Let $ H_g$ be the Siegel upper-half plance of genus $ g$ , and $ \Gamma=Sp_{2g}(\mathbb{Z})$ the full modular group of genus $ g$ . As usual, we can define non-holomorphic Eisenstein and Poincare series as $ $ E^{(g)}_k(Z,s)=\sum_\gamma \delta(\gamma Z)^s j(\gamma,Z)^{-k}$ $ $ $ P^{(g)}_{A,k}(Z,s)=\sum_\gamma \delta(\gamma Z)^s j(\gamma,Z)^{-k}e^{2\pi i \sigma(A\gamma Z)}$ $ where $ \delta(Z)=\det(\Im(Z))$ , $ j(\gamma,Z)=\det(C_\gamma Z+D_\gamma)$ , $ A$ a half-integral positive semidefinite symmetric matrix, $ \sigma$ is the trace and the sum is over $ \Gamma_{g,0}\backslash\Gamma_g$ , for $ \gamma\in\Gamma_{g,0}\iff C_\gamma=0$ .

Is $ P^{(g)}_{A,k}(Z,s)$ decreasing exponentially fast for $ \Im(Z)$ “going up”, for any $ A>0$ ?

To explain what “going up” means, think of $ g=1$ and fix $ s=0$ : then we know that $ P^{(1)}_{A,k}(Z,0)$ is a cusp form for any $ A\in\mathbb{N}^+$ and, like every cusp form, decreases exponentially fast for $ \Im(Z)\to\infty$ . For higher genus $ g>1$ , I think the “height” of $ Z$ should be $ \delta(Z).$

So, when $ s=0$ , the answer to my question is yes for any genus. I think the same should be true for any $ s$ such that the series are nicely convergent (so for $ \Re(s)\ge$ some negative quantity depending on $ k$ and $ g$ ), because of the exponential factors in the definition: in fact, $ $ P^{(g)}_{A,k}(Z,s)=\delta(Z)^s e^{2\pi i\sigma(AZ)}+\sum_{\gamma\not=I} \delta(\gamma Z)^s j(\gamma,Z)^{-k}e^{2\pi i \sigma(A\gamma Z)}$ $ so my wishful thinking here suggests that the dominant term “going up” is the first exponential summand, and the remaining part is irrelevant.

But, by taking absolute values on that series, we see that it is majorated by $ $ \delta(Z)^{-k/2}\delta(Z)^{s+k/2}\sum_{\gamma\not=I} |j(\gamma,Z)|^{-k-2s}=\delta(Z)^{-k/2}\left(E^{(g)}_{0}(Z,s+k/2)-\delta(Z)^{s+k/2}\right)$ $ and the last factor is known to behave polynomially: hence, this approach does not tell us that the remaining part of the Poincare series is irrelevant; on the other hand, the exact same reasoning applies for $ s=0$ where we do know the result is true.

Any idea/result to shed some light here? Thanks!