In [22], the authors proved an explicit formula for the arithmetic intersection number $\left(CM\left(K\right).G_1\right)$ on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field $K$. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus $2$ curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [21] generalizing the singular moduli formula of Gross...

The purpose of the present article is the study of duals of functional codes on algebraic surfaces. We give a direct geometrical description of them, using differentials. Even if this description is less trivial, it can be regarded as a natural extension to surfaces of the result asserting that the dual of a functional code $C_L(D,G)$ on a curve is the differential code $C_{\Omega }(D,G)$ . We study the parameters of such codes and state a lower bound for their minimum distance. Using this bound, one can study some examples...