2
LECTURE 1. BINARY QUADRATIC FORMS
Let
x
=
m
1
v
+
m
2
w
∈
Λ. The length of
x
is given by the formula
k
x

2
=

m
1
v
+
m
2
w

2
= (
m
1
,m
2
)
±
v
·
v
v
·
w
v
·
w w
·
w
²±
m
1
m
2
²
=
am
2
1
+ 2
bm
1
m
2
+
cm
2
2
,
where
a
=
v
·
v
,
b
=
v
·
w
,
c
=
w
·
w
.
(1.1)
Let us consider the (binary) quadratic form (the
distance quadratic form
of Λ)
f
=
ax
2
+ 2
bxy
+
cy
2
.
Notice that its discriminant satisﬁes
D
= 4(
b
2

ac
) =

4
A
(
v
,
w
)
2
<
0
.
(1.2)
Thus
f
is positive deﬁnite. Given a positive integer
n
one may ask about integral
solutions of the equation
f
(
x,y
) =
n.
If there is an integral solution (
m
1
,m
2
) of this equation, we say that the binary
form
f
represents
the number
n
. Geometrically this means that the circle of
radius
√
n
centered at the origin contains one of the points
x
=
m
1
v
+
m
2
w
of the lattice Λ. Notice that the solution of this problem depends only on the
lattice Λ but not on the form
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/08/2012 for the course MATH 300 taught by Professor Ontonkong during the Fall '09 term at SUNY Stony Brook.
 Fall '09
 ONTONKONG

Click to edit the document details