The Faltings Heights of CM Elliptic Curves and Special Gamma Values - - PowerPoint PPT Presentation

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Faltings Heights of CM Elliptic Curves and Special Gamma Values

Lindsay Cadwallader, Olivia Cannon, Tyler Genao July 18, 2016

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Elliptic curves

An elliptic curve E/Q is given by an equation of the form y2 + a1xy + a3y = x3 + a2x2 + a4x + a6, where the a1, a3, a2, a4, a6 are rational numbers. It is convenient to change variables to an equation of the form y x Ax B for some A B . Define the discriminant of E by

E

A B

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Elliptic curves

An elliptic curve E/Q is given by an equation of the form y2 + a1xy + a3y = x3 + a2x2 + a4x + a6, where the a1, a3, a2, a4, a6 are rational numbers. It is convenient to change variables to an equation of the form y2 = x3 + Ax + B for some A, B ∈ Q. Define the discriminant of E by

E

A B

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Elliptic curves

An elliptic curve E/Q is given by an equation of the form y2 + a1xy + a3y = x3 + a2x2 + a4x + a6, where the a1, a3, a2, a4, a6 are rational numbers. It is convenient to change variables to an equation of the form y2 = x3 + Ax + B for some A, B ∈ Q. Define the discriminant of E/Q by ∆E := −16(4A3 + 27B2) ̸= 0.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some examples

E/Q : y2 = x3 − x

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some examples

E/Q : y2 = x3 − x + 1

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Further definitions

Define the j-invariant of E/Q by j(E) := 1728 4A3 4A3 + 27B2 . Define the differential form

E by E

dx y

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Further definitions

Define the j-invariant of E/Q by j(E) := 1728 4A3 4A3 + 27B2 . Define the differential form ωE by ωE := dx 2y.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Lattices

For two complex numbers ω1 and ω2, define the lattice generated by ω1 and ω2 by L = L(ω1, ω2) := Zω1 + Zω2 = {aω1 + bω2 : a, b ∈ Z}. Define the fundamental parallelogram associated to L by PL PL a b a b

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Lattices

For two complex numbers ω1 and ω2, define the lattice generated by ω1 and ω2 by L = L(ω1, ω2) := Zω1 + Zω2 = {aω1 + bω2 : a, b ∈ Z}. Define the fundamental parallelogram associated to L by PL = PL(ω1,ω2) := {aω1 + bω2 : a, b ∈ [0, 1)}.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

A fundamental parallelogram

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Uniformization

One can prove that for any elliptic curve E/C : y2 = x3 + Ax + B, there exists τ ∈ C with Im(τ) > 0 so that E(C) ∼ = C/(Z + Zτ). For convenience, define L

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Uniformization

One can prove that for any elliptic curve E/C : y2 = x3 + Ax + B, there exists τ ∈ C with Im(τ) > 0 so that E(C) ∼ = C/(Z + Zτ). For convenience, define Lτ := Z + Zτ = [1, τ].

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Faltings height of E/Q

Definition The Faltings height of E/Q is defined by hFal(E/Q) := 1 12 log |∆E| − 1 2 log ( i 2 ∫

E(C)

ωE ∧ ωE ) . If E L , then i

E

Area PL

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Faltings height of E/Q

Definition The Faltings height of E/Q is defined by hFal(E/Q) := 1 12 log |∆E| − 1 2 log ( i 2 ∫

E(C)

ωE ∧ ωE ) . If E(C) ∼ = C/Lτ, then i 2 ∫

E(C)

ω ∧ ¯ ω ∼Q× Area(PLτ ).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Imaginary quadratic orders

Let d ∈ Z be a negative squarefree integer and define the imaginary quadratic field K := Q( √ d) = {a + b √ d : a, b ∈ Q}. Define the discriminant of K by D d if d mod d if d mod

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Imaginary quadratic orders

Let d ∈ Z be a negative squarefree integer and define the imaginary quadratic field K := Q( √ d) = {a + b √ d : a, b ∈ Q}. Define the discriminant of K by D := { d if d ≡ 1 (mod 4), 4d if d ≡ 2, 3 (mod 4).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Imaginary quadratic orders

