On the level of modular curves that give rise to isolated j-invariants

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We say a closed point x on a curve C is sporadic if C has only finitely many closed points of degree at most deg⁡(x) and that x is isolated if it is not in a family of effective degree d divisors parametrized by P1 or a positive rank abelian variety (see Section 4 for more precise definitions and a proof that sporadic points are isolated). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic and isolated points on the modular curves X1(N). In particular, we show that any non-cuspidal non-CM sporadic, respectively isolated, point x∈X1(N) maps down to a sporadic, respectively isolated, point on a modular curve X1(d), where d is bounded by a constant depending only on j(x). Conditionally, we show that d is bounded by a constant depending only on the degree of Q(j(x)), so in particular there are only finitely many j-invariants of bounded degree that give rise to sporadic or isolated points.