论文标题
$αp^θ$的等式分布，具有Chebotarev条件和极端素数的申请
Equidistribution of $αp^θ$ with a Chebotarev condition and applications to extremal primes
论文作者
论文摘要
我们建立了有关$αp^θ$的分数部分的联合分布结果，$θ\ in（0,1），\α> 0 $，其中$ p $是满足chebotarev条件的质量，以固定有限的galois扩展名，超过$ \ mathbb {q} $。作为一个应用程序，对于固定的非CM椭圆曲线$ e/\ mathbb {q} $，在Sato-Tate Measure Modulo的极端数量中给出了一个渐近公式的数量。这些正是Frobenius跟踪$ a_p（e）$满足一致性$ a_p（e）\ equiv [2 \ sqrt {p}] \ bmod \ ell $的Primes $ p $。我们假设对数字字段的Dedekind Zeta函数假设为零区域假设。
We establish a joint distribution result concerning the fractional part of $αp^θ$ for $θ\in (0,1), \ α>0$, where $p$ is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over $\mathbb{Q}$. As an application, for a fixed non-CM elliptic curve $E/\mathbb{Q}$, an asymptotic formula is given for the number of primes at the extremes of the Sato-Tate measure modulo a large prime $\ell$. These are precisely the primes $p$ for which the Frobenius trace $a_p(E)$ satisfies the congruence $a_p(E)\equiv [2\sqrt{p}] \bmod \ell$. We assume a zero-free region hypothesis for Dedekind zeta functions of number fields.
