Explicit Construction of Abelian Extensions of Number Fields Jared Asuncion 21 November 2019 Jared Asuncion Lambda Seminar Talk 21 November 2019 1 / 25
Definition (algebraic number) An algebraic number is a complex number which is a root of a polynomial with coefficients in Z . Definition (algebraic integer) An algebraic integer is an algebraic number which is a root of a monic polynomial with coefficients in Z . Example √− 5 is a root of x 2 + 5 . Hence, it is an algebraic integer. 3 . 14 is a root of 50 x − 157 . Hence, it is an algebraic number. π is NOT an algebraic number. Jared Asuncion Lambda Seminar Talk 21 November 2019 2 / 25
Definition Let L / K be a field extension. The degree [ L : K ] of a field extension L / K is defined to be the dimension of L as a K-vector space. Definition (algebraic number field) An algebraic number field is a field extension of Q of finite degree. Any element of a number field is algebraic. The set of algebraic integers O K of K form a subring of K . Q ( √− 5) is a number field of degree 2 since dim Q ( K ) = 2. Q ( π ) and C are not number fields since they are not finite extensions of Q . Jared Asuncion Lambda Seminar Talk 21 November 2019 3 / 25
Definition (Galois extension) A field extension L / K is Galois if the group Aut( L / K ) of automorphisms of L fixing K is equal to the degree of the extension. Notation If L / K is Galois, we will denote Aut( L / K ) by Gal( L / K ) . K = Q ( i ) is a Galois extension of Q since the automorphisms of K fixing Q are given by: a + bi �→ a + bi a + bi �→ a − bi √ 3 K = Q ( 2) is not a Galois extension of Q since the only automorphism of K fixing Q is the identity automorphism. Jared Asuncion Lambda Seminar Talk 21 November 2019 4 / 25
Definition (abelian extension) A Galois extension L / K is abelian if the group Gal( L / K ) of automorphisms of L fixing K is abelian. K = Q ( i ) is an abelian extension since | Gal( L / K ) | = 2 and all groups of order 2 are abelian. K = Q ( ζ n ), where ζ n = exp(2 π i / n ), is an abelian extension since its Galois group is Gal( K / Q ) ∼ = ( Z / n Z ) × Jared Asuncion Lambda Seminar Talk 21 November 2019 5 / 25
Definition (abelian extension) A Galois extension L / K is abelian if the group Gal( L / K ) of automorphisms of L fixing K is abelian. K = Q ( i ) is an abelian extension since | Gal( L / K ) | = 2 and all groups of order 2 are abelian. K = Q ( ζ n ), where ζ n = exp(2 π i / n ), is an abelian extension since its Galois group is Gal( K / Q ) ∼ = ( Z / n Z ) × Problem (Hilbert’s 12th Problem) Given a number field K, construct all abelian extensions of K by adjoining special values of particular analytic functions. Jared Asuncion Lambda Seminar Talk 21 November 2019 5 / 25
Problem (Hilbert’s 12th Problem) Given a number field K, construct all (finite) abelian extensions of K by adjoining special values of particular functions. Theorem (Kronecker-Weber Theorem) Every finite abelian extension of Q is contained in a field Q (exp(2 π iz )) for some z ∈ Q . ‘particular function’ ‘special values’ e : R → S 1 ( C ) Append e ( z ) such that z ∈ Q . z → exp(2 π iz ) Jared Asuncion Lambda Seminar Talk 21 November 2019 6 / 25
e = exp(2 π iz ) R S 1 ( C ) Observations Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25
e = exp(2 π iz ) R S 1 ( C ) Observations The kernel of the map e is Z . Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25
R / Z e = exp(2 π iz ) S 1 ( C ) Observations The kernel of the map e is Z . Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25
R / Z e = exp(2 π iz ) S 1 ( C ) Observations The kernel of the map e is Z . The image of the map is a geometric object, a circle. Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25
R / Z e = exp(2 π iz ) S 1 ( C ) Observations The kernel of the map e is Z . The image of the map is a geometric object, a circle. Both domain and codomain have a group structure. Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25
R / Z e = exp(2 π iz ) S 1 ( C ) 0 1 2 3 4 5 6 6 6 6 6 6 Observations The kernel of the map e is Z . The image of the map is a geometric object, a circle. Both domain and codomain have a group structure. The torsion points of the domain are easy to determine. Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25
R / Z e = exp(2 π iz ) S 1 ( C ) Observations The kernel of the map e is Z . The image of the map is a geometric object, a circle. Both domain and codomain have a group structure. The torsion points of the domain are easy to determine. The torsion points of the codomain are what we append to Q . Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25
