An Introduction to Iwasawa Theory the Main Conjecture

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 ∃∞ly 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 ∃∞ly 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)

We know ∃∞ly 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 ε) 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 ε) 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 ε) 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 ε) leads to a contradiction. There are ∞ly many irregular primes: 37, 59, 67, 101,...

but it’s unknown if there are ∞ly many regular primes

Theorem (Kummer’s Criterion)

An odd prime p is regular if and only if p - numerator of Bj x P∞ 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 P∞ 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 ∞ly many irregular The Main primes: 37, 59, 67, 101,... Conjecture but it’s unknown if there are ∞ly 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 ∞ly many irregular The Main primes: 37, 59, 67, 101,... Conjecture but it’s unknown if there are ∞ly many regular primes

Theorem (Kummer’s Criterion)

An odd prime p is regular if and only if p - numerator of Bj x P∞ 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 P∞ 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 P∞ 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−1|2n 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−1|2n 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−1|2n 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 + ··· = |B2n|(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 + ··· = |B2n|(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 ∃!meromorphic continuation of ζ s.t.

ζ(s) = (2π)s−12 sin(sπ/2)ζ(1 − s) (−s)!

Bernhard Riemann (1826–1866)

Cyclotomic Fields

Zp-Extensions For <(s) > 1, Iwasawa Modules ∞ X 1 Y The Main ζ(s) := = (1 − p−s)−1 Conjecture ns n=1 p Bernhard Riemann (1826–1866)

Cyclotomic Fields

Zp-Extensions For <(s) > 1, Iwasawa Modules ∞ X 1 Y The Main ζ(s) := = (1 − p−s)−1 Conjecture ns n=1 p

∃!meromorphic continuation of ζ s.t.

ζ(s) = (2π)s−12 sin(sπ/2)ζ(1 − s) (−s)! Theorem (Kummer Criterion) Then A 6= 0 if and only if p|ζ(−n) for some odd n.

E.g., we have 691|ζ(−11). What does the −11 mean here?

Restatement of Kummer Criterion

Cyclotomic Fields

-Extensions Zp Let A denote the p-primary part of the class group of Q(ζp). Iwasawa Modules

The Main Conjecture E.g., we have 691|ζ(−11). What does the −11 mean here?

Restatement of Kummer Criterion

Cyclotomic Fields

-Extensions Zp Let A denote the p-primary part of the class group of Q(ζp). Iwasawa Modules

The Main Conjecture Theorem (Kummer Criterion) Then A 6= 0 if and only if p|ζ(−n) for some odd n. Restatement of Kummer Criterion

Cyclotomic Fields

-Extensions Zp Let A denote the p-primary part of the class group of Q(ζp). Iwasawa Modules

The Main Conjecture Theorem (Kummer Criterion) Then A 6= 0 if and only if p|ζ(−n) for some odd n.

E.g., we have 691|ζ(−11). What does the −11 mean here? Deﬁne ω : Gal(Q(ζp)/Q) → Zp via

ω(σ) ≡ φ(σ)(mod p) and ω(σ)p−1 = 1

The Teichmuller¨ Character

Cyclotomic Fields × Zp-Extensions ∃isomorphism φ: Gal(Q(ζp)/Q) → (Z/(p)) given by Iwasawa Modules φ(σ) The Main σ(ζp) = ζp Conjecture The Teichmuller¨ Character

Cyclotomic Fields × Zp-Extensions ∃isomorphism φ: Gal(Q(ζp)/Q) → (Z/(p)) given by Iwasawa Modules φ(σ) The Main σ(ζp) = ζp Conjecture

Deﬁne ω : Gal(Q(ζp)/Q) → Zp via

ω(σ) ≡ φ(σ)(mod p) and ω(σ)p−1 = 1 Regard A as a ZpG-module (written additively):

p−1 M −n A = Aω n=1 where −n α ∈ Aω ⇔ σα = ω(σ)−nα ∀σ ∈ G

‘Eigenespace’ Decomposition

Cyclotomic Fields G := Gal(Q(ζp)/Q) acts on A = p-primary part of class grp. Zp-Extensions Iwasawa Modules

The Main Conjecture ‘Eigenespace’ Decomposition

Cyclotomic Fields G := Gal(Q(ζp)/Q) acts on A = p-primary part of class grp. Zp-Extensions Iwasawa Modules

The Main Conjecture Regard A as a ZpG-module (written additively):

p−1 M −n A = Aω n=1 where −n α ∈ Aω ⇔ σα = ω(σ)−nα ∀σ ∈ G − E.g., we have Aω 11 6= 0 for p = 691.

