Mock modular forms and (indefinite) quadratic forms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:33:18Z http://mathoverflow.net/feeds/question/120324 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120324/mock-modular-forms-and-indefinite-quadratic-forms Mock modular forms and (indefinite) quadratic forms Richard Eager 2013-01-30T14:12:03Z 2013-02-01T15:28:16Z <p>Define the function $$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$ where $c(n,m,l)$ is defined by $$c(n,m,l) = \begin{cases} (-1)^{s+l} &amp; \mbox{if } 4n - m^2 + l^2 = 2s(s+1) \end{cases}$$ $$\begin{cases} 0 &amp; otherwise \;\;\;\;\;\;\;\;\;\;\;\;\;\;<br> \end{cases}$$ for some integer $s$ and $c(n,m,l) = 0$ unless $4n - m^2 -l^2 \ge 0.$ $f(q,z,1)$ is known to be related to a Mock modular form. I conjecture that $$f(q,1,-1) = \sum_{n \ge 0} (-1)^n (2n + 1) q^{n(n+1)/2}.$$ Is there an elementary proof of the above conjecture? Is the function $f(q,z,y)$ a known mathematical object, perhaps related to a Siegel modular form?</p> <p>Update: $f(q,z,y)$ is a product of Jacobi theta functions and $\mu(q;z,y),$ where $\mu(q;z,y)$ is a Lerch sum studied by Zweger in his thesis. Zweger's thesis also relates mock modular forms to indefinite quadratic forms of signature $(1,n),$ so perhaps it isn't too unsurprising that $f(q,z,y)$ takes a "nice" form.</p>