Faltings Heights of CM Elliptic Curves Tyler Genao Florida Atlantic - - PowerPoint PPT Presentation

Faltings Heights of CM Elliptic Curves

Tyler Genao

Florida Atlantic University

In collaboration with Adrian Barquero-Sanchez, Lindsay Cadwallader, Olivia Cannon, Riad Masri

Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves

◮ Consider an equation of the form

y2 = x3 + Ax + B for some A, B in a field F.

Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves

◮ Consider an equation of the form

y2 = x3 + Ax + B for some A, B in a field F.

◮ If 4A3 + 27B2 = 0, we call this equation an elliptic curve, and

write it as E/F.

Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves

◮ Consider an equation of the form

y2 = x3 + Ax + B for some A, B in a field F.

◮ If 4A3 + 27B2 = 0, we call this equation an elliptic curve, and

write it as E/F.

◮ Define the discriminant of E/Q to be

∆E := −16(4A3 + 27B2).

Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves (Examples)

E/R : y2 = x3 − x

Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves (Examples)

E/R : y2 = x3 − x + 1

Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves (Group Law)

◮ One can view the curve E(R) as a group. If P and Q are two

points on the curve, define the operation for P add Q like so: take the line intersecting both P and Q; it will intersect the curve at another point, say R. Then reflect that point over the y axis, and call this point P + Q.

Tyler Genao Faltings Heights of CM Elliptic Curves

Lattices

◮ For two complex numbers ω1 and ω2, define the Lattice

generated by ω1 and ω2 to be L(ω1, ω2) := Zω1 + Zω2 = {aω1 + bω2 : a, b ∈ Z}.

Tyler Genao Faltings Heights of CM Elliptic Curves

Lattices

◮ For two complex numbers ω1 and ω2, define the Lattice

generated by ω1 and ω2 to be L(ω1, ω2) := Zω1 + Zω2 = {aω1 + bω2 : a, b ∈ Z}.

◮ The parallelogram PL(ω1,ω2) defined by ω1 and ω2 defines a

fundamental parallelogram for C/L(ω1, ω2).

Tyler Genao Faltings Heights of CM Elliptic Curves

A Fundamental Parallelogram

Tyler Genao Faltings Heights of CM Elliptic Curves

Uniformization

◮ It turns out that for any elliptic curve

E/C : y2 = x3 + Ax + B, there exists τ ∈ C with Im(τ) > 0 so that E(C) ∼ = C/L(1, τ).

Tyler Genao Faltings Heights of CM Elliptic Curves

The Faltings Height of an Elliptic Curve

Definition

hFal(E/Q) := 1 12 log |∆E/Q| − 1 2 log

• i

2

• E(C)

ω ∧ ¯ ω

• .

Tyler Genao Faltings Heights of CM Elliptic Curves

The Faltings Height of an Elliptic Curve

Definition

hFal(E/Q) := 1 12 log |∆E/Q| − 1 2 log

• i

2

• E(C)

ω ∧ ¯ ω

• .

Remark

i 2

• E(C)

ω ∧ ¯ ω ∼Q× Area(PL(1,τ)).

Tyler Genao Faltings Heights of CM Elliptic Curves

Imaginary Quadratic Orders

◮ For any negative squarefree d ∈ Z, call the field

K = Q( √ d) := {a + b √ d : a, b ∈ Q} an imaginary quadratic number field.

Tyler Genao Faltings Heights of CM Elliptic Curves

Imaginary Quadratic Orders

◮ For any negative squarefree d ∈ Z, call the field

K = Q( √ d) := {a + b √ d : a, b ∈ Q} an imaginary quadratic number field.

◮ Define the discriminant of K to be

DK :=

• d

if d ≡ 1 (mod 4), 4d if d ≡ 2, 3 (mod 4).

Tyler Genao Faltings Heights of CM Elliptic Curves

Imaginary Quadratic Orders

◮ Define the number

ωK := DK + √DK 2 .

Tyler Genao Faltings Heights of CM Elliptic Curves

Imaginary Quadratic Orders

◮ Define the number

ωK := DK + √DK 2 .

◮ For an integer f > 0, the ring

Of = [1, f ωK] := {a + bf ωK : a, b ∈ Z} is called an imaginary quadratic order of conductor f in K.

◮ the order O1 of K is called the maximal order of K.

Tyler Genao Faltings Heights of CM Elliptic Curves

The Endomorphism Ring

◮ Recall that for an elliptic curve E/Q, there is a lattice

L = [1, τ] such that E(C) ∼ = C/L.

