论文标题
非CM椭圆形曲线，无限多个几乎是Frobenius痕迹
Non-CM elliptic curves with infinitely many almost prime Frobenius traces
论文作者
论文摘要
令$ e $为$ \ mathbb {q} $而没有复杂乘法的椭圆曲线。对于$ e $的Prime $ p $，我们编写$ \＃e_p（\ Mathbb {f} _p）= p + 1 -a_p（e）$，用于$ \ mathbb {f} _p {f} _p $ - 合理点的减少$ e_p $ e_p $ e_p $ e_p $ e_p $ e $ modulo $ p $。根据普遍的Riemann假设（GRH），我们研究了整数$ | a_p（e）| $是素数。特别是，我们证明了以下结果：（i）$ | a_p（e）| $的primes $ p <x $的数量是prime的元素，从上面限制了$ c_1（e）\ frac {x} {（\ log x）^2} $，对于某些常数$ c_1（e）$; （ii）$ | a_p（e）| $的总数$ p <x $是最多4个不同的素数的产物，无数次数，从下面限制了$ c_2（e）\ frac {x} {（\ log log x）{（\ log log x）^2} $对于某些常数$ c_2（e）$; （iii）$ | a_p（e）| $的数量$ p <x $是最多5个不同的素数的产物，以多种多数计数，从下面限制了$ c_3（e）\ frac {x} {（\ log log x）{（\ log x）^2} $，对于某些正常常数$ c_3（e）> 0 $ 0 $ 0 $。在GRH下，我们还证明了Primes $ p $的互惠总和为$ | a_p（e）| $是素数。此外，在Grh的领导下，以及Artin的Holomorphy猜想和一个对Artin L功能的相关性猜想，我们证明，Primes $ p <x $的数量$ | a_p（e）| $是最多2个独特的素质的产物，与$ c_4（e）c_ frac（x）c_ frac（x）对于某些常数$ C_4（e）$。常数$ c_i（e）$，$ 1 \ leq i \ leq 4 $，按$ e $进行明确定义，是另一个明确的常数$ c（e）$的因素，在猜想中出现$ \ \＃\ {p <x：| a_p（e | a_p（e）| \ \ text {is prime} \} \ sim c（e）\ frac {x} {（\ log x）^2} $。
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. For a prime $p$ of good reduction for $E$, we write $\#E_p(\mathbb{F}_p) = p + 1 - a_p(E)$ for the number of $\mathbb{F}_p$-rational points of the reduction $E_p$ of $E$ modulo $p$. Under the Generalized Riemann Hypothesis (GRH), we study the primes $p$ for which the integer $|a_p(E)|$ is a prime. In particular, we prove the following results: (i) the number of primes $p < x$ for which $|a_p(E)|$ is a prime is bounded from above by $C_1(E) \frac{x}{(\log x)^2}$ for some constant $C_1(E)$; (ii) the number of primes $p < x$ for which $|a_p(E)|$ is the product of at most 4 distinct primes, counted without multiplicity, is bounded from below by $C_2(E) \frac{x}{(\log x)^2}$ for some constant $C_2(E)$; (iii) the number of primes $p < x$ for which $|a_p(E)|$ is the product of at most 5 distinct primes, counted with multiplicity, is bounded from below by $C_3(E) \frac{x}{(\log x)^2}$ for some positive constant $C_3(E) > 0$. Under GRH, we also prove the convergence of the sum of the reciprocals of the primes $p$ for which $|a_p(E)|$ is a prime. Furthermore, under GRH, together with Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we prove that the number of primes $p < x$ for which $|a_p(E)|$ is the product of at most 2 distinct primes, counted with multiplicity, is bounded from below by $C_4(E) \frac{x}{(\log x)^2}$ for some constant $C_4(E)$. The constants $C_i(E)$, $1 \leq i \leq 4$, are defined explicitly in terms of $E$ and are factors of another explicit constant $C(E)$ that appears in the conjecture that $\#\{p < x: |a_p(E)| \ \text{is prime}\} \sim C(E) \frac{x}{(\log x)^2}$.
