科研之友
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摘要
For the study of theMordell-Weil group of an elliptic curve E over a complex function field of a projective curve B, the first author introduced the use of differential-geometric methods arising from Kahler metrics on HxC invariant under the action of the semidirect product SL(2, R) proportional to R-2. To a properly chosen geometric model pi : E -> B of E as an elliptic surface and a non-torsion holomorphic section sigma : B -> E there is an associated "verticality" eta(sigma) of s related to the locally defined Betti map. The first-order linear differential equation satisfied by.s, expressed in terms of invariant metrics, is made use of to count the zeros of.s, in the case when the regular locus B-0 subset of B of pi : E -> B admits a classifying map f(0) into a modular curve for elliptic curves with level-k structure, k >= 3, explicitly and linearly in terms of the degree of the ramification divisor R (f0) of the classifying map, and the degree of the log-canonical line bundle of B-0 in B. Our method highlights deg( R-f0) in the estimates, and recovers the effective estimate obtained by a different method of Ulmer-Urzua on the multiplicities of the Betti map associated to a non-torsion section, noting that the finiteness of zeros of.s was due to Corvaja-Demeio-Masser-Zannier. The role of R (f0) 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.
