Related papers: A note on asymptotic class number upper bounds in …

Malle proposed a conjecture for counting the number of $G$-extensions $L/K$ with discriminant bounded above by $X$, denoted $N(K,G;X)$, where $G$ is a fixed transitive subgroup $G\subset S_n$ and $X$ tends towards infinity. We introduce a…

Elementary abelian groups are finite groups in the form of $A=(\mathbb{Z}/p\mathbb{Z})^r$ for a prime number $p$. For every integer $\ell>1$ and $r>1$, we prove a non-trivial upper bound on the $\ell$-torsion in class groups of every…

In this paper we interpret the solutions to a particular Galois embedding problem over an extension K/F whose Galois group is a finite, cyclic p group in terms of certain Galois submodules within the parameterizing space of elementary…

Fix two distinct odd primes $p$ and $q$. We study "$p\ne q$" Iwasawa theory in two different settings. Let $K$ be an imaginary quadratic field of class number 1 such that both $p$ and $q$ split in $K$. We show that under appropriate…

Let $p$ be an odd prime and $e_p$ be its irregularity index. If $4e_p+8 <\frac{p-1}{2}$ we construct a Galois representation with image in the diagonal torus of $\op{GSp}_4(\Fp)$ that lifts to a characteristic $0$ representation unramified…

Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K = Q(zeta) be the p-cyclotomic field and let O_K be the ring of integers of K. Let pi be the prime ideal of K lying over p. An integer B \in O_K is…

In this paper, we consider infinite Galois extensions of number fields and study the relation between their local degrees and the structure of their Galois groups. It is known that, if $K$ is a number field and $L/K$ is an infinite Galois…

Let p be a prime number and M a quadratic number field, M not equal to Q(\sqrt{p}) if p is congruent to 1 modulo 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D_{2p} and an elliptic…

We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first…

Let $L/K$ be a Galois extension of number fields and let $G=\mathrm{Gal}(L/K)$. We show that under certain hypotheses on $G$, for a fixed prime number $p$, Leopoldt's conjecture at $p$ for certain proper intermediate fields of $L/K$ implies…

Let p be a prime and F(p) the maximal p-extension of a field F containing a primitive p-th root of unity. We give a new characterization of Demuskin groups among Galois groups Gal(F(p)/F) when p=2, and, assuming the Elementary Type…

An important function attached to a complex simple Lie group $G$ is its asymptotic character $X(\lambda,x)$ (where $\lambda,x$ are real (co)weights of $G$) - the Fourier transform in $x$ of its Duistermaat-Heckman function $DH_\lambda(p)$…

In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group $G$ is an almost simple group in which the socle $A$ has prime index $p$. Each Hopf-Galois structure is associated to a group $N$ of the same…

We study the theory of finite-order p-adic functions and distributions on ray class groups of number fields, and apply this to the construction of (possibly unbounded) p-adic L-functions for automorphic forms on GL(2) which may be…

For $d \geq 4$ and $p$ a sufficiently large prime, we construct a lattice $\Gamma \leq {\rm PSp}_{2d}(\mathbb Q_p),$ such that its universal central extension cannot be sofic if $\Gamma$ satisfies some weak form of stability in…

Let $K$ be a number field and $G$ a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified $G$-extensions of $K$ with ring of integers of fixed realisable class as a Galois module.

Let $n=2g+2$ be a positive even integer, $f(x)$ a degree $n$ complex polynomial without multiple roots and $C_f: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over the field $\C$ of complex numbers. Let a $(g-1)$-dimensional…

We study the growth of the Galois invariants of the $p$-Selmer group of an elliptic curve in a degree $p$ Galois extension. We show that this growth is determined by certain local cohomology groups and determine necessary and sufficient…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption…

For each prime $p$ and each positive integer $d$, we construct the first examples of second countable, topologically simple, $p$-adic Lie groups of dimension $d$ whose Lie algebras are abelian. This answers several questions of Gl\"ockner…