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Configurations of Extremal Even Unimodular Lattices Scott D. Kominers Harvard Mathematics Department Brown University SUMS March 8, 2008 Configurations of ExtremalEven Unimodular Lattices p. 1/33 Key Terms Configurations of ExtremalEven

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SLIDE 1

Configurations of Extremal Even Unimodular Lattices

Scott D. Kominers Harvard Mathematics Department

Brown University SUMS March 8, 2008

Configurations of ExtremalEven Unimodular Lattices – p. 1/33

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Key Terms

Configurations of ExtremalEven Unimodular Lattices – p. 2/33

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Key Terms

Lattice L

Configurations of ExtremalEven Unimodular Lattices – p. 2/33

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Key Terms

Lattice L — “integer vector space”

Configurations of ExtremalEven Unimodular Lattices – p. 2/33

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Key Terms

Lattice L — “integer vector space”

free Z-module

Configurations of ExtremalEven Unimodular Lattices – p. 2/33

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Key Terms

Lattice L — “integer vector space”

free Z-module with an inner product ·, · : L × L → R

Configurations of ExtremalEven Unimodular Lattices – p. 2/33

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Key Terms

Lattice L — “integer vector space”

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Key Terms

Lattice L — “integer vector space”

rank

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Key Terms

Lattice L — “integer vector space”

rank

number of basis vectors

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Key Terms

Lattice L — “integer vector space”

even

Configurations of ExtremalEven Unimodular Lattices – p. 4/33

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Key Terms

Lattice L — “integer vector space”

even

x, x ∈ 2Z for all x ∈ L

Configurations of ExtremalEven Unimodular Lattices – p. 4/33

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Key Terms

Lattice L — “integer vector space”

even

x, x ∈ 2Z for all x ∈ L all vectors have even squared length

Configurations of ExtremalEven Unimodular Lattices – p. 4/33

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Key Terms

Lattice L — “integer vector space”

unimodular

Configurations of ExtremalEven Unimodular Lattices – p. 5/33

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Key Terms

Lattice L — “integer vector space”

unimodular

L is “self-dual”

Configurations of ExtremalEven Unimodular Lattices – p. 5/33

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SLIDE 15

Key Terms

Lattice L — “integer vector space”

unimodular

L is “self-dual” L’s matrix has determinant 1

Configurations of ExtremalEven Unimodular Lattices – p. 5/33

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Key Terms

Review: even unimodular lattice L

Configurations of ExtremalEven Unimodular Lattices – p. 6/33

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Key Terms

Review: even unimodular lattice L all vectors have even squared length

Configurations of ExtremalEven Unimodular Lattices – p. 7/33

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Key Terms

Review: even unimodular lattice L “self-dual”

Configurations of ExtremalEven Unimodular Lattices – p. 8/33

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Key Terms

Review: even unimodular lattice L “integer vector space”

Configurations of ExtremalEven Unimodular Lattices – p. 9/33

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Key Terms

even unimodular lattice L...

Configurations of ExtremalEven Unimodular Lattices – p. 10/33

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Key Terms

even unimodular lattice L...

“rare”

Configurations of ExtremalEven Unimodular Lattices – p. 10/33

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Key Terms

even unimodular lattice L...

“rare”

rank(L) = 8n

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Key Terms

even unimodular lattice L...

Configurations of ExtremalEven Unimodular Lattices – p. 11/33

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Key Terms

even unimodular lattice L...

extremal

Configurations of ExtremalEven Unimodular Lattices – p. 11/33

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Key Terms

even unimodular lattice L...

extremal

shortest vector

Configurations of ExtremalEven Unimodular Lattices – p. 11/33

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Key Terms

even unimodular lattice L...

extremal

shortest vector ↔ as long as possible

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We care about...

extremal even unimodular lattices

Configurations of ExtremalEven Unimodular Lattices – p. 12/33

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We care because...

extremal even unimodular lattices

Configurations of ExtremalEven Unimodular Lattices – p. 13/33

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We care because...

extremal even unimodular lattices are useful

Configurations of ExtremalEven Unimodular Lattices – p. 13/33

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We care because...

extremal even unimodular lattices are useful

Sphere-packing problems

Configurations of ExtremalEven Unimodular Lattices – p. 13/33

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SLIDE 31

We care because...

extremal even unimodular lattices are useful

Sphere-packing problems

Storing n-dimensional oranges

Configurations of ExtremalEven Unimodular Lattices – p. 13/33

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SLIDE 32

We care because...

