Abelian extensions of number fields Jared Asuncion ALGANT Symposium Jared Asuncion Abelian extensions of number fields ALGANT Symposium 1 / 18
Definition A number field is a field extension of Q of finite degree. Definition An abelian extension is a Galois extension in which the Galois group G is abelian. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 2 / 18
Theorem (Kronecker-Weber Theorem (KWT)) The abelian extensions of K = Q are generated by values at rational arguments τ of the exponential function τ �→ exp(2 π i τ ) . Jared Asuncion Abelian extensions of number fields ALGANT Symposium 3 / 18
Theorem (Kronecker-Weber Theorem (KWT)) The abelian extensions of K = Q are generated by values at rational arguments τ of the exponential function τ �→ exp(2 π i τ ) . Hilbert’s twelfth problem Given a number field K , construct all abelian extensions of K by adjoining special values of particular analytic functions. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 3 / 18
Theorem (Kronecker-Weber Theorem (KWT)) The abelian extensions of K = Q are generated by values at rational arguments τ of the exponential function τ �→ exp(2 π i τ ) . Hilbert’s twelfth problem Given a number field K , construct all abelian extensions of K by adjoining special values of particular analytic functions. Class field theory Class field theory tells us that every finite abelian extension L of a number field K is contained in some ray class field extension H K ( m ) of K . gives us the structure of Gal( H K ( m ) / K ). Jared Asuncion Abelian extensions of number fields ALGANT Symposium 3 / 18
Theorem For any integer m ∈ Z : H Q (1) = Q H Q ( m ) = Q (exp(2 π in / m )) for any n ∈ Z coprime to m. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 4 / 18
Theorem For any integer m ∈ Z : H Q (1) = Q H Q ( m ) = Q (exp(2 π in / m )) for any n ∈ Z coprime to m. R / Z exp(2 π i • ) S 1 ( C ) Jared Asuncion Abelian extensions of number fields ALGANT Symposium 4 / 18
Theorem For any integer m ∈ Z : H Q (1) = Q H Q ( m ) = Q (exp(2 π in / m )) for any n ∈ Z coprime to m. R / Z exp(2 π i • ) S 1 ( C ) 0 1 2 3 4 5 6 6 6 6 6 6 6 6 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 4 / 18
Theorem For any integer m ∈ Z : H Q (1) = Q H Q ( m ) = Q (exp(2 π in / m )) for any n ∈ Z coprime to m. R / Z exp(2 π i • ) S 1 ( C ) 0 1 2 3 4 5 6 6 6 6 6 6 6 6 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 4 / 18
For K = Q , we have the following situation: R / Z exp(2 π i • ) S 1 ( C ) 0 1 We have an analogue for when K is an imaginary quadratic number field. √ i.e. K = Q ( D ) with D < 0 ??? ???????? ??? Jared Asuncion Abelian extensions of number fields ALGANT Symposium 5 / 18
Definition An elliptic curve over k ( char k � = 2 , 3 ) is a smooth projective curve given by an equation of the form y 2 = f ( x ) = x 3 + ax + b where a , b ∈ k and f ( x ) has no double roots in k. An elliptic curve E over C is isomorphic to a complex torus. That is, there exists an isomorphism C / Λ ∼ = E ( C ) for some lattice Λ. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 6 / 18
Definition The j-invariant of an elliptic curve E : y 2 = x 3 + ax + b is defined to be 4 a 3 j ( E ) = 1728 · 4 a 3 + 27 b 2 . Consider an elliptic curve over C , isomorphic to the complex torus C / Λ. Then 60 G 4 (Λ) 3 j ( E ) = (60 G 4 (Λ)) 3 − (140 G 6 (Λ)) 2 ω ∈ Λ \ 0 ω − k , the k th Eisenstein series. where G k (Λ) = � Remark Two elliptic curves over C are isomorphic if and only if they have the same j-invariant. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 7 / 18
An elliptic curve has a group structure. P y 2 = x 3 + 1 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18
An elliptic curve has a group structure. P y 2 = x 3 + 1 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18
An elliptic curve has a group structure. P 2 P y 2 = x 3 + 1 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18
An elliptic curve has a group structure. P 2 P y 2 = x 3 + 1 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18
An elliptic curve has a group structure. P 2 P 3 P y 2 = x 3 + 1 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18
