论文标题
$ k3 $表面家庭的明确卫星式类型分销
Explicit Sato-Tate type distribution for a family of $K3$ surfaces
论文作者
论文摘要
在1960年代，桦木证明了弗罗贝尼乌斯（Frobenius）在大小据有限场上随机拍摄的椭圆形曲线的痕迹是通过半圆形分布来建模的（即非CM椭圆曲线的常见SATO-TATE）。与Birch的结果相比，作者和Saikia最近发表的一篇论文证明，限制了frobenius痕迹的限制性分布$ k3 $x_λ$的某个家族的$a_λ$a_λ$，the Picard Rank $ 19 $是$ O（3）$分布。我们用$ \ frac {1} {4π} f（t）表示的分布，$与半圆形分布完全不同。它在$ [-3,3] $上得到支持，并在$ t = \ pm1的垂直渐近线。$在这里我们将此结果显式。我们证明，如果$ p \ geq 5 $是prime，$ -3 \ leq a <b \ leq 3，$ then $$ \ left | \ frac {\＃\ {λ\ {λ\ in \ mathbb {f} _p {f} _p ：a_λ（p）\在[a，b] \}}} {p} - \ frac {1} {4π} \ int_a^b f（t）dt \ right | \ leq \ leq \ frac {110.84} {p^{1/4}}}}}}。结果，我们能够确定何时有限场$ \ mathbb {f} _p $足够大，以使离散直方图达到任何给定的高度接近$ t = \ pm1。$为了获得这些结果，我们利用兰克金 - 卡恩·托括号的理论中的谐音理论。
In the 1960's, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato-Tate for non-CM elliptic curves). In analogy with Birch's result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces $A_λ(p)$ of a certain family of $K3$ surfaces $X_λ$ with generic Picard rank $19$ is the $O(3)$ distribution. This distribution, which we denote by $\frac{1}{4π}f(t),$ is quite different from the semicircular distribution. It is supported on $[-3,3]$ and has vertical asymptotes at $t=\pm1.$ Here we make this result explicit. We prove that if $p\geq 5$ is prime and $-3\leq a<b\leq 3,$ then $$ \left|\frac{\#\{λ\in\mathbb{F}_p :A_λ(p)\in[a,b]\}}{p}-\frac{1}{4π}\int_a^b f(t)dt\right|\leq \frac{110.84}{p^{1/4}}. $$ As a consequence, we are able to determine when a finite field $\mathbb{F}_p$ is large enough for the discrete histograms to reach any given height near $t=\pm1.$ To obtain these results, we make use of the theory of Rankin-Cohen brackets in the theory of harmonic Maass forms.
