论文标题
在椭圆曲线的高度扭转场中完全分裂的素数
On totally split primes in high-degree torsion fields of elliptic curves
论文作者
论文摘要
类似于算术进展中的素数，我们可以研究高级$ \ mathbb {q} $的扩展的素数。由Kowalski的问题激发，我们专注于扩展$ \ Mathbb {q}（e [d]）$，通过毗邻非CM椭圆曲线$ E/\ Mathbb {Q} $的$ d $ torsion点获得的坐标。如果在$ \ mathbb {q}（e [d]）$中，$ e $的质量$ p $对于某些$ d $，$ p <| \ text {gal}（\ mathbb {q}（e [e [e [d]）/\ mathbb {q}）/\ mathb {q}） = d^{4-o（1）} $（因此，Chebotarev密度定理中的预期主术语不考虑$ p $）。我们证明，对于几乎所有整数$ d $，都存在一个非CM椭圆曲线$ e/\ mathbb {q} $和prime $ p <| \ text {gal}（\ Mathbb {q}（e [d]）/\ mathbb {q}）|此外，我们证明，对于几乎所有可以适当分解的$ d $，都存在一个非CM椭圆曲线$ e/\ mathbb {q} $和$ p^{0.2694} <d $的prime $ p $，该<d $完全分为$ \ mathbb {q}（e [e [d]）$。 为了证明这一点，我们使用Kowalski的工作将问题与某些残基类中的素数分布联系起来。$ d^2 $。因此，障碍物$ p <d^4 $与古典Bombieri-Vinogradov定理的极限有关。为了摆脱这一问题，我们利用了$ d $方便地分配的假设，就像在Bombieri，Friedlander，Fouvry和Iwaniec的算术发展中的素数一样，以及Zhang，Polymath和Polymath和作者的最新作品中。与这些作品相反，我们不需要任何深度指数的总和（即Kloosterman和Weil/Deligne Bonde的总和）。取而代之的是，我们只需要用于乘法字符的经典大筛。我们使用Harman的筛子方法来获取用于素数的组合分解。
Analogously to primes in arithmetic progressions to large moduli, we can study primes that are totally split in extensions of $\mathbb{Q}$ of high degree. Motivated by a question of Kowalski we focus on the extensions $\mathbb{Q}(E[d])$ obtained by adjoining the coordinates of $d$-torsion points of a non-CM elliptic curve $E/\mathbb{Q}$. A prime $p$ is said to be an outside prime of $E$ if it is totally split in $\mathbb{Q}(E[d])$ for some $d$ with $p<|\text{Gal}(\mathbb{Q}(E[d])/\mathbb{Q})| = d^{4-o(1)}$ (so that $p$ is not accounted for by the expected main term in the Chebotarev Density Theorem). We show that for almost all integers $d$ there exists a non-CM elliptic curve $E/\mathbb{Q}$ and a prime $p<|\text{Gal}(\mathbb{Q}(E[d])/\mathbb{Q})|$ which is totally split in $\mathbb{Q}(E[d])$. Furthermore, we prove that for almost all $d$ that factorize suitably there exists a non-CM elliptic curve $E/\mathbb{Q}$ and a prime $p$ with $p^{0.2694} < d$ which is totally split in $\mathbb{Q}(E[d])$. To show this we use work of Kowalski to relate the question to the distribution of primes in certain residue classes modulo $d^2$. Hence, the barrier $p < d^4$ is related to the limit in the classical Bombieri-Vinogradov Theorem. To break past this we make use of the assumption that $d$ factorizes conveniently, similarly as in the works on primes in arithmetic progression to large moduli by Bombieri, Friedlander, Fouvry, and Iwaniec, and in the more recent works of Zhang, Polymath, and the author. In contrast to these works we do not require any of the deep exponential sum bounds (ie. sums of Kloosterman sums or Weil/Deligne bound). Instead, we only require the classical large sieve for multiplicative characters. We use Harman's sieve method to obtain a combinatorial decomposition for primes.
