论文标题
Jacobi形式，Saito-Kurokawa Lifts，它们的回调和平均sup-norms
Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average
论文作者
论文摘要
We formulate a precise conjecture about the size of the $L^\infty$-mass of the space of Jacobi forms on $\mathbb H_n \times \mathbb C^{g \times n}$ of matrix index $S$ of size $g$.这个$ l^\ infty $ - 质量是通过该空间的伯格曼内核的大小来衡量的。我们证明了所有此类$ n，g，s $的猜想下限，并在$ k $ axpect $ n = 1 $，$ g \ ge 1 $时证明了上限。 当$ n = 1 $和$ g = 1 $时，我们对索引（旧和）新空间的尺寸进行了更精致的研究，后者是通过Waldspurger的公式进行的。 Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central $L$-values $L(1/2, f \otimes χ_D)$ with $f$ varying over newforms of level a prime $p$ and even weight $k$ as $k,p \to \infty$ and $D$ being (explicitly) polynomially bounded by $k,p$.这里$χ_d$是一个真正的二次dirichlet字符。 我们还证明，通过三种不同的方法（有或不使用中央$ l $ values），Saito-Kurokawa升降机的大小为$ k^{5/2} $，并表明它们的回头量的大小是对角的bb bbb h $ \ nathbb h \ h $ \ h $ k^2 $ k^2 $ k^2 $ k^2 在附录中，回答了整个空间的回调$ s^2_k $，这里的大小为$ k^3 $。
We formulate a precise conjecture about the size of the $L^\infty$-mass of the space of Jacobi forms on $\mathbb H_n \times \mathbb C^{g \times n}$ of matrix index $S$ of size $g$. This $L^\infty$-mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such $n,g,S$ and prove the upper bound in the $k$ aspect when $n=1$, $g \ge 1$. When $n=1$ and $g=1$, we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger's formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central $L$-values $L(1/2, f \otimes χ_D)$ with $f$ varying over newforms of level a prime $p$ and even weight $k$ as $k,p \to \infty$ and $D$ being (explicitly) polynomially bounded by $k,p$. Here $χ_D$ is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight $k$) is $k^{5/2}$ by three different methods (with or without the use of central $L$-values), and show that the size of their pullbacks to the diagonally embedded $\mathbb H \times \mathbb H$ is $k^2$. In an appendix, the same question is answered for the pullbacks of the whole space $S^2_k$, the size here being $k^3$.
