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Supriya Pisolkar

Publications and source records attributed to Supriya Pisolkar.

13 recordsLinked to original sources
arXiv2026

Power Integral Bases in Polynomial Compositions

In this paper, we study the monogeneity of a special class of composed polynomials of the form $ (f \circ g)(x) = (x^m + c)^n + a(x^m + c)^{n-1} + d(x^m + c)^{n-2} + b,$ where \( f(x) = x^n + a x^{n-1} + d x^{n-2} + b \in \mathbb{Z}[x] \) satisfies \( a^2 = 4d \) and \( g(x) = x^m + c \in \mathbb{Z}[x] \). Assuming that \( (f \circ g)(x) \) is irreducible over \( \mathbb{Q} \), we obtain necessary and sufficient conditions on the parameters \( a, b, c, d, m, n \) for the polynomial to be monogenic. These conditions help to identify when the set \( \{1, θ, \dots, θ^{mn-1}\} \) forms an integral basis of the number field \( \mathbb{Q}(θ) \), where \( θ\) is a root of \( (f \circ g)(x) \). We also provide lower bound for the counting of such monogenic polynomials. Furthermore, we study the behaviour of solutions to certain related differential equations and present a class of polynomials with non-square-free discriminants as an application of the main results.

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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 Teichmüller 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.

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arXiv2026

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 Teichmüller 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.

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arXiv2022

From Fontaine-Mazur conjecture to analytic pro-p groups -- A survey

Fontaine-Mazur Conjecture is one of the core statements in modern arithmetic geometry. Several formulations were given since its original statement in 1993, and various angles have been adopted by numerous authors to try to tackle it. Boston's seminal paper in 1992 gave a range of purely group-theoretic methods rather than representation-theoretic ones to prove some special cases of this conjecture. Such methods have been later successfully carried on by Maire and his co-authors, and brings different informations on the objects involved in the conjecture. This survey article aims to review what is known in this direction and to present some interesting related questions the authors work on.

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arXiv2020

Morphisms between two constructions of Witt vectors of non-commutative rings

Let $A$ be any unital associative, possibly non-commutative ring and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be the abelian group constructed by Hesselholt. In arXiv:1708.04065 it was proved that if $ p=2$ and $A$ is non commutative unital torsion free ring then there is no surjective continuous group homomorphism from $W(A) \to HH_0(E(A)): = E(A)/\overline{[E(A),E(A)]}$ which commutes with the Verschiebung operator and the Teichmüller map. In this paper we generalise this result to all primes $p$ and simplify the arguments used for $p=2$. We also prove that if $A$ a is non-commutative unital ring then there is no continuous map of sets $HH_0(E(A)) \to W(A)$ which commutes with the ghost maps.

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arXiv2017

On the comparison of two constructions of Witt vectors of non-commutative rings

Let $A$ be any associative ring , possibly non-commutative, and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be that constructed by Hesselholt. The goal of this paper is to answer the following question by Hesselholt: Is $HH_0(E(A)) $ isomorphic to $W(A)$? We show that in the case $p=2$, there is no such isomorphism possible if one insists it to be compatible with the Verscheibung operator and the Teichmüller map.

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arXiv2015

On uniform lattices in real semisimple groups

In this article we prove that the co-compactness of the arithmetic lattices in a connected semisimple real Lie group is preserved if the lattices under consideration are representation equivalent. This is in the spirit of the question posed by Gopal Prasad and A. S. Rapinchuk where instead of representation equivalence, the lattices under consideration are weakly commensurable Zariski dense subgroups.

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arXiv2015

On the splitting fields of generic elements in Zariski dense subgroups

Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $Γ$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the commensurability class of the field $\mathcal{F}$ given by the compositum of the splitting fields of characteristic polynomials of generic elements of $Γ$ determines the group $G$ upto isogeny over the algebraic closure of $K$.

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arXiv2012

Commensurability and representation equivalent arithmetic lattices

Gopal Prasad and A. S. Rapinchuk defined a notion of weakly commensurable lattices in a semisimple group, and gave a classification of weakly commensurable Zariski dense subgroups. A motivation was to classify pairs of locally symmetric spaces isospectral with respect to the Laplacian on functions. For this, in higher ranks, they assume the validity of Schanuel's conjecture. In this note, we observe that if we use the stronger notion of representation equivalence of lattices, then Schanuel's conjecture can be avoided. Further, the results are also applicable in a $S$-arithmetic setting. We also introduce a new relation on the class of arithmetic lattices, stronger than weak commensurability, which we call as characteristic equivalence, and show that it simplifies some of the arguments used in Prasad and Rapinchuk (2009) to deduce commensurability type results from weak commensurability.

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arXiv2012

Remark on equicharacteristic analogue of Hesselholt's conjecture on cohomology of Witt vectors

Let $L/K$ be a finite Galois extension of complete discrete valued fields of characteristic $p$. Assume that the induced residue field extension $k_L/k_K$ is separable. For an integer $n\geq 0$, let $W_n(\sO_L)$ denote the ring of Witt vectors of length $n$ with coefficients in $\sO_L$. We show that the proabelian group ${H^1(G,W_n(\sO_L))}_{n\in \N}$ is zero. This is an equicharacteristic analogue of Hesselholt's conjecture.

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arXiv2010

On cohomology of Witt vectors of algebraic integers and a conjecture of Hesselholt

Let $K$ be a complete discrete valued field of characteristic zero with residue field $k_K$ of characteristic $p > 0$. Let $L/K$ be a finite Galois extension with the Galois group $G$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. In his paper, Hesselholt conjectured that $H^1(G,W(\sO_L))$ is zero, where $\sO_L$ is the ring of integers of $L$ and $W(\sO_L)$ is the Witt ring of $\sO_L$ w.r.t. the prime $p$. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt's conjecture for all Galois extensions.

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arXiv2008

Absolute norms of p-primary units

We prove a local analogue of a theorem of J. Martinet about the absolute norm of the relative discriminant ideal of an extension of number fields. The result can be seen as a statement about 2-primary units. We also prove a similar statement about the absolute norms of p-primary units, for all primes p.

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