Introduction
Counting Solutions (mod p k )
Future Work
Representation by Ternary Quadratic Forms
Edna Jones
Rose-Hulman Institute of Technology
Texas A&M Math REU
July 23, 2014
Edna Jones
Representation by Ternary Quadratic Forms
1 / 20
Introduction
Counting Solutions (mod p k )
Future Work
The Quadratic Forms of Interest
Q(~x) = ax 2 + by 2 + cz 2 , where
a, b, c are positive integers
gcd(a, b, c) = 1
 
x

~x = y 
z
Edna Jones
Representation by Ternary Quadratic Forms
2 / 20
Introduction
Counting Solutions (mod p k )
Future Work
The Quadratic Forms of Interest
Q(~x) = ax 2 + by 2 + cz 2 , where
a, b, c are positive integers
gcd(a, b, c) = 1
 
x

~x = y 
z
Examples:
Q(~x) = x 2 + 3y 2 + 5z 2
Q(~x) = 3x 2 + 4y 2 + 5z 2
Q(~x) = x 2 + 5y 2 + 7z 2
Edna Jones
Representation by Ternary Quadratic Forms
2 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Globally Represented
Definition
An integer m is (globally) represented by Q if there exists
~x ∈ Z3 such that Q(~x) = m.
Edna Jones
Representation by Ternary Quadratic Forms
3 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Globally Represented
Definition
An integer m is (globally) represented by Q if there exists
~x ∈ Z3 such that Q(~x) = m.
Example
1 and 9 are globally represented by Q(~x) = x 2 + 5y 2 + 7z 2 ,
because
1 = 12 + 5 · 02 + 7 · 02
9 = 22 + 5 · 12 + 7 · 02
Edna Jones
Representation by Ternary Quadratic Forms
3 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Locally Represented
Definition
Let p be a positive prime integer. An integer m is locally
represented by Q at the prime p if for every nonnegative
integer k there exists ~x ∈ Z3 such that
Q(~x) ≡ m (mod p k ).
Edna Jones
Representation by Ternary Quadratic Forms
4 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Locally Represented
Definition
Let p be a positive prime integer. An integer m is locally
represented by Q at the prime p if for every nonnegative
integer k there exists ~x ∈ Z3 such that
Q(~x) ≡ m (mod p k ).
Definition
An integer m is locally represented (everywhere) by Q if m is
locally represented at p for every prime p and there exists
~x ∈ R3 such that Q(~x) = m.
Edna Jones
Representation by Ternary Quadratic Forms
4 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Locally Represented Example
Example
1 and 3 are locally represented everywhere by
Q(~x) = x 2 + 5y 2 + 7z 2 .
Edna Jones
Representation by Ternary Quadratic Forms
5 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Locally Represented Example
Example
1 and 3 are locally represented everywhere by
Q(~x) = x 2 + 5y 2 + 7z 2 .
12 + 5 · 02 + 7 · 02 ≡ 1 (mod p k ) for any prime p and
integer k ≥ 0
Edna Jones
Representation by Ternary Quadratic Forms
5 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Locally Represented Example
Example
1 and 3 are locally represented everywhere by
Q(~x) = x 2 + 5y 2 + 7z 2 .
12 + 5 · 02 + 7 · 02 ≡ 1 (mod p k ) for any prime p and
integer k ≥ 0
More difficult to see why 3 locally represented everywhere
by Q, because 3 is not globally represented by Q
Edna Jones
Representation by Ternary Quadratic Forms
5 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Difference between Globally and Locally
Represented
m is globally represented by Q
=⇒ m is locally represented everywhere by Q
Edna Jones
Representation by Ternary Quadratic Forms
6 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Difference between Globally and Locally
Represented
m is globally represented by Q
=⇒ m is locally represented everywhere by Q
m is locally represented everywhere by Q
=⇒
6
m is globally represented by Q
Edna Jones
Representation by Ternary Quadratic Forms
6 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Difference between Globally and Locally
Represented
m is globally represented by Q
=⇒ m is locally represented everywhere by Q
m is locally represented everywhere by Q
=⇒
6
m is globally represented by Q
However, for m square-free and sufficiently large,
m is locally represented everywhere by Q
=⇒ m is globally represented by Q
Edna Jones
Representation by Ternary Quadratic Forms
6 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Difference between Globally and Locally
Represented
m is globally represented by Q
=⇒ m is locally represented everywhere by Q