Define the number ωK := D + √ D 2 . For an integer f , the ring

f

f

K

a bf

K

a b is called an imaginary quadratic order of conductor f in K. The order is called the maximal order of K (the ring of integers of K).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Imaginary quadratic orders

Define the number ωK := D + √ D 2 . For an integer f > 0, the ring Of := [1, fωK] := {a + bfωK : a, b ∈ Z} is called an imaginary quadratic order of conductor f in K. The order is called the maximal order of K (the ring of integers of K).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Imaginary quadratic orders

Define the number ωK := D + √ D 2 . For an integer f > 0, the ring Of := [1, fωK] := {a + bfωK : a, b ∈ Z} is called an imaginary quadratic order of conductor f in K. The order O1 is called the maximal order of K (the ring of integers of K).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

CM elliptic curves

For an elliptic curve E/Q and corresponding lattice L = Lτ, define the endomorphism ring of E/Q by EndC(E) := {α ∈ C : αL ⊆ L}.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

CM elliptic curves

Theorem For an elliptic curve E/Q, the endomorphism ring EndC(E) is isomorphic either to Z or to an order Of in some K. If End E is isomorphic to

f, then E

is said to have complex multiplication (or CM) by

f.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

CM elliptic curves

Theorem For an elliptic curve E/Q, the endomorphism ring 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) by Of.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

A result of Deligne

Let E/Q have CM by a maximal order. Deligne (Seminar Bourbaki, 1984) explicitly computed hFal E in terms of Euler’s

• function

s xs e

xdx

at rational numbers. Our main result is an analogous formula for any order

f.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

A result of Deligne

Let E/Q have CM by a maximal order. Deligne (Seminar Bourbaki, 1984) explicitly 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

f.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

A result of Deligne

Let E/Q have CM by a maximal order. Deligne (Seminar Bourbaki, 1984) explicitly 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.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Preliminary notation

Let K be an imaginary quadratic field of discriminant D. Let

D be the order of the unit group of K, which equals 2, 4, or

6, depending on D. Let h D be the class number of K. For an order

f

K, let

f

f D be the discriminant. Let

D k be the Kronecker symbol, which equals

• r

depending on the integer k.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Preliminary notation

Let K be an imaginary quadratic field of discriminant D. Let ωD be the order of the unit group of K, which equals 2, 4, or 6, depending on D. Let h D be the class number of K. For an order

f

K, let

f

f D be the discriminant. Let

D k be the Kronecker symbol, which equals

• r

depending on the integer k.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Preliminary notation

Let K be an imaginary quadratic field of discriminant D. Let ωD be the order of the unit group of K, which equals 2, 4, or 6, depending on D. Let h(D) be the class number of K. For an order

f

K, let

f

f D be the discriminant. Let

D k be the Kronecker symbol, which equals

• r

depending on the integer k.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Preliminary notation

Let K be an imaginary quadratic field of discriminant D. Let ωD be the order of the unit group of K, which equals 2, 4, or 6, depending on D. Let h(D) be the class number of K. For an order Of ⊆ K, let ∆f := f2D be the discriminant. Let

D k be the Kronecker symbol, which equals

• r

depending on the integer k.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Preliminary notation

Let K be an imaginary quadratic field of discriminant D. Let ωD be the order of the unit group of K, which equals 2, 4, or 6, depending on D. Let h(D) be the class number of K. For an order Of ⊆ K, let ∆f := f2D be the discriminant. Let χD(k) be the Kronecker symbol, which equals −1, 0, or 1 depending on the integer k.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Our Main Theorem

Theorem If E/Q has CM by an order Of ⊂ K, then hFal(E/Q) = − log  |∆E|−1/12 ( π √ |∆f| )1/2 |D| ∏

k=1

Γ ( k |D| )χD(k)

ωD 4h(D) ∏

p|f

pe(p)/2   , where e(p) := − (1 − pordp(f))(1 − χD(p)) pordp(f)−1(1 − p)(χD(p) − p).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Examples of CM elliptic curves

