Soit $k$ un corps parfait de caractéristique 5. Nous démontrons que l’anneau versel de l’action d’un élément d’ordre 5 et de conducteur de Hasse 2 comme automorphisme d’un anneau de séries formelles $k[[ t ]]$ calculé par Bertin et Mézard, est en fait universel. C’est le premier exemple d’anneau non-trivial de déformation universel en conducteur supérieur.

Let $k$ denote a perfect field of characteristic 5. We show that the versal deformation ring of an element of order 5 and Hasse conductor 2 as automorphism of a ring of formal power series $k[[ t ]]$, computed by Bertin and Mézard, is in fact universal. This provides the first example of a non trivial universal deformation ring in higher conductor.

In this paper, we study the congruences of the Fourier coefficients of the Mathieu mock theta function, which appears in the Mathieu moonshine phenomenon discovered by Eguchi, Ooguri, and Tachikawa.

We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fiber if the central fiber is a toric Fano variety admitting a small resolution.