Class Field Theory Class field theory tells us that every finite abelian extension of a number field K is contained in some ray class field extension H K ( m ) of K . Jared Asuncion Lambda Seminar Talk 21 November 2019 8 / 25
Class Field Theory Class field theory tells us that every finite abelian extension of a number field K is contained in some ray class field extension H K ( m ) of K . For the case when the base field is Q , we have: H Q (1) = Q H Q ( m ) = Q (exp(2 π i · 1 / m )) . Jared Asuncion Lambda Seminar Talk 21 November 2019 8 / 25
Class Field Theory Class field theory tells us that every finite abelian extension of a number field K is contained in some ray class field extension H K ( m ) of K . For the case when the base field is Q , we have: H Q (1) = Q H Q ( m ) = Q (exp(2 π i · 1 / m )) . Hilbert’s 12th Problem What about other base fields? Jared Asuncion Lambda Seminar Talk 21 November 2019 8 / 25
Class Field Theory Class field theory tells us that every finite abelian extension of a number field K is contained in some ray class field extension H K ( m ) of K . For the case when the base field is Q , we have: H Q (1) = Q H Q ( m ) = Q (exp(2 π i · 1 / m )) . Hilbert’s 12th Problem What about other base fields? √ The case K = Q ( − D ), totally imaginary quadratic number fields is explicitly solved using elliptic curves. No other case is completely solved. Jared Asuncion Lambda Seminar Talk 21 November 2019 8 / 25
Definition An elliptic curve defined over k ( char k � = 2 , 3 ) is a smooth projective curve given by an equation of the form E : Y 2 Z = X 3 + aXZ 2 + bZ 3 where a , b ∈ k and f ( x ) has no double roots in the algebraic closure of k. Jared Asuncion Lambda Seminar Talk 21 November 2019 9 / 25
Definition An elliptic curve defined over k ( char k � = 2 , 3 ) is a smooth projective curve given by an equation of the form E : Y 2 Z = X 3 + aXZ 2 + bZ 3 where a , b ∈ k and f ( x ) has no double roots in the algebraic closure of k. y x Jared Asuncion Lambda Seminar Talk 21 November 2019 9 / 25
Definition An elliptic curve defined over k ( char k � = 2 , 3 ) is a smooth projective curve given by an equation of the form E : Y 2 Z = X 3 + aXZ 2 + bZ 3 where a , b ∈ k and f ( x ) has no double roots in the algebraic closure of k. y It has exactly one point at infinity, which we denote by ∞ = (0 : 1 : 0). x Jared Asuncion Lambda Seminar Talk 21 November 2019 9 / 25
Definition An elliptic curve defined over k ( char k � = 2 , 3 ) is a smooth projective curve given by an equation of the form E : Y 2 Z = X 3 + aXZ 2 + bZ 3 where a , b ∈ k and f ( x ) has no double roots in the algebraic closure of k. y It has exactly one point at infinity, which we denote by ∞ = (0 : 1 : 0). x We will usually write the affine equation y 2 = x 3 + ax + b to define elliptic curves and remember that there is an extra point at infinity. E : y 2 = x 3 + 1 Jared Asuncion Lambda Seminar Talk 21 November 2019 9 / 25
Notation Let k ⊆ K. The set of K-rational points of an elliptic curve E is given by E ( K ) = {∞} ∪ { ( x , y ) ∈ K × K : y 2 = x 3 + ax + b } . Theorem Let E be an elliptic curve over k. For each k ⊆ K, the set E ( K ) has a group structure with ∞ as the identity element. Jared Asuncion Lambda Seminar Talk 21 November 2019 10 / 25
Notation Let k ⊆ K. The set of K-rational points of an elliptic curve E is given by E ( K ) = {∞} ∪ { ( x , y ) ∈ K × K : y 2 = x 3 + ax + b } . Theorem Let E be an elliptic curve over k. For each k ⊆ K, the set E ( K ) has a group structure with ∞ as the identity element. For each integer m ∈ Z , there is a corresponding group homomorphism (i.e. an endomorphism) from E ( K ) to E ( K ): [ − 1] : E ( K ) → E ( K ) [2] : E ( K ) → E ( K ) � f ′ ( x ) 2 � ( x , y ) �→ ( x , − y ) ( x , y ) �→ 4 f ( x ) − a − 2 x , · · · Jared Asuncion Lambda Seminar Talk 21 November 2019 10 / 25
Example The elliptic curve E : y 2 = x 3 + x over Q has an endomorphism [ i ] : ( x , y ) �→ ( − x , iy ) . Hence Z � End E. Jared Asuncion Lambda Seminar Talk 21 November 2019 11 / 25
Example The elliptic curve E : y 2 = x 3 + x over Q has an endomorphism [ i ] : ( x , y ) �→ ( − x , iy ) . Hence Z � End E. Definition Let K be an imaginary quadratic number field and let O K be its ring of integers. Let End E be the ring of endomorphisms of E. If End E ∼ = O K , then E is said to have complex multiplication by O K . Jared Asuncion Lambda Seminar Talk 21 November 2019 11 / 25
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