Of D f E/Q [ 1, 1+√−3

2

] −3 1 y2 + y = x3 [1, √−3] −3 2 y2 = x3 − 15x + 22 [ 1, 3+3√−3

2

] −3 3 y2 + y = x3 − 30x + 63 [1, i] −4 1 y2 = x3 − x [1, 2i] −4 2 y2 = x3 − 11x − 14 [ 1, 1+√−7

2

] −7 1 y2 + xy = x3 − x2 − 2x − 1 [1, √−7] −7 2 y2 = x3 − 595x − 5586 [1, √−2] −8 1 y2 = x3 − x2 − 3x − 1 [ 1, 1+√−11

2

] −11 1 y2 + y = x3 − x2 − 7x + 10 [ 1, 1+√−19

2

] −19 1 y2 + y = x3 − 38x + 90 [ 1, 1+√−43

2

] −43 1 y2 + y = x3 − 860x + 9707 [ 1, 1+√−67

2

] −67 1 y2 + y = x3 − 7370x + 243528 [ 1, 1+√−163

2

] −163 1 y2 + y = x3 − 2174420x + 1234136692

Faltings Heights of CM Elliptic Curves and Special Gamma Values

An example of our Main Theorem

Consider the elliptic curve E/Q : y2 = x3 − 11x + 14, which has CM by the order O2 = Z + Z[2i] ⊂ Q(i). We have f , D , , h D , . The discriminant of E is

E

Faltings Heights of CM Elliptic Curves and Special Gamma Values

An example of our Main Theorem

Consider the elliptic curve E/Q : y2 = x3 − 11x + 14, which has CM by the order O2 = Z + Z[2i] ⊂ Q(i). We have f = 2, D = −4, ∆2 = −16, h(D) = 1, ω−4 = 4. The discriminant of E is

E

Faltings Heights of CM Elliptic Curves and Special Gamma Values

An example of our Main Theorem

Consider the elliptic curve E/Q : y2 = x3 − 11x + 14, which has CM by the order O2 = Z + Z[2i] ⊂ Q(i). We have f = 2, D = −4, ∆2 = −16, h(D) = 1, ω−4 = 4. The discriminant of E/Q is ∆E = −16(4(−11)3 + 27(14)2) = 512 = 29.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

From this, our Main Theorem gives hFal(E/Q) = − log ( 2−3/4 (π 4 )1/2

4

∏

k=1

Γ (k 4 )χ−4(k) 2e(2)/2 ) . We also have , and . Moreover, e . This gives hFal E log

Faltings Heights of CM Elliptic Curves and Special Gamma Values

From this, our Main Theorem gives hFal(E/Q) = − log ( 2−3/4 (π 4 )1/2

4

∏

k=1

Γ (k 4 )χ−4(k) 2e(2)/2 ) . We also have χ−4(1) = 1, χ−4(2) = 0, χ−4(3) = −1, and χ−4(4) = 0. Moreover, e . This gives hFal E log

Faltings Heights of CM Elliptic Curves and Special Gamma Values

From this, our Main Theorem gives hFal(E/Q) = − log ( 2−3/4 (π 4 )1/2

4

∏

k=1

Γ (k 4 )χ−4(k) 2e(2)/2 ) . We also have χ−4(1) = 1, χ−4(2) = 0, χ−4(3) = −1, and χ−4(4) = 0. Moreover, e(2) = 1/2. This gives hFal E log

Faltings Heights of CM Elliptic Curves and Special Gamma Values

From this, our Main Theorem gives hFal(E/Q) = − log ( 2−3/4 (π 4 )1/2

4

∏

k=1

Γ (k 4 )χ−4(k) 2e(2)/2 ) . We also have χ−4(1) = 1, χ−4(2) = 0, χ−4(3) = −1, and χ−4(4) = 0. Moreover, e(2) = 1/2. This gives hFal(E/Q) = − log ( π1/2 23/2 Γ (1 4 ) Γ (3 4 )−1) .

