arXiv subjects

Biswanath Samanta

Publications and source records attributed to Biswanath Samanta.

4 recordsLinked to original sources

arXiv2026

On the Vanishing of the Brauer-Manin Obstruction for Normic Bundles

We study the behaviour of the Brauer--Manin obstruction to the existence of rational points under finite field extensions. For $(p, mp)$-normic bundles over number fields, we prove that the Brauer--Manin obstruction vanishes after base change to finite extensions whose degrees satisfy suitable $p$-divisibility conditions depending on $m$. We further show that, for $(p,2p)$-normic bundles with $p=2$ or $3$, it is enough to assume that the extension degree is divisible by $p$. We also prove that the divisibility hypothesis is, in general optimal, by constructing a conic bundle for which the Brauer--Manin obstruction persists over a quadratic extension.

math.NT↗

arXiv2026

A universal construction of $p$-typical Witt vectors of associative rings

For a prime $p$ and an associative ring $R$ with unity, there are various constructions of $p$-typical Witt vectors of $R$, all of which specialize to the classical $p$-typical Witt vectors when $R$ is commutative. These constructions are endowed with a Verschiebung operator $V$ and a Teichm\"{u}ller map $\langle \cdot \rangle$, and they satisfy the property that the map $x \mapsto V\langle x^p\rangle - p\langle x \rangle$ is additive. In this paper, we adapt the group-theoretic universal characterization of classical $p$-typical Witt vectors proposed in arXiv:2405.12680 to the non-commutative setting. Our main result is that this approach yields a construction of Witt vectors for associative rings, denoted $E$, which specializes correctly to the classical Witt functor in the commutative case. The construction of $E$ is inspired by the Witt functor of Cuntz--Deninger, and we show that $E$ is a universal pre-Witt functor, subject to an explicit conjecture concerning non-commutative polynomials. We further introduce the notion of a Witt functor and construct a universal Witt functor $\hat{E}$, which is closely related to Hesselholt's Witt functor $W_H$. We suspect that $W_H$ is, in fact, the universal Morita-invariant Witt functor.

math.NT↗

arXiv2024

Pullback Method with Applications to Severi--Brauer Fibrations

Given a variety with a suitable Brauer class, we present a general pullback construction that produces varieties that has Brauer--Manin obstruction to the existence of rational points. We then study Severi--Brauer fibrations and their Brauer groups without relying on explicit defining equations. As a key application, we show that there exist Severi--Brauer fibrations with index one that fails Hasse principle.

math.NT↗

arXiv2024

A universal group-theoretic characterisation of $p$-typical Witt vectors

For a prime $p$ and a commutative ring $R$ with unity, let $W(R)$ denote the group of $p$-typical Witt vectors. The group $W(R)$ is endowed with a Verschiebung operator $V: W(R)\to W(R)$ and a Teichm\"{u}ller map $\langle \ \rangle: R\rightarrow W(R)$. One of the properties satisfied by $V, \langle \ \rangle$ is that the map $R \to W(R)$ given by $x\mapsto V\langle x^p \rangle - p\langle x \rangle$ is an additive map. In this paper we show that for $p\neq 2$, this property essentially characterises the functor $W$. Unlike other characterisations, this is a group-theoretic characterisation, in the sense that it does not use the ring structure of $W(R)$. Most constructions of the group of $p$-typical Witt vectors of non-commutative rings do not have a ring structure, and hence the above characterisation is more suitable for generalisation to the non-commutative setup.

math.NT↗