Related papers: p-Rational Fields and the Structure of Some Module…

Let $K\left\langle X \right\rangle$ denote the free associative algebra generated by a set $X = \{x_1, \dots, x_n\}$ over a field $K$ of characteristic $0$. Let $I_p$, for $p \geq 2$, denote the two-sided ideal in $K\left\langle X…

Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. Based on the Gr\"obner basis theory for $A$ and for free modules over $A$, an elimination theory…

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…

We study the p-rationality of real quadratic fields in terms of generalized Fibonacci numbers and their periods modulo positive integers.

Let $p$ be a prime, and $F$ a non-archimedean local field with residue characteristic $p$ and ring of integers $\mathcal{O}_{F}$. Set $G_{S}:={\rm SL}_{2}(F)$and $K_{0}:={\rm SL}_{2}(\mathcal{O}_{F})$ . For a smooth irreducible…

We give a conjectural description for the kernel of the map assigning to each finite $\mathbb Z_p$-free $G\times\mathbb Z_p$-set its rational permutation module where G is a finite p-group. We prove that this conjecture is true when G is an…

Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields. For $p=5$ we prove the existence of infinitely many real quadratic $p$-rational fields.

Let $L$ be a field of characteristic zero, let $h:\mathbb{P}^1\to \mathbb{P}^1$ be a rational map defined over $L$, and let $c\in \mathbb{P}^1(L)$. We show that there exists a finitely generated subfield $K$ of $L$ over which both $c$ and…

For a cubic number field $L$, we consider the $\mathbb{Z}$-order in $L$ of the form $\mathbb{Z}[\alpha]$, where $\alpha$ is a root of a polynomial of the form $x^3-ax+b$ and $a,b\in\mathbb{Z}$ are integers such that $v_p(a)\leq 2$ or…

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 $G$ be a finite $p$-group. We construct a $G$-extension $K/k$ of number fields such that the $p$-adic completion of the unit group of $K$ has a prescribed $\mathbb{Z}_p[G]$-module structure, up to free direct summands.

We study the algebras of hermitian automorphic forms for the lattice $L_n=diag(1,1,\ldots,1,-1)$ and for the field $K=\mathbb{Q}(\sqrt{-d})$ such that $p=2$ is unramified and the ring of integers $\mathcal{O}_K$ is a p.i.d. We prove that…

Let G be a semisimple group over an algebraically closed field of characteristic p>0. We give a (partly conjectural) simple, closed formula for the character of many indecomposable tilting rational G-modules, assuming that p is large.

In this paper, a new criterion is given to determine the $p-$rationality of some complex cubic number fields in terms of $ p-$divisibility of certain terms of a third-order recurrence sequence, several illustrated examples are…

We study the rational torsion subgroup of the modular Jacobian $J_0(N)$ for $N$ a square-free integer. We give a new proof of a result of Ohta on a generalization of Ogg's conjecture: for a prime number $p \nmid 6N$, the $p$-primary part of…

Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also…

Let $E/F$ be an unramified extension of non-archimedean local fields of residual characteristic different than $2$. We provide a simple geometric proof of a variation of a result of Y. Hironaka. Namely we prove that the module…

In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of…

Let $p$ be an odd prime number. For a degree $p$ extension of $p$-adic fields $L/K$, we give a complete characterization of the condition for the ring of integers $\mathcal{O}_L$ to be free as a module over its associated order in the…

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian…