Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Jacobi first noticed the connection between the functions that bear his name and counting the representations of sums-of-squares, \begin{eqnarray} \theta_{3}^{n}(q) = \left( \sum_{k \in \mathbb{Z}} q^{k^{2}} \right)^{n} = \sum_{k \geq 0} r_{2,n}(k) \ q^{k}, \end{eqnarray} where $q = e^{\pi i \tau}$ (or $e^{2 \pi i \tau}$ depending on which text you consult) and $r_{2,n}(k)$ is the number of ways to represent $k$ as a sum of $n$ squares, i.e., the integral solutions of the equation $k = x_{1}^{2} + \dots + x_{n}^{2}$. The Jacobi theta functions, including $\theta_{3}$, are known to satisfy a myriad of symmetries and identities.

Question: Are these identities related in any way to the family of curves $\{ X^{2} + Y^{2} = k \}_{k \in \mathbb{N}}$? If so, how?

Suppose, more generally, I'd like to count $m$-powers instead of squares. The corresponding $q$-series is then \begin{eqnarray} \left( \sum_{k \in \mathbb{Z}} q^{k^{m}} \right)^{n} = \sum_{k \geq 0} r_{m,n}(k) \ q^{k}, \end{eqnarray} If the answer to the first question is yes, then this function (known?) and its symmetries should then be intimately connected to $\{ X_{1}^{m} + X_{2}^{m} + \dots + X_{n}^{m} = k \}_{k \in \mathbb{N}}$, a family of deformations of the homogeneous hypersurface $F = \sum_{i = 1}^{n} X_{i}^{m} = 0$.

Generalizing further still, let $F = F(X_1, \dots, X_n)$ be a $\mathbb{Z}$-polynomial in $n$ indeterminates. Let $a(k)$ count the number of ways that an integer $k$ can be represented by $F$ over the integers. What general characteristics are known about the function $\sum_{k \geq 0} a(k) \ q^{k}$? Can one determine its symmetries by the symmetries of $\{ F(X_1, \dots, X_n) = k \}_{k \in \mathbb{N}}$ as a family of deformations of the hypersurface $F = 0$? Can one define a related $L$-function and presuppose its functional identities similarly?

Any clarification is certainly appreciated!

share|cite|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.