Towards a classification of isolated \(j\)-invariants (Q6622397)

Let \(C\) be a curve, that is a smooth projective geometrically integral 1-dimensional scheme defined over a number field \(k\). The degree \(\deg(x)\) of a closed point \(x\) is defined to be the length of the \(\mathrm{Gal}_k\)-orbit of points in \(C(k)\) corresponding to \(x\). To any closed point \(x\in C\) of degree \(d\) is associated the \(k\)-rational effective divisor \(P_1 + \cdots + P_d\), where \(P_1,\ldots, P_d\) are the points in the \(\mathrm{Gal}_k\)-orbit associated to \(x\). Thus, we have the natural map from the \(d\)th symmetric power of \(C\) to the curve's Jacobian \(\Phi_d : C^{(d)}\rightarrow Jac(C)\) which maps an effective divisor \(D\) of degree \(d\) to the class \([D - dP_0]\).\N\NThe point \(x\) is said to be \(\mathbb{P^1-parametrized}\), if there exists a point \(x^{\prime} \in C^(d)(k)\) with \(x^{\prime} \neq x\) such that \(\Phi_d(x) = \Phi_d(x^{\prime})\). Otherwise, we say \(x\) is \(\mathbb{P^1-isolated}\). The point \(x\) is \(\mathbf{AV-parametrized}\), if there exists a positive rank abelian subvariety \(A/k\) with \(A \subset Jac(C)\) such that \(\Phi_d(x)+A \subset \mathrm{im}(\Phi_d)\). Otherwise, we say \(x\) is \(\mathbf{AV-isolated}\). The point \(x\) is \textbf{isolated} if it is both \(\mathbb{P^1}\)-isolated and AV-isolated. Finally, the point \(x\) is sporadic if there are only finitely many closed points of \(C\) of degree at most deg(x).\N\NIn this paper, the isolated points of the modular curve \(X_1(N)\) are studied. If \(x \in X_1(N)\) is an isolated (resp. sporadic) point, we say \(j(x) \in X_1(1) \cong \mathbb{P^1}\) is an isolated (resp. sporadic) \(j\)-invariant. An algorithm is developed to determine whether a given non-CM \(j\)-invariant in \(\mathbb{Q}\) is isolated. Let \(x = [E, P ]\in X_1(N)\) be a non-CM isolated point with \(j(E)\in \mathbb{Q}\). Fix an equation for \(E/\mathbb{Q}\) and let \(N_E\) denote its conductor. If either \(N_E \leq 500 000\), either \(N_E\) is only divisible by primes \(p \leq 7\), or \(N_E = p \leq 300 000 000\) for some prime number \(p\), Then \(j(E) \in \{-140625/8, -9317, 351/4, -162677523113838677\}\). Moreover, each one of these \(j\)-invariants corresponds to a \(\mathbb{P^1}\)-isolated point on \(X_1(21)\), \(X_1(37)\), \(X_1(28)\), or \(X_1(37)\), respectively.\N\NWe say a point \(x\) of the modular curve \(X_0(N)\) defined over \(\mathbb{Q}\) \textbf{exceptional} if \(X_0(N)(\mathbb{Q})\) is finite and \(x\) corresponds to a non-CM elliptic curve over \(\mathbb{Q}\). It is proved that if \(E\) is an elliptic curve corresponding to an exceptional rational point on \(X_0(N)\) for some positive integer \(N\) and \(j(E)\) is sporadic, then \(j(E) =-140625/8\) or \(-9317\).