Related papers: Determination of Conductors from Galois Module Str…

We prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give…

We prove that the Krull-Schmidt decomposition of the Galois module of the $p$-adic completion of algebraic units is controlled by the primes that are ramified in the Galois extension and the $S$-ideal class group. We also compute explicit…

We prove that function fields of varieties of dimension at least two over an algebraic closure of a finite field are determined, modulo purely inseparable extensions, by the quotient by the second term in the lower central series of their…

We determine the Galois module structure of the ring of integers for all cubic fields using roots of the generic cyclic cubic polynomial $f_n(X)=X^3-nX^2-(n+3)X-1$. Let $L_n=\mathbb Q(\rho_n)$ be a cyclic cubic field with Galois group…

In this paper, we show the existence of a non-solvable Galois extension of $\Q$ which is unramified outside 2. The extension $K$ we construct has degree $2251731094732800=2^{19}(3\cdot 5\cdot 17\cdot 257)^2$ and has root discriminant…

Fontaine and Mazur conjecture that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. We consider this conjecture for the maximal unramified p-extension of a CM-field k.

Let $F$ be a field of characteristic $0$ containing all roots of unity. We construct a functorial compact Hausdorff space $X_F$ whose profinite fundamental group agrees with the absolute Galois group of $F$, i.e. the category of finite…

Let K/k be a finite Galois extension of number fields with Galois group G, S a large set of primes of K, and E the G-module of S-units of K. Previous work has determined the data which is necessary to determine the stable isomorphism class…

We consider Hopf Galois structures on a separable field extension $L/K$ of degree $p^n$, for $p$ an odd prime number, $n\geq 3$. For $p > n$, we prove that $L/K$ has at most one abelian type of Hopf Galois structures. For a nonabelian group…

Given an abelian algebraic group $A$ over a global field $F$, $\alpha \in A(F)$, and a prime $\ell$, the set of all preimages of $\alpha$ under some iterate of $[\ell]$ generates an extension of $F$ that contains all $\ell$-power torsion…

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary…

We determine the representation of the group of automorphisms for cyclotomic function fields in characteristic $p > 0$ induced by the natural action on the space of holomorphic differentials via construction of an explicit basis of…

We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight $3$ newforms will show that there are Galois extensions of $\mathbb{Q}$ with Galois…

This paper was motivated by a recent paper by Krumm and Pollack investigating modulo-$p$ behaviour of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to…

We use Galois cohomology to study the $p$-rank of the class group of $\mathbf{Q}(N^{1/p})$, where $N \equiv 1 \bmod{p}$ is prime. We prove a partial converse to a theorem of Calegari--Emerton, and provide a new explanation of the known…

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

Let $p$ be a prime number, let $K$ be a $p$-field (a local field with finite residue field of characteristic $p$), let $L$ be a finite galoisian tamely ramified extension of $K$, and let $G=\mathrm{Gal}(L|K)$. Suppose that $L$ is split over…

For a rational prime $p \geq 3$ and an integer $n \geq 2$, we study the modularity of continuous 2-dimensional mod $p^n$ Galois representations of $\Gal(\bar{\Q}/\Q)$ whose residual representations are odd and absolutely irreducible. Under…

We complete the results of a previous article. Let $G$ be a split connected reductive group over a finite extension $F$ of $\mathbb{Q}_p$. When $F=\mathbb{Q}_p$, we determine the extensions between unitary continuous $p$-adic and smooth mod…

For a totally real field $F$, a finite extension $\mathbf{F}$ of $\mathbf{F}_p$ and a Galois character $\chi: G_F \to \mathbf{F}^{\times}$ unramified away from a finite set of places $\Sigma \supset \{\mathfrak{p} \mid p\}$ consider the…