ζp ∈ C. Let Q(ζp) be the pth cyclotomic field. It is well-known that Q(ζp) is a CM-field. If p is a Fermat prime then the only CM-subfield of Q(ζp) is Q(ζp) itself, since the Galois group of Q(ζp)/Q is a cyclic 2-group, whose only element of order 2 acts as the complex conjugation. All other subfields of Q(ζp) are totally real. Let f(x) ∈ C[x] be a polynomial of degree n ≥ 5 without multiple ro...

We introduce stable tropical curves, and use these to count covers of the p-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers.

We introduce stable tropical curves, and use these to count covers of the p-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers.

Restricted modules for the restricted toral rank two contact Lie algebra are considered. Contragredients of the simple modules, Cartan invariants, and dimensions of the simple modules and their projective covers are determined. Let L be a finite dimensional restricted Lie algebra. All L-modules in this paper are assumed to be left, restricted and finite dimensional over the defining field. Each...

Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over with three points removed. We investigate which finite groups $G$ can arise as monodromy group \'{e}tale covers that are tamely ramified removed points. This provides new information about tame fundamental line. In particular, we show for each prime $p\ge 5$, there families symmetric $S_n$ or alt...

this paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``advanced school and conference on homological and geometrical methods in representation theory'' at the international centre for theoretical physics (ictp), miramare - trieste, italy, 11-29 january 2010. we show how to associate to a triple of positive integer...

In this paper, we investigate the notions of X ⊥ -projective, id="M2"> -injective, and id="M3"> -flat modules give some characterizations these modules, where id="M4"> is a class left modules. We prove that all id="M5"> -projective Kaplansky. Further, if id="M6"...

In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k 6= 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs...