Related papers: Sur la capitulation des 2-classes d'id\'eaux du co…

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions $L/K$ of squarefree degree $n$. If $E/K$ is the normal closure of $L/K$ then $G=\mathrm{Gal}(E/K)$ can be viewed as a permutation group of degree…

In this paper, we use the theory of genus fields to study the Euclidean ideals of certain real biquadratic fields $K.$ Comparing with the previous works, our methods yield a new larger family of real biquadratic fields $K$ having Euclidean…

In this paper we make an initial study on type D moduli spaces in positive characteristic $p\neq 2$, where we allow $p$ ramified in the definite quaternion algebra. We classify the isogeny classes of $p$-divisible groups with additional…

Let p be a prime and K be a number field with non-trivial p-class group Cl(p,K). A crucial step in identifying the Galois group G=G(p,K) of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G(p,2,K),…

Let K be a number field, and let a be a non-zero element of K. Fix some prime number l. We compute the density of the following set: the primes p of K such that the multiplicative order of the reduction of a modulo p is coprime to l (or,…

Let $P_k$ be the subgroup generated by $k$th powers of primitive elements in $F_r$, the free group of rank $r$. We show that $F_2/P_k$ is finite if and only if $k$ is $1$, $2$, or $3$. We also fully characterize $F_2/P_k$ for $k = 2,3,4$.…

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-$p$ groups and Lie algebras. A study of the set N_p of positive integers which occur as orders of nonsingular…

The present paper studies the homology of the groups $SL_2(k[C])$ and $GL_2(k[C])$ where $C=\overline{C}\setminus\{P_1,\dots,P_s\}$ is a smooth affine curve over an algebraically closed field $k$. It is well-known that these groups act on a…

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. If $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we prove that the admissible smooth representations of…

Let $p$ be an odd prime and $f \geq 1$. We consider a $p$-adic locally algebraic $\text{GL}_2(\mathbb Q_{p^f})$-representation attached to a tuple of $f$ weights $k=(k_i)$ for $0 \leq i \leq f-1$ and a $p$-adic integer $a_p$ with valuation…

Let $\ell>3$ be a prime such that $\ell \equiv 3 \pmod{4}$ and $\mathbb{Q}(\sqrt{\ell})$ has class number 1. Then Hirzebruch and Zagier noticed that the class number of $\mathbb{Q}(\sqrt{-\ell})$ can be expressed as $h(-\ell) =…

Let $K$ be an imaginary quadratic field of discriminant $d_K$ with ring of integers $\mathcal{O}_K$. When $K$ is different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, we consider a certain specific model for the elliptic curve…

Let $p$ be a prime number and let $k\geq 2$ be an integer. In this article we study the semi-simple reductions modulo $p$ of two-dimensional irreducible crystalline $p$-adic Galois representations with Hodge-Tate weights $0$ and $k-1$ and…

Let $K$ be a number field and $\mathcal{C}$ a full class of finite groups. We write $K^{\mathcal{C}}/K$ for the maximal pro-$\mathcal{C}$ Galois extension of $K$, and $G_K^{\mathcal{C}}$ for its Galois group. In this paper, we deal with the…

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard…

We give several families of reasonably small integers $k, \ell \ge 1$ and real positive $\alpha, \beta \le 1$, such that the products $p_1\ldots p_k s$, where $p_1, \ldots, p_k \le m^\alpha$ are primes and $s \le m^\beta$ is a product of at…

Let $H_{\mathbf{k}}$ be a symplectic reflection algebra corresponding to a cyclic subgroup $\Gamma \subseteq SL_2 \C$ of order $n$ and $U_{\mathbf{k}} = eH_{\mathbf{k}} e$ the spherical subalgebra of $H_{\mathbf{k}}$. We show that for…

Let $\ell$ and $p$ be prime numbers and $K_{n,m}=\mathbb{Q}(p^{\frac{1}{\ell^n}},\zeta_{2\ell^{m}})$. We study the $\ell$-class group of $K_{n,m}$ in this paper. When $\ell=2$, we determine the structure of the $2$-class group of $K_{n,m}$…

This paper describes in terms of Artin-Schreier equations field extensions whose Galois group is isomorphic to any of the four non-cyclic groups of order $p^3$ or the ten non-Abelian groups of order $p^4$, $p$ an odd prime, over a field of…

We present a method to compute the geometric Picard rank of a $K3$ surface over $\bbQ$. Contrary to a widely held belief, we show it is possible to verify Picard rank $1$ using reduction only at a single prime. Our method is based on…