Explicit Construction of Abelian Extensions of Number Fields Jared - - PowerPoint PPT Presentation

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 x2 + 5. Hence, it is an algebraic integer. 3.14 is a root of 50x − 157. Hence, it is an algebraic number. π is NOT an algebraic number.

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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 OK of K form a subring of K. Q(√−5) is a number field of degree 2 since dimQ(K) = 2. Q(π) and C are not number fields since they are not finite extensions of Q.

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Definition (Galois extension) A field extension L/K is Galois if the group Aut(L/K) of automorphisms

• f 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 K = Q(

3

√ 2) is not a Galois extension of Q since the only automorphism of K fixing Q is the identity automorphism.

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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(ζn), where ζn = exp(2πi/n), is an abelian extension since its Galois group is Gal(K/Q) ∼ = (Z/nZ)×

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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(ζn), where ζn = exp(2πi/n), is an abelian extension since its Galois group is Gal(K/Q) ∼ = (Z/nZ)× Problem (Hilbert’s 12th Problem) Given a number field K, construct all abelian extensions of K by adjoining special values of particular analytic functions.

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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(2πiz)) for some z ∈ Q. ‘particular function’ e : R → S1(C) z → exp(2πiz) ‘special values’ Append e(z) such that z ∈ Q.

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e = exp(2πiz) R S1(C) Observations

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e = exp(2πiz) R S1(C) Observations The kernel of the map e is Z.

Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25

e = exp(2πiz) R/Z S1(C) Observations The kernel of the map e is Z.

Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25

e = exp(2πiz) R/Z S1(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

e = exp(2πiz) R/Z S1(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.

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e = exp(2πiz) R/Z S1(C)

6 1 6 2 6 3 6 4 6 5 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

e = exp(2πiz) R/Z S1(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.

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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 HK(m) of K.

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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 HK(m) of K. For the case when the base field is Q, we have: HQ(1) = Q HQ(m) = Q(exp(2πi · 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 HK(m) of K. For the case when the base field is Q, we have: HQ(1) = Q HQ(m) = Q(exp(2πi · 1/m)). Hilbert’s 12th Problem What about other base fields?

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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 HK(m) of K. For the case when the base field is Q, we have: HQ(1) = Q HQ(m) = Q(exp(2πi · 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 2Z = 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 2Z = X 3 + aXZ 2 + bZ 3 where a, b ∈ k and f (x) has no double roots in the algebraic closure of k. x y

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 2Z = X 3 + aXZ 2 + bZ 3 where a, b ∈ k and f (x) has no double roots in the algebraic closure of k. x y It has exactly one point at infinity, which we denote by ∞ = (0 : 1 : 0).

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 2Z = X 3 + aXZ 2 + bZ 3 where a, b ∈ k and f (x) has no double roots in the algebraic closure of k. x y E : y2 = x3 + 1 It has exactly one point at infinity, which we denote by ∞ = (0 : 1 : 0). We will usually write the affine equation y2 = x3 + ax + b to define elliptic curves and remember that there is an extra point at infinity.

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Notation Let k ⊆ K. The set of K-rational points of an elliptic curve E is given by E(K) = {∞} ∪ {(x, y) ∈ K × K : y2 = x3 + 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.

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Notation Let k ⊆ K. The set of K-rational points of an elliptic curve E is given by E(K) = {∞} ∪ {(x, y) ∈ K × K : y2 = x3 + 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) (x, y) → (x, −y) (x, y) → f ′(x)2 4f (x) − a − 2x, · · ·

• Jared Asuncion

Lambda Seminar Talk 21 November 2019 10 / 25

Example The elliptic curve E : y2 = x3 + x over Q has an endomorphism [i] : (x, y) → (−x, iy). Hence Z End E.

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Example The elliptic curve E : y2 = x3 + 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 OK be its ring of

• integers. Let End E be the ring of endomorphisms of E. If End E ∼

= OK, then E is said to have complex multiplication by OK.

