Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20...
Transcript of Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20...
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IntroductionCounting Solutions (mod pk )
Future Work
Representation by Ternary Quadratic Forms
Edna Jones
Rose-Hulman Institute of TechnologyTexas A&M Math REU
July 23, 2014
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IntroductionCounting Solutions (mod pk )
Future Work
The Quadratic Forms of Interest
Q(~x) = ax2 + by 2 + cz2, where
a, b, c are positive integers
gcd(a, b, c) = 1
~x =
xyz
Examples:
Q(~x) = x2 + 3y 2 + 5z2
Q(~x) = 3x2 + 4y 2 + 5z2
Q(~x) = x2 + 5y 2 + 7z2
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IntroductionCounting Solutions (mod pk )
Future Work
The Quadratic Forms of Interest
Q(~x) = ax2 + by 2 + cz2, where
a, b, c are positive integers
gcd(a, b, c) = 1
~x =
xyz
Examples:
Q(~x) = x2 + 3y 2 + 5z2
Q(~x) = 3x2 + 4y 2 + 5z2
Q(~x) = x2 + 5y 2 + 7z2
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2,because
1 = 12 + 5 · 02 + 7 · 02
9 = 22 + 5 · 12 + 7 · 02
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2,because
1 = 12 + 5 · 02 + 7 · 02
9 = 22 + 5 · 12 + 7 · 02
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IntroductionCounting Solutions (mod pk )
Future Work
Locally Represented
Definition
Let p be a positive prime integer. An integer m is locallyrepresented by Q at the prime p if for every nonnegativeinteger k there exists ~x ∈ Z3 such that
Q(~x) ≡ m (mod pk).
Definition
An integer m is locally represented (everywhere) by Q if m islocally represented at p for every prime p and there exists~x ∈ R3 such that Q(~x) = m.
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IntroductionCounting Solutions (mod pk )
Future Work
Locally Represented
Definition
Let p be a positive prime integer. An integer m is locallyrepresented by Q at the prime p if for every nonnegativeinteger k there exists ~x ∈ Z3 such that
Q(~x) ≡ m (mod pk).
Definition
An integer m is locally represented (everywhere) by Q if m islocally represented at p for every prime p and there exists~x ∈ R3 such that Q(~x) = m.
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IntroductionCounting Solutions (mod pk )
Future Work
Locally Represented Example
Example
1 and 3 are locally represented everywhere byQ(~x) = x2 + 5y 2 + 7z2.
12 + 5 · 02 + 7 · 02 ≡ 1 (mod pk) for any prime p andinteger k ≥ 0
More difficult to see why 3 locally represented everywhereby Q, because 3 is not globally represented by Q
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IntroductionCounting Solutions (mod pk )
Future Work
Locally Represented Example
Example
1 and 3 are locally represented everywhere byQ(~x) = x2 + 5y 2 + 7z2.
12 + 5 · 02 + 7 · 02 ≡ 1 (mod pk) for any prime p andinteger k ≥ 0
More difficult to see why 3 locally represented everywhereby Q, because 3 is not globally represented by Q
Edna Jones Representation by Ternary Quadratic Forms 5 / 20
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IntroductionCounting Solutions (mod pk )
Future Work
Locally Represented Example
Example
1 and 3 are locally represented everywhere byQ(~x) = x2 + 5y 2 + 7z2.
12 + 5 · 02 + 7 · 02 ≡ 1 (mod pk) for any prime p andinteger k ≥ 0
More difficult to see why 3 locally represented everywhereby Q, because 3 is not globally represented by Q
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IntroductionCounting Solutions (mod pk )
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 Q6=⇒ 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?
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IntroductionCounting Solutions (mod pk )
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 Q6=⇒ 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
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IntroductionCounting Solutions (mod pk )
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 Q6=⇒ 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
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IntroductionCounting Solutions (mod pk )
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 Q6=⇒ 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?
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IntroductionCounting Solutions (mod pk )
Future Work
Questions that Arose
How do you determine that m is locally representedeverywhere by Q?
How do you determine that m is locally represented by Qat a prime p?
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IntroductionCounting Solutions (mod pk )
Future Work
Questions that Arose
How do you determine that m is locally representedeverywhere by Q?
How do you determine that m is locally represented by Qat a prime p?