extremal even unimodular lattices are useful

Sphere-packing problems

Storing n-dimensional oranges Chemical lattices (in dimensions n ≤ 3)

Configurations of ExtremalEven Unimodular Lattices – p. 13/33

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We care because...

extremal even unimodular lattices are useful

Sphere-packing problems

Storing n-dimensional oranges Chemical lattices (in dimensions n ≤ 3) (Continuous) communication decoding

Configurations of ExtremalEven Unimodular Lattices – p. 13/33

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Methods: Theta Functions

Configurations of ExtremalEven Unimodular Lattices – p. 14/33

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Methods: Theta Functions

Slogan:

“The theta function of a lattice L encodes the lengths of L’s vectors.”

Configurations of ExtremalEven Unimodular Lattices – p. 14/33

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Methods: Theta Functions

Slogan:

“The theta function of a lattice L encodes the lengths of L’s vectors.”

How:

“dot product x, x” ⇐ ⇒ “length”

Configurations of ExtremalEven Unimodular Lattices – p. 14/33

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Methods: Theta Functions

Slogan:

“The theta function of a lattice L encodes the lengths of L’s vectors.”

How:

“dot product x, x” ⇐ ⇒ “length”

What:

ΘL(τ) =

  • x∈L

eiπτx,x =

∞

  • k=1

akeiπτ(k)

Configurations of ExtremalEven Unimodular Lattices – p. 14/33

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Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Configurations of ExtremalEven Unimodular Lattices – p. 15/33

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SLIDE 39

Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Example:

ΘZ2(τ)

Configurations of ExtremalEven Unimodular Lattices – p. 15/33

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Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Example:

ΘZ2(τ) =

  • x∈L

eiπτx,x = 1 + 4eiπτ + 4e2iπτ + 4e4iπτ + · · ·

Configurations of ExtremalEven Unimodular Lattices – p. 15/33

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SLIDE 41

Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

Theta functions of even unimodular lattices are examples of modular forms.

Configurations of ExtremalEven Unimodular Lattices – p. 16/33

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SLIDE 42

Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

Theta functions of even unimodular lattices are examples of modular forms. We can write down the spaces of modular forms explicitly.

Configurations of ExtremalEven Unimodular Lattices – p. 16/33

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SLIDE 43

Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

We can write down the spaces of modular forms explicitly.

Configurations of ExtremalEven Unimodular Lattices – p. 17/33

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SLIDE 44

Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

We can write down the spaces of modular forms explicitly. We can therefore study the function ΘL...

Configurations of ExtremalEven Unimodular Lattices – p. 17/33

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SLIDE 45

Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

We can write down the spaces of modular forms explicitly. We can therefore study the function ΘL... ...even if we cannot write down a basis for L.

Configurations of ExtremalEven Unimodular Lattices – p. 17/33

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Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

is a modular form which encodes the lengths of L’s vectors.”

Configurations of ExtremalEven Unimodular Lattices – p. 18/33

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Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

is a modular form which encodes the lengths of L’s vectors.”

What we do:

We study weighted theta functions

x∈L P(x)eiπτx,x

(also modular forms) and obtain linear systems of equations...

Configurations of ExtremalEven Unimodular Lattices – p. 18/33

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SLIDE 48

Methods: Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

is a modular form which encodes the lengths of L’s vectors.”

What we do:

We study weighted theta functions

x∈L P(x)eiπτx,x

(also modular forms) and obtain linear systems of equations... ...which give combinatorial information about lattice vectors.

Configurations of ExtremalEven Unimodular Lattices – p. 18/33

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Results

Configurations of ExtremalEven Unimodular Lattices – p. 19/33

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Results

Theorem.

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Results

Theorem Template.

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Results

Theorem Template. For L even unimodular and extremal

  • f rank n, the minimal-norm vectors of L generate L.