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Example The elliptic curve E : y2 = x3 + 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 OK be its ring of

• integers. Let End E be the ring of endomorphisms of E. If End E ∼

= OK, then E is said to have complex multiplication by OK. Theorem (Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(1) = k0K where k0 is the field of moduli of E.

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Assume all lattices are of the form Λ = τ1Z + τ2Z with Im(τ1/τ2) > 0.

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Assume all lattices are of the form Λ = τ1Z + τ2Z with Im(τ1/τ2) > 0. Definition (Weierstrass ℘-function) The Weierstrass ℘-function for a lattice Λ is defined to be ℘Λ(z) = 1 z2 +

• n∈Λ\(0,0)
• 1

(z + n)2 − 1 n2

• .

Jared Asuncion Lambda Seminar Talk 21 November 2019 12 / 25

Assume all lattices are of the form Λ = τ1Z + τ2Z with Im(τ1/τ2) > 0. Definition (Weierstrass ℘-function) The Weierstrass ℘-function for a lattice Λ is defined to be ℘Λ(z) = 1 z2 +

• n∈Λ\(0,0)
• 1

(z + n)2 − 1 n2

• .

Theorem The Weierstrass ℘Λ-function is an elliptic function. This means that: ℘Λ has countably many poles. meromorphic ℘Λ(z) = ℘Λ(z + τ1) = ℘Λ(z + τ2). doubly-periodic

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Definition (Eisenstein series) The Eisenstein series of weight 2k where k ≥ 2 is an integer is defined by the following series: G2k(Λ) =

• n∈Λ\(0,0)

1 n2k .

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Definition (Eisenstein series) The Eisenstein series of weight 2k where k ≥ 2 is an integer is defined by the following series: G2k(Λ) =

• n∈Λ\(0,0)

1 n2k . Theorem The Weierstrass ℘Λ-function satisfies the ordinary non-linear differential equation: ℘′

Λ = 4℘3 Λ − g2(Λ)℘Λ − g3(Λ)

where g2 = 60G4 and g3 = 140G6.

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Consider the map f : C → P2(C) z → (℘Λ(z) : ℘′

Λ(z) : 1).

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Consider the map f : C → P2(C) z → (℘Λ(z) : ℘′

Λ(z) : 1).

Theorem The map f is well-defined modulo Λ. Proof idea: ℘Λ and ℘′

Λ are periodic with respect to the lattice.

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Consider the map f : C → P2(C) z → (℘Λ(z) : ℘′

Λ(z) : 1).

Theorem The map f is well-defined modulo Λ. Proof idea: ℘Λ and ℘′

Λ are periodic with respect to the lattice.

Theorem The image of f is an elliptic curve over C. Proof idea: It satisfies the differential equation of the form y2 = 4x3 − g2x − g3. With an invertible change of variables, (x, y) → (x, y/2), we find that the image satisfies the equation y2 = x3 + ax + b for some a, b ∈ C.

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Theorem The map f : C/Λ → EΛ(C) z → (℘Λ(z) : ℘′

Λ(z)/2 : 1)

defines an isomorphism between the complex torus C/Λ and the elliptic curve given by the equation EΛ : y2 = x3 − g2(Λ) 4 x − g3(Λ) 4

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Theorem The map f : C/Λ → EΛ(C) z → (℘Λ(z) : ℘′

Λ(z)/2 : 1)

defines an isomorphism between the complex torus C/Λ and the elliptic curve given by the equation EΛ : y2 = x3 − g2(Λ) 4 x − g3(Λ) 4 Observe that z ∈ Λ is sent to ∞.

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Theorem The map f : C/Λ → EΛ(C) z → (℘Λ(z) : ℘′

Λ(z)/2 : 1)

defines an isomorphism between the complex torus C/Λ and the elliptic curve given by the equation EΛ : y2 = x3 − g2(Λ) 4 x − g3(Λ) 4 Observe that z ∈ Λ is sent to ∞. One can check verify that this is a group isomorphism.

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Definition Two complex tori C/Λ and C/Λ′ are said to be isomorphic if their lattices Λ and Λ′ are homothetic (i.e. Λ′ = αΛ for some α ∈ C×).

