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Let $E$ be an elliptic curve defined by an equation $y^2=4x^3+ax+b$ where $a$ and $b$ are algebraic numbers. What is the relation between the Faltings height $h_F(E)$ and the periods

$$ \int_{\gamma} \frac{dx}{y} $$

where $\gamma$ is a topological (nonzero) cycle in $H_1(E(\mathbb{C}), \mathbb{Q})$?

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Since you haven't said anything about the integrality of $a$ and $b$, nor about the minimality of the Weierstrass equation, the value of the integral lacks all of the arithmetic information that is contained in the Faltings height. If you change variables by $(x,y) \to (u^2x,u^3y)$, then your period changes by $u^{-1}$, but the Faltings height is a function of the curve $E$, and does not depend on a choice of Weierstrass equation.

If you start with a global minimal Weierstrass equation, e.g., over $\mathbb Q$, then the archimedean part of $h_F(E)$ is more-or-less, but not quite, Im($\tau$) for an appropriate ratio of periods. You might look at my article in Arithmetic Geometry, Springer, 1986, Gary Cornell and Joseph Silverman, editors. I worked out pretty explicitly how $h_F(E)$ is expressed in terms of local factors, including the periods appearing as the archimedean factors.

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