论文标题
伯格曼内核和全态siegel cusp形式的界限
Bounds for the Bergman kernel and the sup-norm of holomorphic Siegel cusp forms
论文作者
论文摘要
我们在伯格曼内核的大小上证明了“ $ k $”界的“多项式”，用于holomorphic siegel siegel cusp $ n $和重量$ k $的形式。当$ n = 1,2 $时，我们的边界与上述尺寸上的猜想界限一致，而下限的所有$ n \ ge 1 $都匹配。对于$ l^2 $ normalister siegel cusp form $ f $ a $ 2 $，我们对其sup-norm的限制为$o_ε（k^{9/4+ε}）$。此外，我们表明，在Siegel上半场的Siegel基本域中包含的任何紧凑型集合$ω$（不取决于$ k $），$ \ Mathrm {sp}（2，\ Mathbb z）$，$ f $ f $o_Ω（k^3/2-- $ n/undy $ undy $ f $ f $ f $ f $ f $此设置。
We prove `polynomial in $k$' bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. When $n=1,2$ our bounds agree with the conjectural bounds on the aforementioned size, while the lower bounds match for all $n \ge 1$. For an $L^2$-normalised Siegel cusp form $F$ of degree $2$, our bound for its sup-norm is $O_ε(k^{9/4+ε})$. Further, we show that in any compact set $Ω$ (which does not depend on $k$) contained in the Siegel fundamental domain of $\mathrm{Sp}(2, \mathbb Z)$ on the Siegel upper half space, the sup-norm of $F$ is $O_Ω(k^{3/2 - η})$ for some $η>0$, going beyond the `generic' bound in this setting.