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Definition Two complex tori C/Λ and C/Λ′ are said to be isomorphic if their lattices Λ and Λ′ are homothetic (i.e. Λ′ = αΛ for some α ∈ C×). Definition The j-invariant of a lattice can be defined as a function on lattices: j(Λ) = g2(Λ)3 g2(Λ)3 − 27g3(Λ)2

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Definition Two complex tori C/Λ and C/Λ′ are said to be isomorphic if their lattices Λ and Λ′ are homothetic (i.e. Λ′ = αΛ for some α ∈ C×). Definition The j-invariant of a lattice can be defined as a function on lattices: j(Λ) = g2(Λ)3 g2(Λ)3 − 27g3(Λ)2 Theorem Two elliptic curves are isomorphic if and only if they have the same j-invariant. This means that j(Λ) = j(αΛ) for any α ∈ C×.

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Definition A field K is called a field of definition of an elliptic curve E = C/Λ if there exists an elliptic curve E ′ isomorphic to E defined over K. Let Λ be a lattice. For any α ∈ C×, Q(g2(αΛ), g3(αΛ)) is a field of definition for the elliptic curve Λ.

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Definition A field K is called a field of definition of an elliptic curve E = C/Λ if there exists an elliptic curve E ′ isomorphic to E defined over K. Let Λ be a lattice. For any α ∈ C×, Q(g2(αΛ), g3(αΛ)) is a field of definition for the elliptic curve Λ. Definition There exists a unique minimal field of definition k0 for any elliptic curve E over C. This field is called its field of moduli. The field of moduli of an elliptic curve E = C/Λ is Q(j(Λ)). We write j(E) to denote j(Λ) for any Λ such that E ∼ = C/Λ.

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Theorem (Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(1) = k0K = K(j(E)) where k0 is the field of moduli of E.

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Theorem (Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(1) = k0K = K(j(E)) where k0 is the field of moduli of E. Let E be an elliptic curve with complex multiplication by Q(√−5).

Jared Asuncion Lambda Seminar Talk 21 November 2019 18 / 25

Theorem (Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(1) = k0K = K(j(E)) where k0 is the field of moduli of E. Let E be an elliptic curve with complex multiplication by Q(√−5). The ring of integers of K is OK = Z[√−5] = Z + √−5 Z.

Jared Asuncion Lambda Seminar Talk 21 November 2019 18 / 25

Theorem (Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(1) = k0K = K(j(E)) where k0 is the field of moduli of E. Let E be an elliptic curve with complex multiplication by Q(√−5). The ring of integers of K is OK = Z[√−5] = Z + √−5 Z. One can show that E ∼ = C/OK.

Jared Asuncion Lambda Seminar Talk 21 November 2019 18 / 25

Theorem (Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(1) = k0K = K(j(E)) where k0 is the field of moduli of E. Let E be an elliptic curve with complex multiplication by Q(√−5). The ring of integers of K is OK = Z[√−5] = Z + √−5 Z. One can show that E ∼ = C/OK. j(OK) = 320

• 1975 + 884

√ 5

• .

Thus, the field of moduli of E is Q( √ 5)

Jared Asuncion Lambda Seminar Talk 21 November 2019 18 / 25

Theorem (Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(1) = k0K = K(j(E)) where k0 is the field of moduli of E. Let E be an elliptic curve with complex multiplication by Q(√−5). The ring of integers of K is OK = Z[√−5] = Z + √−5 Z. One can show that E ∼ = C/OK. j(OK) = 320

• 1975 + 884

√ 5

• .

Thus, the field of moduli of E is Q( √ 5). Thus, HK(1) = K( √ 5).

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Let E be the same elliptic curve with complex multiplication by Q(√−5) from the previous slide. Recall that we took Λ = Z + √ −5 Z.

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Let E be the same elliptic curve with complex multiplication by Q(√−5) from the previous slide. Recall that we took Λ = Z + √ −5 Z. There exists α such that EαΛ is defined over Q( √ 5). One example for α is: α ≈ 1.480525.

