Added. (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical variety has an isolated minimum. For example, Zagier has shown (Algebraic numbers close to both $0$ and $1$) that for the subvariety $Z : 1+x+y =0$ of the linear torus $\mathbb{G}_m^2$, the minimum, away from the few torsion points on $Z$, of the standard Weil height is $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big)$, with equality if and only if $x$ or $y$ is a primitive $10$th root of unity, and this minimum is isolated.

Another example, though of somewhat different flavor: the minimum height of a totally real algebraic number is, again, $\frac{1}{2} \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.2406059\ldots$, and this minimum is isolated. The lim inf of the height of a totally real algebraic number is at most $0.2732831\ldots$, and indeed it has been suggested that this is the lowest possible accumulation point for heights of totally real algebraic numbers. One realizes this accumulation point by taking, iteratively, $\xi_0 := 1$ and $\xi_n - \xi_n^{-1} := \xi_{n-1}$.

I wonder whether it is a general feature of both (a) Arakelov heights on arithmetical varieties (apart from the obvious examples); and (b) totally real or totally $p$-adic points on semiabelian varieties, to have an isolated minimum for the heights of their algebraic points.

Original post. By a polarized arithmetical variety I will mean a pair $(X,L)$ of a finite-type proper regular integral scheme $X$ flat and generically smooth over $\mathbb{Z}$, and a relatively ample invertible sheaf $L \in \mathrm{PIC}(X)$ equipped with an $F_{\infty}$-invariant hermitian metric $\| \cdot\|$ on the associated holomorphic line bundles $L_{\mathbb{C}}$, such that $\|\cdot\|$ is the uniform limit of positive $C^{\infty}$ metrics.

There is then an Arakelov height function $h_L$ on the algebraic points $X(\bar{\mathbb{Q}})$, given by the arithmetic intersection number of the associated multisection with $\hat{c}_1(L)$, divided by $[\mathbb{Q}(x):\mathbb{Q}]$. Let me call such a polarization $(X,L)$ strict harmonic if the set $$ \{ x \in X(\bar{\mathbb{Q}}) \quad | \quad h_L(x) = \inf_{X(\bar{\mathbb{Q}})} h_L \} $$ is Zariski-dense. Let me call $(X,L)$ harmonic (or non-strict harmonic) if $$ \inf_{X(\bar{\mathbb{Q}})} h_L = \liminf_{X(\bar{\mathbb{Q}})} h_L, $$ where the lim inf is under the Zariski topology. One may ask whether or not the two conditions are in fact equivalent.

One can show, as a consequence of the arithmetic Riemann-Roch theorem, that if $(X,L)$ is harmonic, the infimum equals the normalized arithmetic self-intersection (or arithmetic volume) $L^{\dim{X}} \Big/ L_{\mathbb{Q}}^{\dim{X_{\mathbb{Q}}}} \cdot \dim{X}$, and that moreover, the points of minimum height have their Galois orbits equidistributed in $c_1(L)$ (which, by definition, is a uniform limit of Chern forms of smooth metrics). Examples of strict harmonic arithmetical varieties include the canonical symmetric polarizations of abelian schemes over the full ring of integers of a number field (in which case the height $h_L$ is just the Neron-Tate canonical height, and the points of minimum height are precisely the torsion points); and, on the other hand, projective space with its standard Weil height, or more generally, with the canonical heights of Call-Silverman. A setup which generalizes both these examples is to have a self-map $f : X \to X$, an isomorphism $f^*L \cong L^{\otimes q}$ (over $\mathbb{Z}$, not just generically!) with some $q > 1$, and the height $\hat{h}_f(x) := \lim q^{-n}h(f^nx)$. This is an honest Arakelov (i.e., geometric) height precisely when both $f$ and the isomorphism $f^*L \cong L^{\otimes q}$ are defined over globally over $\mathbb{Z}$, rather than just generically over $\mathbb{Q}$.

Two questions:

  1. Is it true that semistable elliptic curves over $\mathbb{Q}$ (this case being the simplest), or more generally abelian varieties with non-integral moduli, are never harmonic in the above sense, with respect to any symmetric canonical polarization? with respect to any (ample, symmetric) polarization?

  2. Is it true that an arithmetical surface of genus $> 1$ is never harmonic in the above sense? In particular, is the canonical polarization $\omega = \omega_{X/\mathbb{Z}}$ ever harmonic?

Remark on question 1. For an elliptic curve $E/\mathbb{Q}$ with minimal discriminant $\Delta$ and the canonical polarization with $L := \mathcal{O}_E([O])$ -- note that this polarization involves the canonical compactification of the Neron model, as well as the canonical metric on $L$ from Arakelov theory -- everything reduces to the question of whether there exists a sequence of points with Arakelov height $h_L(x)$ converging to $-\log{|\Delta|}/24$.

Obviously, I am interested in this broader question: what are the arithmetical varieties for which admit a generic sequence of points of minimum height?


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