论文标题
从差分几何的角度来
Multiplicities of the Betti map associated to a section of an elliptic surface from a differential-geometric perspective
论文作者
论文摘要
For the study of the Mordell-Weil group of an elliptic curve ${\bf E}$ 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 ${\rm sl}（2，\ mathbb r）\ ltimes \ mathbb r^2 $。对于正确选择的几何模型$π：\ Mathcal e \ to $ {\ bf e} $作为椭圆表面和非反应全态$σ：b \ to \ Mathcal e $，有一个相关的``forticality''$η__σ$ of $ ferticality'$η_的$η_σ$与Local capties capt capt bet capt cectit capt bet。 $η_σ$满足的一阶线性微分方程是根据不变度量表示的，用于计数$η_σ$的零，如果常规位点$ b^0 \ b^0 \ b^0 \ subset b $ of $π： \ ge 3 $，就分类映射的分类分数$ r_ {f_0} $以及$ b^0 $ in $ b $的log-canonical Line Bundle的程度明确线性地线性。 Our method highlights ${\rm deg}(R_{f_0})$ 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 $η_σ$ was due to Corvaja-Demeio-Masser-Zannier. $ r_ {f_0} $的作用在主题上是很自然的，因为在椭圆形的模块化表面的情况下，shioda定理没有非转移部分，第一作者为此提供了差异几何形状。我们的方法阐明了对某些阿贝尔方案的非扭转部分的研究。
For the study of the Mordell-Weil group of an elliptic curve ${\bf E}$ 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 ${\rm SL}(2,\mathbb R) \ltimes \mathbb R^2$. To a properly chosen geometric model $π: \mathcal E \to B$ of ${\bf E}$ as an elliptic surface and a non-torsion holomorphic section $σ: B \to \mathcal E$ there is an associated ``verticality'' $η_σ$ of $σ$ related to the locally defined Betti map. The first-order linear differential equation satisfied by $η_σ$, expressed in terms of invariant metrics, is made use of to count the zeros of $η_σ$, in the case when the regular locus $B^0\subset B$ of $π: \mathcal E \to B$ 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 ${\rm deg}(R_{f_0})$ 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 $η_σ$ 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.
