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 y 2 = x 3 + 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 y 2 = x 3 + Ax + B for some A , B in a field F . ◮ If 4 A 3 + 27 B 2 � = 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 y 2 = x 3 + Ax + B for some A , B in a field F . ◮ If 4 A 3 + 27 B 2 � = 0, we call this equation an elliptic curve , and write it as E / F . ◮ Define the discriminant of E / Q to be ∆ E := − 16(4 A 3 + 27 B 2 ) . Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves (Examples) E / R : y 2 = x 3 − x Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves (Examples) E / R : y 2 = x 3 − 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 ω 1 and ω 2 , define the Lattice generated by ω 1 and ω 2 to be L ( ω 1 , ω 2 ) := Z ω 1 + Z ω 2 = { a ω 1 + b ω 2 : a , b ∈ Z } . Tyler Genao Faltings Heights of CM Elliptic Curves

Lattices ◮ For two complex numbers ω 1 and ω 2 , define the Lattice generated by ω 1 and ω 2 to be L ( ω 1 , ω 2 ) := Z ω 1 + Z ω 2 = { a ω 1 + b ω 2 : a , b ∈ Z } . ◮ The parallelogram P L ( ω 1 ,ω 2 ) defined by ω 1 and ω 2 defines a fundamental parallelogram for C / L ( ω 1 , ω 2 ). 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 : y 2 = x 3 + 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 � � h Fal ( E / Q ) := 1 12 log | ∆ E / Q | − 1 i � 2 log ω ∧ ¯ ω . 2 E ( C ) Tyler Genao Faltings Heights of CM Elliptic Curves

The Faltings Height of an Elliptic Curve Definition � � h Fal ( E / Q ) := 1 12 log | ∆ E / Q | − 1 i � 2 log ω ∧ ¯ ω . 2 E ( C ) Remark i � ω ∧ ¯ ω ∼ Q × Area ( P L (1 ,τ ) ) . 2 E ( C ) 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 � d if d ≡ 1 (mod 4) , D K := 4 d if d ≡ 2 , 3 (mod 4) . Tyler Genao Faltings Heights of CM Elliptic Curves

Imaginary Quadratic Orders ◮ Define the number ω K := D K + √ D K . 2 Tyler Genao Faltings Heights of CM Elliptic Curves

Imaginary Quadratic Orders ◮ Define the number ω K := D K + √ D K . 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 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 End C ( E ) := { α ∈ C : α L ⊆ L } . Tyler Genao Faltings Heights of CM Elliptic Curves

CM Elliptic Curves Theorem For an elliptic curve E / Q , End C ( E ) is isomorphic either to Z or to an order O f in some K . Tyler Genao Faltings Heights of CM Elliptic Curves

CM Elliptic Curves Theorem For an elliptic curve E / Q , 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 . Tyler Genao Faltings Heights of CM Elliptic Curves

Motivation ◮ For elliptic curves E / Q with CM by a maximal order, Deligne computed h Fal ( E / Q ) in terms of Euler’s Γ-function � ∞ x s − 1 e − x dx Γ( s ) := 0 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 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 ⊆ K . Tyler Genao Faltings Heights of CM Elliptic Curves

Preliminaries to the Main Theorem Let K be an imaginary quadratic number field with discriminant D K . Tyler Genao Faltings Heights of CM Elliptic Curves

Preliminaries to the Main Theorem Let K be an imaginary quadratic number field with discriminant D K . ◮ Let ω D K 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 D K . ◮ Let ω D K 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 D K . ◮ Let ω D K be 2, 4, or 6, depending on K . ◮ Let h ( K ) < ∞ denote the class number of K . ◮ Let χ D K ( 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 O f of K , with K having discriminant D K . Then  ω DK  � 1 / 2 | D K | � � χ DK ( k ) � π k 4 h ( K ) � �  | ∆ E / Q | − 1 / 12 p e ( p ) / 2  , h Fal ( E / Q ) = − log Γ � | D K | | D K | f k =1 p | f where (1 − p ord p ( f ) )(1 − χ D ( p )) e ( p ) = − p ord p ( f ) − 1 (1 − p )( χ D ( p ) − p ) . Tyler Genao Faltings Heights of CM Elliptic Curves

Elliptic Curves with CM by Orders of Class Number One O f E / Q D f 1 , 1+ √− 3 � � y 2 + y = x 3 − 3 1 2 [1 , √− 3] y 2 = x 3 − 15 x + 22 − 3 2 1 , 3+3 √− 3 � � y 2 + y = x 3 − 30 x + 63 − 3 3 2 y 2 = x 3 − x [1 , i ] − 4 1 y 2 = x 3 − 11 x − 14 [1 , 2 i ] − 4 2 1 , 1+ √− 7 � � y 2 + xy = x 3 − x 2 − 2 x − 1 − 7 1 2 [1 , √− 7] y 2 = x 3 − 595 x − 5586 − 7 2 [1 , √− 2] y 2 = x 3 − x 2 − 3 x − 1 − 8 1 1 , 1+ √− 11 � � y 2 + y = x 3 − x 2 − 7 x + 10 − 11 1 2 1 , 1+ √− 19 � � y 2 + y = x 3 − 38 x + 90 − 19 1 2 1 , 1+ √− 43 � � y 2 + y = x 3 − 860 x + 9707 − 43 1 2 1 , 1+ √− 67 � � y 2 + y = x 3 − 7370 x + 243528 − 67 1 2 1 , 1+ √− 163 � � y 2 + y = x 3 − 2174420 x + 1234136692 − 163 1 2 Tyler Genao Faltings Heights of CM Elliptic Curves

Example of Faltings Height Calculations Consider the elliptic curve E / Q : y 2 = x 3 − 11 x + 14. Tyler Genao Faltings Heights of CM Elliptic Curves

Example of Faltings Height Calculations Consider the elliptic curve E / Q : y 2 = x 3 − 11 x + 14. For the elliptic curve E / Q we have � � 2 � 1 � 1 h Fal ( E / Q ) = − log 4 π 1 / 2 Γ . 4 Tyler Genao Faltings Heights of CM Elliptic Curves

Example of Faltings Height Calculations Consider the elliptic curve E / Q : y 2 = x 3 − 11 x + 14. For the elliptic curve E / Q we have � � 2 � 1 � 1 h Fal ( E / Q ) = − log 4 π 1 / 2 Γ . 4 ◮ E / Q has CM by the order O 2 = Z + 2 Z [ 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 : y 2 = x 3 − 11 x + 14. For the elliptic curve E / Q we have � � 2 � 1 � 1 h Fal ( E / Q ) = − log 4 π 1 / 2 Γ . 4 ◮ E / Q has CM by the order O 2 = Z + 2 Z [ i ] ⊂ Q [ i ], which has f = 2, D = − 4, ∆ 2 = − 16, and h ( D ) = 1. ◮ ∆ E = − 16(4( − 11) 3 + 27(14) 2 ) = 512 = 2 9 . Tyler Genao Faltings Heights of CM Elliptic Curves