Dear Ariyan, the elliptic curve with equation $$y^2=x^3+6$$ has Faltings height
$$-(3/2)\log(\Gamma(1/3)/\Gamma(2/3))+(1/4)\log(3)=-0.748752...;$$ the curve
of genus $2$ with equation $$y^2+y=x^5$$ has Faltings height
$$
h_{\rm Fal}(C_{\bar{\bf Q}})=2\log(2\pi)-
{1\over 2}\log\big(\Gamma(1/5)^5\Gamma(2/5)^3\Gamma(3/5)\Gamma(4/5)^{-1}\big)
$$
$$
\approx
-1.452509239645644650317707042;
$$
For the first example, see Deligne, "Preuve des conjectures de Tate et Shafarevich", Séminaire Bourbaki. For the second one, see Bost, Mestre, Moret-Bailly, "Sur le calcul explicite des 'classes de Chern' des surfaces arithmétiques de genre $2$", Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). Astérisque No. 183 (1990), 69–105.
Another explicit formula that should allow you to produce elliptic curves of arbitrarily large
Faltings height is the inequality
$$
|h(j_E)-12h_{\rm Fal}(E)|\leqslant 6\log(1+h(j_E))+47.15
$$
See paragraph 5. of the article "Serre's uniformity..." by Bilu and Parent for references.
Something else you can do is make numerical experiments with formula in Conj. 3 of the article of Colmez, "Hauteur de Faltings..." (Compositio), which is true (without $\log(2)$ factor !, see A. Obus, arXiv:1107.0684) if the CM field is abelian over $\bf Q$. In that case, the Artin $L$-functions become Dirichlet $L$-functions and can be computed explicitly in terms of values of the Gamma function using the Hurwitz formula.
This is not a complete answer but I hope that it helps.
(This answer has been edited -- it used to say that a finite cover of $\overline M_{g,n}$ gives a counterexample, which no longer seems obvious.)
If you had written "Deligne-Mumford stack" instead of "scheme", then a counterexample would be given by the spaces $\overline M_{g,n}$, which are certainly smooth and proper but far from rational in the large $g$ limit (or, for $g > 0$, in the large $n$ limit). The original references here are, I guess, Deligne (for $\overline M_{1,11}$) and Harris-Mumford (for $\overline M_{25}$).
Kevin Buzzard's hint with the Ramanujan $\Delta$ function is relevant here; indeed, $H^{11,0}(\overline M_{1,11})$ is nonzero, and the $\ell$-adic Galois representation corresponding to $H^{11,0}\oplus H^{0,11}$ is the representation attached to $\Delta$.
My answer to the question Is the moduli space of curves defined over the field with one element? contains some more detailed information about these things.
The natural way to find a scheme instead of a stack would then be to consider finite or generically finite covers of $\overline M_{g,n}$ by smooth proper schemes. There are several closely related constructions of such covers in the literature by Looijenga, Boggi-Pikaart, Pikaart-de Jong, Abramovich-Corti-Vistoli, all using some kind of non-abelian level structure on curves, but as far as I can tell none of them work over the integers.
Best Answer
Hey Bjorn. Let me try for a counterexample. Consider a hypersurface in projective $N$-space, defined by one degree 2 equation with integral coefficients. When is such a gadget smooth? Well the partial derivatives are all linear and we have $N+1$ of them, so we want some $(N+1)$ times $(N+1)$ matrix to have non-zero determinant mod $p$ for all $p$, so we want the determinant to be +-1. The determinant we're taking is that of a symmetric matrix with even entries down the diagonal (because the derivative of $X^2$ is $2X$) and conversely every symmetric integer matrix with even entries down the diagonal comes from a projective quadric hypersurface. So aren't we now looking for a positive-definite (to stop there being any Q-points or R-points) even unimodular lattice?
So in conclusion I think that the hypersurface cut out by the quadratic form associated in this way to e.g. the $E_8$ lattice or the Leech lattice gives a counterexample!