Multiplicities of the Betti map associated to a section of an elliptic surface from a differential-geometric perspective

Abstract

<p>For the study of the Mordell–Weil group of an elliptic curve E over a complex function field of a projective curve <em>B</em>, the first author introduced the use of differential-geometric methods arising from Kähler metrics on �×� invariant under the action of the semi-direct product SL(2,�)⋉�2. To a properly chosen geometric model �:�→� of E as an elliptic surface and a non-torsion holomorphic section σ:�→� there is an associated “verticality” �σ of σ related to the locally defined Betti map. The first-order linear differential equation satisfied by �σ, expressed in terms of invariant metrics, is made use of to count the zeros of �σ, in the case when the regular locus �0⊂� of �:�→� admits a classifying map �0 into a modular curve for elliptic curves with level-<em>k</em> structure, �≥3, explicitly and linearly in terms of the degree of the ramification divisor ��0 of the classifying map, and the degree of the log-canonical line bundle of �0 in <em>B</em>. Our method highlights deg(��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 �σ was due to Corvaja–Demeio–Masser–Zannier. The role of ��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.<br></p>