Tyler Genao Faltings Heights of CM Elliptic Curves

The Endomorphism Ring

◮ Recall that for an elliptic curve E/Q, there is a lattice

L = [1, τ] such that E(C) ∼ = C/L.

◮ For an elliptic curve E/Q corresponding to a lattice L, we

define the endomorphism ring of E/Q to be EndC(E) := {α ∈ C : αL ⊆ L}.

Tyler Genao Faltings Heights of CM Elliptic Curves

CM Elliptic Curves

Theorem

For an elliptic curve E/Q, EndC(E) is isomorphic either to Z or to an order Of in some K.

Tyler Genao Faltings Heights of CM Elliptic Curves

CM Elliptic Curves

Theorem

For an elliptic curve E/Q, EndC(E) is isomorphic either to Z or to an order Of in some K.

◮ If EndC(E) is isomorphic to Of , then E/Q is said to have

complex multiplication, or CM.

Tyler Genao Faltings Heights of CM Elliptic Curves

Motivation

◮ For elliptic curves E/Q with CM by a maximal order, Deligne

computed hFal(E/Q) in terms of Euler’s Γ-function Γ(s) := ∞ xs−1e−xdx at rational numbers.

Tyler Genao Faltings Heights of CM Elliptic Curves

Motivation

◮ For elliptic curves E/Q with CM by a maximal order, Deligne

computed hFal(E/Q) in terms of Euler’s Γ-function Γ(s) := ∞ xs−1e−xdx at rational numbers.

◮ Our main result is an analogous formula for any order

Of ⊆ K.

Tyler Genao Faltings Heights of CM Elliptic Curves

Preliminaries to the Main Theorem

Let K be an imaginary quadratic number field with discriminant DK.

Tyler Genao Faltings Heights of CM Elliptic Curves

Preliminaries to the Main Theorem

Let K be an imaginary quadratic number field with discriminant DK.

◮ Let ωDK be 2, 4, or 6, depending on K.

Tyler Genao Faltings Heights of CM Elliptic Curves

Preliminaries to the Main Theorem

Let K be an imaginary quadratic number field with discriminant DK.

◮ Let ωDK be 2, 4, or 6, depending on K. ◮ Let h(K) < ∞ denote the class number of K.

Tyler Genao Faltings Heights of CM Elliptic Curves

Preliminaries to the Main Theorem

Let K be an imaginary quadratic number field with discriminant DK.

◮ Let ωDK be 2, 4, or 6, depending on K. ◮ Let h(K) < ∞ denote the class number of K. ◮ Let χDK (k) be -1, 0, or 1, depending on the integer k.

Tyler Genao Faltings Heights of CM Elliptic Curves

Theorem

The Faltings Height of an Elliptic Curve E/Q with CM

Let E/Q be an elliptic curve with complex multiplication by an imaginary quadratic order Of of K, with K having discriminant

• DK. Then

hFal(E/Q) = − log  |∆E/Q|−1/12

• π

f

• |DK|

1/2 |DK |

• k=1

Γ

• k

|DK| χDK (k)

ωDK

4h(K)

p|f

pe(p)/2   ,

where

e(p) = − (1 − pordp(f ))(1 − χD(p)) pordp(f )−1(1 − p)(χD(p) − p).

Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves with CM by Orders of Class Number One

Of D f E/Q

• 1, 1+√−3

2

• −3

1 y 2 + y = x3 [1, √−3] −3 2 y 2 = x3 − 15x + 22

• 1, 3+3√−3

2

• −3

3 y 2 + y = x3 − 30x + 63 [1, i] −4 1 y 2 = x3 − x [1, 2i] −4 2 y 2 = x3 − 11x − 14

• 1, 1+√−7

2

• −7

1 y 2 + xy = x3 − x2 − 2x − 1 [1, √−7] −7 2 y 2 = x3 − 595x − 5586 [1, √−2] −8 1 y 2 = x3 − x2 − 3x − 1

• 1, 1+√−11

2

• −11

1 y 2 + y = x3 − x2 − 7x + 10

• 1, 1+√−19

2

• −19

1 y 2 + y = x3 − 38x + 90

• 1, 1+√−43

2

• −43

1 y 2 + y = x3 − 860x + 9707

• 1, 1+√−67

2

• −67

1 y 2 + y = x3 − 7370x + 243528

• 1, 1+√−163

2

• −163

1 y 2 + y = x3 − 2174420x + 1234136692

Tyler Genao Faltings Heights of CM Elliptic Curves

Example of Faltings Height Calculations

Consider the elliptic curve E/Q : y2 = x3 − 11x + 14.