m is locally represented everywhere by Q
=⇒
6
m is globally represented by Q
However, for m square-free and sufficiently large,
m is locally represented everywhere by Q
=⇒ m is globally represented by Q
How large is sufficiently large?
Edna Jones
Representation by Ternary Quadratic Forms
6 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Questions that Arose
How do you determine that m is locally represented
everywhere by Q?
Edna Jones
Representation by Ternary Quadratic Forms
7 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Questions that Arose
How do you determine that m is locally represented
everywhere by Q?
How do you determine that m is locally represented by Q
at a prime p?
Edna Jones
Representation by Ternary Quadratic Forms
7 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Counting Solutions (mod p k )
Let p be a positive prime integer and k a non-negative integer.
Definition
rpk ,Q (m) = # ~x ∈ (Z/p k Z)3 : Q(~x) ≡ m (mod p k )
Edna Jones
Representation by Ternary Quadratic Forms
8 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Counting Solutions (mod p k )
Let p be a positive prime integer and k a non-negative integer.
Definition
rpk ,Q (m) = # ~x ∈ (Z/p k Z)3 : Q(~x) ≡ m (mod p k )
m is locally represented by Q at a prime p if and only if
rpk ,Q (m) > 0 for every nonnegative integer k.
Edna Jones
Representation by Ternary Quadratic Forms
8 / 20
Introduction
Counting Solutions (mod p k )
Future Work
An Abbreviation and a Definition
Abbreviate e 2πiw as e(w ).
Edna Jones
Representation by Ternary Quadratic Forms
9 / 20
Introduction
Counting Solutions (mod p k )
Future Work
An Abbreviation and a Definition
Abbreviate e 2πiw as e(w ).
Definition
n
The quadratic Gauss sum G
over Z/qZ is defined by
q
X
q−1 2 n
nj
G
=
e
.
q
q
j=0
Edna Jones
Representation by Ternary Quadratic Forms
9 / 20
Introduction
Counting Solutions (mod p k )
Future Work
An Abbreviation and a Definition
Abbreviate e 2πiw as e(w ).
Definition
n
The quadratic Gauss sum G
over Z/qZ is defined by
q
X
q−1 2 n
nj
G
=
e
.
q
q
j=0
I have explicit formulas for quadratic Gauss sums.
Edna Jones
Representation by Ternary Quadratic Forms
9 / 20
Introduction
Counting Solutions (mod p k )
Future Work
A Sum Containing e(w )
(
q
X
q,
nt
=
e
q
0,
t=0
Edna Jones
if n ≡ 0 (mod q),
otherwise.
Representation by Ternary Quadratic Forms
10 / 20
Introduction
Counting Solutions (mod p k )
Future Work
A Sum Containing e(w )
(
q
X
q,
nt
=
e
q
0,
t=0
if n ≡ 0 (mod q),
otherwise.
p k −1
(
X (Q(~x) − m)t p k , if Q(~x) ≡ m (mod p k ),
=
e
pk
0, otherwise.
t=0
Edna Jones
Representation by Ternary Quadratic Forms
10 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Counting Solutions (mod p k )
k
p −1 1 X
(Q(~x) − m)t
=
e
p k t=0
pk
Edna Jones
(
1,
0,
if Q(~x) ≡ m (mod p k ),
otherwise.
Representation by Ternary Quadratic Forms
11 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Counting Solutions (mod p k )
k
p −1 1 X
(Q(~x) − m)t
=
e
p k t=0
pk
(
1,
0,
if Q(~x) ≡ m (mod p k ),
otherwise.
rpk ,Q (m) = # ~x ∈ (Z/p k Z)3 : Q(~x) ≡ m (mod p k )
Edna Jones
Representation by Ternary Quadratic Forms
11 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Counting Solutions (mod p k )
k
p −1 1 X
(Q(~x) − m)t
=
e
p k t=0
pk
(
1,
0,
if Q(~x) ≡ m (mod p k ),
otherwise.
rpk ,Q (m) = # ~x ∈ (Z/p k Z)3 : Q(~x) ≡ m (mod p k )
rpk ,Q (m) =
X
~x∈(Z/p k Z)3
p k −1 1 X
(Q(~x) − m)t
e
p k t=0
pk
Edna Jones
Representation by Ternary Quadratic Forms
11 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Counting Solutions (mod p k )
rpk ,Q (m) =
X
~x∈(Z/p k Z)3
p k −1 1 X
(Q(~x) − m)t
e
p k t=0
pk
p k −1 p k −1 p k −1
k
−1 X X X 1 pX
(ax 2 + by 2 + cz 2 − m)t
=
e
p k t=0
pk
x=0 y =0 z=0
p k −1 −mt
at
bt
ct
1 X
= k
e
G k G k G k
k
p t=0
p
p
p
p
Edna Jones
Representation by Ternary Quadratic Forms
12 / 20
Introduction
Counting Solutions (mod p k )
Future Work
A Formula for rpk ,Q (m)
Let Q(~x) = ax 2 + by 2 + cz 2 .
Let p be an odd prime such that p - abc.
Let m be square-free.
Edna Jones
Representation by Ternary Quadratic Forms
13 / 20
Introduction
Counting Solutions (mod p k )
Future Work
A Formula for rpk ,Q (m)
Let Q(~x) = ax 2 + by 2 + cz 2 .
Let p be an odd prime such that p - abc.
Let m be square-free.