Configurations of ExtremalEven Unimodular Lattices – p. 22/33

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Results

Theorem Template. For L even unimodular and extremal

  • f rank n, the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24}

Configurations of ExtremalEven Unimodular Lattices – p. 22/33

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Results

Theorem Template. For L even unimodular and extremal

  • f rank n, the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov: n ∈ {32}

Configurations of ExtremalEven Unimodular Lattices – p. 22/33

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Results

Theorem Template. For L even unimodular and extremal

  • f rank n, the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov: n ∈ {32} Ozeki: n ∈ {32, 48}

Configurations of ExtremalEven Unimodular Lattices – p. 22/33

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Results

Theorem Template. For L even unimodular and extremal

  • f rank n, the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov: n ∈ {32} Ozeki: n ∈ {32, 48}

  • S. D. Kominers:

n ∈ {56, 72, 96}

Configurations of ExtremalEven Unimodular Lattices – p. 22/33

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Results

Theorem Template. For L even unimodular and extremal

  • f rank n, the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov: n ∈ {32} Ozeki: n ∈ {32, 48}

  • S. D. Kominers:

n ∈ {56, 72, 96}

  • N. D. Elkies:

n ∈ {120}

Configurations of ExtremalEven Unimodular Lattices – p. 22/33

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Results

Theorem Template. For L even unimodular and extremal

  • f rank n, the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov: n ∈ {32} Ozeki: n ∈ {32, 48}

  • S. D. Kominers:

n ∈ {56, 72, 96}

  • N. D. Elkies:

n ∈ {120} (much harder!)

Configurations of ExtremalEven Unimodular Lattices – p. 22/33

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Results

Theorem. For L even unimodular and extremal of rank n ∈ {8, 24, 32, 48, 56, 72, 96, 120}, the minimal-norm vectors of L generate L.

Configurations of ExtremalEven Unimodular Lattices – p. 23/33

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Results

Theorem. For L even unimodular and extremal of rank n ∈ {8, 24, 32, 48, 56, 72, 96, 120}, the minimal-norm vectors of L generate L. Such lattices are known to exist for n ∈ {8, 24, 32, 48, 56}.

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Results

Theorem. For L even unimodular and extremal of rank n ∈ {8, 24, 32, 48, 56, 72, 96, 120}, the minimal-norm vectors of L generate L. Such lattices are known to exist for n ∈ {8, 24, 32, 48, 56}. Such lattices are not known to exist for n ∈ {72, 96, 120}.

Configurations of ExtremalEven Unimodular Lattices – p. 23/33

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Results

Theorem. For L even unimodular and extremal of rank n ∈ {8, 24, 32, 48, 56, 72, 96, 120}, the minimal-norm vectors of L generate L.

Configurations of ExtremalEven Unimodular Lattices – p. 24/33

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Results

Theorem. For L even unimodular and extremal of rank n ∈ {8, 24, 32, 48, 56, 72, 96, 120}, the minimal-norm vectors of L generate L. Note: The analogous result is false for n = 16.

Configurations of ExtremalEven Unimodular Lattices – p. 24/33

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Results

Theorem. For L even unimodular and extremal of rank n ∈ {8, 24, 32, 48, 56, 72, 96, 120}, the minimal-norm vectors of L generate L. Note: The analogous result is false for n = 16. Now: What about n = 40?

Configurations of ExtremalEven Unimodular Lattices – p. 24/33

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Results

Theorem. For L even unimodular and extremal of rank n ∈ {8, 24, 32, 48, 56, 72, 96, 120}, the minimal-norm vectors of L generate L. Note: The analogous result is false for n = 16. Now: What about n = 40? Bad news: The analogous result fails for n = 40.

Configurations of ExtremalEven Unimodular Lattices – p. 24/33

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Results

Theorem. For L even unimodular and extremal of rank n ∈ {8, 24, 32, 48, 56, 72, 96, 120}, the minimal-norm vectors of L generate L. Note: The analogous result is false for n = 16. Now: What about n = 40? Bad news: The analogous result fails for n = 40. Good news: It is almost true.

Configurations of ExtremalEven Unimodular Lattices – p. 24/33

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Results

Theorem. Let m(L) be the minimal norm of vectors in an extremal even unimodular lattice L of rank n. The vectors in L having norms m(L) and (m(L) + 2) generate L.

Configurations of ExtremalEven Unimodular Lattices – p. 25/33

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Results

Theorem. Let m(L) be the minimal norm of vectors in an extremal even unimodular lattice L of rank n. The vectors in L having norms m(L) and (m(L) + 2) generate L. Folklore: n ∈ {16}

Configurations of ExtremalEven Unimodular Lattices – p. 25/33

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Results

Theorem. Let m(L) be the minimal norm of vectors in an extremal even unimodular lattice L of rank n. The vectors in L having norms m(L) and (m(L) + 2) generate L. Folklore: n ∈ {16} Ozeki: n ∈ {40}

