arXiv subjects

Rony A. Bitan

Publications and source records attributed to Rony A. Bitan.

7 recordsLinked to original sources

arXiv2025

Geometric Gauss-Dedekind

Gauss and Dedekind have shown a bijection between the set of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of primitive positive definite binary quadratic $\mathbb{Z}$-forms of the discriminant of $\mathbb{Q}(\sqrt{Δ<0})$ and the class group of its ring of integers. Using étale cohomology we show an analogue of this correspondence in the positive characteristic. This leads to the description of the set of genera and to another result analogous to Gauss' one by which any form composed with itself belongs to the principal genus.

math.AG↗

arXiv2021

The twisted forms of a semisimple group over an $\mathbb{F}_q$-curve

Let $C$ be a smooth, projective and geometrically connected curve defined over a finite field $\mathbb{F}_q(C)$. Given a semisimple $C-S$-group scheme $\underline{G}$ where $S$ is a finite set of closed points of $C$, we describe the set of ($\mathcal{O}_S$-classes of) twisted forms of $\underline{G}$ in terms of geometric invariants of its fundamental group $F(\underline{G})$.

math.AG↗

arXiv2019

On the flat cohomology of binary norm forms

Let $\mathcal{O}$ be an order of index $m$ in the maximal order of a quadratic number field $k=\mathbb{Q}(\sqrt{d})$. Let $\underline{\mathbf{O}}_{d,m}$ be the orthogonal $\mathbb{Z}$-group of the associated norm form $q_{d,m}$. We describe the structure of the pointed set $H^1_{\mathrm{fl}}(\mathbb{Z},\underline{\mathbf{O}}_{d,m})$, which classifies quadratic forms isomorphic (properly or improperly) to $q_{d,m}$ in the flat topology. Gauss classified quadratic forms of fundamental discriminant and showed that the composition of any binary $\mathbb{Z}$-form of discriminant $Δ_k$ with itself belongs to the principal genus. Using cohomological language, we extend these results to forms of certain non-fundamental discriminants.

math.NT↗

arXiv2018

On the genera of semisimple groups defined over an integral domain of a global function field

Let $K=\mathbb{F}_q(C)$ be the global function field of rational functions over a smooth and projective curve $C$ defined over a finite field $\mathbb{F}_q$. The ring of regular functions on $C-S$ where $S \neq \emptyset$ is any finite set of closed points on $C$ is a Dedekind domain $\mathcal{O}_S$ of $K$. For a semisimple $\mathcal{O}_S$-group $\underline{G}$ with a smooth fundamental group $\underline{F}$, we aim to describe both the set of genera of $\underline{G}$ and its principal genus (the latter if $\underline{G} \otimes_{\mathcal{O}_S} K$ is isotropic at $S$) in terms of abelian groups depending on $\mathcal{O}_S$ and $\underline{F}$ only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain $\underline{G}$. We also use it to express the Tamagawa number $τ(G)$ of a semisimple $K$-group $G$ by the Euler Poincaré invariant. This facilitates the computation of $τ(G)$ for twisted $K$-groups.

math.AG↗

arXiv2017

Between the genus and the $Γ$-genus of an integral quadratic $Γ$-form

Let Γbe a finite group and (V,q) be a regular quadratic Γ-form defined over an integral domain $\mathcal{O}_S$ of a global function field (of odd characteristic). We use flat cohomology to classify the quadratic Γ-forms defined over $\mathcal{O}_S$ that are locally Γ-isomorphic for the flat topology to (V,q) and compare between the genus c(q) and the Γ-genus c_Γ(q) of q. We show that c_Γ(q) should not inject in c(q). The suggested obstruction arises from the failure of the Witt cancellation theorem for $\mathcal{O}_S$.

math.AG↗

arXiv2017

On the classification of quadratic forms over an integral domain of a global function field

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Any finite set $S$ of closed points of $C$ gives rise to an integral domain $\mathcal{O}_S:=\mathbb{F}_q[C-S]$ in $K$. We show that given an $\mathcal{O}_S$-regular quadratic space $(V,q)$ of rank $n \geq 3$, the set of genera in the proper classification of quadratic $\mathcal{O}_S$-spaces isomorphic to $(V,q)$ in the flat or étale topology, is in $1:1$ correspondence with ${_2\text{Br}}(\mathcal{O}_S)$, thus there are $2^{|S|-1}$ such. If $(V,q)$ is isotropic, then $\text{Pic}(\mathcal{O}_S)/2$ classifies the forms in the genus of $(V,q)$. For $n \geq 5$ this is true for all genera, hence the full classification is via the abelian group $H^2_{\text{ét}}(\mathcal{O}_S,\underlineμ_2)$.

math.AG↗

arXiv2016

The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Removing one closed point $C^\text{af} = C-\{\infty\}$ results in an integral domain $\mathcal{O}_{\{\infty\}} = \mathbb{F}_q[C^\text{af}]$ of $K$, over which we consider a non-degenerate bilinear and symmetric form $f$ with orthogonal group $\underline{\textbf{O}}_V$. We show that the set $\text{Cl}_\infty(\underline{\textbf{O}}_V)$ of $\mathcal{O}_{\{\infty\}}$-isomorphism classes in the genus of $f$ of rank $n>2$, is bijective as a pointed set to the abelian groups $H^2_{\text{ét}}(\mathcal{O}_{\{\infty\}},\underlineμ_2) \cong \text{Pic}(C^\text{af})/2$, i.e. is an invariant of $C^\text{af}$. We then deduce that any such $f$ of rank $n>2$ admits the local-global Hasse principal if and only if $|\text{Pic}(C^\text{af})|$ is odd. For rank $2$ this principle holds if the integral closure of $\mathcal{O}_{\{\infty\}}$ in the splitting field of $\underline{\textbf{O}}_V \otimes_{\mathcal{O}_{\{\infty\}}} K$ is a UFD.

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