Related papers: On the $p$-ranks of class groups of certain Galois…

For any odd prime $p$ we consider representations of a group of order $p$ in the symplectic group $Sp(p-1,Z[1/n])$ of $(p-1)\times(p-1)$-matrices over the ring $Z[1/n]$, $0\neq n\in N$. We construct a relation between the conjugacy classes…

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

In the previous work, Lim and the author determined the rank variety of the simple $\mathbb{F}\mathfrak{S}_{kp}$-module $D(p-1)=D^{(kp-p+1,1^{p-1})}$ with respect to some maximal elementary abelian $p$-subgroup $E_k$ and the complexity when…

Let $N, p \geq 5$ be primes such that $N \equiv 1 \bmod p$. We study the rank $r$ of the Hecke algebra that parametrizes modular forms of weight 2 and level $N$ that are Eisenstein modulo $p$. When $r$ is $2$ or $3$, we prove that $r-1$…

Let K be a number field containing the group of n-th roots of unity and S a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal…

In this paper we introduce a new method for finding Galois groups by computer. This is particularly effective in the case of Galois groups of p-extensions ramified at finitely many primes but unramified at the primes above p. Such Galois…

For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(\zeta_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we…

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…

Let $K$ be a local field of characteristic $p>0$ with perfect residue field and let $G$ be a finite $p$-group. In this paper we use Saltman's construction of a generic $G$-extension of rings of characteristic $p$ to construct totally…

Let p be an odd prime, let S be a finite set of primes q congruent to 1 mod p but not mod p^2 and let G_S be the Galois group of the maximal p-extension of Q un-ramified outside of S. If r is a continuous homomorphism of G_S into GL_2(Z_p)…

In this article we survey and examine the realizability of $p$-groups as Galois groups over arbitrary fields. In particular we consider various cohomological criteria that lead to necessary and sufficient conditions for the realizability of…

We prove that for any prime $p$ the finite $p$-groups of fixed coclass have only finitely many different mod-$p$ cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of…

Let p>3 be a prime number and let r be an integer with 1<r<p-1. For each r, let moreover G_r denote the unique quotient of the maximal class pro-p group of size p^{r+1}. We show that the mod-p cohomology ring of G_r has depth one and that,…

Let $p$ be a prime number, and let $K$ be a $p$-adic local field. We study a class of semistable $p$-adic Galois representations of $K$, which we call {\it triangulordinary} because it includes the ordinary ones yet allows non-\'etale…

Given a prime $p$ and $q$ a power of $p$, we study the statistics of $p$-ranks of Artin--Schreier covers of given genus defined over $\mathbb{F}_q$, in the large $q$-limit. We refer to this problem as the geometric problem. We also study an…

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…

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…

For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z\_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct…

This paper is devoted to the study of the $\ell$-adic representations of the absolute Galois group $G$ of ${\mathbb Q}_p$, $p\geq 5$, associated to an elliptic curve over ${\mathbb Q}_p$, as $\ell$ runs through the set of all prime numbers…

Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for…