Configurations of ExtremalEven Unimodular Lattices – p. 25/33

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Results

Theorem. Let m(L) be the minimal norm of vectors in an extremal even unimodular lattice L of rank n. The vectors in L having norms m(L) and (m(L) + 2) generate L. Folklore: n ∈ {16} Ozeki: n ∈ {40}

  • S. D. Kominers & Z. Abel:

n ∈ {40, 80, 120}

Configurations of ExtremalEven Unimodular Lattices – p. 25/33

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Results

Theorem. Let m(L) be the minimal norm of vectors in an extremal even unimodular lattice L of rank n. The vectors in L having norms m(L) and (m(L) + 2) generate L. Folklore: n ∈ {16} Ozeki: n ∈ {40}

  • S. D. Kominers & Z. Abel:

n ∈ {40, 80, 120} (unified method!)

Configurations of ExtremalEven Unimodular Lattices – p. 25/33

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Results

Theorem. Let m(L) be the minimal norm of vectors in an extremal even unimodular lattice L of rank n. The vectors in L having norms m(L) and (m(L) + 2) generate L. Folklore: n ∈ {16} Ozeki: n ∈ {40}

  • S. D. Kominers & Z. Abel:

n ∈ {40, 80, 120} (unified method!) Note: This bound is sharp for n ∈ {16, 40, 80}

Configurations of ExtremalEven Unimodular Lattices – p. 25/33

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Results

Configurations of ExtremalEven Unimodular Lattices – p. 26/33

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Results

Natural Question.

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stluseR

Natural Question.

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stluseR

Natural Question. If we can learn about a (possibly nonexistent) lattice from its theta series, can we learn about a lattice’s theta series from a lattice’s basis?

Configurations of ExtremalEven Unimodular Lattices – p. 29/33

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stluseR

Natural Question. If we can learn about a (possibly nonexistent) lattice from its theta series, can we learn about a lattice’s theta series from a lattice’s basis? Answer: In principle, yes.

Configurations of ExtremalEven Unimodular Lattices – p. 29/33

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stluseR

Natural Question. If we can learn about a (possibly nonexistent) lattice from its theta series, can we learn about a lattice’s theta series from a lattice’s basis? Answer: In principle, yes. Problem: Computationally intensive....

Configurations of ExtremalEven Unimodular Lattices – p. 29/33

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stluseR

Theorem. If L is the unique rank-72 even unimodular lattice with automorphisms given by SL2(Z/71Z) then ΘL = 1 + 71712q3 + 6213012336q4 + 15281966487168q5 + · · · .

N. D. Elkies, Z. Ab el, & S. D. K
  • miners

Configurations of ExtremalEven Unimodular Lattices – p. 30/33

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stluseR

Theorem. If L is the unique rank-72 even unimodular lattice with automorphisms given by SL2(Z/71Z) then ΘL = 1 + 71712q3 + 6213012336q4 + 15281966487168q5 + · · · . (Just proven by

N. D. Elkies, Z. Ab el, & S. D. K
  • miners.)

Configurations of ExtremalEven Unimodular Lattices – p. 30/33

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Where do we go from here?

Configurations of ExtremalEven Unimodular Lattices – p. 31/33

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SLIDE 82

Where do we go from here?

Configuration results for non-extremal even unimodular lattices of ranks n ∈ {56, 72, 96, 120}?

Configurations of ExtremalEven Unimodular Lattices – p. 31/33

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SLIDE 83

Where do we go from here?

Configuration results for non-extremal even unimodular lattices of ranks n ∈ {56, 72, 96, 120}? Theta series of the unique rank-80 even unimodular lattice with automorphisms given by SL2(Z/79Z)?

Configurations of ExtremalEven Unimodular Lattices – p. 31/33

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SLIDE 84

Where do we go from here?

Configuration results for non-extremal even unimodular lattices of ranks n ∈ {56, 72, 96, 120}? Theta series of the unique rank-80 even unimodular lattice with automorphisms given by SL2(Z/79Z)? Rank-72 extremal even unimodular lattices?

Configurations of ExtremalEven Unimodular Lattices – p. 31/33

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Acknowledgements

Harvard Mathematics Department

  • Prof. Noam D. Elkies

Zachary Abel

  • Prof. Benedict Gross

Center for Excellence in Education / Research Science Institute

  • Mr. Christopher Mihelich
  • Dr. John Rickert

Harvard College PRISE Brown University SUMS Keith Conrad

Configurations of ExtremalEven Unimodular Lattices – p. 32/33

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QED

Questions?

Configurations of ExtremalEven Unimodular Lattices – p. 33/33