Weakly bounded height on modular curves
Let \(C\subset \mathbb{A}^2\) be an irreducible algebraic curve defined over \(\overline{\mathbb{Q}}\). The topic of this paper is the height of algebraic points in the intersection of \(C\) with modular curves of varying level in the spirit of the Zilber-Pink conjecture (although the results proved here are not implied by the latter) and work of \textit{E. Bombieri}, \textit{D. Masser} and \textit{U. Zannier} [Int. Math. Res. Not. 2007, No. 19, Article ID rnm057, 33 p. (2007; Zbl 1145.11049)] on unlikely intersections. Let \(S=\bigcup_{N\geq1} Y_0(N)\cup\bigcup_{j_0} (\{j_0\}\times\mathbb{A}^1)\cup(\mathbb{A}^1\times\{j_0\})\) be the union of all special subvarieties of \(\mathbb{A}^2\) of codimension at least 1, where \(j_0\) runs through the \(j\)-invariants of elliptic curves with complex multiplication. Let \(C\) be irreducible and not special. The author conjectures that the height of a point \(P\in C(\overline{\mathbb{Q}})\cap S\) is weakly bounded. More precisely, for a point \(P\in \mathbb{A}^2(\overline{\mathbb{Q}})\), let \(S_\mathbb{Q}(P)\) be the minimal variety over \(\mathbb{Q}\) containing the intersection of all special subvarieties \(V\subset \mathbb{A}^2\) such that \(P\in V\). The main conjecture of this article predicts that if \(C\) is irreducible and not special, then there is a constant \(c(C)\) such that every \(P\in C(\overline{\mathbb{Q}})\cap S\) satisfies \[ h(P)\leq c(C)(1+\deg S_\mathbb{Q}(P)), \] where the degree is taken via pullback with respect to the natural embedding \(\mathbb{A}^2\hookrightarrow\mathbb{P}^1\times\mathbb{P}^1\). Several results supporting this conjecture for curves such that the coordinate functions \(X\) and \(Y\) restricted to \(C\) satisfy \(\deg(X)\neq\deg(Y)\) are also proved: It is shown in Theorem~1.1 that if \(P\in C(\overline{\mathbb{Q}})\cap Y_0(N)\), then \(h(P)\leq c(C)\log(1+N)\), where \(c(C)\) is a constant. After complementing this in Theorem~1.2 by showing that for any irreducible curve \(C\) that is not special the height of points \(P\in C(\overline{\mathbb{Q}})\cap Y_0(N)\) is unbounded when ranging over all \(N\), the author deduces a finiteness statement for curves as in Theorem~1.1: For any \(D\in\mathbb{R}\), there are only finitely points \(P\in C(\overline{\mathbb{Q}})\cap\bigcup_{N\geq1} Y_0(N)\) of degree bounded by \(D\) (Corollary~1.1). Moreover, it is shown that for all such curves \(C\) the conjecture follows from the Generalized Riemann Hypothesis for all odd, real Dirichlet \(L\)-functions.
scientific article; zbMATH DE number 5723514