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Let E be the same elliptic curve with complex multiplication by Q(√−5) from the previous slide. Recall that we took Λ = Z + √ −5 Z. There exists α such that EαΛ is defined over Q( √ 5). One example for α is: α ≈ 1.480525. We have g2(αΛ) = g3(αΛ) = −4a a = 1461375 349448 + 805545 698896 √ 5

• Jared Asuncion

Lambda Seminar Talk 21 November 2019 19 / 25

Let E be the same elliptic curve with complex multiplication by Q(√−5) from the previous slide. Recall that we took Λ = Z + √ −5 Z. There exists α such that EαΛ is defined over Q( √ 5). One example for α is: α ≈ 1.480525. We have g2(αΛ) = g3(αΛ) = −4a a = 1461375 349448 + 805545 698896 √ 5

• Then

EαΛ : x3 − ax − a Notice that a ∈ Q( √ 5).

Jared Asuncion Lambda Seminar Talk 21 November 2019 19 / 25

Recall that C/Λ ∼ = E(C) via the map f : z → (℘Λ(z) : ℘′

Λ(z) : 1):

f C/Λ E(C) Observations

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Recall that C/Λ ∼ = E(C) via the map f : z → (℘Λ(z) : ℘′

Λ(z) : 1):

f C/Λ E(C) Observations The kernel of the (original) map was Λ ∼ = Z2.

Jared Asuncion Lambda Seminar Talk 21 November 2019 20 / 25

Recall that C/Λ ∼ = E(C) via the map f : z → (℘Λ(z) : ℘′

Λ(z) : 1):

f C/Λ E(C) Observations The kernel of the (original) map was Λ ∼ = Z2. The image of the map is a geometric object, an elliptic curve.

Jared Asuncion Lambda Seminar Talk 21 November 2019 20 / 25

Recall that C/Λ ∼ = E(C) via the map f : z → (℘Λ(z) : ℘′

Λ(z) : 1):

f C/Λ E(C) Observations The kernel of the (original) map was Λ ∼ = Z2. The image of the map is a geometric object, an elliptic curve. Both domain and codomain have a group structure.

Jared Asuncion Lambda Seminar Talk 21 November 2019 20 / 25

Recall that C/Λ ∼ = E(C) via the map f : z → (℘Λ(z) : ℘′

Λ(z) : 1):

f C/Λ E(C) Observations The kernel of the (original) map was Λ ∼ = Z2. The image of the map is a geometric object, an elliptic curve. 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 20 / 25

Recall that C/Λ ∼ = E(C) via the map f : z → (℘Λ(z) : ℘′

Λ(z) : 1):

f C/Λ E(C) Observations The kernel of the (original) map was Λ ∼ = Z2. The image of the map is a geometric object, an elliptic curve. 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 20 / 25

Recall that C/Λ ∼ = E(C) via the map f : z → (℘Λ(z) : ℘′

Λ(z) : 1):

f C/Λ E(C) Observations The kernel of the (original) map was Λ ∼ = Z2. The image of the map is a geometric object, an elliptic curve. Both domain and codomain have a group structure. The torsion points of the domain are easy to determine. The torsion points of the codomain have two coordinates!

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Definition (normalized Kummer variety) Let E be an elliptic curve over C and let E0 be an elliptic curve defined

• ver its field of moduli k0. The normalized Kummer variety of E is the

pair (W , h) such that W is the quotient of E0 by its group Aut E0 of automorphisms. h is a morphism of varieties defined over k0.

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Definition (normalized Kummer variety) Let E be an elliptic curve over C and let E0 be an elliptic curve defined

• ver its field of moduli k0. The normalized Kummer variety of E is the

pair (W , h) such that W is the quotient of E0 by its group Aut E0 of automorphisms. h is a morphism of varieties defined over k0. Recall E0 : x3 − ax − a where a = 1461375

349448 + 805545 698896

√ 5. E0 has complex multiplication by OK with K = Q(√−5).

