Let C be a curve over Q provided with an integral model, an ample line bundle on the model and a semipositive metric. To these data we can associate the height of the curve and the height of every algebraic point of the curve. The essential minimum of the curve is the minimal accumulation point of the height of the algebraic points. The essential minimum is a mysterious and elusive invariant. A result of Zhang shows that the essential minimum has a lower bound in terms of the height of the curve, and an example of Zagier shows that there can be several isolated values of the height below the essential minimum. When C is the modular curve, the line bundle agrees with the bundle of modular forms and the metric is the Weil-Petersson metric, then the height of an algebraic point agrees with the stable Faltings height of the corresponding elliptic curve. In this talk we will discuss methods of proving lower and upper bounds for the essential minimum and apply them to the modular curve, giving a partial description of the spectrum of the stable Faltings height of elliptic curves. This is joint with with Ricardo Menares and Juan Rivera-Letelier.

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