Cyclotomic Fields Zp-Extensions Iwasawa Modules An Introduction to Iwasawa Theory The Main Conjecture Jordan Schettler University of California, Santa Barbara ?/?/2012 Outline Cyclotomic Fields Zp-Extensions Iwasawa 1 Cyclotomic Fields Modules The Main Conjecture 2 Zp-Extensions 3 Iwasawa Modules 4 The Main Conjecture Cyclotomic Fields Zp-Extensions Iwasawa Modules The Main Conjecture Cyclotomic Fields Throughout m; n are positive integers. Notation Cyclotomic Fields Zp-Extensions Iwasawa Modules The Main Unless noted otherwise, p is an odd prime: 3, 5, 7, 11, 13, Conjecture 17, 19 ::: Notation Cyclotomic Fields Zp-Extensions Iwasawa Modules The Main Unless noted otherwise, p is an odd prime: 3, 5, 7, 11, 13, Conjecture 17, 19 ::: Throughout m; n are positive integers. E.g., Diophantus noticed there are no tuples of integers (x; y; z) s.t. x2 + y 2 = 4z + 3 We know 91ly many tuples of relatively prime positive integers (x; y; z) s.t. x2 + y 2 = z2 Diophantus of Alexandria (≈ AD 207–291) Cyclotomic Arithmetica: integer or Fields rational solutions to -Extensions Zp polynomial equations w/ Iwasawa Modules integer coeff.s The Main Conjecture We know 91ly many tuples of relatively prime positive integers (x; y; z) s.t. x2 + y 2 = z2 Diophantus of Alexandria (≈ AD 207–291) Cyclotomic Arithmetica: integer or Fields rational solutions to -Extensions Zp polynomial equations w/ Iwasawa Modules integer coeff.s The Main Conjecture E.g., Diophantus noticed there are no tuples of integers (x; y; z) s.t. x2 + y 2 = 4z + 3 Diophantus of Alexandria (≈ AD 207–291) Cyclotomic Arithmetica: integer or Fields rational solutions to -Extensions Zp polynomial equations w/ Iwasawa Modules integer coeff.s The Main Conjecture E.g., Diophantus noticed there are no tuples of integers (x; y; z) s.t. x2 + y 2 = 4z + 3 We know 91ly many tuples of relatively prime positive integers (x; y; z) s.t. x2 + y 2 = z2 Fermat proved this conjecture in the case n = 4 by the method of infinite descent. Thus the conjecture boils down to the case n = p. Pierre de Fermat (1601–1665) Cyclotomic Fields Fermat claimed that if n ≥ 3, there Zp-Extensions are no tuples of positive integers Iwasawa n n n Modules (x; y; z) s.t. x + y = z The Main Conjecture Thus the conjecture boils down to the case n = p. Pierre de Fermat (1601–1665) Cyclotomic Fields Fermat claimed that if n ≥ 3, there Zp-Extensions are no tuples of positive integers Iwasawa n n n Modules (x; y; z) s.t. x + y = z The Main Conjecture Fermat proved this conjecture in the case n = 4 by the method of infinite descent. Pierre de Fermat (1601–1665) Cyclotomic Fields Fermat claimed that if n ≥ 3, there Zp-Extensions are no tuples of positive integers Iwasawa n n n Modules (x; y; z) s.t. x + y = z The Main Conjecture Fermat proved this conjecture in the case n = 4 by the method of infinite descent. Thus the conjecture boils down to the case n = p. As ideals in Z[ζp]: (z)p = (xp + y p) p−1 Y j = (x + ζpy) j=0 Note: Z[ζp] is a Dedekind domain but not necessarily a PID. Ernst Kummer (1810–1893) Cyclotomic Fields Suppose (x; y; z) is a tuple of pairwise relatively prime p p p Zp-Extensions integers s.t. x + y = z . Iwasawa Modules The Main Conjecture Note: Z[ζp] is a Dedekind domain but not necessarily a PID. Ernst Kummer (1810–1893) Cyclotomic Fields Suppose (x; y; z) is a tuple of pairwise relatively prime p p p Zp-Extensions integers s.t. x + y = z . Iwasawa Modules The Main Conjecture As ideals in Z[ζp]: (z)p = (xp + y p) p−1 Y j = (x + ζpy) j=0 Ernst Kummer (1810–1893) Cyclotomic Fields Suppose (x; y; z) is a tuple of pairwise relatively prime p p p Zp-Extensions integers s.t. x + y = z . Iwasawa Modules The Main Conjecture As ideals in Z[ζp]: (z)p = (xp + y p) p−1 Y j = (x + ζpy) j=0 Note: Z[ζp] is a Dedekind domain but not necessarily a PID. p Thus (x + ζpy) = J is the pth power of an ideal J in Z[ζp]. If p - class number of Q(ζp), then J = (α) is principal, p but x + ζpy = εα (some unit ") leads to a contradiction. Kummer Continued Cyclotomic Fields j Zp-Extensions If p - xyz, the ideals (x + ζpy) are pairwise relatively prime. Iwasawa Modules The Main Conjecture If p - class number of Q(ζp), then J = (α) is principal, p but x + ζpy = εα (some unit ") leads to a