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I'm currently writing my master thesis about the j-invariant and his q-expansion. Now i have the result that the growth of the coefficients is asymptotically $$c(n) \sim \frac{e^{4\pi \sqrt{n}}}{\sqrt{2}n^{\frac{3}{4}}}.$$

Is there any special reason to investigate this growth? Is it possible to use this result to make some statements about the j-invariant or is this just a "nice-to-know" information?

Thanks in advance.

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  • $\begingroup$ $j(\tau)$ is not holomorphic at the cusp so it is not a modular form. Now for a cusp form $f(\tau)= \sum_{n=1}^\infty a_n e^{2i \pi n \tau}$ for $SL_2(\mathbb{Z})$, you can look at its Mellin transform $F(s) = \int_0^\infty x^{s-1} f(i x)dx = (2\pi)^{-s} \Gamma(s) \sum_{n=1}^\infty a_n n^{-s}$ where the modularity of $f$ means $F(s) = F(k-s)$ a similar functional equation as for $\zeta(s)$. $\endgroup$ – reuns Dec 17 '16 at 19:32
  • $\begingroup$ If also the coefficients $a_n$ are multiplicative (if $f$ is a Hecke eigenform) then there is a Riemann hypothesis for $F(s)$ which is a statement about the growth of the coefficients $\frac{1}{\sum_{n=1}^\infty a_n n^{-s}}=\sum_{n=1}^\infty b_n n^{-s}$ $\endgroup$ – reuns Dec 17 '16 at 19:32
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There are lots of reasons why it's interesting to know the growth rate of modular functions. For the $j$-invariant, the coefficients are closely related to the dimensions of the irreducible representations of the monster group (the largest sporadic finite simple group), so the size of those coefficients tells you something about the representations of $M$. It's also not a coincidence that the growth rate of $c(n)$ looks a lot like the asymptotic formula for the partition function $p(n)\sim e^{\pi\sqrt{2n/3}}/4\sqrt3 n$, and the growth rate of $p(n)$ is of interest in many sorts of combinatorial calculations.

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  • $\begingroup$ This helps me a lot. Do you have an example when it's interesting to know the growth rate of modular forms in general? $\endgroup$ – Tim Vormann Dec 16 '16 at 19:11
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    $\begingroup$ @Tim: look up Ramanujan conjecture proved by Deligne. $\endgroup$ – Abdelmalek Abdesselam Dec 16 '16 at 20:33

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