arXiv subjects

Guillermo Mantilla-Soler

Publications and source records attributed to Guillermo Mantilla-Soler.

21 records · Page 2Linked to original sources

arXiv2011

Power map permutations and symmetric differences in finite groups

Let $G$ be a finite group. For all $a \in \Z$, such that $(a,|G|)=1$, the function $ρ_a: G \to G$ sending $g$ to $g^a$ defines a permutation of the elements of $G$. Motivated by a recent generalization of Zolotarev's proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation $ρ_a$. By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer $d_{G}$ such that $\text{sgn}(ρ_a)=(\frac{d_G}{a})$ for all $G$ in a large class of groups, containing all finite nilpotent and odd order groups.

math.GR↗

arXiv2011

On number fields with equivalent integral trace forms

Let $K$ be a number field. The \textit{integral trace form} is the integral quadratic form given by $\text{tr}_{K/\mathbb{Q}}(x^2)|_{O_{K}}.$ In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields of fundamental discriminant have equivalent trace zero forms if and only if they are conjugated.

math.NT↗

arXiv2011

Integral trace forms associated to cubic extensions

Given a nonzero integer $d$, we know by Hermite's Theorem that there exist only finitely many cubic number fields of discriminant $d$. However, it can happen that two non-isomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form $q_K:\text{tr}_{K/\mathbb{Q}}(x^2)|_{O^{0}_{K}}$ as such a refinement. For a cubic field of fundamental discriminant $d$ we show the existence of an element $T_K$ in Bhargava's class group $\Cl(\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}\otimes\mathbb{Z}^{2}; -3d)$ such that $q_K$ is completely determined by $T_K$. By using one of Bhargava's composition laws, we show that $q_K$ is a complete invariant whenever $K$ is totally real and of fundamental discriminant

math.NT↗