5 research outputs found
Periodicity of free subgroup numbers modulo prime powers
We completely characterise when the sequence of free subgroup numbers of a
finitely generated virtually free group is ultimately periodic modulo a given
prime power.Comment: AmS-LaTeX; 18 page
The sixth Painleve transcendent and uniformization of algebraic curves
We exhibit a remarkable connection between sixth equation of Painleve list
and infinite families of explicitly uniformizable algebraic curves. Fuchsian
equations, congruences for group transformations, differential calculus of
functions and differentials on corresponding Riemann surfaces, Abelian
integrals, analytic connections (generalizations of Chazy's equations), and
other attributes of uniformization can be obtained for these curves. As
byproducts of the theory, we establish relations between Picard-Hitchin's
curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous
differential equation which Apery used to prove the irrationality of Riemann's
zeta(3).Comment: Final version. Numerous improvements; English, 49 pages, 1 table, no
figures, LaTe