We prove that the functional equation f​⋆​f(2 t)=λf(t)2, for t in Z/dZ with d odd, admits a nonzero solution f if λ=a​+​ib a, b positive integers such a​+​b​=​d and a​≡(d+1)24 mod 4. The proof involves theta functions on elliptic curves complex multiplication.

We consider normal covers of CP with abelian deck group, branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces. We compute all individual Lyapunov exponents for abelian squaretiled surfaces, a...

In this paper, we try to determine when the derived category of an abelian category is the homotopy category of a model structure on the category of chain complexes. We prove that this is always the case when the abelian category is a Grothendieck category, as has also been done by Morel. But this model structure is not very useful for defining derived tensor products. We therefore consider ano...

The dynamics of abelian vector and antisymmetric tensor gauge fields can be described in terms of twisted self-duality equations. The latter play an important role in exposing the duality symmetries of the theory. I this contribution I review recent progress in the construction of action principles for these equations and in particular their non-abelian generalizations.

We consider the general problem when local regularity implies the global one in the setting where local regularity means the existence of a square of certain length in every position of an infinite word. The square can occur as centered or to the left or to the right from each position. In each case there are three variants of the problem depending on whether the square is that of words, that o...