Arithmetic discriminants and morphisms of curves (Q2701687)

Let \(C\) be a curve over a number field~\(k\), with regular model~\(X\) over the ring of integers~\(R\) of~\(k\). For an algebraic point \(P \in C\), the arithmetic discriminant is defined by a formula analogous to the adjunction formula for the genus of a curve on a surface, NEWLINE\[NEWLINEd_a(P) = (H_P . (\omega_{X/\text{Spec} R} + H_P))/[k(P) : k]NEWLINE\]NEWLINE where \(H_P\) is the arithmetic curve on~\(X\) corresponding to~\(P\). NEWLINENEWLINENEWLINELet \(h\) be a Weil height on~\(C\) associated to a degree~\(1\) divisor; then \textit{P.~Vojta} [J. Am.\ Math.\ Soc.\ 5, 763-804 (1992; Zbl 0778.11037)] proves that \(h_K(P) \leq d_a(P) + \epsilon h(P) + O_\epsilon(1)\) for all \(\epsilon > 0\). Here \(h_K\) is a Weil height associated to the canonical divisor~\(K\) on~\(C\). Vojta conjectures that one may replace \(d_a(P)\) by \(d(P) = \log |N_{k/{\mathbb Q}} D_{k(P)/k}|/[k(P) : {\mathbb Q}]\), the normalised field discriminant of~\(k(P)\). NEWLINENEWLINENEWLINEVojta's inequality provides a lower bound for~\(d_a(P)\) in terms of \(h_K(P)\) and~\(h(P)\). In the paper under review, the authors provide an upper bound, namely NEWLINE\[NEWLINEd_a(P) \leq h_K(P) + 2[k(P) : k] h(P) + O(1).NEWLINE\]NEWLINE This is then used to generalise a result of Vojta's on the finiteness of the set of points on~\(C\) of given degree mapping to points with the same field of definition under a dominant morphism \(f : C \to C'\). Vojta's original result is the special case \(C' = {\mathbb P}^1\). NEWLINENEWLINENEWLINEThe authors proceed to sharpen their upper bound for points in a dense open subset of~\(C\) to get \(d_a(P) \leq h_K(P) + (2[k(P) : k] - 2 + \epsilon) h(P) + O_\epsilon(1)\) if \([k(P) : k] \leq g(C)\). This leads to corresponding variants of the results derived from the upper bound. NEWLINENEWLINENEWLINEIn a concluding section, the authors show that on bi-elliptic genus~\(2\) curves, there are families of quadratic points satisfying \(d_a(P) = h_K(P) + 2 h(P) + O(1)\) (besides other families, where \(2h(P)\) is replaced by~\(0\) or by~\(4h(P)\)). NEWLINENEWLINENEWLINEThe reader should be warned that there are a few misprints in this paper, namely on page~1925, where the `\(\geq\)' sign in Corollary~2.1 should be a `\(>\)' sign, and on page~1926, where the formulas involving the ramification divisor~\(R_f\) are erroneous.