Abelian extensions of number fields Jared Asuncion ALGANT Symposium - - PowerPoint PPT Presentation

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(2πiτ).

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(2πiτ). 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(2πiτ). 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 HK(m) of K. gives us the structure of Gal(HK(m)/K).

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 3 / 18

Theorem For any integer m ∈ Z: HQ(1) = Q HQ(m) = Q(exp(2πin/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: HQ(1) = Q HQ(m) = Q(exp(2πin/m)) for any n ∈ Z coprime to m. exp(2πi•) R/Z S1(C)

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 4 / 18

Theorem For any integer m ∈ Z: HQ(1) = Q HQ(m) = Q(exp(2πin/m)) for any n ∈ Z coprime to m. exp(2πi•) R/Z

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S1(C)

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Jared Asuncion Abelian extensions of number fields ALGANT Symposium 4 / 18

Theorem For any integer m ∈ Z: HQ(1) = Q HQ(m) = Q(exp(2πin/m)) for any n ∈ Z coprime to m. exp(2πi•) R/Z

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S1(C)

6 1 6 2 6 3 6 4 6 5 6

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 4 / 18

For K = Q, we have the following situation: exp(2πi•) R/Z 1 S1(C) We have an analogue for when K is an imaginary quadratic number field. i.e. K = Q( √ D) with D < 0 ???

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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 y2 = f (x) = x3 + 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 : y2 = x3 + ax + b is defined to be j(E) = 1728 · 4a3 4a3 + 27b2 . Consider an elliptic curve over C, isomorphic to the complex torus C/Λ. Then j(E) = 60G4(Λ)3 (60G4(Λ))3 − (140G6(Λ))2 where Gk(Λ) =

ω∈Λ\0 ω−k, the kth Eisenstein series.

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. y2 = x3 + 1 P

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18

An elliptic curve has a group structure. y2 = x3 + 1 P

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18

An elliptic curve has a group structure. y2 = x3 + 1 P 2P

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18

An elliptic curve has a group structure. y2 = x3 + 1 P 2P

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 8 / 18

An elliptic curve has a group structure. y2 = x3 + 1 P 2P 3P

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 OK if there exists an inclusion OK ֒ → End E. The elliptic curve E : y2 = x3 + 1

• ver Q has an endomorphism w : (x, y) → (ζ3x, y). In fact,

Z[ζ3] ⊂ 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 OK then HK(1) = K(j(E)) HK(m) = K(j(E), ???)

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 10 / 18

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Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18

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Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18

??? C/(Z + Zτ) E(C)

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18

??? C/(Z + Zτ) E(C)

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18

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C/(Z + Zτ) E(C)

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18

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C/(Z + Zτ) E(C)

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 11 / 18

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C/(Z + Zτ) E(C)

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 OK, where K is an imaginary quadratic number field. Then HK(1) = K(j(E)) HK(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 K0. 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 Cg/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 OK, where K is a quartic CM-field with cyclic Galois group. Then HK(1) ⊇ K(i(A)) and HK(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 OK, where K is a quartic CM-field with cyclic Galois group. Then HK(1) ⊇ K(i(A)) =: CMK(1) and HK(m) ⊇ K(i(A), F(t)) =: CMK(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 OK, where K is a quartic CM-field with cyclic Galois group. Then HK(1) ⊇ K(i(A)) =: CMK(1) and HK(m) ⊇ K(i(A), F(t)) =: CMK(m) where t ∈ A(C) is a proper m-torsion point. How to find HK(m)?

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 15 / 18

K K0 Q 2 2

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 16 / 18

K K0 Q CMK(m) 2 2 We use the Main Theorem of CM and find CMK(m).

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 16 / 18

K K0 Q CMK(m) HK0(m) 2 2 We use the Main Theorem of CM and find CMK(m). We use Stark’s conjectures to find HK0(m) for the totally real quadratic field.

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 16 / 18

K K0 Q CMK(m) HK0(m) CMK(m)HK0(m) 2 2 We use the Main Theorem of CM and find CMK(m). We use Stark’s conjectures to find HK0(m) for the totally real quadratic field.

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 16 / 18

K K0 Q CMK(m) HK0(m) CMK(m)HK0(m) HK(m) 2 2 We use the Main Theorem of CM and find CMK(m). We use Stark’s conjectures to find HK0(m) for the totally real quadratic field.

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 16 / 18

K K0 Q CMK(m) HK0(m) CMK(m)HK0(m) HK(m) 2 2 We use the Main Theorem of CM and find CMK(m). We use Stark’s conjectures to find HK0(m) for the totally real quadratic field. Theorem (Streng, 2010) Let K be a quartic CM-field with cyclic Galois group over Q. The extension HK(m)/ CMK(m)HK0(m) is of at most exponent 2.

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 16 / 18

The extension having exponent 2 means that we need to take square roots of elements from the base field CMK(1)HK0(1) to find HK(1). Are square roots from K sufficient? No. Use Kummer theory? Not feasible. Base field too big to compute class fields, etc.

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 17 / 18

Theorem (Shimura) The Hilbert class field HK(1) is a subfield of CMK(m)HK0(m) for some integer m. I have a theorem that gives an upper bound for such an m. For m = 2, one can use Shimura reciprocity to make the computations feasible. For m > 2, more work to be done.

Jared Asuncion Abelian extensions of number fields ALGANT Symposium 18 / 18