− Conjecturally, Aω n = 0 for all p and all even n.

J. Herbrand (1908–1931), K. Ribet (1948–)

Cyclotomic Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture

Theorem (Herbrand showed ⇒, Ribet showed ⇐) − Let n be odd. Then Aω n 6= 0 ⇔ p|ζ(−n). − Conjecturally, Aω n = 0 for all p and all even n.

J. Herbrand (1908–1931), K. Ribet (1948–)

Cyclotomic Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture

Theorem (Herbrand showed ⇒, Ribet showed ⇐) − Let n be odd. Then Aω n 6= 0 ⇔ p|ζ(−n).

− E.g., we have Aω 11 6= 0 for p = 691. J. Herbrand (1908–1931), K. Ribet (1948–)

Cyclotomic Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture

Theorem (Herbrand showed ⇒, Ribet showed ⇐) − Let n be odd. Then Aω n 6= 0 ⇔ p|ζ(−n).

− E.g., we have Aω 11 6= 0 for p = 691.

− Conjecturally, Aω n = 0 for all p and all even n. Theorem (Kubota and Leopoldt, 1964) j ∃!continuous function Lp(s, ω ) from Zp to Qp s.t. ζ(−n) L (−n, ωj ) = p (1 − pn)−1

whenever n ≡ j − 1 (mod p − 1).

p-Adic Interpolation

Theorem (Higher Kummer Congruences) Cyclotomic Fields If n ≡ m 6≡ −1 (mod p − 1) and n 6= m Zp-Extensions Iwasawa Modules ζ(−n) ζ(−m) − < |n − m|p The Main (1 − pn)−1 (1 − pm)−1 Conjecture p

where | · |p is the normalized p-adic absolute value. p-Adic Interpolation

where | · |p is the normalized p-adic absolute value.

Theorem (Kubota and Leopoldt, 1964) j ∃!continuous function Lp(s, ω ) from Zp to Qp s.t. ζ(−n) L (−n, ωj ) = p (1 − pn)−1

whenever n ≡ j − 1 (mod p − 1). Study A = p-primary part of the class group of Q(ζp)

p−1 M ω−n ‘Eigenspace’ decomposition A = A n=1

n+1 ω−n For n odd, Lp(s, ω ) contains information about A

Summary So Far

Cyclotomic Fields Fermat’s Last Theorem for exponent p (an odd prime) Zp-Extensions Iwasawa Modules

The Main Conjecture p−1 M ω−n ‘Eigenspace’ decomposition A = A n=1

n+1 ω−n For n odd, Lp(s, ω ) contains information about A

Summary So Far

Cyclotomic Fields Fermat’s Last Theorem for exponent p (an odd prime) Zp-Extensions Iwasawa Modules The Main Study A = p-primary part of the class group of (ζ ) Conjecture Q p n+1 ω−n For n odd, Lp(s, ω ) contains information about A

Summary So Far

Cyclotomic Fields Fermat’s Last Theorem for exponent p (an odd prime) Zp-Extensions Iwasawa Modules The Main Study A = p-primary part of the class group of (ζ ) Conjecture Q p

p−1 M ω−n ‘Eigenspace’ decomposition A = A n=1 Summary So Far

Cyclotomic Fields Fermat’s Last Theorem for exponent p (an odd prime) Zp-Extensions Iwasawa Modules The Main Study A = p-primary part of the class group of (ζ ) Conjecture Q p

p−1 M ω−n ‘Eigenspace’ decomposition A = A n=1

n+1 ω−n For n odd, Lp(s, ω ) contains information about A Cyclotomic Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture Zp-Extensions ∼ × = Zp = (roots of unity) × (1 + pZp) =∼ Z ⊕ (p − 1) Zp

so (for p odd)

n × ( (ζ , ζ 2 , ζ 3 ,...)/ ) ∼ ( /( )) Gal Q p p p Q = ←−lim Z p n

First Examples of Zp-Extensions

Cyclotomic Fields We have Zp-Extensions n × Gal( (ζ n )/ ) =∼ ( /(p )) Iwasawa Q p Q Z Modules

