Related papers: The local lifting problem for dihedral groups

For a prime $p$, a cyclic-by-$p$ group $G$ and a $G$-extension $L|K$ of complete discrete valuation fields of characteristic $p$ with algebraically closed residue field, the local lifting problem asks whether the extension $L|K$ lifts to…

We are investigating the lifting problem for local actions involving semidirect products of a cyclic $p$-group with a cyclic group prime to $p$, where $p$ represents the characteristic of the special fiber. We establish a criterion based on…

Let $k$ be an algebraically closed field of characteristic $2$. In this paper we describe the $(\mathbb{Z}/2\mathbb{Z})^3$-actions on $k[[z]]$ for which there is a discrete valuation ring $R$, a finite extension of the ring of Witt vectors…

Let k be an algebraically closed field of characteristic p and let G be a subgroup of Aut(k[[t]]) be a faithful action on a local power series ring over k. Let R be a discrete valuation ring of characteristic 0 with residue field k. One…

Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort…

The lifting problem that we consider asks: given a smooth curve in characteristic p and a group of automorphisms, can we lift the curve, along with the automorphisms, to characteristic zero? One can reduce this to a local question (the…

The local Oort conjecture states that, if G is cyclic and k is an algebraically closed field of characteristic p, then all G-extensions of k[[t]] should lift to characteristic zero. We prove a critical case of this conjecture. In…

We solve the local lifting problem for the alternating group A_4, thus showing that it is a local Oort group. Specifically, if k is an algebraically closed field of characteristic 2, we prove that every A_4-extension of k[[s]] lifts to…

We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all…

Let $k$ be an algebraically closed field of characteristic $p > 0$. We study obstructions to lifting to characteristic 0 the faithful continuous action $\phi$ of a finite group $G$ on $k[[t]]$. To each such $\phi$ a theorem of Katz and…

It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero $p$-adic representation, if local lifting problems at places above $p$ are…

We study irreducible odd mod $p$ Galois representations $\bar{\rho} \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show…

We describe integral lifts K(L), indexed by local fields L of degree n = [L:\Q_p], of the extraordinary cohomology theories K(n), and apply the generalized character theory of Hopkins, Kuhn and Ravenel to identify K(L)(BG) \otimes \Q$, for…

Given $\mathbb P^4_k$, with $k$ algebraically closed field of characteristic $p>0$, and $X\subset \mathbb P^4_k$ integral surface of degree $d$, let $Y=X\cap H$ be the general hyperplane section of $X$. We suppose that $h^0\mathscr…

We study irreducible mod p representations, valued in general reductive groups, of the Galois group of a number field. When the number field is totally real, we show that odd representations satisfying local ramification hypotheses and a…

We give a necessary and sufficient condition for a modular representation of a group $G=C_{p^h} \rtimes C_m$ in a field of characteristic zero to be lifted to a representation over local principal ideal domain of characteristic zero…

For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete…

Let A be an excellent integral henselian local noetherian ring, k its residue field of characteristic p>0 and K its fraction field. Using an algebraization technique introduced by the first named author, and the one-dimension case already…

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$…

We prove an algebraic version of a classical theorem in topology, asserting that an abelian p-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove…