Generalizing MacPherson-Vilonen’s method [2] to arbitrary plane curve singularities, we provide a classification of perverse sheaves on the neighborhood of the origin in the complex plane, which are adapted to a germ of a complex analytic plane curve. We rely on the presentation of the fundamental group of the complement of the curve as obtained by Neto and Silva [5]. The main result is an equivalence of categories between the category of perverse sheaves on $ \mathbf{C}^{2}$ stratified with respect to a singular plane curve and the category of $n$-tuples of finite dimensional vector spaces and linear maps satisfying a finite number of suitable relations. As an application, we classify perverse sheaves with no vanishing cycles at the origin for a special case.

We establish congruences for higher order Euler polynomials modulo 720. We apply this result for constructing analogues of Stern congruences for Euler secant numbers $E_{4n}\equiv 5(\mathrm{mod}\ 60), E_{4n+2}\equiv -1(\mathrm{mod}\ 60)$ to Euler tangent numbers and Genocchi numbers. We prove that Euler tangent numbers satisfy the following congruences $E_{4n+1}\equiv 16(\mathrm{mod}\ 720)$, and $E_{4n+3}\equiv -272(\mathrm{mod}\ 720)$. We establish 12-periodic property of Genocchi numbers modulo 45.

In [8] the first author classified automorphism groups of smooth plane curves of degree not less than four into five types. If the curve has a unique outer Galois point, then the quotient group of its automorphism group by the Galois group at the point, which is called the reduced automorphism group, is a finite subgroup of one-dimensional projective linear group. This article is a sequel of [10] and [9]. In this article, we shall determine the defining equation of the curve when the reduced automorphism group is an icosahedral group and give a description of the full automorphism group.