# Weil height of an Abelian Variety with everywhere (potentially) good reduction

Background: Suppose that $E$ is an elliptic curve over $\mathbb{Q}$ with everywhere (potentially) good reduction. there are many ways to define the height of $E$, and I will be concerned with the height where one chooses an ample bundle on $X(1)$ and looks at the height of the point representing $E$. This is then well defined up to an $O(1)$ additive factor, and I am only concerned with asymptotics so this is fine.

Now, the $j$ function gives a map $j:X(1)\cong \mathbb{P}^1$, and so we have the formula

$$h(E) = \sum_{p\not\mid \infty} Max(|j(E)|_p,0) + Max(1,\log|j(E)|)$$

and since $E$ has potentially good reduction, $j(E)$ is an integer, so that (as long as its non-zero), $$h(E) = \log |j(E)|.$$

This is really nice, because it means we can compute the height from purely the infinite place.

Question: Is there an analogue for heights of Abelian varieties with everywhere (potentially) good reduction?

It can't quite hold as stated since $\mathcal{A}_g$ is not affine for any $g\geq 1$, but I am still wondering if one can use good reduction somehow...

Thanks!

• If $j(E)=0$, your formula $h(E)=log j(E)$ is problematic! Indeed, ditto if $j(E)<0$. So you really do need to write it as $h(E)=\max\bigl\{1,\log|j(E)|\bigr\}$. And also, although you must realize this, your height ignores twisting, so $y^2=x^3+1$ and $y^2=x^3+1234567890$ have the same height, although arithmetically they may be quite different. – Joe Silverman Dec 16 '14 at 20:52
• Thanks Joe, corrected. Indeed, I only care bout the `stable height', so quadratic twists don't come into it. Likewise, I only care about large height, which is why I neglected $j(E)=0$. – jacob Dec 18 '14 at 13:55
• Consider a Shimura cure parameterizing abelian surfaces with quaternion multiplication. These will have good reduction everywhere but some finite set of bad primes, and I think potentially good reduction on those bad primes - since otherwise we would get a quaternion algebra action on a torus of dimension $\leq 2$. So you have some random curve and you want a formula for the Weil height of the curve using only, or primarily, Archimedean information. This will be hard because the Archimedean place is compact so it can only explain a finite amount of the height, or at least it seems so. – Will Sawin Mar 31 '16 at 13:20

Here are two (almost three) comments that you might find useful. I don't know the answer to your question in general unfortunately.

First, Autissier proved that for abelian varieties over $\overline{\mathbb Q}$ with good reduction, the Faltings height of $A$ equals the theta height of $A$ plus some contribution at infinity. See the main theorem on p. 1 of his paper Hauteur de Faltings et hauteur de Néron-Tate du diviseur thêta. So the difference of these two heights can be computed by a contribution at infinity (as you desire).

Secondly, if you consider only Jacobians then you can say a little bit more. In fact, if $A = Jac(X)$ is the Jacobian of a curve $X$ over $\overline{\mathbb Q}$, then the arithmetic Noether formula (due to Faltings and Moret-Bailly, see for instance Section 1.6 in https://www.math.leidenuniv.nl/scripties/RdeJong.pdf) states that $$12 h_{Fal}(X) = \omega^2(X) + \Delta(X) + \delta_{Fal}(X) - 4g\log(2\pi).$$ If you assume the Jacobian to have good reduction over $\overline{\mathbb Q}$, then this doesn't imply $X$ has good reduction over $\overline{\mathbb Q}$. Nevertheless, it might be instructive to look at Jacobians of curves with good reduction over $\overline{\mathbb Q}$. So let's assume that $X$ has good reduction over $\overline{\mathbb Q}$ as well. Then, by definition of the discriminant $\Delta(X)$, the Noether formula reduces to

$$12 h_{Fal}(X) = \omega^2(X) + \delta_{Fal}(X) - 4g\log(2\pi).$$

You now see that the Faltings height of $A$ is given by $\omega^2(X) +\delta_{Fal}(X) - 4g\log(2\pi)$, as $h_{Fal}(A) = h_{Fal}(X)$. The self-intersection of the dualizing sheaf is never zero (Bogomolov conjecture), so that the Faltings height will always involve $\omega^2$ (in some sense). You can go a bit further now and rewrite the above formula in terms of the height of the Weierstrass divisor and purely analytic contributions. To do so, use Prop. 2.5.2 in loc. cit. The end product is an expression for the Faltings height of $A$ in terms of the height of the Weierstrass divisor and the analytic invariants $R(X)$ and $\delta_{Fal}(X)$. Now your question reduces to:

Can the height of the Weierstrass divisor with respect to the relative dualizing sheaf endowed with the Arakelov metric on the minimal regular semi-stable model of $X$ be computed in analytic terms solely?

Thirdly, in the case of CM abelian varieties of abelian type (a case you most likely are interested in) there are also formulas relating the Faltings height to periods due to Colmez. I can also comment on this if you'd like.

Note: I used some notation and terminology that I find convenient. Don't hesitate to let me know if elaboration needed.