Computing Hilbert class fields of quartic fields using complex - - PowerPoint PPT Presentation

Computing Hilbert class fields of quartic fields using complex multiplication

Jared Asuncion supervisors: Andreas Enge, Marco Streng 28 November 2019

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 1 / 14

Definition (degree of a field extension) 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.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) 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. For example, consider the field extension Q(√−5)/Q. Q(√−5) is a Q-vector space with basis {1, √−5}. Hence [Q[√−5] : Q] = 2.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) 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.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) 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. [Q(√−5) : Q] = 2. Hence it is a number field.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) 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. Definition (group of automorphisms of L fixing K) Aut(L/K) = {σ ∈ Aut L : σ(x) = x for each x ∈ K} σ ∈ Aut L is an invertible ring homomorphism from L to L.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) 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. Definition (group of automorphisms of L fixing K) Aut(L/K) = {σ ∈ Aut L : σ(x) = x for each x ∈ K} σ ∈ Aut L is an invertible ring homomorphism from L to L. Aut(Q( √ −5)/Q) = { a+b √ −5 → a + b √ −5, a+b √ −5 → a − b √ −5 }. It is a group of order 2.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) 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. Definition (group of automorphisms of L fixing K) Aut(L/K) = {σ ∈ Aut L : σ(x) = x for each x ∈ K} Definition (abelian extension) A number field extension L/K is said to be an abelian extension if: | Aut(L/K)| = [L : K] i.e. a Galois extension Aut(L/K) is commutative.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 3 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Example Q( √ −5) ⊆ Q(ζ20) where ζ20 = exp(2πi · 1/20). √ −5 = 2ζ7

20 − ζ5 20 + 2ζ3 20.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 3 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 4 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Class Field Theory Assume K = Q or K is a number field such that there does NOT exist an injective ring homomorphism σ : K ֒ → R. i.e. K has no real embeddings

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 4 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Class Field Theory Assume K = Q or K is a number field such that there does NOT exist an injective ring homomorphism σ : K ֒ → R. i.e. K has no real embeddings Class field theory tells us: The existence of a ray class field HK(m). Every finite degree abelian extension of K is contained in a ray class field HK(m). Information on the structure of Aut(HK(m)/K).

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 4 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Class Field Theory Assume K = Q or K is a number field such that there does NOT exist an injective ring homomorphism σ : K ֒ → R. i.e. K has no real embeddings Class field theory tells us: The existence of a ray class field HK(m). Every finite degree abelian extension of K is contained in a ray class field HK(m). Information on the structure of Aut(HK(m)/K). HQ(m) = Q(ζm) Aut(HQ(m)/Q) ∼ = (Z/mZ)× ζm := exp(2πi · 1/m) ζm → ζn

m ↔ n ∈ (Z/mZ)×

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 4 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. HQ(m) = Q(exp(2πi · 1/m)) e = exp(2πiz) R S1(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. HQ(m) = Q(exp(2πi · 1/m)) e = exp(2πiz) R/Z S1(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. HQ(m) = Q(exp(2πi · 1/m)) e = exp(2πiz) R/Z S1(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. HQ(m) = Q(exp(2πi · 1/m)) e = exp(2πiz) R/Z S1(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. HQ(m) = Q(exp(2πi · 1/m)) e = exp(2πiz) R/Z S1(C)

6 1 6 2 6 3 6 4 6 5 6

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L/Q with finite degree is contained in a field Q(exp(2πiz)) for some z ∈ Q. Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. HQ(m) = Q(exp(2πi · 1/m)) e = exp(2πiz) R/Z S1(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Only other explicitly solved case: imaginary quadratic number fields: K = Q( √ −D) D > 0. Instead of a circle, the geometric object is an elliptic curve.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 6 / 14

Only other explicitly solved case: imaginary quadratic number fields: K = Q( √ −D) D > 0. Instead of a circle, the geometric object is an elliptic curve. Definition (eliptic curve) An elliptic curve defined over C is a smooth projective curve given by an equation of the form E : Y 2Z = X 3 + aXZ 2 + bZ 3 where a, b ∈ C and x3 + ax + b has no double roots.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 6 / 14

