Lecture 9: Theta functions and lattices May 12, 2020 1 / 7
Lattices in R n A lattice is an additive subgroup Λ Ă R n such that § Λ – Z n ( ô Λ is free on n generators) § Λ b Z R “ R n (Λ spans the whole real vector space) # n + ˇ ÿ v n a basis for R n Λ “ ˇ m 1 , . . . , m n P Z m i � v i , � v 1 , . . . , � ˇ i “ 1 Λ 1 – Λ (lattices are isomorphic) if one can be obtained from the other one by a rotation : Λ 1 “ U p Λ q , U P O p n q . The covolume covol p Λ q “ vol p � v 1 , . . . , � v n q “ | det p � v 1 , . . . , � v n q| is an invariant. 2 / 7
Gram matrix # r + ˇ ÿ Λ “ m i � v i ˇ m 1 , . . . , m n P Z , ˇ i “ 1 v n a basis for R n � v 1 , . . . , � A “ tp � v i , � v j qu Gram matrix (Gramian) v n q 2 “ covol p Λ q 2 Recall: A ą 0, det p A q “ det p � v 1 , . . . , � Given A ą 0, one can reconstruct the lattice (up to rotation) by v i = i th column of A 1 { 2 . taking � x q “ 1 x “ 1 ´ÿ ¯ ÿ x T A � Q p � 2 � x i � v i , x i � v i 2 “ 1 ´ÿ ¯ ÿ x i � v i , y i � v i 2 p Q p � x ` � y q ´ Q p � x q ´ Q p � y qq 3 / 7
Integral lattices & quadratic forms # r + ˇ ÿ Λ “ m i � v i ˇ m 1 , . . . , m n P Z , ˇ i “ 1 v n a basis for R n � v 1 , . . . , � Λ is called integral if p � u , � v q P Z for all � u , � v P Λ ô the Gram matrix A “ tp � v j qu is integral v i , � x q : “ 1 x “ 1 ´ÿ ¯ x T A � ÿ Q p � 2 � x i � v i , x i � v i 2 “ 1 ´ÿ ¯ ÿ 2 p Q p � x ` � y q ´ Q p � x q ´ Q p � y qq x i � v i , y i � v i A positive-definite quadratic form Q is called integral if Q p Z n q Ă Z . Lemma. Q is integral ô A is even integral ô Λ is integral & p � u , � u q P 2 Z for � u P Λ 4 / 7
Integral lattices ù modular forms Λ even integral r m p Λ q “ # t � u P Λ : Q p � u q “ m u u P Λ : 1 “ # t � 2 p � u , � u q “ m u 8 r m p Λ q q m P M n { 2 p Γ 0 p N q , χ D q ÿ Θ Λ p z q “ (Hecke–Schoenberg) m “ 0 5 / 7
Unimodular lattices An integral lattice Λ is called unimodular if its covolume equals 1: covol p Λ q “ 1 In Lecture 9 we proved that even unimodular lattices only exist in dimensions divisible by 8: n “ 8 , 16 , 24 , . . . The smallest example is: # 8 + x P Z 8 Y p Z ` 1 2 q 8 ˇ ÿ Ă R 8 Γ 8 “ x i P 2 Z � ˇ ˇ i “ 1 6 / 7
The “smallest” unimodular lattice # 8 + u P Z 8 Y p Z ` 1 2 q 8 ˇ ÿ Ă R 8 Γ 8 “ � u i P 2 Z ˇ ˇ i “ 1 u q “ ř u 2 § even integral: p � u , � i P 2 Z § unimodular: with Λ : “ Z 8 Y p Z ` 1 2 q 8 we have r Λ : Z 8 s “ 2 and r Λ : Γ 8 s “ 2 , so covol p Γ 8 q “ covol p Z 8 q “ 1. d | m d 3 ñ # t � u P Γ 8 : p � u , � u q “ 2 m u “ 240 ř In Lecture 9 we have seen the integral quadratic form corresponding to the following choice of basis in Γ 8 : v 8 “ p 1 2 , ´ 1 2 , 1 2 , 1 2 , ´ 1 2 , 1 2 , ´ 1 2 , 1 � v i “ � e i ` � e i ` 1 , 1 ď i ď 7 , � 2 q ˜ 8 8 ¸ 8 7 1 x 2 ÿ ÿ ÿ ÿ Q p � x q “ “ i ` x i ´ 1 x i ` x 3 x 8 x i v i , x i v i 2 i “ 1 i “ 1 i “ 1 i “ 2 7 / 7
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