Representation number via quad forms of theta quadratic form

So the number of solutions is the same. In particular, if they give the same series, because they yield the same number for each ##n## (which is a result you can look up in any book, or prove yourself), this means that they yield the same number for each ##n## because they yield the same series.
  • #1
binbagsss
1,278
11
##\theta(\tau, A) = \sum\limits_{\vec{x}\in Z^{m}} e^{\pi i A[x] \tau } ##

##=\sum\limits^{\infty}_{n=0} r_{A}(n)q^{n} ##,

where ## r_{A} = No. [ \vec{x} \in Z^{m} ; A[\vec{x}] =n]##

where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive definite, of rank ##m## and even. (and I think symmetric?)

So I thought that this meant to solve the quadratic ##A[x]= \vec{x^t} A \vec{x} = n ##, for each ##n##, and the representation number is then given by the number of solutions to this?, subject to ## \vec{x} \in Z^{m} ## ,

What is ##Z^{m}## here please? ( z the integer symbol)

Many thanks
 
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  • #2
## \vec{x} \in Z^{m} ## is simply ##\vec{x} = (x_1, \ldots , x_n)^\tau \in \underbrace{\mathbb{Z} \times \ldots \times \mathbb{Z}}_{n-\ times}##.
Was that your question?
 
  • #3
fresh_42 said:
## \vec{x} \in Z^{m} ## is simply ##\vec{x} = (x_1, \ldots , x_n)^\tau \in \underbrace{\mathbb{Z} \times \ldots \times \mathbb{Z}}_{n-\ times}##.
Was that your question?

no not quite, i needed to check this before i can post my full question, and that my interpretation of what the representation number is correct? (otherwise the question I am about to post may not make sense)
 
  • #4
binbagsss said:
##\theta(\tau, A) = \sum\limits_{\vec{x}\in Z^{m}} e^{\pi i A[x] \tau } ##

##=\sum\limits^{\infty}_{n=0} r_{A}(n)q^{n} ##,

where ## r_{A} = No. [ \vec{x} \in Z^{m} ; A[\vec{x}] =n]##

where ##A[x]= x^t A x ##, is the associated quadratic from to the matrix ##A##, where here ##A## is positive definite, of rank ##m## and even. (and I think symmetric?)

So I thought that this meant to solve the quadratic ##A[x]= \vec{x^t} A \vec{x} = n ##, for each ##n##, and the representation number is then given by the number of solutions to this?, subject to ## \vec{x} \in Z^{m} ## ,

What is ##Z^{m}## here please? ( z the integer symbol)

Many thanks

Okay so on the attachment of extract from my book, I'm not understanding the comment '##Q_{1}(x,y) ## and ##Q_{2}(x,y) ## yeild the same series since they represent the same integers.'

So as I said above my interpretation of how to compute the ##r(n)## was to :

set ##2 Q(x,y) = A(x,y) = n ## , for each ##n## in turn and count the number of solutions to this for each ##n##.

So looking at ##Q_{0}(x,y)##, should find ##2(x^{2}+xy+6y^2)=0## has one solution (i.e ##(x,y)=0##) , ##2(x^{2}+xy+6y^2)=1## should find 2 solutions and ##2(x^{2}+xy+6y^2)=2,3## has no solutions for ##x \in Z^m ##
Is my understanding correct here?

So then looking at ##Q_1 (x,y)## and ##Q_{2} (x,y) ## which differ only on the sign of the ##xy## term, I don't see how it is obvious that these will have the same number of solutions for ##Q(x,y) = n## for each ##n##?

Many thanks in advance.
 

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  • #5
If you replace ##x## by ##-x## (or equivalently ##y## by ##-y\,##) you get the same number of pairs ##(x,y) \in \mathbb{Z}^2## with ##Q_i(x,y)=n##. Different pairs though, but equally many.
 
  • #6
fresh_42 said:
If you replace ##x## by ##-x## (or equivalently ##y## by ##-y\,##) you get the same number of pairs ##(x,y) \in \mathbb{Z}^2## with ##Q_i(x,y)=n##. Different pairs though, but equally many.

oh right thanks,
how is this obvious? e.g how do you know you won't end up getting complex solutions for the corresponding sign change, ta
 
  • #7
binbagsss said:
oh right thanks,
how is this obvious? e.g how do you know you won't end up getting complex solutions for the corresponding sign change, ta
$$\#\{(x,y)\in \mathbb{Z}^2\,\vert \,Q_1(x,y)=0\}=\#\{(-x,y)\in \mathbb{Z}^2\,\vert \,Q_1(-x,y)=0\}=\#\{(w,y)\in \mathbb{Z}^2\,\vert \,Q_1(-x,y)=0\, \wedge \, w=-x\,\}=\#\{(w,y)\in \mathbb{Z}^2\,\vert \,Q_2(x,y)=0\, \wedge \, w=-x\,\}=\#\{(w,y)\in \mathbb{Z}^2\,\vert \,Q_2(x,y)=0\, \wedge \, w=x\,\}=\#\{(x,y)\in \mathbb{Z}^2\,\vert \,Q_2(x,y)=0\}$$ because we consider all pairs in ##\mathbb{Z}^2##, so the sign doesn't make any difference in the total number of solutions, only in the way we write, resp. notate them: ##\#\{(x,y)\in \mathbb{Z}^2\,\vert \,x=1 \wedge y=2\}=\#\{(x,y)\in \mathbb{Z}^2\,\vert \,x=-1 \wedge y=2\}##.
 

FAQ: Representation number via quad forms of theta quadratic form

1. What is a representation number via quad forms?

A representation number via quad forms is a mathematical concept used in number theory and algebraic geometry. It refers to the number of ways a given number can be written as the sum of squares of integers, also known as quadratic forms.

2. How is the representation number via quad forms related to theta quadratic forms?

The representation number via quad forms is closely related to theta quadratic forms, as it is used to calculate the number of ways a particular number can be represented as a sum of squares using theta quadratic forms. Theta quadratic forms are a special type of quadratic form that arise in the study of modular forms.

3. What are some applications of representation number via quad forms?

Representation number via quad forms have several applications in mathematics, including in number theory, algebraic geometry, and modular forms. They can also be used to solve problems related to lattice points, integer partitions, and the class number problem.

4. Can a number have multiple representation numbers via quad forms?

Yes, a number can have multiple representation numbers via quad forms. For example, the number 5 has 2 representation numbers via quad forms: 5 = 1^2 + 2^2 and 5 = 2^2 + 1^2. This is because there are multiple ways to express a number as a sum of squares using different theta quadratic forms.

5. Is there a formula for calculating representation number via quad forms?

Yes, there is a formula for calculating the representation number via quad forms of a given number. It involves using the theta function, which is a mathematical function that is closely related to theta quadratic forms. However, this formula can be quite complex and may require advanced mathematical knowledge to understand and apply.

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