First steps of local symplectic algebra V.I.Arnold The goal of this paper is the solution of a very special problem of symplectic singularity theory -- that of the classification of the simplest singularities of curves in a symplectic manifold. The classical Darboux--Givental theorem claims, that the germ of a smooth submanifold of a symplectic manifold is defined (up to a symplectomorphism) by the restriction of the symplectic form to the tangent space of the submanifold. In the case of a smooth curve this restriction vanishes. The results of the present paper suggest that something nontrivial remains from the symplectic structure at the singular points of the curve. It would be interesting to describe this ghost of the symplectic structure in terms of the local algebra of the singularirty. In this paper such a formula is missing: I just provide the classification of the curves with simplest singularities in a symplectic manifold. The principal result of the present paper is the classification of the singularities $A_{2k}$ in the symplectic space (up to symplectomorphisms). All such singularities are simple. This means that each curve singularity $A_{2k}$ has only a finite number of symplectic forms.