Elliptic curves with Galois-stable cyclic subgroups of order 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).

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