contradiction. Kummer Continued Cyclotomic Fields j Zp-Extensions If p - xyz, the ideals (x + ζpy) are pairwise relatively prime. Iwasawa Modules The Main Conjecture p Thus (x + ζpy) = J is the pth power of an ideal J in Z[ζp]. p but x + ζpy = εα (some unit ") leads to a contradiction. Kummer Continued Cyclotomic Fields j Zp-Extensions If p - xyz, the ideals (x + ζpy) are pairwise relatively prime. Iwasawa Modules The Main Conjecture p Thus (x + ζpy) = J is the pth power of an ideal J in Z[ζp]. If p - class number of Q(ζp), then J = (α) is principal, Kummer Continued Cyclotomic Fields j Zp-Extensions If p - xyz, the ideals (x + ζpy) are pairwise relatively prime. Iwasawa Modules The Main Conjecture p Thus (x + ζpy) = J is the pth power of an ideal J in Z[ζp]. If p - class number of Q(ζp), then J = (α) is principal, p but x + ζpy = εα (some unit ") leads to a contradiction. There are 1ly many irregular primes: 37; 59; 67; 101;::: but it’s unknown if there are 1ly many regular primes Theorem (Kummer’s Criterion) An odd prime p is regular if and only if p - numerator of Bj x P1 xn for all j = 2; 4;:::; p − 3 where ex −1 = n=0 Bn n! . Regular versus Irregular Primes Cyclotomic We say a prime p is regular if p - class number of Q(ζp). Fields Zp-Extensions Iwasawa Modules The Main Conjecture Theorem (Kummer’s Criterion) An odd prime p is regular if and only if p - numerator of Bj x P1 xn for all j = 2; 4;:::; p − 3 where ex −1 = n=0 Bn n! . Regular versus Irregular Primes Cyclotomic We say a prime p is regular if p - class number of Q(ζp). Fields Zp-Extensions Iwasawa Modules There are 1ly many irregular The Main primes: 37; 59; 67; 101;::: Conjecture but it’s unknown if there are 1ly many regular primes Regular versus Irregular Primes Cyclotomic We say a prime p is regular if p - class number of Q(ζp). Fields Zp-Extensions Iwasawa Modules There are 1ly many irregular The Main primes: 37; 59; 67; 101;::: Conjecture but it’s unknown if there are 1ly many regular primes Theorem (Kummer’s Criterion) An odd prime p is regular if and only if p - numerator of Bj x P1 xn for all j = 2; 4;:::; p − 3 where ex −1 = n=0 Bn n! . Theorem (Bernoulli’s/Faulhaber’s Formula) n+1 n m X n + 1 Bj 1n + 2n + 3n + ··· + mn = n + 1 j (−m)j j=0 x P1 x n Bernoulli Numbers: ex −1 = n=0 Bn n! Cyclotomic Fields 1 B0 = 1; B1 = − 2 while B2n+1 = 0 for all n and Zp-Extensions Iwasawa 1 1 1 Modules B = ; B = − ; B = ; 2 6 4 30 6 42 The Main Conjecture 1 5 691 B = − ; B = ; B = − 8 30 10 66 12 2730 x P1 x n Bernoulli Numbers: ex −1 = n=0 Bn n! Cyclotomic Fields 1 B0 = 1; B1 = − 2 while B2n+1 = 0 for all n and Zp-Extensions Iwasawa 1 1 1 Modules B = ; B = − ; B = ; 2 6 4 30 6 42 The Main Conjecture 1 5 691 B = − ; B = ; B = − 8 30 10 66 12 2730 Theorem (Bernoulli’s/Faulhaber’s Formula) n+1 n m X n + 1 Bj 1n + 2n + 3n + ··· + mn = n + 1 j (−m)j j=0 B2n has numerator = ??? Theorem (Kummer Congruences) If n ≡ m 6≡ −1 (mod p − 1), B B n+1 ≡ m+1 (mod p) n + 1 m + 1 More About Bernoulli Numbers In lowest terms, Cyclotomic Fields Y B has denominator = p Zp-Extensions 2n Iwasawa p−1j2n Modules The Main Conjecture Theorem (Kummer Congruences) If n ≡ m 6≡ −1 (mod p − 1), B B n+1 ≡ m+1 (mod p) n + 1 m + 1 More About Bernoulli Numbers In lowest terms, Cyclotomic Fields Y B has denominator = p Zp-Extensions 2n Iwasawa p−1j2n Modules The Main Conjecture B2n has numerator = ??? More About Bernoulli Numbers In lowest terms, Cyclotomic Fields Y B has denominator = p Zp-Extensions 2n Iwasawa p−1j2n Modules The Main Conjecture B2n has numerator = ??? Theorem (Kummer Congruences) If n ≡ m 6≡ −1 (mod p − 1), B B n+1 ≡ m+1 (mod p) n + 1 m + 1 In 1913, Ramanujan suggested 1 1 + 2 + 3 + ··· = − 12 In general, n n n Bn+1 1 + 2 + 3 + ··· = − n+1 Zeta Values In 1735, Euler showed 2 Cyclotomic 1 + 1 + 1 + ··· = π Fields 12 22 32 6 Zp-Extensions Iwasawa In general, Modules 2n 1 + 1 + ··· = jB2nj(2π) The Main 12n 22n 2(2n)! Conjecture Zeta Values In 1735, Euler showed 2 Cyclotomic 1 + 1 + 1 + ··· = π Fields 12 22 32 6 Zp-Extensions Iwasawa In general, Modules 2n 1 + 1 + ··· = jB2nj(2π) The Main 12n 22n 2(2n)! Conjecture In 1913, Ramanujan suggested 1 1 + 2 + 3 + ··· = − 12 In general, n n n Bn+1 1 + 2 + 3 + ··· = − n+1 9!meromorphic continuation of ζ s.t.