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Now, we can use the Gamma reflection formula Γ(z)Γ(1 − z) = π sin(πz) to compute that Γ (3 4 )−1 = 1 π √ 2Γ (1 4 ) . Substituting this into the preceding expression gives hFal E log

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Now, we can use the Gamma reflection formula Γ(z)Γ(1 − z) = π sin(πz) to compute that Γ (3 4 )−1 = 1 π √ 2Γ (1 4 ) . Substituting this into the preceding expression gives hFal(E/Q) = − log ( 1 4π1/2 Γ (1 4 )2) .

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some additional definitions

An ideal of Of ⊂ K is a subgroup of Of under addition which is also closed under multiplication by any element of Of. A fractional ideal is a multiple of an ideal of

f by an element of K .

A fractional ideal is invertible if there exists a fractional ideal such that

• f. We call the group of invertible fractional ideals I

f

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some additional definitions

An ideal of Of ⊂ K is a subgroup of Of under addition which is also closed under multiplication by any element of Of. A fractional ideal is a multiple of an ideal of Of by an element of K×. A fractional ideal is invertible if there exists a fractional ideal such that

• f. We call the group of invertible fractional ideals I

f

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some additional definitions

An ideal of Of ⊂ K is a subgroup of Of under addition which is also closed under multiplication by any element of Of. A fractional ideal is a multiple of an ideal of Of by an element of K×. A fractional ideal a is invertible if there exists a fractional ideal b such that ab = Of. We call the group of invertible fractional ideals I(Of)

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some Additional Definitions

The class group Cl(

f) is the set of equivalence classes of I f

modulo principal fractional ideals. Given invertible ideals and , for some K The class number of

f, denoted h f , is the number of such

equivalence classes.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some Additional Definitions

The class group Cl(Of) is the set of equivalence classes of I(Of) modulo principal fractional ideals. Given invertible ideals and , for some K The class number of

f, denoted h f , is the number of such

equivalence classes.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some Additional Definitions

The class group Cl(Of) is the set of equivalence classes of I(Of) modulo principal fractional ideals. Given invertible ideals a and b, [a] = [b] ⇐ ⇒ a = λb for some λ ∈ K×. The class number of

f, denoted h f , is the number of such

equivalence classes.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Some Additional Definitions

The class group Cl(Of) is the set of equivalence classes of I(Of) modulo principal fractional ideals. Given invertible ideals a and b, [a] = [b] ⇐ ⇒ a = λb for some λ ∈ K×. The class number of Of, denoted h(Of), is the number of such equivalence classes.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

More background

Given a class A ∈ Cl(Of), we can choose an invertible ideal a of Of in A such that a = Za + Z ( −b + √∆f 2 ) , where ax2 + bx + c is a primitive positive definite quadratic form of discriminant ∆f, with a = N(a).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

More background

We have a−1 = Z + Z ( b + √∆f 2a ) = Z + Zza−1, where za−1 := b + √∆f 2a is the root in the upper-half plane of the quadratic form ax2 − bx + c.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Dedekind eta function

The Dedekind eta function is the weight 1/2 modular form for SL2(Z) defined by η(z) := q1/24

∞

∏

n=1

(1 − qn), where q := e2πiz. For convenience, we define the SL

• invariant function

F z Im z z

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Dedekind eta function

The Dedekind eta function is the weight 1/2 modular form for SL2(Z) defined by η(z) := q1/24

∞

∏

n=1

(1 − qn), where q := e2πiz. For convenience, we define the SL2(Z)-invariant function F(z) := √ Im(z)|η(z)|2.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

A crucial Proposition

Proposition If E/Q has CM by Of, then hFal(E/Q) = − log  2π|∆E|−1/12 ∏

[a]∈Cl(Of)

F(za−1)1/h(Of)   .