Jared Asuncion Lambda Seminar Talk 21 November 2019 21 / 25

Definition (normalized Kummer variety) Let E be an elliptic curve over C and let E0 be an elliptic curve defined

• ver its field of moduli k0. The normalized Kummer variety of E is the

pair (W , h) such that W is the quotient of E0 by its group Aut E0 of automorphisms. h is a morphism of varieties defined over k0. Recall E0 : x3 − ax − a where a = 1461375

349448 + 805545 698896

√ 5. E0 has complex multiplication by OK with K = Q(√−5). The invertible elements of OK are {±1}. Thus, Aut E0 = {[±1]}.

Jared Asuncion Lambda Seminar Talk 21 November 2019 21 / 25

Definition (normalized Kummer variety) Let E be an elliptic curve over C and let E0 be an elliptic curve defined

• ver its field of moduli k0. The normalized Kummer variety of E is the

pair (W , h) such that W is the quotient of E0 by its group Aut E0 of automorphisms. h is a morphism of varieties defined over k0. Recall E0 : x3 − ax − a where a = 1461375

349448 + 805545 698896

√ 5. E0 has complex multiplication by OK with K = Q(√−5). The invertible elements of OK are {±1}. Thus, Aut E0 = {[±1]}. Since x(P) = x(Q) if and only if P = [±1]Q then we can parametrize the points of W := E0/ Aut E0 by their x-coordinates.

Jared Asuncion Lambda Seminar Talk 21 November 2019 21 / 25

Definition (normalized Kummer variety) Let E be an elliptic curve over C and let E0 be an elliptic curve defined

• ver its field of moduli k0. The normalized Kummer variety of E is the

pair (W , h) such that W is the quotient of E0 by its group Aut E0 of automorphisms. h is a morphism of varieties defined over k0. Recall E0 : x3 − ax − a where a = 1461375

349448 + 805545 698896

√ 5. E0 has complex multiplication by OK with K = Q(√−5). The invertible elements of OK are {±1}. Thus, Aut E0 = {[±1]}. Since x(P) = x(Q) if and only if P = [±1]Q then we can parametrize the points of W := E0/ Aut E0 by their x-coordinates. Take h : (x, y) → x.

Jared Asuncion Lambda Seminar Talk 21 November 2019 21 / 25

Definition (normalized Kummer variety) Let E be an elliptic curve over C and let E0 be an elliptic curve defined

• ver its field of moduli k0. The normalized Kummer variety of E is the

pair (W , h) such that W is the quotient of E0 by its group Aut E0 of automorphisms. h is a morphism of varieties defined over k0. Recall E0 : x3 − ax − a where a = 1461375

349448 + 805545 698896

√ 5. E0 has complex multiplication by OK with K = Q(√−5). The invertible elements of OK are {±1}. Thus, Aut E0 = {[±1]}. Since x(P) = x(Q) if and only if P = [±1]Q then we can parametrize the points of W := E0/ Aut E0 by their x-coordinates. Take h : (x, y) → x. (W , h) is a normalized Kummer variety of E.

Jared Asuncion Lambda Seminar Talk 21 November 2019 21 / 25

Theorem (Second Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over HK(1) with complex multiplication by OK. Then HK(m) = HK(1)(h(t) : t ∈ ker[m]) where (W , h) is a normalized Kummer variety for E.

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Theorem (Second Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over HK(1) with complex multiplication by OK. Then HK(m) = HK(1)(h(t) : t ∈ ker[m]) where (W , h) is a normalized Kummer variety for E. We have solved for (W , h), a normalized Kummer variety for E. Recall W is a quotient of E0 = C/αΛ with αΛ = ω1Z + ω2Z.

Jared Asuncion Lambda Seminar Talk 21 November 2019 22 / 25

Theorem (Second Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over HK(1) with complex multiplication by OK. Then HK(m) = HK(1)(h(t) : t ∈ ker[m]) where (W , h) is a normalized Kummer variety for E. We have solved for (W , h), a normalized Kummer variety for E. Recall W is a quotient of E0 = C/αΛ with αΛ = ω1Z + ω2Z. Find the set T of 3-torsion points in C/Λ. Easy! ex: ω1/3 ∈ T.

