Quadratic Number Fields – Lecture 6

TU Kaiserslautern – Summer term 2017

Tommy Hofmann

June 12, 2017

Lattices and J -sets

I If α, β ∈ C× are complex numbers with αβ 6∈ R, then

Λ = {aα + bβ | a, b ∈ Z}

is called a lattice and (α, β) is called a basis of Λ.

I A basis (α, β) of a lattice Λ ⊆ C is called normalized, ifIm(βα) > 0.

I The J -set of Λ is defined to be the set

J (Λ) =

{β

α

∣∣∣∣ (α, β) normalized basis

}.

I Λ ∼ Λ′ ⇔ J (Λ) = J (Λ′) ⇔ J (Λ) ∩ J (Λ′) 6= ∅.

The lattice Z[i ]

R

iR

−4

−4i

−3

−3i

−2

−2i

−1

−1i

1

1i

2

2i

3

3i

4

4i

J (Z[i ])100 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

J (Z[i ])500 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

J (Z[i ])1000 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

J (Z[i ])5000 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

J (Z[i ])10000 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

The lattice 〈2, 1 +√−5〉

R

iR

−4

−4i

−3

−3i

−2

−2i

−1

−1i

1

1i

2

2i

3

3i

4

4i

J (〈2, 1 +√−5〉)

100 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

J (〈2, 1 +√−5〉)

500 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

J (〈2, 1 +√−5〉)

1000 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

J (〈2, 1 +√−5〉)

5000 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1

J (〈2, 1 +√−5〉)

10000 points

R

iR

−2 −1 0 1 2

0:25

0:5

0:75

1