Definition The multiplication-by-m map is a morphism which sends a point P ∈ E to the point mP ∈ E. The multiplication-by- m map is an endomorphism for any m . Hence Z ⊆ End E . Definition Let K be an imaginary quadratic number field. We say E has complex multiplication (CM) by O K if there exists an inclusion O K ֒ → End E. The elliptic curve E : y 2 = x 3 + 1 over Q has an endomorphism w : ( x , y ) �→ ( ζ 3 x , y ). In fact, Z [ ζ 3 ] ⊂ End E . Jared Asuncion Abelian extensions of number fields ALGANT Symposium 9 / 18
Theorem For an elliptic curve E over C with complex multiplication by O K then H K (1) = K ( j ( E )) H K ( m ) = K ( j ( E ) , ???) Jared Asuncion Abelian extensions of number fields ALGANT Symposium 10 / 18
??? ???????? ??? Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18
??? ??? C / ( Z + Z τ ) Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18
??? C / ( Z + Z τ ) E ( C ) Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18
??? C / ( Z + Z τ ) E ( C ) Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18
??? C / ( Z + Z τ ) E ( C ) 1 2 3 4 5 6 6 6 6 6 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18
h C / ( Z + Z τ ) E ( C ) 1 2 3 4 5 6 6 6 6 6 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18
h C / ( Z + Z τ ) E ( C ) 1 2 3 4 5 6 6 6 6 6 Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18
An m -torsion point P on an elliptic curve E is said to be proper if nP = 0 if and only if n is a multiple of m . Jared Asuncion Abelian extensions of number fields ALGANT Symposium 12 / 18
An m -torsion point P on an elliptic curve E is said to be proper if nP = 0 if and only if n is a multiple of m . Theorem (Main Theorem of Complex Multiplication for EC) Let E be an elliptic curve E over C with complex multiplication by O K , where K is an imaginary quadratic number field. Then H K (1) = K ( j ( E )) H K ( m ) = K ( j ( E ) , h ( t )) where t ∈ E ( C ) is a proper m-torsion point. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 12 / 18
Hilbert’s twelfth problem Given a number field K , construct all abelian extensions of K by adjoining special values of particular analytic functions. Hilbert’s 12th is solved only for these fields: K = Q K , imaginary quadratic number field ??? Then what? The Main Theorem of Complex Multiplication has a version that deals with particular higher dimensional number fields. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 13 / 18
Definition A CM-field K is a totally imaginary number field which is a quadratic extension of a totally real number field K 0 . An imaginary quadratic number field K is a degree 2 CM-field. Definition An abelian variety is a projective group variety. A complex elliptic curve E is an abelian variety of dimension 1. A complex abelian variety of dimension g is isomorphic to a g -dimensional complex torus C g / Λ. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 14 / 18
j-invariant j � Igusa invariant i Weber function h � F Theorem (Main Theorem of Complex Multiplication for AS) Jared Asuncion Abelian extensions of number fields ALGANT Symposium 15 / 18
j-invariant j � Igusa invariant i Weber function h � F Theorem (Main Theorem of Complex Multiplication for AS) Let A be an abelian surface over C with complex multiplication by O K , where K is a quartic CM-field with cyclic Galois group. Then H K (1) ⊇ K ( i ( A )) and H K ( m ) ⊇ K ( i ( A ) , F ( t )) where t ∈ A ( C ) is a proper m-torsion point. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 15 / 18
j-invariant j � Igusa invariant i Weber function h � F Theorem (Main Theorem of Complex Multiplication for AS) Let A be an abelian surface over C with complex multiplication by O K , where K is a quartic CM-field with cyclic Galois group. Then H K (1) ⊇ K ( i ( A )) =: CM K (1) and H K ( m ) ⊇ K ( i ( A ) , F ( t )) =: CM K ( m ) where t ∈ A ( C ) is a proper m-torsion point. Jared Asuncion Abelian extensions of number fields ALGANT Symposium 15 / 18
j-invariant j � Igusa invariant i Weber function h � F Theorem (Main Theorem of Complex Multiplication for AS) Let A be an abelian surface over C with complex multiplication by O K , where K is a quartic CM-field with cyclic Galois group. Then H K (1) ⊇ K ( i ( A )) =: CM K (1) and H K ( m ) ⊇ K ( i ( A ) , F ( t )) =: CM K ( m ) where t ∈ A ( C ) is a proper m-torsion point. How to find H K ( m )? Jared Asuncion Abelian extensions of number fields ALGANT Symposium 15 / 18
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