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IntroductionCounting Solutions (mod pk )
Future Work
Counting Solutions (mod pk)
Let p be a positive prime integer and k a non-negative integer.
Definition
rpk ,Q(m) = #{~x ∈ (Z/pkZ)3 : Q(~x) ≡ m (mod pk)
}
m is locally represented by Q at a prime p if and only ifrpk ,Q(m) > 0 for every nonnegative integer k .
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IntroductionCounting Solutions (mod pk )
Future Work
Counting Solutions (mod pk)
Let p be a positive prime integer and k a non-negative integer.
Definition
rpk ,Q(m) = #{~x ∈ (Z/pkZ)3 : Q(~x) ≡ m (mod pk)
}m is locally represented by Q at a prime p if and only ifrpk ,Q(m) > 0 for every nonnegative integer k .
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IntroductionCounting Solutions (mod pk )
Future Work
An Abbreviation and a Definition
Abbreviate e2πiw as e(w).
Definition
The quadratic Gauss sum G
(n
q
)over Z/qZ is defined by
G
(n
q
)=
q−1∑j=0
e
(nj2
q
).
I have explicit formulas for quadratic Gauss sums.
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IntroductionCounting Solutions (mod pk )
Future Work
An Abbreviation and a Definition
Abbreviate e2πiw as e(w).
Definition
The quadratic Gauss sum G
(n
q
)over Z/qZ is defined by
G
(n
q
)=
q−1∑j=0
e
(nj2
q
).
I have explicit formulas for quadratic Gauss sums.
Edna Jones Representation by Ternary Quadratic Forms 9 / 20
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IntroductionCounting Solutions (mod pk )
Future Work
An Abbreviation and a Definition
Abbreviate e2πiw as e(w).
Definition
The quadratic Gauss sum G
(n
q
)over Z/qZ is defined by
G
(n
q
)=
q−1∑j=0
e
(nj2
q
).
I have explicit formulas for quadratic Gauss sums.
Edna Jones Representation by Ternary Quadratic Forms 9 / 20
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IntroductionCounting Solutions (mod pk )
Future Work
A Sum Containing e(w)
q∑t=0
e
(nt
q
)=
{q, if n ≡ 0 (mod q),
0, otherwise.
pk−1∑t=0
e
((Q(~x)−m)t
pk
)=
{pk , if Q(~x) ≡ m (mod pk),
0, otherwise.
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IntroductionCounting Solutions (mod pk )
Future Work
A Sum Containing e(w)
q∑t=0
e
(nt
q
)=
{q, if n ≡ 0 (mod q),
0, otherwise.
pk−1∑t=0
e
((Q(~x)−m)t
pk
)=
{pk , if Q(~x) ≡ m (mod pk),
0, otherwise.
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IntroductionCounting Solutions (mod pk )
Future Work
Counting Solutions (mod pk)
1
pk
pk−1∑t=0
e
((Q(~x)−m)t
pk
)=
{1, if Q(~x) ≡ m (mod pk),
0, otherwise.
rpk ,Q(m) = #{~x ∈ (Z/pkZ)3 : Q(~x) ≡ m (mod pk)
}rpk ,Q(m) =
∑~x∈(Z/pkZ)3
1
pk
pk−1∑t=0
e
((Q(~x)−m)t
pk
)
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IntroductionCounting Solutions (mod pk )
Future Work
Counting Solutions (mod pk)
1
pk
pk−1∑t=0
e
((Q(~x)−m)t
pk
)=
{1, if Q(~x) ≡ m (mod pk),
0, otherwise.
rpk ,Q(m) = #{~x ∈ (Z/pkZ)3 : Q(~x) ≡ m (mod pk)
}
rpk ,Q(m) =∑
~x∈(Z/pkZ)3
1
pk
pk−1∑t=0
e
((Q(~x)−m)t
pk
)
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IntroductionCounting Solutions (mod pk )
Future Work
Counting Solutions (mod pk)
1
pk
pk−1∑t=0
e
((Q(~x)−m)t
pk
)=
{1, if Q(~x) ≡ m (mod pk),
0, otherwise.
