Torsion Points on Elliptic Curves With Complex Multiplication (With an Appendix by Alex Rice)

Abstract

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X 1 (N ). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X 1 (N ), we deduce that, for sufficiently large N , X 1 (N ) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

Computation on elliptic curves with complex multiplication

“…Computing this degree sequence takes 0.03 seconds † . If we recursively perform this sieve, it takes 0.24 seconds to find that the torsion exponents which occur for j = 0 over a number field of degree 2 are [2,3,4,6,7].…”

“…This may seem like a relatively short amount of time to be worried about, but for j = 0 and a number field of degree 6 it takes about a minute to find [2,3,4,6,7,9,14,19] as the list of torsion exponents. For degree 12 it takes over an hour.…”

Uniform boundedness in terms of ramification

“…Note that δ(ψ) = δ(ψ ) = 1 because p = 37 ≡ 1 mod 4. The fact that the elliptic curves with j = −7 · 11 3 and j = −7 · 137 3 · 2083 3 have a Q-rational isogeny of degree 37 was dis-cussed in Sect. 3.…”

“…It remains to consider the two j-invariants with a Q-rational isogeny of degree 37, namely j 0 = −7 · 11 3 and j 0 = −7 · 137 3 · 2083 3 . Let E/Q and E /Q be elliptic curves with j-invariants j(E) = j 0 and j(E ) = j 0 .…”

“…Since R has exact order p n , it follows that one of λ or μ is non-zero modulo p. Let us assume that λ ≡ 0 mod p. By Lemma 15 of [3], the quadratic imaginary field F is contained in…”