Multiplicities of the Betti Map Associated to a Section of an Elliptic Surface from a Differential-Geometric Perspective

Abstract

For the study of the Mordell–Weil group of an elliptic curve $\textbf{E}$ over a complex function field of a projective curve B, the first author introduced the use of differential-geometric methods arising from Kähler metrics on ${\mathcal {H}} \times {\mathbb {C}}$ invariant under the action of the semi-direct product $\textrm{SL}(2,{\mathbb {R}}) \ltimes {\mathbb {R}}^2$. To a properly chosen geometric model $\pi : {\mathcal {E}} \rightarrow B$ of $\textbf{E}$ as an elliptic surface and a non-torsion holomorphic section $\sigma : B \rightarrow {\mathcal {E}}$ there is an associated “verticality” $\eta _\sigma$ of $\sigma$ related to the locally defined Betti map. The first-order linear differential equation satisfied by $\eta _\sigma$, expressed in terms of invariant metrics, is made use of to count the zeros of $\eta _\sigma$, in the case when the regular locus $B^0\subset B$ of $\pi : {\mathcal {E}} \rightarrow B$ admits a classifying map $f_0$ into a modular curve for elliptic curves with level-k structure, $k \ge 3$, explicitly and linearly in terms of the degree of the ramification divisor $R_{f_0}$ of the classifying map, and the degree of the log-canonical line bundle of $B^0$ in B. Our method highlights $\textrm{deg}(R_{f_0})$ in the estimates, and recovers the effective estimate obtained by a different method of Ulmer–Urzúa on the multiplicities of the Betti map associated to a non-torsion section, noting that the finiteness of zeros of $\eta _\sigma$ was due to Corvaja–Demeio–Masser–Zannier. The role of $R_{f_0}$ is natural in the subject given that in the case of an elliptic modular surface there is no non-torsion section by a theorem of Shioda, for which a differential-geometric proof had been given by the first author. Our approach sheds light on the study of non-torsion sections of certain abelian schemes.