rpk ,Q(m) = #{~x ∈ (Z/pkZ)3 : Q(~x) ≡ m (mod pk)
}rpk ,Q(m) =
∑~x∈(Z/pkZ)3
1
pk
pk−1∑t=0
e
((Q(~x)−m)t
pk
)
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IntroductionCounting Solutions (mod pk )
Future Work
Counting Solutions (mod pk)
rpk ,Q(m) =∑
~x∈(Z/pkZ)3
1
pk
pk−1∑t=0
e
((Q(~x)−m)t
pk
)
=
pk−1∑x=0
pk−1∑y=0
pk−1∑z=0
1
pk
pk−1∑t=0
e
((ax2 + by 2 + cz2 −m)t
pk
)
=1
pk
pk−1∑t=0
e
(−mt
pk
)G
(at
pk
)G
(bt
pk
)G
(ct
pk
)
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IntroductionCounting Solutions (mod pk )
Future Work
A Formula for rpk ,Q(m)
Let Q(~x) = ax2 + by 2 + cz2.Let p be an odd prime such that p - abc .Let m be square-free.
rpk ,Q(m) =
1, if k = 0,
p2k(
1 +1
p
(−abcm
p
)), if p - m or k = 1,
p2k(
1− 1
p2
), if p | m and k > 1,
where
(·p
)is the Legendre symbol.
Under the above conditions, rpk ,Q(m) > 0 for every k ≥ 0.
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IntroductionCounting Solutions (mod pk )
Future Work
A Formula for rpk ,Q(m)
Let Q(~x) = ax2 + by 2 + cz2.Let p be an odd prime such that p - abc .Let m be square-free.
rpk ,Q(m) =
1, if k = 0,
p2k(
1 +1
p
(−abcm
p
)), if p - m or k = 1,
p2k(
1− 1
p2
), if p | m and k > 1,
where
(·p
)is the Legendre symbol.
Under the above conditions, rpk ,Q(m) > 0 for every k ≥ 0.
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IntroductionCounting Solutions (mod pk )
Future Work
A Formula for rpk ,Q(m)
Let Q(~x) = ax2 + by 2 + cz2.Let p be an odd prime such that p - abc .Let m be square-free.
rpk ,Q(m) =
1, if k = 0,
p2k(
1 +1
p
(−abcm
p
)), if p - m or k = 1,
p2k(
1− 1
p2
), if p | m and k > 1,
where
(·p
)is the Legendre symbol.
Under the above conditions, rpk ,Q(m) > 0 for every k ≥ 0.
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
3 is square-free
5 and 7 are the only odd primes that divide 1 · 5 · 7Now only need to check if 3 is locally represented at theprimes 2, 5, and 7
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
3 is square-free
5 and 7 are the only odd primes that divide 1 · 5 · 7Now only need to check if 3 is locally represented at theprimes 2, 5, and 7
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
3 is square-free
5 and 7 are the only odd primes that divide 1 · 5 · 7Now only need to check if 3 is locally represented at theprimes 2, 5, and 7
Edna Jones Representation by Ternary Quadratic Forms 14 / 20
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 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 theprimes 2, 5, and 7
Edna Jones Representation by Ternary Quadratic Forms 14 / 20
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
3 is square-free
5 and 7 are the only odd primes that divide 1 · 5 · 7Now only need to check if 3 is locally represented at theprimes 2, 5, and 7
Edna Jones Representation by Ternary Quadratic Forms 14 / 20
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IntroductionCounting Solutions (mod pk )
Future Work
Another Formula for rpk ,Q(m)
Let Q(~x) = ax2 + by 2 + cz2.Let p be an odd prime such that p divides exactly one ofa, b, c .
Without loss of generality, say p | c but p - ab.If p - m,
rpk ,Q(m) =
1, k = 0,
p2k(
1− 1
p
(−abp
)), k ≥ 1,
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 15 / 20
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IntroductionCounting Solutions (mod pk )
Future Work
Another Formula for rpk ,Q(m)
Let Q(~x) = ax2 + by 2 + cz2.Let p be an odd prime such that p divides exactly one ofa, b, c .Without loss of generality, say p | c but p - ab.