1, 

if k = 0,

 2k
1 −abcm

p
1+
, if p - m or k = 1,
rpk ,Q (m) =
p
p



1

2k

1− 2 ,
if p | m and k > 1,
p
p
where
·
p
is the Legendre symbol.
Edna Jones
Representation by Ternary Quadratic Forms
13 / 20
Introduction
Counting Solutions (mod p k )
Future Work
A Formula for rpk ,Q (m)
Let Q(~x) = ax 2 + by 2 + cz 2 .
Let p be an odd prime such that p - abc.
Let m be square-free.


1, 

if k = 0,

 2k
1 −abcm

p
1+
, if p - m or k = 1,
rpk ,Q (m) =
p
p



1

2k

1− 2 ,
if p | m and k > 1,
p
p
where
·
p
is the Legendre symbol.
Under the above conditions, rpk ,Q (m) > 0 for every k ≥ 0.
Edna Jones
Representation by Ternary Quadratic Forms
13 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
m square-free, p odd, and p - abc
=⇒ m is locally represented by Q at the prime p
Edna Jones
Representation by Ternary Quadratic Forms
14 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
m square-free, p odd, and p - abc
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
Edna Jones
Representation by Ternary Quadratic Forms
14 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
m square-free, p odd, and p - abc
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
3 is square-free
Edna Jones
Representation by Ternary Quadratic Forms
14 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
m square-free, p odd, and p - abc
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
3 is square-free
5 and 7 are the only odd primes that divide 1 · 5 · 7
Edna Jones
Representation by Ternary Quadratic Forms
14 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
m square-free, p odd, and p - abc
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
3 is square-free
5 and 7 are the only odd primes that divide 1 · 5 · 7
Now only need to check if 3 is locally represented at the
primes 2, 5, and 7
Edna Jones
Representation by Ternary Quadratic Forms
14 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Another Formula for rpk ,Q (m)
Let Q(~x) = ax 2 + by 2 + cz 2 .
Let p be an odd prime such that p divides exactly one of
a, b, c.
Edna Jones
Representation by Ternary Quadratic Forms
15 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Another Formula for rpk ,Q (m)
Let Q(~x) = ax 2 + by 2 + cz 2 .
Let p be an odd prime such that p divides exactly one of
a, b, c.
Without loss of generality, say p | c but p - ab.
Edna Jones
Representation by Ternary Quadratic Forms
15 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Another Formula for rpk ,Q (m)
Let Q(~x) = ax 2 + by 2 + cz 2 .
Let p be an odd prime such that p divides exactly one of
a, b, c.
Without loss of generality, say p | c but p - ab.
If p - m,

1, k = 0,
rpk ,Q (m) =
1
−ab
p 2k 1 −
, k ≥ 1,
p
p
·
where
is the Legendre symbol.
p
Edna Jones
Representation by Ternary Quadratic Forms
15 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Another Formula for rpk ,Q (m)
Let Q(~x) = ax 2 + by 2 + cz 2 .
Let p be an odd prime such that p divides exactly one of
a, b, c.
Without loss of generality, say p | c but p - ab.
If p - m,

