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

Abstract

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). Motivated by well-known classification problems concerning rational torsion of elliptic curves, we study sporadic points on the modular curves X 1 (N ). In particular, we show that any non-cuspidal non-CM sporadic point x ∈ X 1 (N ) maps down to a sporadic point on a modular curve X 1 (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 points. 1In this paper, we study sporadic points of arbitrary degree, focusing particularly on sporadic points corresponding to non-CM elliptic curves. We prove that non-CM non-cuspidal sporadic points on X 1 (n) map to sporadic points on X 1 (d), for d some bounded divisor of n.Theorem 1.1. Fix a non-CM elliptic curve E over k := Q(j(E)), and let m be an integer divisible by 2, 3 and all primes ℓ where the ℓ-adic Galois representation is not surjective. Let M = M(E, m) be the level of the m-adic Galois representation of E. If x ∈ X 1 (n) is a sporadic point with j(x) = j(E), then π(x) ∈ X 1 (gcd(n, M)) is a sporadic point, where π denotes the natural map X 1 (n) → X 1 (gcd(n, M)).For many elliptic curves, we may take both m and M to be quite small. For instance, let E be the set of elliptic curves over Q where the ℓ-adic Galois representation is surjective for all ℓ > 3 and where the 6-adic Galois representation has level dividing 24. Note that E contains all Serre curves [Ser72, Proof of Prop. 22] (that is, elliptic curves over Q whose adelic Galois representation is as large as possible) and hence contains almost all elliptic curves over Q [Jon10]. For E ∈ E, we may apply Theorem 1.1 with m = 6 and M|24.The curve X 1 (24) has infinitely many quartic points, but no rational or quadratic points, nor cubic points corresponding to elliptic curves over Q [Maz77, KM88, Mor]. Therefore X 1 (24) has no sporadic points with Q-rational j-invariant. For M a proper divisor of 24, the curves X 1 (M) have genus 0, and so also have no sporadic points. Hence Theorem 1.1 yields the following corollary.Corollary 1.2. For all n, there are no sporadic points on X 1 (n) corresponding to elliptic curves in E. In particular, there are no sporadic points corresponding to Serre curves.

The least degree of a CM point on a modular curve

“…The following result is a direct generalization of [9, Lemma 6.2]. Theorem Let $$H_0$$ be a subgroup of $$\operatorname{GL}_2(\mathbb {Z}/N\mathbb {Z})/\lbrace \pm 1\rbrace$$.…”

“…By [23, Theorem 3.1] we have $$J_1(N)(\mathbb {Q})$$ is finite. From this it follows — see, for example, [9, Theorem 4.2] — that $$\delta (X_1(N)) = \gamma _{\mathbb {Q}}(X_1(N))$$. Comparing the work of [24] to our own calculations, we find that $$d_{\operatorname{CM}}(X_1(N)) < \gamma _{\mathbb {Q}}(X_1(N))$$.$$\Box$$…”

On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one

“…To emphasize that a resolution of Fermat equation with prime exponent over the fields K considered above is a task worth pursuing, we will show that such a resolution is possible if we assume a folklore conjecture (see [2]) motivated by a question of Serre.…”

“…Let K = Q( √ −d), where d ∈ {1, 2, 7}. Assume Conjecture 2.2 holds for K. If p ≥ 5 is a rational prime number, then the equation (2) a p + b p + c p = 0 has no solutions a, b, c ∈ K \ {0}.…”

Ramification in Division Fields and Sporadic Points on Modular Curves

“…Thus, in order to answer Question 1.1 more generally, it seems likely we will need to have a better understanding of sporadic points on modular curves. To this end, Bourdon, Ejder, Liu, Odumodu, and Viray [2] have recently shown that, assuming Serre's uniformity conjecture, the number of sporadic j-invariants (j-invariants corresponding to sporadic points on some modular curve X 1 (N )) in a given number field is finite. Regarding Question 1.2, Merel [31] answered it in the affirmative.…”