If p - m,
rpk ,Q(m) =
1, k = 0,
p2k(
1− 1
p
(−abp
)), k ≥ 1,
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 15 / 20
![Page 38: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/38.jpg)
IntroductionCounting Solutions (mod pk )
Future Work
Another Formula for rpk ,Q(m)
Let Q(~x) = ax2 + by 2 + cz2.Let p be an odd prime such that p divides exactly one ofa, b, c .Without loss of generality, say p | c but p - ab.If p - m,
rpk ,Q(m) =
1, k = 0,
p2k(
1− 1
p
(−abp
)), k ≥ 1,
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 15 / 20
![Page 39: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/39.jpg)
IntroductionCounting Solutions (mod pk )
Future Work
Another Formula for rpk ,Q(m)
Let Q(~x) = ax2 + by 2 + cz2.Let p be an odd prime such that p divides exactly one ofa, b, c .Without loss of generality, say p | c but p - ab.If p - m,
rpk ,Q(m) =
1, k = 0,
p2k(
1− 1
p
(−abp
)), k ≥ 1,
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 15 / 20
![Page 40: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/40.jpg)
IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
5 divides exactly one of the coefficients of Q5 - 33 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 theprime 2
Edna Jones Representation by Ternary Quadratic Forms 16 / 20
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
5 divides exactly one of the coefficients of Q5 - 33 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 theprime 2
Edna Jones Representation by Ternary Quadratic Forms 16 / 20
![Page 42: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/42.jpg)
IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
5 divides exactly one of the coefficients of Q
5 - 33 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 theprime 2
Edna Jones Representation by Ternary Quadratic Forms 16 / 20
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
5 divides exactly one of the coefficients of Q5 - 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 theprime 2
Edna Jones Representation by Ternary Quadratic Forms 16 / 20
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
5 divides exactly one of the coefficients of Q5 - 33 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 theprime 2
Edna Jones Representation by Ternary Quadratic Forms 16 / 20
![Page 45: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/45.jpg)
IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
5 divides exactly one of the coefficients of Q5 - 33 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 theprime 2
Edna Jones Representation by Ternary Quadratic Forms 16 / 20
![Page 46: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/46.jpg)
IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 and m = 3
5 divides exactly one of the coefficients of Q5 - 33 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 theprime 2
Edna Jones Representation by Ternary Quadratic Forms 16 / 20
![Page 47: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/47.jpg)
IntroductionCounting Solutions (mod pk )
Future Work
Locally Represented at the Prime 2
Theorem
If 2 - abcm and there exists a solution to
Q(~x) = ax2 + by 2 + cz2 ≡ m (mod 8),
then m is locally represented by Q at the prime 2.
Edna Jones Representation by Ternary Quadratic Forms 17 / 20
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 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
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IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 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
![Page 50: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/50.jpg)
IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 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
![Page 51: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/51.jpg)
IntroductionCounting Solutions (mod pk )
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) = x2 + 5y 2 + 7z2 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
![Page 52: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/52.jpg)
IntroductionCounting Solutions (mod pk )
Future Work
Future Work
Try to find a lower bound on the largest integer m that islocally but not globally represented by Q
computationally (using Sage)
theoretically (using theta series, Eisenstein series, andcusp forms)
Edna Jones Representation by Ternary Quadratic Forms 19 / 20
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IntroductionCounting Solutions (mod pk )
Future Work
Future Work
Try to find a lower bound on the largest integer m that islocally but not globally represented by Q
computationally (using Sage)
theoretically (using theta series, Eisenstein series, andcusp forms)
Edna Jones Representation by Ternary Quadratic Forms 19 / 20
![Page 54: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/54.jpg)
IntroductionCounting Solutions (mod pk )
Future Work
Future Work
Try to find a lower bound on the largest integer m that islocally but not globally represented by Q
computationally (using Sage)
theoretically (using theta series, Eisenstein series, andcusp forms)
Edna Jones Representation by Ternary Quadratic Forms 19 / 20
![Page 55: Representation by Ternary Quadratic FormsEdna Jones Representation by Ternary Quadratic Forms 4/20 Introduction Counting Solutions (mod pk) Future Work Locally Represented De nition](https://reader036.fdocuments.us/reader036/viewer/2022071516/6138a7b80ad5d206764963e8/html5/thumbnails/55.jpg)
IntroductionCounting Solutions (mod pk )
Future Work
Thank you for listening!
Edna Jones Representation by Ternary Quadratic Forms 20 / 20