arXiv subjects

Piotr Krasoń

Publications and source records attributed to Piotr Krasoń.

9 recordsLinked to original sources

arXiv2026

Homological Methods in the Generalization of Drinfeld Modules

We introduce and study a natural class of Anderson t- modules, called triangular t-modules, characterized by having Drinfeld modules as their $τ$-composition factors. They form a homologically meaningful generalization of Drinfeld modules and exhibit rich arithmetic structure.\smallskip We establish criteria for purity, strict and almost strict, and develop a reduction procedure that lowers the degrees of the defining biderivations. As a consequence, every almost strictly pure triangular t-module becomes strictly pure after a finite base extension. We then investigate morphisms and isogenies between triangular t-modules, provide a characterization of triangular isogenies, and describe the algebra of endomorphisms, including a criterion for commutativity. On the analytic side, we show that all triangular t- modules are uniformizable and establish finiteness and purity criteria with consequences for Taelman's conjecture. Finally, we develop a duality theory for triangular t- modules and their biderivations, proving compatibility with $τ$-composition series and establishing analogues of the Cartier-Nishi theorem and the Weil-Barsotti formula.

math.NT↗

arXiv2026

Algorithms for determination of t-module structures on some extension groups

In \cite{kk04} the second and third author extended the methods of \cite{pr} and determined the \tm module structure on $\Ext^1(Φ,Ψ)$ where $Φ$ and $Ψ$ were Anderson \tm modules over $A={\mathbf F}_q[t]$ of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper we generalize the results of \cite{pr} and \cite{kk04} and present complete algorithm for computation of \tm module structure on $\Ext^1(Φ,Ψ)$ for \tm modules $Φ$ and $Ψ$ such that ${\mathrm{deg}}Φ> {\mathrm{deg}} Ψ.$ The last condition is not sufficient for our algorithm to be executable. We show that it can be applied when the matrix at the biggest power of $τ$ in $Φ_t$ is invertible. We also introduce a notion of $τ$-composition series which we find suitable for the additive category of \tm modules and show that under certain assumptions on the composition series of $Φ$ and $Ψ$ our algorithm is also executable.

math.NT↗

arXiv2026

Duality for t- modules: The Difficult Cases

This paper continues our previous work on duality for Anderson $t$-modules. We study two dimensional triangular t-modules in a specific reduce form. Computer - assisted symbolic computations with matrices over a skew field revealed a reduction pattern for t-modules satisfying the ALD condition. Using this pattern, we prove that such t-modules are isomorphic to their double duals and extend the validity of the Cartier-Nishi theorem and the Weil-Barsotti formula to a substantially broader class of t-modules.

math.AG↗

arXiv2025

Weil-Barsotti formula for $\mathbf{T}$-modules

In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the Weil-Barsotti formula for the function field case concerning $\Ext_τ^1(E,C)$ where $E$ is a Drinfeld module and $C$ is the Carlitz module was proved. We generalize this formula to the case where $E$ is a strictly pure \tm module $Φ$ with the zero nilpotent matrix $N_Φ.$ For such a \tm module $Φ$ we explicitly compute its dual \tm module $Φ^{\vee}$ as well as its double dual $Φ^{{\vee}{\vee}}.$ This computation is done in a a subtle way by combination of the \tm reduction algorithm developed by F. Głoch, D.E. K{\k e}dzierski, P. Kraso{ń} [ Algorithms for determination of \tm module structures on some extension groups , arXiv:2408.08207] and the methods of the work of D.E. K{\k e}dzierski and P. Kraso{ń} [On $\Ext^1$ for Drinfeld modules, Journal of Number Theory 256 (2024) 97-135]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix $N_Φ$ is non-zero.

math.NT↗

arXiv2020

Note on linear relations in Galois cohomology and {é}tale $K$-theory of curves

In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In \cite{bk13} G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for {é}tale $K$-theory of a curve . This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases this result is the best possible i.e if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for {é}tale $K$-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S. Bara{ń}czuk in \cite{b17}. We show that all our results remain valid for Quillen $K$-theory of ${\cal X}$ if the Bass and Quillen-Lichtenbaum conjectures hold true for ${\cal X}.$

math.KT↗

arXiv2020

Linear relations for Lauricella $F_D$ functions and symmetric polynomial

In this paper we develop an algorithm for obtaining some new linear relations among the Lauricella $F_D$ functions. Relations we obtain, generalize those hinted in the work of B. C. Carlson. The coefficients of these relations are contained in the ring of polynomials in the variables $x_1,\dots,x_N$ or in some exceptional cases in the field of rational functions ${\mathbb R}(x_1,\dots,x_N,p)$. The method is based on expressing suitably chosen Euler type indefinite integrals associated with these functions recursively as linear combination of some other Euler type integrals and elementary functions and then integrating over the interval $[0,1].$ We describe the complete algorithm for obtaining these relations. We believe that such relations might be useful in computations.

math.CA↗

arXiv2020

On eigenproblem for inverted harmonic oscillators

We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\infty}(\mathbb R)$. The eigenfunctions are described in terms of the confluent hypergeometric functions, the spectrum is ${\mathbb C}$. The spectrum of the differential operator $-{\frac{d}{dx^2}}-ω^{2}{x^2}$ is continuous and has physical significance only for the states which are in $L^{2}(\mathbb R)$ and correspond to real eigenvalues. To identify them we use two approaches. First we define a unitary operator between $L^{2}(\mathbb R)$ and $L^{2}$ for two copies of $\mathbb R$. This operator has the property that the spectrum of the image of the inverted harmonic oscillator corresponds to the spectrum of the operator $-i{\frac{d}{dx}}$. This shows that the (generalized) spectrum of the inverted harmonic operator is real. The second approach uses rigged Hilbert spaces.

math-ph↗

arXiv2019

On a reduction map for Drinfeld modules

In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\cal O}_K$ of $t$-modules that are products of the Drinfeld modules ${\widehatφ}=ϕ_{1}^{e_1}\times \dots \times ϕ_{t}^{e_{t}}.$ Here $K$ is a finite extension of the field of fractions of $A={\mathbb F}_{q}[t].$ We assume that the ${\mathrm{rank}}(ϕ)_{i})=d_{i}$ and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. ${\mathrm{End}}(ϕ_{i})=A$ for $ i=1,\dots , t.$ Our main result is the following numeric criterion. Let ${N}={N}_{1}^{e_1}\times\dots\times {N}_{t}^{e_t}$ be a finitely generated $A$ submodule of the Mordell-Weil group ${\widehatφ}({\cal O}_{K})=ϕ_{1}({\cal O}_{K})^{e_{1}}\times\dots\times ϕ_{t}({\cal O}_{K})^{{e}_{t}},$ and let $Λ\subset N$ be an $A$ - submodule. If we assume $d_{i}\geq e_{i}$ and $P\in N$ such that $r_{\cal W}(P)\in r_{\cal W}(Λ) $ for almost all primes ${\cal W}$ of ${\cal O}_{K},$ then $P\in Λ+N_{tor}.$ We also build on the recent results of S.Bara{ń}czuk \cite{b17} concerning the dynamical local to global principle in Mordell-Weil type groups and the solvability of certain dynamical equations to the aforementioned $t$-modules.

math.NT↗

arXiv2019

Arithmetic of Heisenberg ring and cyclic group actions

In this paper we compute in some new cases the cardinalities of the fibers of certain natural fibrations that appear in the analysis of the configuration space of the Heisenberg ring. This is done by means of certain cyclic group actions on some subsets of restricted partitions.

math-ph↗