Lecture 9: Theta functions and lattices May 12, 2020 1 / 7 - - PowerPoint PPT Presentation

Lecture 9: Theta functions and lattices

May 12, 2020

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Lattices in Rn

A lattice is an additive subgroup Λ Ă Rn such that

§ Λ – Zn (ô Λ is free on n generators) § Λ bZ R “ Rn (Λ spans the whole real vector space)

Λ “ # n ÿ

i“1

mi vi ˇ ˇ ˇ m1, . . . , mn P Z + , v1, . . . , vn a basis for Rn Λ1 – Λ (lattices are isomorphic) if one can be obtained from the

• ther one by a rotation : Λ1 “ UpΛq,

U P Opnq. The covolume covolpΛq “ volp v1, . . . , vnq “ | detp v1, . . . , vnq| is an invariant.

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Gram matrix

Λ “ # r ÿ

i“1

mi vi ˇ ˇ ˇ m1, . . . , mn P Z + ,

• v1, . . . ,

vn a basis for Rn A “ tp vi, vjqu Gram matrix (Gramian) Recall: A ą 0, detpAq “ detp v1, . . . , vnq2 “ covolpΛq2 Given A ą 0, one can reconstruct the lattice (up to rotation) by taking vi = ith column of A1{2. Qp xq “ 1 2 xTA x “ 1 2 ´ÿ xi vi, ÿ xi vi ¯ ´ÿ xi vi, ÿ yi vi ¯ “ 1 2 pQp x ` yq ´ Qp xq ´ Qp yqq

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Integral lattices & quadratic forms

Λ “ # r ÿ

i“1

mi vi ˇ ˇ ˇ m1, . . . , mn P Z + ,

• v1, . . . ,

vn a basis for Rn Λ is called integral if p u, vq P Z for all u, v P Λ ô the Gram matrix A “ tp vi, vjqu is integral Qp xq :“ 1 2 xTA x “ 1 2 ´ÿ xi vi, ÿ xi vi ¯ ´ÿ xi vi, ÿ yi vi ¯ “ 1 2 pQp x ` yq ´ Qp xq ´ Qp yqq A positive-definite quadratic form Q is called integral if QpZnq Ă Z.

• Lemma. Q is integral ô A is even integral

ô Λ is integral & p u, uq P 2Z for u P Λ

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Integral lattices ù modular forms

Λ even integral rmpΛq “ #t u P Λ : Qp uq “ mu “ #t u P Λ : 1 2p u, uq “ mu ΘΛpzq “

8

ÿ

m“0

rmpΛqqm P Mn{2pΓ0pNq, χDq (Hecke–Schoenberg)

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Unimodular lattices

An integral lattice Λ is called unimodular if its covolume equals 1: covolpΛq “ 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: Γ8 “ #

• x P Z8 Y pZ ` 1

2q8 ˇ ˇ ˇ

8

ÿ

i“1

xi P 2Z + Ă R8

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The “smallest” unimodular lattice

Γ8 “ #

• u P Z8 Y pZ ` 1

2q8 ˇ ˇ ˇ

8

ÿ

i“1

ui P 2Z + Ă R8

§ even integral:

p u, uq “ ř u2

i P 2Z § unimodular: with Λ :“ Z8 Y pZ ` 1 2q8 we have

rΛ : Z8s “ 2 and rΛ : Γ8s “ 2, so covolpΓ8q “ covolpZ8q “ 1. ñ #t u P Γ8 : p u, uq “ 2mu “ 240 ř

d|m d3

In Lecture 9 we have seen the integral quadratic form corresponding to the following choice of basis in Γ8:

• vi “

ei ` ei`1, 1 ď i ď 7,

• v8 “ p 1

2 , ´ 1 2 , 1 2 , 1 2 , ´ 1 2 , 1 2 , ´ 1 2 , 1 2 q

Qp xq “ 1 2 ˜ 8 ÿ

i“1

xi vi ,

8

ÿ

i“1

xi vi ¸ “

8

ÿ

i“1

x2

i ` 7

ÿ

i“2

xi´1xi ` x3x8 7 / 7