Jared Asuncion Lambda Seminar Talk 21 November 2019 22 / 25

Theorem (Second Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over HK(1) with complex multiplication by OK. Then HK(m) = HK(1)(h(t) : t ∈ ker[m]) where (W , h) is a normalized Kummer variety for E. We have solved for (W , h), a normalized Kummer variety for E. Recall W is a quotient of E0 = C/αΛ with αΛ = ω1Z + ω2Z. Find the set T of 3-torsion points in C/Λ. Easy! ex: ω1/3 ∈ T. The x-coordinates of E0(C) are ℘αΛ(z) for z ∈ T.

Jared Asuncion Lambda Seminar Talk 21 November 2019 22 / 25

Theorem (Second Main Theorem of Complex Multiplication) Let K be an imaginary quadratic number field and let E be an elliptic curve over HK(1) with complex multiplication by OK. Then HK(m) = HK(1)(h(t) : t ∈ ker[m]) where (W , h) is a normalized Kummer variety for E. We have solved for (W , h), a normalized Kummer variety for E. Recall W is a quotient of E0 = C/αΛ with αΛ = ω1Z + ω2Z. Find the set T of 3-torsion points in C/Λ. Easy! ex: ω1/3 ∈ T. The x-coordinates of E0(C) are ℘αΛ(z) for z ∈ T. Hence, to get HK(3), we add to HK(1) the values of ℘αΛ(z) for each z ∈ T.

Jared Asuncion Lambda Seminar Talk 21 November 2019 22 / 25

We have: ℘αΛ ω1 3

• = 11

486

• 845 + 477

√ 15 +

• 4120880 + 807270

√ 15

• ℘αΛ

ω2 3

• = 11

486

• 845 − 477

√ 15 −

• 4120880 − 807270

√ 15

• ℘αΛ

ω1 + ω2 3

• = 11

486

• −845 − 379

√ 5 −

• −14070 − 6290

√ 5

• ℘αΛ

ω1 + 2ω2 3

• = 11

486

• −845 − 379

√ 5 +

• −14070 − 6290

√ 5

• We find that HK(1) (those algebraic numbers above) = HK(1)(

√ 3). Using the second main theorem of multiplication, we find that HK(3) = HK(1)( √ 3) = K( √ 5, √ 3) K = Q( √ −5)

Jared Asuncion Lambda Seminar Talk 21 November 2019 23 / 25

Theorem (EC case) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(m) = K(j(E), h(t)) where (E/ Aut E, h) is a normalized Kummer variety of E and t is a proper m-torsion point. Theorem (PPAV case) Let K be a CM field of degree 2g and let A be a simple g-dimensional principally polarized abelian variety over C with complex multiplication by

• OK. Then

HK r (m) ⊇ K r(i(A), h(t))

• CMKr (m)

where K r is the reflex field of K and (A/ Aut A, h) is a normalized Kummer variety of A and t is a proper m-torsion point.

Jared Asuncion Lambda Seminar Talk 21 November 2019 24 / 25

Theorem (EC case) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(m) = K(j(E), h(t)) where (E/ Aut E, h) is a normalized Kummer variety of E and t is a proper m-torsion point. Theorem (PPAV case) Let K be a CM field of degree 2g and let A be a simple g-dimensional principally polarized abelian variety over C with complex multiplication by

• OK. Then

HK r (m) ⊇ K r(i(A), h(t))

• CMKr (m)

where K r is the reflex field of K and (A/ Aut A, h) is a normalized Kummer variety of A and t is a proper m-torsion point.

