mystery that L ( s , χ 1 ) university-logo knows the answer! Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory The cyclotomic character Gauß: κ N � ( Z / N Z ) × G ( Q ( ζ N ) / Q ) ∼ = with g ( ζ N ) = ζ κ N ( g ) for all g ǫ G ( Q ( ζ N ) / Q ) N N = 4 : university-logo ⇒ χ 1 is character of Galois group G ( Q ( i ) / Q ) ⇒ L ( s , χ 1 ) (analytic) invariant of Q ( i ) . Otmar Venjakob Are ζ -functions able to solve Diophantine equations?

ζ -functions and Diophantine equations L -functions The function field case Diophantine Equations Classical Iwasawa theory The analytic class number formula Non-commutative Iwasawa Theory Analytic class number formula for imaginary quadratic number fields: √ # µ ( Q ( i )) N h Q ( i ) = L ( 1 , χ 1 ) 2 π 4 · 2 = 2 π L ( 1 , χ 1 ) university-logo 4 = π L ( 1 , χ 1 ) = 1 (by Leibniz’ formula) ⇒ Z [ i ] is a UFD. Otmar Venjakob Are ζ -functions able to solve Diophantine equations?