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

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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


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Explicit Construction of Abelian Extensions of Number Fields

Jared Asuncion 21 November 2019

Jared Asuncion Lambda Seminar Talk 21 November 2019 1 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 2 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 3 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 4 / 25

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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)×

Jared Asuncion Lambda Seminar Talk 21 November 2019 5 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 5 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 6 / 25

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e = exp(2πiz) R S1(C) Observations

Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25

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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

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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

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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

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25

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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

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 7 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 8 / 25

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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

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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?

Jared Asuncion Lambda Seminar Talk 21 November 2019 8 / 25

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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

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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

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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

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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

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 9 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 10 / 25

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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

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 11 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 11 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 11 / 25

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

Jared Asuncion Lambda Seminar Talk 21 November 2019 12 / 25

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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

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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

  • .

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

Jared Asuncion Lambda Seminar Talk 21 November 2019 12 / 25

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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 .

Jared Asuncion Lambda Seminar Talk 21 November 2019 13 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 13 / 25

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

Λ(z) : 1).

Jared Asuncion Lambda Seminar Talk 21 November 2019 14 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 14 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 14 / 25

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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

Jared Asuncion Lambda Seminar Talk 21 November 2019 15 / 25

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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 ∞.

Jared Asuncion Lambda Seminar Talk 21 November 2019 15 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 15 / 25

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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×).

Jared Asuncion Lambda Seminar Talk 21 November 2019 16 / 25

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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

Jared Asuncion Lambda Seminar Talk 21 November 2019 16 / 25

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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×.

Jared Asuncion Lambda Seminar Talk 21 November 2019 16 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 17 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 17 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 18 / 25

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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

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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). The ring of integers of K is OK = Z[√−5] = Z + √−5 Z.

Jared Asuncion Lambda Seminar Talk 21 November 2019 18 / 25

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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). 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

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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). 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

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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). 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).

Jared Asuncion Lambda Seminar Talk 21 November 2019 18 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 19 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 19 / 25

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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

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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

  • Then

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

Jared Asuncion Lambda Seminar Talk 21 November 2019 19 / 25

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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

Jared Asuncion Lambda Seminar Talk 21 November 2019 20 / 25

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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

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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. The image of the map is a geometric object, an elliptic curve.

Jared Asuncion Lambda Seminar Talk 21 November 2019 20 / 25

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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. 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

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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. 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

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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. 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

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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. 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!

Jared Asuncion Lambda Seminar Talk 21 November 2019 20 / 25

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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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 21 / 25

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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

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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). The invertible elements of OK are {±1}. Thus, Aut E0 = {[±1]}.

Jared Asuncion Lambda Seminar Talk 21 November 2019 21 / 25

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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). 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

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SLIDE 65

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

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SLIDE 66

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

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SLIDE 67

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.

Jared Asuncion Lambda Seminar Talk 21 November 2019 22 / 25

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SLIDE 68

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

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SLIDE 69

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

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SLIDE 70

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

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SLIDE 71

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

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SLIDE 72

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

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SLIDE 73

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

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SLIDE 74

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

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SLIDE 75

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

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SLIDE 76

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

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SLIDE 77

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

slide-78
SLIDE 78

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

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SLIDE 79

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

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SLIDE 80

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

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SLIDE 81

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