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Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. Inthis paper we study the elliptic and the strongly elliptic ideals of $A$ withthe aim to characterize elliptic and strongly elliptic singularities, accordingto the definitions given by Wagreich and by Yau. In analogy with the rationalsingularities, in the main result we characterize a strongly ellipticsingularity in terms of the normal Hilbert coefficients of the integrallyclosed $\mathfrak m$-primary ideals of $A$. Unlike $p_g$-ideals, ellipticideals and strongly elliptic ideals are not necessarily normal and necessaryand sufficient conditions for being normal are given. In the last section wediscuss the existence (and the effective construction) of strongly ellipticideals in any two-dimensional normal local ring.

Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities