arXiv subjects
Publications and source records attributed to M. Archita.
We show that the group of proper projective similitudes of a totally decomposable algebra with orthogonal or symplectic involution of index at most 2 over a field of characteristic different from 2 is R-trivial. We further give an example showing that the statement does not extend to index 4 in the orthogonal case.
Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum of 2-dimensional torsion forms. This is obtained from a more general study of weakly isotropic forms over henselian valued fields and over function fields in one variable.
Over an arbitrary field of characteristic different from $2$ admitting an anisotropic torsion $3$-fold Pfister form, we apply a construction due to Merkurjev to produce an algebra with orthogonal involution of degree $6$ which admits proper projective similitudes that are not $R$-trivial. In particular, such examples exist over every finitely generated transcendental extension of a local or global number field, as well as over every finitely generated extension of transcendence degree $3$ of $\mathbb{R}$.
For a central simple algebra with a symplectic involution (A,s) over a field of characteristic different from 2, we show that its group of projective similitudes PSim(A,s) is R-trivial in two new cases.
This article studies the set of R-equivalence classes of the group of proper projective similitudes of an algebra with involution of the first kind. The main results concern base fields of characteristic different from 2 over which every 9-dimensional quadratic form has a nontrivial zero. This includes function fields of p-adic curves and extensions of transcendence degree 3 of C. Main results of [28] and [29] are extended by relaxing the condition on the base field as well as on the Clifford invariant for orthogonal involutions.
For a quadratic form $φ$ over a field of characteristic different from $2$, we study whether its group of proper projective similitudes ${\bf PSim}^+(φ)$ is rationally connected (i.e. $R$-trivial). We obtain new sufficient conditions in terms of structure properties of $φ$. We further provide new examples of quadratic forms $φ$ belonging to a given power of the fundamental ideal in the Witt ring and such that ${\bf PSim}^+(φ)$ is not rationally connected.
Given a finite group $G$ of order $n.$ Denote the sum of the inverse-power of element orders in $G$ by $m(G).$ Let $\mathbb{Z}_n$ be the cyclic group of order $n.$ Suppose $G$ is a non-cyclic group of order $n$ then we show that $m(G)\geq \frac{5}{4}m(\mathbb{Z}_n).$ Our result improves the inequality $m(G)>m(\mathbb{Z}_n)$ obtained by Baniasad Azad, M., and Khorsravi B. Moreover, this bound is best as for $n=4l, l$ odd, there exists a group $G$ of order $n$ satisfying $m(G)=\frac{5}{4}m(\mathbb{Z}_n)$. Moreover, we will establish that $\frac{1}{q-1}m(G)< m(\mathbb{Z}_n)\leq \frac{4}{5}m(G),$ where $G$ is a non-cyclic group of odd order.
Let $F$ be the function field of a smooth, geometrically integral curve over a $p$-adic field with $p\neq 2.$ Let $G$ be a classical adjoint group of type $^1D_n$ defined over $F$. We show that $G(F) / R$ is trivial, where $R$ denotes {\it rational equivalence} on $G(F)$.