The Main Conjecture ∼ × = Zp = (roots of unity) × (1 + pZp) =∼ Z ⊕ (p − 1) Zp

First Examples of Zp-Extensions

Cyclotomic Fields We have Zp-Extensions n × Gal( (ζ n )/ ) =∼ ( /(p )) Iwasawa Q p Q Z Modules so (for p odd) The Main Conjecture n × ( (ζ , ζ 2 , ζ 3 ,...)/ ) ∼ ( /( )) Gal Q p p p Q = ←−lim Z p n = (roots of unity) × (1 + pZp) =∼ Z ⊕ (p − 1) Zp

First Examples of Zp-Extensions

Cyclotomic Fields We have Zp-Extensions n × Gal( (ζ n )/ ) =∼ ( /(p )) Iwasawa Q p Q Z Modules so (for p odd) The Main Conjecture n × ( (ζ , ζ 2 , ζ 3 ,...)/ ) ∼ ( /( )) Gal Q p p p Q = ←−lim Z p n ∼ × = Zp =∼ Z ⊕ (p − 1) Zp

First Examples of Zp-Extensions

Cyclotomic Fields We have Zp-Extensions n × Gal( (ζ n )/ ) =∼ ( /(p )) Iwasawa Q p Q Z Modules so (for p odd) The Main Conjecture n × ( (ζ , ζ 2 , ζ 3 ,...)/ ) ∼ ( /( )) Gal Q p p p Q = ←−lim Z p n ∼ × = Zp = (roots of unity) × (1 + pZp) First Examples of Zp-Extensions

Cyclotomic Fields S n R n Q(ζp ) Zp-Extensions v vv Iwasawa vv Modules vv vv The Main ∞ Q(ζp3 ) Conjecture Q vv vv p vv Z vv vv Q2 Q(ζp2 ) uu Zp uu uu uu uu Q1 Q(ζp) tt tt tt tt Z/(p−1) tt Q There is at least one Zp-extension of any number ﬁeld F.

Namely, there is the cyclotomic Zp-extension FQ∞/F.

Note: If ζp ∈ F, then FQ∞ = F(ζp, ζp2 ,...).

Cyclotomic Zp-Extensions

Cyclotomic Fields Zp-Extensions The ﬁeld Q∞ above is the only Zp-extension of Q. Iwasawa Modules

The Main Conjecture Namely, there is the cyclotomic Zp-extension FQ∞/F.

Note: If ζp ∈ F, then FQ∞ = F(ζp, ζp2 ,...).

Cyclotomic Zp-Extensions

Cyclotomic Fields Zp-Extensions The ﬁeld Q∞ above is the only Zp-extension of Q. Iwasawa Modules

The Main Conjecture There is at least one Zp-extension of any number ﬁeld F. Note: If ζp ∈ F, then FQ∞ = F(ζp, ζp2 ,...).

Cyclotomic Zp-Extensions

Cyclotomic Fields Zp-Extensions The ﬁeld Q∞ above is the only Zp-extension of Q. Iwasawa Modules

The Main Conjecture There is at least one Zp-extension of any number ﬁeld F.

Namely, there is the cyclotomic Zp-extension FQ∞/F. Cyclotomic Zp-Extensions

Cyclotomic Fields Zp-Extensions The ﬁeld Q∞ above is the only Zp-extension of Q. Iwasawa Modules

The Main Conjecture There is at least one Zp-extension of any number ﬁeld F.

Namely, there is the cyclotomic Zp-extension FQ∞/F.

Note: If ζp ∈ F, then FQ∞ = F(ζp, ζp2 ,...). The subﬁelds of F∞ which contain F lie in a tower

F ⊂ F1 ⊂ F2 ⊂ ... ⊂ F∞

s.t. for all n ∼ n Gal(Fn/F) = Z/(p )

General Zp-Extensions

Cyclotomic Fields Now let p be any prime, and suppose F is a number ﬁeld s.t. Zp-Extensions Iwasawa ∼ Modules Gal(F∞/F) = Zp The Main Conjecture General Zp-Extensions

Cyclotomic Fields Now let p be any prime, and suppose F is a number ﬁeld s.t. Zp-Extensions Iwasawa ∼ Modules Gal(F∞/F) = Zp The Main Conjecture

The subﬁelds of F∞ which contain F lie in a tower

F ⊂ F1 ⊂ F2 ⊂ ... ⊂ F∞

s.t. for all n ∼ n Gal(Fn/F) = Z/(p ) Theorem (Iwasawa’s Growth Formula)

Suppose F ⊂ F1 ⊂ F2 ... is a Zp-extension of number ﬁelds. Let An denote the p-primary part of the class group of Fn. Then ∃integers λ, µ, ν s.t.

