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Let $A$ be a semiabelian variety over $\bar{\mathbb{Q}}$ and $B$ a semiabelian subvariety of $A$. Let $\pi:A\to A/B$ be the canonical morphism. Let $h:A(\bar{\mathbb{Q}})\to\mathbb{R}$ and $h':(A/B)(\bar{\mathbb{Q}})\to\mathbb{R}$ be canonical heights. (see POONEN B., "Mordell-Lang plus Bogomolov").

Question : $\exists{\alpha,\beta\geq 0},~~\forall a'\in(A/B)(\bar{\mathbb{Q}}),~~\exists a\in A(\bar{\mathbb{Q}})\text{ such that }\left \{ \begin{array}{rcl} \pi(a)&=&a' \\ h(a)&\leq& \alpha h'(a') +\beta \end{array} \right.$

Remark : If $A$ is split, it is true with $\beta=0$. (Because $A/B$ is isogenous to a semiabelian subvariety of $A$).

Thanks.

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I think the answer is yes. More generally, suppose $\pi:X\rightarrow Y$ is a surjective map of irreducible schemes over $\mathbb{Q}$ with compactifications $\overline{X},\overline{Y}$ respectively. Pick very ample line bundles $\mathcal{L}_X, \mathcal{L}_Y$ and use them to define heights $h_X,h_Y$. Then there exist $\alpha,\beta>0$ such that for any $y\in Y(\bar{\mathbb{Q}})$ there exists $x$ over it with $h_X(x)\leq \alpha h_Y(y) + \beta$. I believe these heights are comparable to the canonical heights on a semi-abelian variety, so this should answer your question.

After a finite etale base change the generic point of $Y$ will have a section back to $X$, so by possibly replacing $Y$ by an open subset of it, and making this base change we can assume that $\pi$ has a section $\psi:Y\rightarrow X$. Note that etale base changes won't affect the height by more than a constant. Moreover, replacing $Y$ by an open subset is justified because we can then repeat the argument for the complement of this open set and this process will eventually end by Noetherian induction. We now shrink a bit further to assume that $Y$ is an affine scheme $Y=\rm{spec}(R)$. I claim that one can take $x=\psi(y)$.

Since $\mathcal{L}_Y$ is very ample, we can take a basis of sections $(y_1,\dots,y_n)$ such that $(y_i/y_j)_{i,j}$ generate the fraction field of $R$. Letting $(x_1,\dots,x_m)$ be the corresponding sections of $\mathcal{L}_X$ it follows that there are rational functions $Q_{i,j}(y_1/y_2,\dots,y_1/y_n)$ which are equal to the restriction of $x_i/x_j$ to $\psi(Y)$. The claim follows.

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  • $\begingroup$ To define heights on a noncompact variety you need more than a very ample line bundle, I think. For instance on $\mathbb G_m$ all line bundles are trivial, but I don't think we mean to take the zero height. $\endgroup$
    – Will Sawin
    Commented Dec 9, 2013 at 1:18
  • $\begingroup$ Hey Will, you're right, I meant for $\mathcal{L}_X$ to be a line bundle on $\overline{X}$, which is why I took a compactification in the first place. It turns out that no matter how you compactify, you get comparable height functions (i.e. asymptotically within constants of each other). You can prove this similarly to the above argument. $\endgroup$
    – jacob
    Commented Dec 10, 2013 at 4:19

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