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
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 A B E 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 y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 , where the a 1 , a 3 , a 2 , a 4 , a 6 are rational numbers.
Define the discriminant of E by A B E 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 y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 , where the a 1 , a 3 , a 2 , a 4 , a 6 are rational numbers. It is convenient to change variables to an equation of the form y 2 = x 3 + Ax + B for some A , B ∈ Q .
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 y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 , where the a 1 , a 3 , a 2 , a 4 , a 6 are rational numbers. It is convenient to change variables to an equation of the form y 2 = x 3 + Ax + B for some A , B ∈ Q . Define the discriminant of E / Q by ∆ E := − 16(4 A 3 + 27 B 2 ) ̸ = 0 .
Faltings Heights of CM Elliptic Curves and Special Gamma Values Some examples E / Q : y 2 = x 3 − x
Faltings Heights of CM Elliptic Curves and Special Gamma Values Some examples E / Q : y 2 = x 3 − x + 1
Define the differential form E by dx E y Faltings Heights of CM Elliptic Curves and Special Gamma Values Further definitions Define the j -invariant of E / Q by 4 A 3 j ( E ) := 1728 4 A 3 + 27 B 2 .
Faltings Heights of CM Elliptic Curves and Special Gamma Values Further definitions Define the j -invariant of E / Q by 4 A 3 j ( E ) := 1728 4 A 3 + 27 B 2 . Define the differential form ω E by ω E := dx 2 y .
Define the fundamental parallelogram associated to L by P L P L a b a b Faltings Heights of CM Elliptic Curves and Special Gamma Values Lattices For two complex numbers ω 1 and ω 2 , define the lattice generated by ω 1 and ω 2 by L = L ( ω 1 , ω 2 ) := Z ω 1 + Z ω 2 = { a ω 1 + b ω 2 : a , b ∈ Z } .
Faltings Heights of CM Elliptic Curves and Special Gamma Values Lattices For two complex numbers ω 1 and ω 2 , define the lattice generated by ω 1 and ω 2 by L = L ( ω 1 , ω 2 ) := Z ω 1 + Z ω 2 = { a ω 1 + b ω 2 : a , b ∈ Z } . Define the fundamental parallelogram associated to L by P L = P L ( ω 1 ,ω 2 ) := { a ω 1 + b ω 2 : a , b ∈ [0 , 1) } .
Faltings Heights of CM Elliptic Curves and Special Gamma Values A fundamental parallelogram
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 : y 2 = x 3 + Ax + B , there exists τ ∈ C with Im ( τ ) > 0 so that E ( C ) ∼ = C /( Z + Z τ ) .
Faltings Heights of CM Elliptic Curves and Special Gamma Values Uniformization One can prove that for any elliptic curve E / C : y 2 = x 3 + Ax + B , there exists τ ∈ C with Im ( τ ) > 0 so that E ( C ) ∼ = C /( Z + Z τ ) . For convenience, define L τ := Z + Z τ = [1 , τ ] .
If E L , then i Area P L E 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 ( ) i h Fal ( E / Q ) := 1 12 log | ∆ E | − 1 ∫ 2 log ω E ∧ ω E . 2 E ( C )
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 ( ) i h Fal ( E / Q ) := 1 12 log | ∆ E | − 1 ∫ 2 log ω E ∧ ω E . 2 E ( C ) If E ( C ) ∼ = C / L τ , then i ∫ ω ∼ Q × Area ( P L τ ) . ω ∧ ¯ 2 E ( C )
Define the discriminant of K by d if d mod D 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 } .
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 if d ≡ 1 ( mod 4) , D := 4 d if d ≡ 2 , 3 ( mod 4) .
For an integer f , the ring f a bf a b K K f 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 √ ω K := D + D . 2
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 √ ω K := D + D . 2 For an integer f > 0 , the ring O f := [1 , f ω K ] := { a + bf ω K : a , b ∈ Z } is called an imaginary quadratic order of conductor f in K .
Faltings Heights of CM Elliptic Curves and Special Gamma Values Imaginary quadratic orders Define the number √ ω K := D + D . 2 For an integer f > 0 , the ring O f := [1 , f ω K ] := { a + bf ω K : a , b ∈ Z } is called an imaginary quadratic order of conductor f in K . The order O 1 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 = L τ , define the endomorphism ring of E / Q by End C ( E ) := { α ∈ C : α L ⊆ L } .
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 End C ( E ) is isomorphic either to Z or to an order O f in some K.
Faltings Heights of CM Elliptic Curves and Special Gamma Values CM elliptic curves Theorem For an elliptic curve E / Q , the endomorphism ring End C ( E ) is isomorphic either to Z or to an order O f in some K. If End C ( E ) is isomorphic to O f , then E / Q is said to have complex multiplication (or CM) by O f .
Deligne (Seminar Bourbaki, 1984) explicitly computed h Fal E in terms of Euler’s -function x s x dx s e 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.
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 h Fal ( E / Q ) in terms of Euler’s Γ -function ∫ ∞ x s − 1 e − x dx Γ( s ) := 0 at rational numbers.
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 h Fal ( E / Q ) in terms of Euler’s Γ -function ∫ ∞ x s − 1 e − x dx Γ( s ) := 0 at rational numbers. Our main result is an analogous formula for any order O f .
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 K , let f D be the discriminant. f f Let D k be the Kronecker symbol, which equals or 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 h D be the class number of K . For an order K , let f D be the discriminant. f f Let D k be the Kronecker symbol, which equals or 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 ω D be the order of the unit group of K , which equals 2, 4, or 6, depending on D .
For an order K , let f D be the discriminant. f f Let D k be the Kronecker symbol, which equals or 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 ω 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 .
Let D k be the Kronecker symbol, which equals or 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 ω 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 O f ⊆ K , let ∆ f := f 2 D be the discriminant.
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