1, k = 0,
rpk ,Q (m) =
1
−ab
p 2k 1 −
, k ≥ 1,
p
p
·
where
is the Legendre symbol.
p
Under the above conditions, rpk ,Q (m) > 0 for every k ≥ 0.
Edna Jones
Representation by Ternary Quadratic Forms
15 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
p odd, p - m, and p divides exactly one of a, b, c
=⇒ m is locally represented by Q at the prime p
Edna Jones
Representation by Ternary Quadratic Forms
16 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
p odd, p - m, and p divides exactly one of a, b, c
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
Edna Jones
Representation by Ternary Quadratic Forms
16 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
p odd, p - m, and p divides exactly one of a, b, c
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
5 divides exactly one of the coefficients of Q
Edna Jones
Representation by Ternary Quadratic Forms
16 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
p odd, p - m, and p divides exactly one of a, b, c
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
5 divides exactly one of the coefficients of Q
5-3
Edna Jones
Representation by Ternary Quadratic Forms
16 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
p odd, p - m, and p divides exactly one of a, b, c
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
5 divides exactly one of the coefficients of Q
5-3
3 is locally represented at the prime 5
Edna Jones
Representation by Ternary Quadratic Forms
16 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
p odd, p - m, and p divides exactly one of a, b, c
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
5 divides exactly one of the coefficients of Q
5-3
3 is locally represented at the prime 5
Similar case holds for the prime 7
Edna Jones
Representation by Ternary Quadratic Forms
16 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
p odd, p - m, and p divides exactly one of a, b, c
=⇒ m is locally represented by Q at the prime p
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
5 divides exactly one of the coefficients of Q
5-3
3 is locally represented at the prime 5
Similar case holds for the prime 7
Now only need to check if 3 is locally represented at the
prime 2
Edna Jones
Representation by Ternary Quadratic Forms
16 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Locally Represented at the Prime 2
Theorem
If 2 - abcm and there exists a solution to
Q(~x) = ax 2 + by 2 + cz 2 ≡ m (mod 8),
then m is locally represented by Q at the prime 2.
Edna Jones
Representation by Ternary Quadratic Forms
17 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
2 - abcm and solution to Q(~x) ≡ m (mod 8) exists
=⇒ m is locally represented by Q at the prime 2
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
Edna Jones
Representation by Ternary Quadratic Forms
18 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
2 - abcm and solution to Q(~x) ≡ m (mod 8) exists
=⇒ m is locally represented by Q at the prime 2
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
2 - (1 · 5 · 7 · 3)
Edna Jones
Representation by Ternary Quadratic Forms
18 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
2 - abcm and solution to Q(~x) ≡ m (mod 8) exists
=⇒ m is locally represented by Q at the prime 2
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
2 - (1 · 5 · 7 · 3)
22 + 5 · 02 + 7 · 12 = 11 ≡ 3 (mod 8)
Edna Jones
Representation by Ternary Quadratic Forms
18 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Back to an Example
2 - abcm and solution to Q(~x) ≡ m (mod 8) exists
=⇒ m is locally represented by Q at the prime 2
Example
Q(~x) = x 2 + 5y 2 + 7z 2 and m = 3
2 - (1 · 5 · 7 · 3)
22 + 5 · 02 + 7 · 12 = 11 ≡ 3 (mod 8)
3 is locally represented everywhere by Q
Edna Jones
Representation by Ternary Quadratic Forms
18 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Future Work
Try to find a lower bound on the largest integer m that is
locally but not globally represented by Q
Edna Jones
Representation by Ternary Quadratic Forms
19 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Future Work
Try to find a lower bound on the largest integer m that is
locally but not globally represented by Q
computationally (using Sage)
Edna Jones
Representation by Ternary Quadratic Forms
19 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Future Work
Try to find a lower bound on the largest integer m that is
locally but not globally represented by Q
computationally (using Sage)
theoretically (using theta series, Eisenstein series, and
cusp forms)
Edna Jones
Representation by Ternary Quadratic Forms
19 / 20
Introduction
Counting Solutions (mod p k )
Future Work
Thank you for listening!
Edna Jones
Representation by Ternary Quadratic Forms
20 / 20