arXiv subjects
Publications and source records attributed to Panagiotis Paramantzoglou.
The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.
We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ in the case of cyclic covers of the projective line.
The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally we give examples motivated from the theory of Galois coverings of curves.
Let $K$ be an algebraically closed field of characteristic $p \geq 0$. A generalized Fermat curve of type $(k,n)$, where $k,n \geq 2$ are integers (for $p \neq 0$ we also assume that $k$ is relatively prime to $p$), is a non-singular irreducible projective algebraic curve $F_{k,n}$ defined over $K$ admitting a group of automorphisms $H \cong {\mathbb Z}_{k}^{n}$ so that $F_{k,n}/H$ is the projective line with exactly $(n+1)$ cone points, each one of order $k$. Such a group $H$ is called a generalized Fermat group of type $(k,n)$. If $(n-1)(k-1)>2$, then $F_{k,n}$ has genus $g_{n,k}>1$ and it is known to be non-hyperelliptic. In this paper, we prove that every generalized Fermat curve of type $(k,n)$ has a unique generalized Fermat group of type $(k,n)$ if $(k-1)(n-1)>2$ (for $p>0$ we also assume that $k-1$ is not a power of $p$). Generalized Fermat curves of type $(k,n)$ can be described as a suitable fiber product of $(n-1)$ classical Fermat curves of degree $k$. We prove that, for $(k-1)(n-1)>2$ (for $p>0$ we also assume that $k-1$ is not a power of $p$), each automorphism of such a fiber product curve can be extended to an automorphism of the ambient projective space. In the case that $p>0$ and $k-1$ is a power of $p$, we use tools from the theory of complete projective intersections in order to prove that, for $k$ and $n+1$ relatively prime, every automorphism of the fiber product curve can also be extended to an automorphism of the ambient projective space. In this article we also prove that the set of fixed points of the non-trivial elements of the generalized Fermat group coincide with the hyper-osculating points of the fiber product model under the assumption that the characteristic $p$ is either zero or $p>k^{n-1}$.