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 $\mathbf{E}$ 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 $\mathrm{SL}(2,\mathbb{R})⋉{\mathbb{R}}^2$. To a properly chosen geometric model $\pi :\mathcal{E}\to B$ of $\mathbf{E}$ as an elliptic surface and a non-torsion holomorphic section $\sigma :B\to \mathcal{E}$ 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 $\pi :\mathcal{E}\to B$ 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 $\mathrm{deg}(R_{f_0})$ 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.