Related papers: Note on some p-invariants of Q(N^{1/p}) using refl…

Let $K$ be an algebraic number field with ring of integers $\Cal{O}_{K}$, $p>2$ be a rational prime and $G$ be the cyclic group of order $p $. Let $\Lambda$ denote the order $\Cal{O}_{K}[G].$ Let $Cl(\Lambda)$ denote the locally free class…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…

Let $E$ be an elliptic curve over $\mathbb{Q}$, $p$ an odd prime number and $n$ a positive integer. In this article, we investigate the ideal class group $\mathrm{Cl}(\mathbb{Q}(E[p^n]))$ of the $p^n$-division field $\mathbb{Q}(E[p^n])$ of…

We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, the class number of $\mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N…

For a prime p and natural number n with p greater than or equal to n, we establish the existence of a non-functorial one-to-one correspondence between isomorphism classes of groups of order p^n whose derived subgroup has exponent dividing…

We determine the $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and $K=\mathbb{Q}(\sqrt[3]{p},\sqrt{-3})$ when $p\equiv 4,7\bmod 9$ is a prime and $3$ is a cubic modulo $p$. This confirms a conjecture made by Barrucand-Cohn, and proves the…

Let $p$ be an odd prime, let $N$ be a prime with $N \equiv 1 \pmod{p}$, and let $\zeta_p$ be a primitive $p$-th root of unity. We study the $p$-rank of the class group of $\mathbb{Q}(\zeta_p, N^{1/p})$ using Galois cohomological methods and…

For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set $K=\mathbb{Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group $\operatorname{Cl}_K$ is a module over…

Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over $\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the group of rational torsion points on $J_0(3p)$…

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group $G(\ell,p,n)$. In particular, we: (1) show a relationship to the Coulomb branch construction of…

Let $n,p,k$ be three positive integers. We prove that the rational fractions of $q$: $${n \brack k}_{q} {}_3\phi_{2} [ . {matrix}q^{1-k},q^{-p},q^{p-n} q,q^{1-n} {matrix}| q;q^{k+1}]\quad\textrm{and}\quad q^{(n-p)p}\qbi{n}{k}{q} {}_3\phi_2[…

Let $p$ be a prime number. Let $C_p$, the cyclic group of order $p$, permute transitively a set of indeterminates $\{ x_1,\ldots ,x_p \}$. We prove that the invariant field $\mathbb{Q}(x_1,\ldots ,x_p)^{C_p}$ is rational over $\mathbb{Q}$…

We represent the rational and mod $p$ cohomology groups of classifying spaces of rank 3 Kac-Moody groups by a direct sum of the invariants of Weyl groups and their quotients. As an application, the authors conclude that there is a…

We examine the phenomenon of capitulation of the $p$-class group $H_K$ of a real number field $K$ in totally ramified cyclic p-extensions $L/K$ of degree $p^N$. Using an elementary property of the algebraic norm $\nu_{L/K}$, we show that…

This paper is a contribution to the description of some congruences on the odd prime factors of the class number of the number fields. An example of results obtained is: Let L/Q be a finite Galois solvable extension with [L:Q]=N, where N >…

The class of abelian $p$-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory $T_p$ whose models are each bi-interpretable with the disjoint union of an…

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…

Let p be a prime number. We classify all smooth irreducible mod-p representations of the unramified unitary group U(1,1)(Q_p^2/Q_p) in two variables. We then investigate Langlands parameters in characteristic p associated to…

For a prime \(p\ge 2\) and a number field K with p-class group of type (p,p) it is shown that the class, coclass, and further invariants of the metabelian Galois group \(G=Gal(F_p^2(K) | K)\) of the second Hilbert p-class field \(F_p^2(K)\)…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a quadratic number field $K$ and an odd prime number $p$, let $L$ be a $\mathbb{Z}_p$-extension of $K$. We prove that $E(L)_{\text{tors}}=E(K)_{\text{tors}}$ when $p>5$. It enables…