Tyler Genao Faltings Heights of CM Elliptic Curves

Example of Faltings Height Calculations

Consider the elliptic curve E/Q : y2 = x3 − 11x + 14. For the elliptic curve E/Q we have hFal(E/Q) = − log

• 1

4π1/2 Γ 1 4 2 .

Tyler Genao Faltings Heights of CM Elliptic Curves

Example of Faltings Height Calculations

Consider the elliptic curve E/Q : y2 = x3 − 11x + 14. For the elliptic curve E/Q we have hFal(E/Q) = − log

• 1

4π1/2 Γ 1 4 2 .

◮ E/Q has CM by the order O2 = Z + 2Z[i] ⊂ Q[i], which has

f = 2, D = −4, ∆2 = −16, and h(D) = 1.

Tyler Genao Faltings Heights of CM Elliptic Curves

Example of Faltings Height Calculations

Consider the elliptic curve E/Q : y2 = x3 − 11x + 14. For the elliptic curve E/Q we have hFal(E/Q) = − log

• 1

4π1/2 Γ 1 4 2 .

◮ E/Q has CM by the order O2 = Z + 2Z[i] ⊂ Q[i], which has

f = 2, D = −4, ∆2 = −16, and h(D) = 1.

◮ ∆E = −16(4(−11)3 + 27(14)2) = 512 = 29.

Tyler Genao Faltings Heights of CM Elliptic Curves

Example of Faltings Height Calculations

Consider the elliptic curve E/Q : y2 = x3 − 11x + 14. For the elliptic curve E/Q we have hFal(E/Q) = − log

• 1

4π1/2 Γ 1 4 2 .

◮ E/Q has CM by the order O2 = Z + 2Z[i] ⊂ Q[i], which has

f = 2, D = −4, ∆2 = −16, and h(D) = 1.

◮ ∆E = −16(4(−11)3 + 27(14)2) = 512 = 29. ◮ #O× K = 4.

Tyler Genao Faltings Heights of CM Elliptic Curves

hFal(E/Q) = − log

• 2−3/4 π

4 1/2

4

• k=1

Γ k 4 χ−4(k) 2e(2)/2

• .

Tyler Genao Faltings Heights of CM Elliptic Curves

hFal(E/Q) = − log

• 2−3/4 π

4 1/2

4

• k=1

Γ k 4 χ−4(k) 2e(2)/2

• .

◮ χ−4(1) = 1, χ−4(2) = 0, χ−4(3) = −1, and χ−4(4) = 0.

Tyler Genao Faltings Heights of CM Elliptic Curves

hFal(E/Q) = − log

• 2−3/4 π

4 1/2

4

• k=1

Γ k 4 χ−4(k) 2e(2)/2

• .

◮ χ−4(1) = 1, χ−4(2) = 0, χ−4(3) = −1, and χ−4(4) = 0. ◮ e(2) = 1/2.

Tyler Genao Faltings Heights of CM Elliptic Curves

hFal(E/Q) = − log

• 2−3/4 π

4 1/2

4

• k=1

Γ k 4 χ−4(k) 2e(2)/2

• .

◮ χ−4(1) = 1, χ−4(2) = 0, χ−4(3) = −1, and χ−4(4) = 0. ◮ e(2) = 1/2.

So we have hFal(E/Q) = − log

• π1/2

23/2 Γ 1 4

• Γ

3 4 −1 .

Tyler Genao Faltings Heights of CM Elliptic Curves

Furthermore, we can use the Gamma reflection formula Γ(z)Γ(1 − z) = π sin(πz) to compute that Γ 3 4 −1 = 1 π √ 2 Γ 1 4

• .

Tyler Genao Faltings Heights of CM Elliptic Curves

Furthermore, we can use the Gamma reflection formula Γ(z)Γ(1 − z) = π sin(πz) to compute that Γ 3 4 −1 = 1 π √ 2 Γ 1 4

• .

Substituting this into the expression gives us that hFal(E/Q) = − log

• 1

4√πΓ 1 4 2 .

Tyler Genao Faltings Heights of CM Elliptic Curves

Thank you!

Tyler Genao Faltings Heights of CM Elliptic Curves