Only other explicitly solved case: imaginary quadratic number fields: K = Q( √ −D) D > 0. Instead of a circle, the geometric object is an elliptic curve. Definition (eliptic curve) An elliptic curve defined over C is a smooth projective curve given by an equation of the form E : Y 2Z = X 3 + aXZ 2 + bZ 3 where a, b ∈ C and x3 + ax + b has no double roots. There is a unique point ∞ = (0 : 1 : 0) when we set Z = 0.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 6 / 14

Only other explicitly solved case: imaginary quadratic number fields: K = Q( √ −D) D > 0. Instead of a circle, the geometric object is an elliptic curve. Definition (eliptic curve) An elliptic curve defined over C is a smooth projective curve given by an equation of the form E : Y 2Z = X 3 + aXZ 2 + bZ 3 where a, b ∈ C and x3 + ax + b has no double roots. There is a unique point ∞ = (0 : 1 : 0) when we set Z = 0. We usually write y2 = x3 + ax + b instead. Z = 0.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 6 / 14

Only other explicitly solved case: imaginary quadratic number fields: K = Q( √ −D) D > 0. Instead of a circle, the geometric object is an elliptic curve. Definition (eliptic curve) An elliptic curve defined over C is a smooth projective curve given by an equation of the form E : Y 2Z = X 3 + aXZ 2 + bZ 3 where a, b ∈ C and x3 + ax + b has no double roots. There is a unique point ∞ = (0 : 1 : 0) when we set Z = 0. We usually write y2 = x3 + ax + b instead. Z = 0. E(C) = {∞} ∪ {(x, y) ∈ C × C : y2 = x3 + ax + b} has a group

• structure. The identity element is ∞.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 6 / 14

Observation 1 Since E(C) is a group, for any m ∈ Z>0, the multiplication-by-m is a group homomorphism from E to E (i.e. an endomorphism). Z ⊆ End E. [−1] : E(C) → E(C) (x, y) → (x, −y)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 7 / 14

Observation 1 Since E(C) is a group, for any m ∈ Z>0, the multiplication-by-m is a group homomorphism from E to E (i.e. an endomorphism). Z ⊆ End E. Observation 2 Some elliptic curves, such as ˜ E : y2 = x3 + x, have more endomorphisms. [−1] : ˜ E(C) → ˜ E(C) [i] : ˜ E(C) → ˜ E(C) (x, y) → (x, −y) (x, y) → (−x, iy)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 7 / 14

Observation 1 Since E(C) is a group, for any m ∈ Z>0, the multiplication-by-m is a group homomorphism from E to E (i.e. an endomorphism). Z ⊆ End E. Observation 2 Some elliptic curves, such as ˜ E : y2 = x3 + x, have more endomorphisms. [−1] : ˜ E(C) → ˜ E(C) [i] : ˜ E(C) → ˜ E(C) (x, y) → (x, −y) (x, y) → (−x, iy) End ˜ E ∼ = Z + Zi = OK K = Q(i)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 7 / 14

Fact Let E be an elliptic curve over C. Then C/Λ ∼ = E(C) for some lattice Λτ = Z + τZ with τ ∈ H1 = {x + yi ∈ C : x, y ∈ R, y > 0}.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 8 / 14

Fact Let E be an elliptic curve over C. Then C/Λ ∼ = E(C) for some lattice Λτ = Z + τZ with τ ∈ H1 = {x + yi ∈ C : x, y ∈ R, y > 0}. There exists a map j : H1 → C such that j(τ) = j(τ ′) ⇔ C/Λτ ∼ = C/Λτ ′ This map is called the j-invariant. We will write j(E) := j(τ) for the j-invariant of E(C) ∼ = C/Λτ.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 8 / 14

f C/Λ E(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 9 / 14

f C/Λ E(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 9 / 14

f C/Λ E(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 9 / 14

Theorem Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be a complex elliptic curve such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). f C/Λ E(C)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 9 / 14

Definition (CM field) A CM-field K is a totally imaginary number field (no embeddings σ : K ֒ → R exist) which is a degree 2 extension of a totally real number field K0 (image of all embeddings σ : K0 ֒ → C lie in R).

