Elliptic curves with Galois-stable cyclic subgroups of order 4.

Infinitely many elliptic curves over 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 j (X) denote the number of elliptic curves over Q with at least j 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 → ∞ , 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 ... , c 1 , 2 = - 0.87125 ... , and c 2 , 1 = 0.035515 ... are calculable constants. Lastly, we show no elliptic curve over Q has more than 2 pairs of Galois-stable cyclic subgroups of order 4. [ABSTRACT FROM AUTHOR]