论文标题
椭圆形曲线具有GALOIS稳定的循环亚组4
Elliptic curves with Galois-stable cyclic subgroups of order 4
论文作者
论文摘要
超过$ {\ bf q} $的无限多个椭圆曲线具有4个命令的稳定循环亚组。此类子组成对成对，与$ n_i（x）的子组相交。让$ n_i（x）$表示椭圆形曲线的数量$ {\ bf q} $ $ i $ i $ i $ i $ i $ i $ i，和高度最多$ x $。在本文中，我们表明$ n_1（x）= c_ {1,1} x^{1/3}+c_ {1,2} x^{1/6}+o（x^{0.105}）$。我们还显示为$ x \ to \ infty $，$ n_2（x）= c_ {2,1} x^{1/6}+o（x^{1/12}）$，错误项的确切性质与riemann Zeta Zeta-Zeta-Zeta-Zeta-Zeta-Zeta-Zeta-Zeta-prientation。在这里，$ c_ {1,1} = 0.95740 \ ldots $，$ c_ {1,2} = - 0.87125 \ ldots $，和$ c_ {2,1} = 0.035515 \ ldots $是可计算的常数。最后，我们证明$ n_i（x）= 0 $ for $ i> 2 $（鉴于椭圆曲线具有6个循环子组4的订单4）。
Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the number of elliptic curves over ${\bf Q}$ with at least $i$ pairs of Galois-stable cyclic subgroups of order 4, and height at most $X$. In this article we show that $N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$. We also show, as $X\to \infty$, that $N_2(X)=c_{2,1}X^{1/6}+o(X^{1/12})$, the precise nature of the error term being related to the prime number theorem and the zeros of the Riemann zeta-function in the critical strip. Here, $c_{1,1}= 0.95740\ldots$, $c_{1,2}=- 0.87125\ldots$, and $c_{2,1}= 0.035515\ldots$ are calculable constants. Lastly, we show that $N_i(X)=0$ for $i > 2$ (the result being trivial for $i>3$ given that an elliptic curve has 6 cyclic subgroups of order 4).