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 10 / 14

Definition (CM field) A CM-field K is a totally imaginary number field (no embeddings σ : K ֒ → R exist) which is a degree 2 extension of a totally real number field K0 (image of all embeddings σ : K0 ֒ → C lie in R). For degree 4 (quartic) CM-fields K: cyclic K = K r K0 Q

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 10 / 14

Definition (CM field) A CM-field K is a totally imaginary number field (no embeddings σ : K ֒ → R exist) which is a degree 2 extension of a totally real number field K0 (image of all embeddings σ : K0 ֒ → C lie in R). For degree 4 (quartic) CM-fields K: cyclic K = K r K0 Q bicyclic K = K r K1 K0 K2 Q

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 10 / 14

Definition (CM field) A CM-field K is a totally imaginary number field (no embeddings σ : K ֒ → R exist) which is a degree 2 extension of a totally real number field K0 (image of all embeddings σ : K0 ֒ → C lie in R). For degree 4 (quartic) CM-fields K: cyclic K = K r K0 Q bicyclic K = K r K1 K0 K2 Q not Galois L L0 K ′ K K r K r ′ K r K0 L00 Q

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 10 / 14

Definition (CM field) A CM-field K is a totally imaginary number field (no embeddings σ : K ֒ → R exist) which is a degree 2 extension of a totally real number field K0 (image of all embeddings σ : K0 ֒ → C lie in R). For degree 4 (quartic) CM-fields K: cyclic K = K r K0 Q bicyclic K = K r K1 K0 K2 Q not Galois L L0 K ′ K K r K r ′ K r K0 L00 Q primitive ⇔ cyclic or non-Galois

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 10 / 14

Theorem (Main Theorem of CM (ec)) Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be an elliptic curve over C such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). Theorem (Main Theorem of CM (ppas), Shimura-Taniyama, 1950s) Let K be a primitive quartic CM field. Let m ∈ Z>0. Let A be a principally polarized abelian surface over C such that End A ∼ = OK. Then: HK r (m) ⊇ K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) where h is an analogue of the Weber function.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 11 / 14

Theorem (Main Theorem of CM (ec)) Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be an elliptic curve over C such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). Theorem (Main Theorem of CM (ppas), Shimura-Taniyama, 1950s) Let K be a primitive quartic CM field. Let m ∈ Z>0. Let A be a principally polarized abelian surface over C such that End A ∼ = OK. Then: HK r (m) ⊇ K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) where h is an analogue of the Weber function.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 11 / 14

Theorem (Main Theorem of CM (ec)) Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be an elliptic curve over C such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). Theorem (Main Theorem of CM (ppas), Shimura-Taniyama, 1950s) Let K be a primitive quartic CM field. Let m ∈ Z>0. Let A be a principally polarized abelian surface over C such that End A ∼ = OK. Then: HK r (m) ⊇ K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) where h is an analogue of the Weber function.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 11 / 14

Theorem (Main Theorem of CM (ec)) Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be an elliptic curve over C such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). Theorem (Main Theorem of CM (ppas), Shimura-Taniyama, 1950s) Let K be a primitive quartic CM field. Let m ∈ Z>0. Let A be a principally polarized abelian surface over C such that End A ∼ = OK. Then: HK r (m) ⊇ K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) where h is an analogue of the Weber function.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 11 / 14

Theorem (Main Theorem of CM (ec)) Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be an elliptic curve over C such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). Theorem (Main Theorem of CM (ppas), Shimura-Taniyama, 1950s) Let K be a primitive quartic CM field. Let m ∈ Z>0. Let A be a principally polarized abelian surface over C such that End A ∼ = OK. Then: HK r (m) ⊇ K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) where h is an analogue of the Weber function.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 11 / 14

