On the normalized arithmetic Hilbert function

 Let X be a projective variety of dimension n defined over a number field.  Philippon and Sombra introduced   the normalized arithmetic Hilbert function of X. This function measures the binary complexity of X. When X is toric, a result of Philippon and Sombra shows that the asymptotic of this function is related to the normalized height of X. After a brief overview of
the theory of heights, I will show that the normalized arithmetic Hilbert function admits an asymptotic expansion the leading coefficient is the normalized height of X . This gives a positive answer to a question raised by Philippon and Sombra.