λn+µpn+ν |An| = p

for all sufﬁciently large n.

Kenkichi Iwasawa (1917–1998)

Cyclotomic Fields Iwasawa spoke about the next result in his

Zp-Extensions talk A theorem on Abelian groups and its Iwasawa application to algebraic number theory Modules at the 1956 summer meeting of the AMS. The Main Conjecture Kenkichi Iwasawa (1917–1998)

Cyclotomic Fields Iwasawa spoke about the next result in his

Zp-Extensions talk A theorem on Abelian groups and its Iwasawa application to algebraic number theory Modules at the 1956 summer meeting of the AMS. The Main Conjecture

Theorem (Iwasawa’s Growth Formula)

Suppose F ⊂ F1 ⊂ F2 ... is a Zp-extension of number ﬁelds. Let An denote the p-primary part of the class group of Fn. Then ∃integers λ, µ, ν s.t.

λn+µpn+ν |An| = p

for all sufﬁciently large n. In fact, p - class number of Fn for all n in this case.

E.g., consider F = Q, or F = Q(ζp) for a regular prime p.

Trivial Iwasawa Invariants λ, µ, ν

Cyclotomic Fields

Zp-Extensions Theorem Iwasawa Modules Let F be a number ﬁeld with exactly one prime above p. If The Main p - class number of F, then λ = µ = ν = 0 for any Conjecture Zp-extension F∞/F. E.g., consider F = Q, or F = Q(ζp) for a regular prime p.

Trivial Iwasawa Invariants λ, µ, ν

Cyclotomic Fields

Zp-Extensions Theorem Iwasawa Modules Let F be a number ﬁeld with exactly one prime above p. If The Main p - class number of F, then λ = µ = ν = 0 for any Conjecture Zp-extension F∞/F.

In fact, p - class number of Fn for all n in this case. Trivial Iwasawa Invariants λ, µ, ν

Cyclotomic Fields

Zp-Extensions Theorem Iwasawa Modules Let F be a number ﬁeld with exactly one prime above p. If The Main p - class number of F, then λ = µ = ν = 0 for any Conjecture Zp-extension F∞/F.

In fact, p - class number of Fn for all n in this case.

E.g., consider F = Q, or F = Q(ζp) for a regular prime p. √ E.g., consider F = Q( −1). If p ≡ 1 (mod 4), then p splits in F/Q, so here λ ≥ 1 for Q∞(i)/Q(i).

What about r2 = 0 (F is totally real)? Can we have λ > 0?

Nontrivial Iwasawa Invariants λ, µ, ν

Cyclotomic Fields

Zp-Extensions Theorem Iwasawa Modules Suppose F is a number ﬁeld in which p splits completely. The Main Then λ ≥ r2 for the cyclotomic Zp-extension FQ∞/F where Conjecture r2 = # complex primes of F. What about r2 = 0 (F is totally real)? Can we have λ > 0?

Nontrivial Iwasawa Invariants λ, µ, ν

Cyclotomic Fields

√ E.g., consider F = Q( −1). If p ≡ 1 (mod 4), then p splits in F/Q, so here λ ≥ 1 for Q∞(i)/Q(i). Nontrivial Iwasawa Invariants λ, µ, ν

Cyclotomic Fields

√ E.g., consider F = Q( −1). If p ≡ 1 (mod 4), then p splits in F/Q, so here λ ≥ 1 for Q∞(i)/Q(i).