Theorem (Main Theorem of CM (ec)) Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be an elliptic curve over C such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). Theorem (Main Theorem of CM (ppas), Shimura-Taniyama, 1950s) Let K be a primitive quartic CM field. Let m ∈ Z>0. Let A be a principally polarized abelian surface over C such that End A ∼ = OK. Then: HK r (m) ⊇ K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) where h is an analogue of the Weber function.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 11 / 14

Theorem (Main Theorem of CM (ec)) Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be an elliptic curve over C such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). Theorem (Main Theorem of CM (ppas), Shimura-Taniyama, 1950s) Let K be a primitive quartic CM field. Let m ∈ Z>0. Let A be a principally polarized abelian surface over C such that End A ∼ = OK. Then: HK r (m) ⊇ K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) where h is an analogue of the Weber function.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 11 / 14

Theorem (Main Theorem of CM (ec)) Let K be an imaginary quadratic number field. Let m ∈ Z>0. Let E be an elliptic curve over C such that End E ∼ = OK. Then: HK(m) = K(j(E), h(P) : P ∈ E(C) , [m](P) = ∞) where h is a Weber function (a ‘normalized’ x-coordinate function). Theorem (Main Theorem of CM (ppas), Shimura-Taniyama, 1950s) Let K be a primitive quartic CM field. Let m ∈ Z>0. Let A be a principally polarized abelian surface over C such that End A ∼ = OK. Then: HK r (m) ⊇ K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) where h is an analogue of the Weber function.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 11 / 14

CMK r = K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) Theorem (Shimura, 1962) Let K r be the reflex field of a primitive quartic CM field K. There exists m ∈ Z>0 such that HK r (1) ⊆ HK r

0(m) · CMK r (m)

(⋆)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 12 / 14

CMK r = K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) Theorem (Shimura, 1962) Let K r be the reflex field of a primitive quartic CM field K. There exists m ∈ Z>0 such that HK r (1) ⊆ HK r

0(m) · CMK r (m)

(⋆) (Crespo 1989) Embedding problems with ramification conditions (Cohen 1999) Advanced topics in computational number theory Theorem (A.) Let S be a finite set of prime ideals of OK r such that: S contains all prime ideals above 2 and |ClK r (1)/S| is odd. Let P = {p : p is a rational prime below p for some p ∈ S}. Let m = 4

p∈P p. Then ⋆ is satisfied.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 12 / 14

CMK r = K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) What We Know For a primitive quartic CM field K: how to find an m such that HK r (1) ⊆ HK r

0(m) · CMK r (m).

how to decide if HK r (1) ⊆ HK r

0(m) · CMK r (m) for any m ∈ Z>0.

how to compute HK r

0(m) (using Stark’s conjectures)

how to compute a defining polynomial for CMK r (m) for m = 1, 2. how to compute a defining polynomial for HK r (1) by viewing it as a subfield of CMK r (2)

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 13 / 14

CMK r = K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) What We Know For a primitive quartic CM field K: how to find an m such that HK r (1) ⊆ HK r

0(m) · CMK r (m).

how to decide if HK r (1) ⊆ HK r

0(m) · CMK r (m) for any m ∈ Z>0.

how to compute HK r

0(m) (using Stark’s conjectures)

how to compute a defining polynomial for CMK r (m) for m = 1, 2. how to compute a defining polynomial for HK r (1) by viewing it as a subfield of CMK r (2) What We Have working implementation of the above computations example for which our implementation is faster than Kummer theory implementations in Magma and PARI

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 13 / 14

CMK r = K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) Things to Work On how to find CMK r,Φr (m) for any other m. statistics on how often an integer m ∈ Z>0 is the minimal m such that HK r (1) ⊆ HK r

0(m) · CMK r (m). Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 14 / 14

CMK r = K r(i(A), h(P) : P ∈ A(C) , [m](P) = ∞) Things to Work On how to find CMK r,Φr (m) for any other m. statistics on how often an integer m ∈ Z>0 is the minimal m such that HK r (1) ⊆ HK r

0(m) · CMK r (m).

Dank u voor uw aandacht.

Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 14 / 14