Jared Asuncion Lambda Seminar Talk 21 November 2019 24 / 25

Theorem (EC case) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(m) = K(j(E), h(t)) where (E/ Aut E, h) is a normalized Kummer variety of E and t is a proper m-torsion point. Theorem (PPAV case) Let K be a CM field of degree 2g and let A be a simple g-dimensional principally polarized abelian variety over C with complex multiplication by

• OK. Then

HK r (m) ⊇ K r(i(A), h(t))

• CMKr (m)

where K r is the reflex field of K and (A/ Aut A, h) is a normalized Kummer variety of A and t is a proper m-torsion point.

Jared Asuncion Lambda Seminar Talk 21 November 2019 24 / 25

Theorem (EC case) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(m) = K(j(E), h(t)) where (E/ Aut E, h) is a normalized Kummer variety of E and t is a proper m-torsion point. Theorem (PPAV case) Let K be a CM field of degree 2g and let A be a simple g-dimensional principally polarized abelian variety over C with complex multiplication by

• OK. Then

HK r (m) ⊇ K r(i(A), h(t))

• CMKr (m)

where K r is the reflex field of K and (A/ Aut A, h) is a normalized Kummer variety of A and t is a proper m-torsion point.

Jared Asuncion Lambda Seminar Talk 21 November 2019 24 / 25

Theorem (EC case) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(m) = K(j(E), h(t)) where (E/ Aut E, h) is a normalized Kummer variety of E and t is a proper m-torsion point. Theorem (PPAV case) Let K be a CM field of degree 2g and let A be a simple g-dimensional principally polarized abelian variety over C with complex multiplication by

• OK. Then

HK r (m) ⊇ K r(i(A), h(t))

• CMKr (m)

where K r is the reflex field of K and (A/ Aut A, h) is a normalized Kummer variety of A and t is a proper m-torsion point.

Jared Asuncion Lambda Seminar Talk 21 November 2019 24 / 25

Theorem (EC case) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(m) = K(j(E), h(t)) where (E/ Aut E, h) is a normalized Kummer variety of E and t is a proper m-torsion point. Theorem (PPAV case) Let K be a CM field of degree 2g and let A be a simple g-dimensional principally polarized abelian variety over C with complex multiplication by

• OK. Then

HK r (m) ⊇ K r(i(A), h(t))

• CMKr (m)

where K r is the reflex field of K and (A/ Aut A, h) is a normalized Kummer variety of A and t is a proper m-torsion point.

Jared Asuncion Lambda Seminar Talk 21 November 2019 24 / 25

Theorem (EC case) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(m) = K(j(E), h(t)) where (E/ Aut E, h) is a normalized Kummer variety of E and t is a proper m-torsion point. Theorem (PPAV case) Let K be a CM field of degree 2g and let A be a simple g-dimensional principally polarized abelian variety over C with complex multiplication by

• OK. Then

HK r (m) ⊇ K r(i(A), h(t))

• CMKr (m)

where K r is the reflex field of K and (A/ Aut A, h) is a normalized Kummer variety of A and t is a proper m-torsion point.

Jared Asuncion Lambda Seminar Talk 21 November 2019 24 / 25

Theorem (EC case) Let K be an imaginary quadratic number field and let E be an elliptic curve over C with complex multiplication by OK. Then HK(m) = K(j(E), h(t)) where (E/ Aut E, h) is a normalized Kummer variety of E and t is a proper m-torsion point. Theorem (PPAV case) Let K be a CM field of degree 2g and let A be a simple g-dimensional principally polarized abelian variety over C with complex multiplication by

• OK. Then

HK r (m) ⊇ K r(i(A), h(t))

• CMKr (m)

where K r is the reflex field of K and (A/ Aut A, h) is a normalized Kummer variety of A and t is a proper m-torsion point.

Jared Asuncion Lambda Seminar Talk 21 November 2019 24 / 25

What I (Try To) Do Take g = 2. i.e. K is a quartic CM field, A is a ppav of dimension 2 Find h(t) to compute CMK r (m). Figure out which are the proper m-torsion points. Determine which m satisfies HK r (1) ⊆ CMK r (m). Compute HK r (1) as a subfield of CMK r (m).

Jared Asuncion Lambda Seminar Talk 21 November 2019 25 / 25