Faltings Heights of CM Elliptic Curves and Special Gamma Values

A Chowla-Selberg formula for imaginary quadratic

• rders

Theorem ∏

[a]∈Cl(Of)

F(za−1) = ( 1 4π √ |∆f| )h(Of)/2 |D| ∏

k=1

Γ ( k |D| )χD(k)

ωDh(Of) 4h(D) ∏

p|f

pe(p)h(Of)/2, where e(p) := − (1 − pordp(f))(1 − χD(p)) pordp(f)−1(1 − p)(χD(p) − p).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Proof of Main Theorem

The majority of our REU was devoted to proving the preceding theorem. In the early 1990s, Nakkajima and Taguchi used arithmetic geometry to prove a similar formula. We take an analytic approach by a direct study of the zeta function of a quadratic order.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Proof of Main Theorem

The majority of our REU was devoted to proving the preceding theorem. In the early 1990s, Nakkajima and Taguchi used arithmetic geometry to prove a similar formula. We take an analytic approach by a direct study of the zeta function of a quadratic order.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Proof of Main Theorem

The majority of our REU was devoted to proving the preceding theorem. In the early 1990s, Nakkajima and Taguchi used arithmetic geometry to prove a similar formula. We take an analytic approach by a direct study of the zeta function of a quadratic order.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Zeta function

The zeta function of Of is defined by ζOf(s) := ∑

I∈I(Of) 0̸=I⊆Of

1 N(I)s , Re(s) > 1.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The fundamental identity

For convenience, define the function gOf(s) := #O×

f

2 (√ |∆f| 2 ) s+1

2 ζOf

( s+1

2

) ζ(s + 1) . Recall that the SL Eisenstein series is defined by E z s

M SL

Im Mz s Im z Re s We show by an elaborate calculation that g

f s

Cl

f

E z s

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The fundamental identity

For convenience, define the function gOf(s) := #O×

f

2 (√ |∆f| 2 ) s+1

2 ζOf

( s+1

2

) ζ(s + 1) . Recall that the SL2(Z) Eisenstein series is defined by E(z, s) := ∑

M∈Γ∞\SL2(Z)

Im(Mz)s, Im(z) > 0, Re(s) > 1. We show by an elaborate calculation that g

f s

Cl

f

E z s

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The fundamental identity

For convenience, define the function gOf(s) := #O×

f

2 (√ |∆f| 2 ) s+1

2 ζOf

( s+1

2

) ζ(s + 1) . Recall that the SL2(Z) Eisenstein series is defined by E(z, s) := ∑

M∈Γ∞\SL2(Z)

Im(Mz)s, Im(z) > 0, Re(s) > 1. We show by an elaborate calculation that gOf(s) = ∑

[a]∈Cl(Of)

E ( za−1, s + 1 2 ) .

Faltings Heights of CM Elliptic Curves and Special Gamma Values

A renormalized Kronecker limit formula

Proposition E ( z, s + 1 2 ) = 1 + log(F(z))(s + 1) + O((s + 1)2).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Taylor series expansion of gOf(s) at s = −1 is, formally, gOf(−1) + g′

Of(−1)(s + 1) + O((s + 1)2).

Thus, we have g

f

Cl

f

log F z It remains to calculate g

f

.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Taylor series expansion of gOf(s) at s = −1 is, formally, gOf(−1) + g′

Of(−1)(s + 1) + O((s + 1)2).

Thus, we have g′

Of(−1) =

∑

[a]∈Cl(Of)

log(F(za−1)). It remains to calculate g

f

.

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Taylor series expansion of gOf(s) at s = −1 is, formally, gOf(−1) + g′

Of(−1)(s + 1) + O((s + 1)2).

Thus, we have g′

Of(−1) =

∑

[a]∈Cl(Of)

log(F(za−1)). It remains to calculate g′

Of(−1).

Faltings Heights of CM Elliptic Curves and Special Gamma Values

The Taylor expansion of gOf(s)

We have the factorization ζOf(s) = Lf(s)ζ(s)L(χD, s), where Lf(s) := ∏

p|f

(1 − p−s)(1 − χD(p)p−s) − pordp(f)(1−2s)−1(1 − p1−s)(χD(p) − p1−s) 1 − p1−2s .

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Therefore by an elaborate series of calculations we get that g′

Of = log

   ( 1 4π √ |∆| ) h(Of)

2

|D|

∏

k=1

Γ ( k |D| )χD(k)

ωD·h(Of) 4h(D)

∏

p|f

pe(p)·

h(Of) 2

  

Faltings Heights of CM Elliptic Curves and Special Gamma Values

Acknowledgements

We would like to thank Dr. Masri and Adrian for their help and guidance on this project, as well as the NSF for their generous support.