What about r2 = 0 (F is totally real)? Can we have λ > 0? Cyclotomic Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture Iwasawa Modules ∼ We have Gal(Ln/Fn) = An by class ﬁeld theory.

g ∈ Γ := Gal(F∞/F) acts on x ∈ X := Gal(L∞/F∞) as −1 g · x = gx˜ g˜ where g˜ extends g to L∞

If F has only one prime above p, and this prime is totally ramiﬁed in F∞/F, then pn ∼ X/(γ − 1)X = An where hγi = Γ

Iwasawa’s Idea to Derive Growth Formula

S Let Ln = p-Hilbert class ﬁeld of Fn. Take L∞ = n Ln. Cyclotomic Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture g ∈ Γ := Gal(F∞/F) acts on x ∈ X := Gal(L∞/F∞) as −1 g · x = gx˜ g˜ where g˜ extends g to L∞

If F has only one prime above p, and this prime is totally ramiﬁed in F∞/F, then pn ∼ X/(γ − 1)X = An where hγi = Γ

Iwasawa’s Idea to Derive Growth Formula

S Let Ln = p-Hilbert class ﬁeld of Fn. Take L∞ = n Ln. Cyclotomic Fields

Zp-Extensions ∼ Iwasawa We have Gal(Ln/Fn) = An by class ﬁeld theory. Modules

The Main Conjecture If F has only one prime above p, and this prime is totally ramiﬁed in F∞/F, then pn ∼ X/(γ − 1)X = An where hγi = Γ

Iwasawa’s Idea to Derive Growth Formula

S Let Ln = p-Hilbert class ﬁeld of Fn. Take L∞ = n Ln. Cyclotomic Fields

Zp-Extensions ∼ Iwasawa We have Gal(Ln/Fn) = An by class ﬁeld theory. Modules

The Main Conjecture

g ∈ Γ := Gal(F∞/F) acts on x ∈ X := Gal(L∞/F∞) as −1 g · x = gx˜ g˜ where g˜ extends g to L∞ Iwasawa’s Idea to Derive Growth Formula

S Let Ln = p-Hilbert class ﬁeld of Fn. Take L∞ = n Ln. Cyclotomic Fields

Zp-Extensions ∼ Iwasawa We have Gal(Ln/Fn) = An by class ﬁeld theory. Modules

The Main Conjecture

g ∈ Γ := Gal(F∞/F) acts on x ∈ X := Gal(L∞/F∞) as −1 g · x = gx˜ g˜ where g˜ extends g to L∞

If F has only one prime above p, and this prime is totally ramiﬁed in F∞/F, then pn ∼ X/(γ − 1)X = An where hγi = Γ Then use the structure theory for Λ-modules to compute

|X/((T + 1)pn − 1)X|

Jean-Pierre Serre (1926–)

Cyclotomic Fields Serre’s 1959 Seminaire Bourbaki: let Zp-Extensions Iwasawa T act on X as γ − 1. Then X becomes Modules a ﬁnitely generated, torsion Λ-module The Main Conjecture where Λ = Zp[[T ]]. Jean-Pierre Serre (1926–)

Then use the structure theory for Λ-modules to compute

|X/((T + 1)pn − 1)X| The prime ideals of Λ are (0), (p), (p, T ) and (f (T )) where f (T ) ≡ xdeg(f ) (mod p) is an irreducible polynomial.

Theorem Suppose M is a ﬁnitely gen’d Λ-module. Then ∃homomorphism with ﬁnite kernel and cokernel

s t r M Λ M Λ M −→ Λ ⊕ m ⊕ n (p i ) (fi (T ) j ) i=1 j=1

deg(f ) where each fj (T ) ≡ x j (mod p) is an irred. poly.

Structure Theorem for Λ-Modules

Λ = [[ ]] Cyclotomic Zp T is a local Noetherian UFD, but not a PID. Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture Theorem Suppose M is a ﬁnitely gen’d Λ-module. Then ∃homomorphism with ﬁnite kernel and cokernel

s t r M Λ M Λ M −→ Λ ⊕ m ⊕ n (p i ) (fi (T ) j ) i=1 j=1

deg(f ) where each fj (T ) ≡ x j (mod p) is an irred. poly.

Structure Theorem for Λ-Modules

Λ = [[ ]] Cyclotomic Zp T is a local Noetherian UFD, but not a PID. Fields Zp-Extensions The prime ideals of Λ are (0), (p), (p, T ) and (f (T )) where Iwasawa deg(f ) Modules f (T ) ≡ x (mod p) is an irreducible polynomial.

