3
$\begingroup$

Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, having good reduction outside of a finite set $S$ of primes in $K$. A singly branched cover $C'$ of $C$ is a curve such that there exists a map $\pi : C' \rightarrow C$ and a point $x \in C(K)$ such that $\pi$ is only branched at $x$.

Is there a way to relate the Faltings height of $C$ and $C'$? In particular, is the height of $C'$ necessarily dependent on $x$?

Improve this question
$\endgroup$
2
  • $\begingroup$ When you say "necessarily dependent on $x$", are you thinking of something like $h(C') \gg h(x)$, where the implied constant depends on $C$? $\endgroup$
    – Joe Silverman
    Commented Dec 19, 2018 at 12:51
  • $\begingroup$ @JoeSilverman yes, except that the function on the right hand side need not be linear in $h(x)$ $\endgroup$
    – Stanley Yao Xiao
    Commented Dec 19, 2018 at 12:58

1 Answer 1

Reset to default
2
$\begingroup$

If $f:C'\to C$ is a singly branched cover, the (Faltings) heights of $C$ and $C'$ are certainly related.

First, by Lemma 6.1 in [1], the inequality $$h(C) \leq h(C') + \log(2\pi) g(C') \log(\deg f)$$ holds. Since $g(C') \leq 3 g(C) \deg f$, this implies that $$ h(C) \leq h(C') + 100 g(C) \log(\deg f) \deg f.$$

Presumably, you are more interested in an inequality the other way around. Using the main result of [2], one can obtain an explicit upper bound for $h(C')$ in terms of the Belyi degree of $C$, the degree of $f$, and the height of $x$. I can explain this in more detail if you'd like.

[1] Javanpeykar, A. An effective Arakelov-theoretic version of the hyperbolic isogeny theorem. Math. Proc. Cambridge Phil. Soc., (2016) Vol. 160, Issue 03, 463-476.

[2] Javanpeykar, A. Polynomial bounds for Arakelov invariants of Belyi curves, with an appendix by Peter Bruin. Algebra and Number Theory, Vol. 8 (2014), No. 1, 89-140.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thank you for the answer! I am familiar with both of your papers, and indeed I am looking for a bound in the other direction of your Lemma 6.1. Is the dependence on the point $x$ intrinsic? For example it seems to me that one can take any isomorphism of the ambient projective space sending $C$ to another isomorphic curve $X$, and which sends $x$ to a point of arbitrarily large height $\endgroup$
    – Stanley Yao Xiao
    Commented Dec 23, 2018 at 20:12

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .