Lecture 11: Elliptic curves and their moduli May 26, 2020 1 / 9 - PowerPoint PPT Presentation

Lecture 11: Elliptic curves and their moduli May 26, 2020 1 / 9

Elliptic curves and complex tori E a , b : y 2 “ x 3 ` ax ` b , 4 a 3 ` 27 b 2 ‰ 0 4 a 3 j p E a , b q “ 1728 4 a 3 ` 27 b 2 Theorem. The map z ÞÑ r ℘ p z q : 1 2 ℘ 1 p z q : 1 s C { Λ Ñ E a , b , where a “ ´ 15 G 4 p Λ q , b “ ´ 35 G 6 p Λ q , ℘ p z q “ 1 ˆ p z ´ λ q 2 ´ 1 1 ˙ ÿ z 2 ` λ 2 λ P Λ zt 0 u is an isomorphism of Riemann surfaces and a homomorphism of abelian groups. j p C {p Z τ ` Z qq “ j p τ q “ 1 q “ e 2 π i τ q ` 744 ` . . . , 2 / 9

Families of elliptic curves: an example E t : y 2 “ x p x ´ 1 qp x ´ t q t ‰ 0 , 1 “ x 3 ´ p 1 ` t q x 2 ` tx ˘ 3 ` a x ´ 1 ` t x ´ 1 ` t ` ` ˘ “ ` b 3 3 a “ ´ 1 b “ ´ 1 3 p t 2 ´ t ` 1 q , 27 p t ´ 2 qp t ` 1 qp 2 t ´ 1 q 4 a 3 ` 27 b 2 “ 256 p t 2 ´ t ` 1 q 3 4 a 3 j p t q “ 1728 t 2 p t ´ 1 q 3 256 p t 2 ´ t ` 1 q 3 “ j p t q ¨ t 2 p t ´ 1 q 2 ñ j p t q is generically 6 : 1 each isomorphism class of elliptic curves over C occurs in this family exactly 6 times, with a few exceptions 3 / 9

An example: Legendre family E t : y 2 “ x p x ´ 1 qp x ´ t q t P C zt 0 , 1 u j p t q “ 256 p t 2 ´ t ` 1 q 3 t 2 p t ´ 1 q 2 Problem: find a modular function t p z q such that j p t p z qq “ j p z q . We expect t p z q to be modular on a subgroup of index 6. 4 / 9

An example: Legendre family 256 p t 2 ´ t ` 1 q 3 “ j ¨ t 2 p t ´ 1 q 2 a t p z q “ q α ` . . . a , α ? ˙ ˆ a 4 256 a 6 ˆ 1 ˙ q 6 α ` . . . “ q ` 744 ` . . . q 4 α ` . . . 6 α “ 4 α ` 1 ñ α “ 1 2 256 a 2 “ 1 ñ a “ ˘ 1 16 16 q ´ 1 1 2 ` c ` . . . t p z q “ c ? . . . 16 q ´ 1 1 3 5 1 2 ` 1 2 ` 5 2 ´ 31 2 ` O p q 2 q t p z q “ 4 q 8 q 5 / 9

An example: Legendre family E t : y 2 “ x p x ´ 1 qp x ´ t q t P C zt 0 , 1 u 16 q ´ 1 1 3 5 1 2 ` 1 2 ` 5 2 ´ 31 2 ` O p q 2 q t p z q “ 4 q 8 q § We expect t p z q to be modular on a subgroup of index 6. § t p z q ‰ 0 for z P H . Try to guess it as an η -product? 6 / 9

An example: Legendre family E t : y 2 “ x p x ´ 1 qp x ´ t q t P C zt 0 , 1 u “ P 1 p C qzt 0 , 1 , 8u 16 q ´ 1 1 3 5 1 2 ` 1 2 ` 5 2 ´ 31 2 ` O p q 2 q t p z q “ 4 q 8 q 16 q ´ 1 ´ 2 ` 20 q ´ 62 q 2 ` O p q 3 q 1 ¯ 1 “ 1 ` 8 q 2 η p z q 24 ? 1 “ 16 η p z 2 q 8 η p 2 z q 16 We can use PARI { GP to confirm this guess: one checks that 256 p t 2 ´ t ` 1 q 3 “ j ¨ t 2 p t ´ 1 q 2 with any high precision O p q N q . η p z q 24 The above given t p z q “ 1 2 q 8 η p 2 z q 16 is a modular function for η p z 16 Γ p 2 q . It plays the same role as j -invariant does for SL 2 p Z q : t : X p Γ p 2 qq – P 1 p C q , H Y P 1 p Q q ` ˘ X p Γ q “ Γ z t : Γ z P 1 p Q q – t 0 , 1 , 8u t p8q “ 8 , t p 0 q “ 1 , t p 1 q “ 0 Such a function for a subgroup of genus 0 is called a Hauptmodul . 7 / 9

Another example Exercise: Play the above game with the family E t : y 2 “ x 3 ´ 2 x 2 ` p 1 ´ t q x , t ‰ 0 , 1 . You should discover a modular function t p z q on a subgroup of index 3. The answer is given on the next page, don’t turn over! 8 / 9

The answer t p z q “ 64 q ´ 2560 q 2 ` 84736 q 3 ` . . . 64∆ p 2 z q “ ∆ p z q ` 64∆ p 2 z q is a Hauptmodul for Γ 0 p 2 q . 9 / 9