# growth of coefficients of half-integral weight modular forms

What is the order of growth of coefficients of half-integral weight modular forms on congruence subgroup with character ? There is the usual Hecke trick to compute bounds on the coefficients of integral weights on $$SL_2(\mathbb{Z})$$ but I don't know how to adapt the argument. I've looked online at many articles but didn't find anything. My guess is that it should be $$O(n^{k/2})$$ where $$k$$ is the half-integral weight.

• @DietrichBurde Done Commented Sep 14, 2022 at 19:57
• See this MO question. Commented Sep 15, 2022 at 12:52

It seems that we should have essentially the same bound as in the full integral weight case. That is, we should have that $$a(n) \ll n^{\frac{k-1}{2} + \epsilon}.$$ In the full integral weight case, one can say a bit more about the $$\epsilon$$ factor in terms of the divisor function, but that might not be true for half-integral weight.
One reason to suspect that this is the truth is because it holds on average. The standard Rankin--Selberg technique of studying the Dirichlet series $$D(s) = \sum_{n \geq 1} \frac{\lvert a(n) \rvert^2}{n^{s + k - 1}}$$ applies. One can show that $$D(s)$$ has meromorphic continuation to $$\mathbb{C}$$ with polynomial growth in vertical strips, and that the rightmost pole is $$s = 1$$. This implies that $$\sum_{n \leq X} \lvert a(n)/n^{\frac{k-1}{2}} \rvert^2 = c X + o(X),$$ and thus that $$\lvert a(n) \rvert \approx n^{\frac{k-1}{2}}$$ on average.
• And of course by Waldspurger's theorem, if we knew such bounds for $n$ a fundamental discriminant, then we would know the Lindelof hypothesis for certain $L$-functions. So this is much harder than Deligne's theorem. Commented Sep 20, 2022 at 20:36