The Main Conjecture Structure Theorem for Λ-Modules

The Main Conjecture Theorem Suppose M is a ﬁnitely gen’d Λ-module. Then ∃homomorphism with ﬁnite kernel and cokernel

s t r M Λ M Λ M −→ Λ ⊕ m ⊕ n (p i ) (fi (T ) j ) i=1 j=1

deg(f ) where each fj (T ) ≡ x j (mod p) is an irred. poly. We have a well-deﬁned ‘characteristic polynomial’

s t Y mi Y nj char(X) = p fi (T ) i=1 j=1

Characteristic Polynomial

Cyclotomic In particular, ∃homom. with ﬁnite kernel and cokernel Fields

Zp-Extensions s t Iwasawa M Λ M Λ Modules X −→ m ⊕ n (p i ) (fi (T ) j ) The Main i=1 j=1 Conjecture deg(f ) where each fj (T ) ≡ x j (mod p) is an irred. poly. Characteristic Polynomial

Cyclotomic In particular, ∃homom. with ﬁnite kernel and cokernel Fields

Zp-Extensions s t Iwasawa M Λ M Λ Modules X −→ m ⊕ n (p i ) (fi (T ) j ) The Main i=1 j=1 Conjecture deg(f ) where each fj (T ) ≡ x j (mod p) is an irred. poly.

We have a well-deﬁned ‘characteristic polynomial’

s t Y mi Y nj char(X) = p fi (T ) i=1 j=1 Connection to Iwasawa Invariants

Cyclotomic Fields

Zp-Extensions Iwasawa In fact, if λ, µ, ν are as in the growth formula for F∞/F, Modules

The Main Conjecture

µ = ordp(char(X)) = m1 + ··· + ms

λ = deg(char(X)) = n1 deg(f1) + ··· + nt deg(ft ) Cyclotomic Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture The Main Conjecture Again, we have a decomposition

p−1 M −n X = X ω n=1

− and each X ω n is a ﬁnitely gen’d torsion Λ-module.

n+1 ω−n Again, Lp(s, ω ) contains info about X .

‘Eigenspaces’ of X

Cyclotomic Fields If F = Q(ζp), then G = Gal(Q(ζp)/Q) acts on X. Zp-Extensions Iwasawa Modules

The Main Conjecture n+1 ω−n Again, Lp(s, ω ) contains info about X .

‘Eigenspaces’ of X

Cyclotomic Fields If F = Q(ζp), then G = Gal(Q(ζp)/Q) acts on X. Zp-Extensions Iwasawa Modules The Main Again, we have a decomposition Conjecture p−1 M −n X = X ω n=1

− and each X ω n is a ﬁnitely gen’d torsion Λ-module. ‘Eigenspaces’ of X

Cyclotomic Fields If F = Q(ζp), then G = Gal(Q(ζp)/Q) acts on X. Zp-Extensions Iwasawa Modules The Main Again, we have a decomposition Conjecture p−1 M −n X = X ω n=1

− and each X ω n is a ﬁnitely gen’d torsion Λ-module.

n+1 ω−n Again, Lp(s, ω ) contains info about X . Theorem (Mazur and Wiles, 1984) Suppose n is odd and n 6≡ 1 (mod p − 1). Then

ω−n ˜ n+1 (char(X )) = (Lp(T , ω ))

as ideals in Λ.

Main Conjecture

Cyclotomic ˜ n+1 Fields There is a power series Lp(T , ω ) ∈ Λ = Zp[[T ]] s.t. Zp-Extensions ˜ s n+1 n+1 Iwasawa Lp((1 + p) − 1, ω ) = Lp(s, ω ) Modules

The Main Conjecture for all s ∈ Zp. Main Conjecture

Cyclotomic ˜ n+1 Fields There is a power series Lp(T , ω ) ∈ Λ = Zp[[T ]] s.t. Zp-Extensions ˜ s n+1 n+1 Iwasawa Lp((1 + p) − 1, ω ) = Lp(s, ω ) Modules

The Main Conjecture for all s ∈ Zp.

Theorem (Mazur and Wiles, 1984) Suppose n is odd and n 6≡ 1 (mod p − 1). Then

ω−n ˜ n+1 (char(X )) = (Lp(T , ω ))

as ideals in Λ. Cyclotomic Fields

Zp-Extensions Iwasawa Modules

The Main Conjecture Thank You!