Approximating algebraic numbers by j-invariants of elliptic curves

Abstract

is very close to 1728. Viewed in another light, then, this result says that j(EA,B) cannot be “too close” to 1728 (relative to the sizes of A and B) if EA,B(Q) contains a torsion point of order at least 5. In [5], the author proved similar results bounding |B| by a power of |A|, again for EA,B with certain torsion/isogeny structure. One may similarly view these results as saying that j(EA,B) cannot be “too close” to 0 for curves EA,B with given torsion structure. This prompts us to formulate here a general result on the approximation of a fixed algebraic number by the j-invariants of elliptic curves with certain torsion structure. The main results are stated over arbitrary number fields and contain, as special cases, most of the results in [2, 5]. It is easy to show that for each N , {j(E) : E/Q admits a rational N -isogeny}