Let F be a finite extension of Qp and K a quadratic extension of F . If (Π, V ) is a representation of GL2(K), H a subgroup of GL2(K) and μ a character of the image subgroup det(H) of K∗, then Π is said to be μ-distinguished with respect to H if there exists a nonzero linear form l on V such that l(Π(g)v) = μ(det g)l(v) for g ∈ H and v ∈ V . We provide new proofs, using entirely local methods, ...

In this note, we prove that a random extension of either the free group FN of rank N ě 3 or of the fundamental group of a closed, orientable surface Sg of genus g ě 2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either OutpFN q or ModpSgq generated by k independent random walks. Our main theorem is that a k–generated random subgroup of ModpSgq or OutpFN ...

We return to the general AKLB setup: A is a Dedekind domain with fraction field K, L is a finite separable extension of K, and B is the integral closure of A in L. But now we add the condition that the extension L/K is normal, hence Galois. We will see shortly that the Galois assumption imposes a severe constraint on the numbers ei and fi in the ram-rel identity (4.1.6). Throughout this chapter...

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, we write $\mathfrak{p}$ for either of the primes $K$ above $2$. $K_\infty$ unique $\mathbb{Z}_2$-extension unramified outside $\mathfrak{p}$. For certain quadratic biquadratic extensions $\mathfrak{F}/K$, prove simple exact formula $\lambda$-invariant Galois group maximal abelian 2-extension ...

Consider a cubic extension K := Q(α), when the minimal polynomial f(x) of α does not totally split in K. The normal closure L := (K) is a sextic extension of Q, with Gal(L/Q) = S3. Now we fix notation and pick one embedding of K as K1, say K1 is fixed by (2, 3) ∈ S3. Here (2, 3) has the explicit description that if I choose one root u1 of f(x), (2, 3) permute the other two conjugate roots of u1...

Let $k'/k$ be a finite purely inseparable field extension and let $G'$ reductive $k'$-group. We denote by $G=\R_{k'/k}(G')$ the Weil restriction of across $k'/k$, pseudo-reductive group. This article gives bounds for exponent geometric unipotent radical $\RR_{u}(G_{\bar{k}})$ in terms invariants starting with case $G'=\GL_n$ applying these results to where is simple

Let p be prime. Let L/K be a finite, totally ramified, purely inseparable extension of local fields, [L : K] = p, n ≥ 2. It is known that L/K is Hopf Galois for numerous Hopf algebras H, each of which can act on the extension in numerous ways. For a certain collection of such H we construct “Hopf Galois scaffolds” which allow us to obtain a Hopf analogue to the Normal Basis Theorem for L/K. The...

Let us consider an algebraic function field defined over a finite Galois extension K of a perfect field k. We give some conditions allowing the descent of the definition field of the algebraic function field from K to k. We apply these results to the descent of the definition field of a tower of function fields. We give explicitly the equations of the intermediate steps of an Artin-Schreier typ...

Let K be a finite tamely ramified extension of Qp and let L/K be a totally ramified (Z/pnZ)-extension. Let πL be a uniformizer for L, let σ be a generator for Gal(L/K), and let f(X) be an element of ØK [X] such that σ(πL) = f(πL). We show that the reduction of f(X) modulo the maximal ideal of ØK determines a certain subextension of L/K up to isomorphism. We use this result to study the field ex...

If K/k is a finite Galois extension of number fields with Galois group G, then the kernel of the capitulation map Clk ~ ClK of ideal class groups is isomorphic to the kernel X(H) of the transfer map H/H’ ~ A, where H = Gal(K/k), A = Gal(K/K) and K is the Hilbert class field of K. H. Suzuki proved that when G is abelian, |G| divides |X(H)|. We call a finite abelian group X a transfer kernel for ...