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Let $Y$ be a smooth projective connected curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $h_{\textrm{Fal}}(Y)$ be the Faltings height of $Y$.

Question 1. Can one classify or describe the curves $Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$?

Question 2. For any $g>0$, does there exist a curve $Y$ of genus $g$ such that $h_{\textrm{Fal}}(Y) <1$?

Essentially, I would like to know which curves one is excluding by looking at curves $Y$ such that $h_{\textrm{Fal}}(Y) \geq 1$.

A result of Bost says that the stable Faltings height of an abelian variety $A$ over $\overline{\mathbf{Q}}$ of dimension $g$ is bounded from below by $-\frac{1}{2}\log(2\pi)g$.

By the Northcott property of the Faltings height, the set of curves of genus $g$ with $h_{\textrm{Fal}}(Y) <1$ is finite. This means that I'm looking at the finite set of curves of genus $g$ with Faltings height not in the interval $$[-\frac{1}{2}\log(2\pi)g,1)\subset[-2/5g, 1).$$

Added: To answer Junkie's question, I'm aware of only one definition of the Faltings height of a curve over $\overline{\mathbf{Q}}$. There are several equivalent definitions, though.

Let $X$ be a smooth projective curve of genus $g>0$ over $\overline{\mathbf{Q}}$. Let $K$ be a number field such that $X$ has a semi-stable regular model $p:\mathcal{X}\to \mathrm{Spec} O_K$ over the ring of integers $O_K$ of $K$. Then, the Faltings height $h_{\mathrm{Fal}}(X)$ of $X$ is the arithmetic degree $$h_{\mathrm{Fal}}(X):=\frac{\widehat{\mathrm{deg}} Rp_\ast \mathcal{O}_{\mathcal{X}}}{[K:\mathbf{Q}]},$$ where we endow the determinant of cohomology with the Arakelov-Faltings metric. This is well-defined, i.e., independent of the field $K$. By Serre duality, it coincides with $$h_{\mathrm{Fal}}(X)=\frac{\widehat{\mathrm{deg}} p_\ast \mathcal{\omega}_{\mathcal{X}/O_K}}{[K:\mathbf{Q}]}.$$ It also coincides with the Faltings height of the Jacobian. All of this is explained in Section 4.4 of

http://www.math.univ-toulouse.fr/~couveig/book.htm

For a curve over a number field, there is also the important relative Faltings height. This invariant depends on the number field $K$, though.

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What is "the Faltings height" to you? There are so many normalisations around, that it can be hard to juggle them. –  Junkie Oct 14 '11 at 3:40
    
For an elliptic curve, I've seen the Faltings height defined by Mazur as $-{1\over 2}\log\Omega$, maybe with a $2\pi$. math.arizona.edu/~swc/notes/files/98MazurLN.ps Maybe this is the "logarithmic" Faltings height, as I've seen $\sqrt{2\pi/\Omega}$. I think none match Deligne's calculation (I recall going thru this many years ago, but forget what adjustments need to be made) of below. Another example is Tonghai Yang throws in factors to make $(\Lambda'/\Lambda)(0)$ come out nice. msri.org/~levy/files/Book49/09yang.pdf But it don't matter if you don't care about constants. :) –  Junkie Oct 14 '11 at 11:40
    
In short, it is natty for me personally to take your definition, and compare it to any of the ones for an elliptic curve. Likely a tedious exercise (I can never track down a source where it is done, though Deligne might come close), but can you say what your normalisation is in terms of $\Omega$ (the fundamental volume of the parallelogram associated to the Neron model - let's keep it easy and over Q at first)? If you don't want to bother with this, I understand too, for as I say, it's just a constant. If nothing else, I've warned you to be careful when the term "Faltings height" is used. :) –  Junkie Oct 14 '11 at 11:49
    
Hey Junkie, thanks alot for your comments. I wansn't aware of any of these ambiguities in the litarature. For an elliptic curve over K with semi-stable reduction, the Faltings height (as I defined above) can be computed explicitly. See Proposition 5.5.1 in R. de Jong's thesis. math.leidenuniv.nl/~rdejong/publications/Thesiswebversion.pdf –  Ari Oct 14 '11 at 17:32
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Looking at these sources I conclude this. The unstable Faltings height is given by ${-1\over 2}\log|{i\over 2}\int\omega\wedge\bar\omega=-{1\over 2}\log(Volume)$, see (14) of Silverman. The stable Faltings height differs from this, naturally, from contributions at places of additive reduction. Deligne differs from this stable Faltings height by an addition of $\log\sqrt\pi$ somewhere. Yang uses an normalisation suitable to his analytic number theory purposes. The "Parshin-Faltings height" more probably means exp of one of the above, like $\sqrt{1/Vol}$, possibly with $\pi$. –  Junkie Oct 15 '11 at 5:51
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1 Answer

up vote 10 down vote accepted

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.

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I get that the fundamental volume of $\Omega={\Gamma(1/3)^2\over \Gamma(2/3)^4}{\pi^2\over 3^{11/6}}\approx 2.8111$ for Deligne's curve, and thus from (say) Colmez's definition of the Faltings height as $-{1\over 2}\log\int \omega\wedge\omega$, a different answer. I suspect the normalisations differ, though perhaps I missing something more subtle? –  Junkie Oct 14 '11 at 12:11
    
Taking $e^{-2D}$ where $D$ is Deligne's result, I get some thing that differs is bigger than (what I call) the fundamental volume by a factor of $3^{5/6}\over{\pi/2}$. –  Junkie Oct 14 '11 at 12:28
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Looking at de Jong's thesis, he has an extra $\log\sqrt{\pi}$ in the formula. He has, for $y^2=x^3-1$ a result of $-{1\over 2}\log[({\Gamma(1/3)\over\Gamma(2/3)})^3{\pi\over\sqrt 3}]$, and Deligne has this without the $\log\sqrt\pi$. –  Junkie Oct 15 '11 at 5:15
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