Elliptic curves with Galois-stable cyclic subgroups of order 4

2020/04/30 by Pomerance, Carl, Schaefer, Edward F. · 2 citations
#11G05 (Primary) 14H52 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2004.14947

Abstract

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 N₁(X) = c_1,1X^1/3+c_1,2X^1/6+O(X^0.105). We also show, as X→ ∞, that N₂(X)=c_2,1X^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 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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