Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups A. Angelakis ♯♭ , P. Stevenhagen ♯ Universiteit Leiden ♯ , Universit´ e Bordeaux 1 ♭ July 12, 2012 - UCSD - X - A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
� Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End a Question Let K be a number field and G K = Gal( K/K ) the absolute Galois group. Question : Does G K determine K ? meaning, if G K 1 ∼ = G K 2 then K 1 ∼ = K 2 ? Answer : Yes! Neukirch, Ikeda, Iwasawa & Uchida (around 1969 − 75) ∃ ! α ∈ Aut( Q ) : α [ K 1 ] = K 2 inducing G K 1 ∼ = G K 2 α Q � ��������� � � � � � � � � α � K 2 K 1 A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End more Questions 1/2 Let K be a number field and G solv = Gal( K solv /K ) the K maximal prosolvable quotient of G K . Question : Does G solv determine K ? K K 1 ∼ K 2 then K 1 ∼ meaning, if G solv = G solv = K 2 ? Answer : Yes! Neukirch, Ikeda, Iwasawa & Uchida (around 1969 − 75) A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End more Questions 2/2 Let K be a number field and A K = G K / [ G K , G K ] the maximal abelian quotient of G K . Question : Does A K determine K ? meaning, if A K 1 ∼ = A K 2 then K 1 ∼ = K 2 ? Answer : No!!! First examples found by Onabe (1976) Uses Kubota’s (1957) description of the character group of G K in terms of Ulm invariants A K is not explicitly given but characterized by the number of Ulm invariants A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End summary K ab is not “explicit”, but A K is explicit (class field theory) Theorem Many imaginary quadratic fields K have the same minimal absolute abelian Galois group � Z 2 × A K ∼ = M = � Z /n Z n ≥ 1 here � Z = lim − Z /n Z , the profinite completion of Z ← 2291 out of 2348, meaning, more than 97 . 5% of K of prime class number < 100 have this minimal group Conjecture: there are infinitely many such K A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End modules over � Z The topological group A K is a module over � Z : The exponentiation of elements of this group with ordinary integers extends to exponentiation with elements of � Z For any σ ∈ A K , we have n →∞ σ n ! = id ∈ A K lim Describe infinite abelian Galois groups as modules over � Z A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End example 1/2 Toy example: K = Q By Kronecker-Weber theorem Q ab = ∪ ∞ n =1 Q ( ζ n ) is the maximal cyclotomic extension of Q This yields the well-known isomorphism − ( Z /n Z ) ∗ = � Z ∗ A Q = lim ← Z ∗ = � � p Z ∗ p , (Chinese Remainder Theorem) ∼ Z ∗ Z / ( p − 1) Z × (1 + p Z p ) = p ∼ = Z / ( p − 1) Z × Z p , ( p � = 2) = � = � T Q ∼ p ( Z / ( p − 1) Z ) ∼ n ≥ 1 Z /n Z This is “the” countable product of finite cyclic groups having infinitely many cyclic components of prime power order for every prime A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End example 2/2 Taking the product over all p , we obtain � ∼ � A Q Z × Z /n Z = n ≥ 1 ∼ � Z × T Q = T Q is the closure of the torsion subgroup of A Q It is not a torsion group itself A Q /T Q is a free � Z -module of rank 1 The subfield of Q ab left invariant by the subgroup T Q ⊂ A Q = Gal( Q ab / Q ) is the unique � Z -extension of Q A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End using cft K ab “unknown” for arbitrary K Class field theory: A K for arbitrary K � ′ K ∗ p ) /K ∗ ] / (conn. comp. of unit element) A K = [( p ≤∞ From now on we take K to be imaginary quadratic. Then, � ′ K ∗ p ) /K ∗ A K = ( p < ∞ A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End inertial part H is the Hilbert class field K ab Cl K is the class group of K µ K is the group of roots of unity in K (of order ≤ 6) O ∗ = � � p O ∗ p unit group of the profinite U K = � completion of the ring of integers O ∗ /µ K O = O K A K � ′ K ∗ p ) /K ∗ A K = ( H p < ∞ ∪ Cl K � p ) / O ∗ = � O ∗ O ∗ /µ K U K = ( K p A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End the � O ∗ & T K K ab K ∗ p ⊃ µ p local roots of unity O ∗ = � p ⊃ � � p O ∗ p µ p = T K T K is the closure of the torsion U K = � O ∗ /µ K subgroup of � O ∗ A K Lemma (1) O ∗ ∼ Z [ K : Q ] × T K � = � H O ∗ ∼ Z 2 × T K � = � Cl K K A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End the T K & T K /µ K Lemma (2) Let w K be the number of roots of unity in K . Then we have a non-canonical isomorphism of profinite groups � T K ∼ Z /nw K Z = n ≥ 1 If w K is squarefree, then T K ∼ = T Q Lemma (3) We have a non-canonical isomorphism � T K /µ K ∼ Z /nw K Z = n ≥ 1 A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End structure of the Inertial part Theorem � Z 2 × � ∞ � n =1 Z /n Z , K � = Q ( i ) O ∗ /µ K ∼ Z 2 × T K /µ K ∼ U K = � = � Z 2 × � ∞ = � n =1 Z / 4 n Z , K = Q ( i ) (isomorphisms of profinite groups) Corollary All imaginary quadratic K � = Q ( i ) of class number 1 have ∞ � Z 2 × A K ∼ = � Z /n Z n =1 This implies Onabe’s observation: there are 8 such K ! A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End diagrams 1/2 K ab � ������������ � � � � � T � � U K � � � � � � L � � � � � � � � � A K H Z 2 � ��������������� Cl K K The invariant field L of the closure T of the torsion subgroup of U K is an extension of H with group � Z 2 A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End diagrams 2/2 class number 1: K ab ������������ T L � � U K = A K � � � � � � � � � Z 2 � � � � � � � H = K The invariant field L of the closure T of the torsion subgroup of U K is an extension of K with group � Z 2 A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End Question: A K ∼ = M for h K > 1 ? K � = Q ( i ) imaginary quadratic 1 → U K − → A K − → Cl K → 1 (1) Z 2 × T U K ∼ = � Does not depend on K It is isomorphic to the “minimal” Galois group � ∞ Z 2 × M = � Z /n Z n =1 Question : Can A K be isomorphic to M for h K > 1? If (1) splits, then A K is isomorphic to M A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
Intro Describing the Inertial part Splitting the Sequence Down to Gaia The End the main sequence does not split Theorem For every imaginary quadratic field K , the sequence ψ 1 → U K − → A K − → Cl K → 1 is totally non-split, i.e., there is no non-trivial subgroup C ⊂ Cl K for which the associated subextension 1 → U K → ψ − 1 [ C ] → C → 1 is split A. Angelakis ♯♭ , P. Stevenhagen ♯ Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups
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