Author: Eiko

Time: 2025-02-28 17:07:20 - 2025-02-28 17:07:20 (UTC)

References:

• Bounding Tangencies of Sections On Elliptic Surfaces by Douglas Ulmer and Giancarlo Urzua

Introduction

Given elliptic surface $\mathcal{E}\to \mathcal{C}$ over a field $k$ of $\mathrm{char}k=0$, there is a zero section $O$, if we have another section $P$ of infinite order, in [Ulmer-Urzua] they gave an explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.

Here $k=\mathbb{C}$ is complex number, $\mathcal{C}$ an irreducible smooth projective curve of genus $g$ and $\pi :\mathcal{E}\to \mathcal{C}$ a Jacobian elliptic surface over $\mathcal{C}$.

The Simple Case of Constant Family

Consider for now that $\mathcal{E}\cong \mathcal{C}\times E\to \mathcal{C}$ is a constant family, then a section $P:\mathcal{C}\to \mathcal{E}=\mathcal{C}\times E$ is equivalent to a map $f_P:\mathcal{C}\to E$ since all the fibres are the same. $P$ and $f_P$ are assumed to be non-constant.

The torsion $\mathcal{E}[n]$ consists of $n^2$ constant sections $\mathcal{C}\times \{p\}\subset \mathcal{C}\times E[n]$, and we can consider the set for any constant section

$$T_c:=\bigcup _{p\in E}\{t\in \mathcal{C}:P\text{ tangent }\mathcal{C}\times \{p\}\text{ at }t\}$$

which is a subset of

$$T_{tors}=\bigcup _{p\in E_{tors}}\{t\in \mathcal{C}:P\text{ tangent }\mathcal{C}\times \{p\}\text{ at }t\}\subset \mathcal{C}.$$

We can see that $T_c$ is intuitively viewed as the set of points where $f_P$ is ‘entirely horizontal’. Therefore, if the multiplicity is denoted as $I(P,t)$, we naturally have $I(P,t)\ge 1$ and $t\in T_c$ if and only if $I(P,t)\ge 2$.

This means, if we pick a non-zero invariant differential on $E$, the pull back of this differential along $f_P$, ${\eta }_P=f_P^{\ast }(\omega )$, has the order of vanishing

$${\mathrm{ord}}_t({\eta }_P)=e_f(t)-1=I(P,t)-1.$$

This immediately gives

$$|T_{tors}|\le |T_c|=\sum _{t\in \mathcal{C}}{1}_{I(P,t)>1}\le \sum _{t\in \mathcal{C}}(I(P,t)-1)=\sum _{t\in \mathcal{C}}{\mathrm{ord}}_t({\eta }_P)=\mathrm{deg}{\eta }_P=2g-2.$$

In the general case, we can define a Betti foliation on a open subset $U$ of $\mathcal{E}$, generalizing the foliation of $\mathcal{C}\times E$ by its leaves $\mathcal{C}\times \{p\}$ that has the subsets $\mathcal{E}[n]$ among its closed leaves. Therefore we can consider the tangencies set $T_{Betti}\subset \mathcal{C}$ and obtain a similar estimate via a real-analytic $1$-form ${\eta }_P$ that generically satisfy $J({\eta }_P,t)=I(P,t)-1$ at all good reduction.

$$\sum (I(P,t)-1)=\sum _{t\in \mathcal{C}}J({\eta }_P,t)=2g-2-d$$