arXiv subjects

Andreas Maurischat

Publications and source records attributed to Andreas Maurischat.

At least 19 recordsLinked to original sources

arXiv2026

On purity of Anderson t-modules

In the theory of abelian Anderson $t$-modules, pure Anderson $t$-modules play an important role. Namoijam and Papanikolas also introduced the notions of strictly pure and almost strictly pure $t$-modules which are special classes of pure $t$-modules. Whereas it is well known that not all pure $t$-modules are isomorphic to strictly pure ones (e.g.~the tensor powers of the Carlitz module), the same question for almost strictly pure $t$-modules was not answered yet. In this article, we show that every pure Anderson $t$-module defined over a field $K$ is indeed isomorphic to an almost strictly pure Anderson $t$-module after base change to a suitable finite algebraic extension of $K$. We also give a necessary and sufficient criterion when such an isomorphism to a strictly pure Anderson $t$-module is possible.

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arXiv2025

Determining $t$-motives and dual $t$-motives in Anderson's theory

Anderson t-modules are analogs of abelian varieties in positive characteristic. Associated to such a t-module, there are its t-motive and its dual t-motive. When dealing with these objects, several questions occur which one would like to solve algorithmically. For example, for a given t-module one would like to decide whether its t-motive is indeed finitely generated free, and determine a basis. Reversely, for a given object in the category of t-motives one would like to decide whether it is the t-motive associated to a t-module, and determine that t-module. In this article, we positively answer such questions by providing the corresponding algorithms. As it turned out, the main part of all these algorithms stem from a single algorithm in non-commutative algebra, and hence the first part of this article doesn't deal with Anderson's objects at all, but are results on finitely generated modules over skew polynomial rings.

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arXiv2025

Pairing Anderson motives via formal residues in the Frobenius endomorphism

Anderson modules form a generalization of Drinfeld modules and are commonly understood as the counterpart of abelian varieties but with function field coefficients. In an attempt to study their ``motivic theory'', two objects of semilinear algebra are attached to an Anderson module: its motive and its dual motive. While the former is better suited to follow the analogy with Grothendieck motives, the latter has proven much useful in the study of transcendence questions in positive characteristic. Despite sharing similar definitions, the relationship between motives and dual motives has remained nebulous. Over perfect fields, it was only proved recently by the second author that the finite generation of the motive is equivalent to the finite generation of the dual motive, answering a long-standing open question in function field arithmetic (the ``abelian equals $A$-finite'' theorem). This work constructs a perfect pairing among the motive and the dual motive of an Anderson module, with values in a module of differentials, thus answering a question raised by Hartl and Juschka. Our construction involves taking the residue of certain formal power series in the Frobenius endomorphism. Although it may seem peculiar, this pairing is natural and compatible with base change. It also comes with several new consequences in function field arithmetic; for example, we generalize the ``abelian equals A-finite'' theorem to a large class of algebras, including fields, perfect algebras and noetherian regular domains.

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arXiv2024

Non-abelian Anderson A-modules: Comparison isomorphisms and Galois representations

In this manuscript, we consider non-abelian Anderson $A$-modules $E$ (of generic characteristic). The main results are on the structure of their motives, and on comparison isomorphisms between their cohomological realizations. In the center of these comparison isomorphisms, there is the space of special functions $\mathfrak{sf}(E)$ as defined by Gazda and the author in arXiv:1903.07302. We also provide a generalization of Anderson's result on the equivalence of uniformizability of the Anderson module and rigid analytic triviality of its associated motive. We contribute results that are new even in the case of abelian Anderson modules. For every non-zero prime ideal $\mathfrak{p}$ of $A$, the relation of the space of special functions to the $\mathfrak{p}$-adic Tate module provides a way to obtain $\mathfrak{p}^{n+1}$-torsion as special values of hyperderivatives of these special functions. Using this result for uniformizable Anderson modules, we are able to describe the $\mathfrak{p}$-adic Galois representation via a rigid analytic trivialization, and hence give a direct link between the image of the $\mathfrak{p}$-adic Galois representation and the motivic Galois group. This generalizes results of various authors.

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arXiv2022

Carlitz twists: their motivic cohomology, regulators, zeta values and polylogarithms

The integral $t$-motivic cohomology and the class module of a (rigid analytically trivial) Anderson $t$-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz $t$-motive, namely, the function field counterpart $\underline{A}(n)$ of Tate twists $\mathbb{Z}(n)$. We find out that these modules are in relation with fundamental objects of function field arithmetic: integral $t$-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denominator of Bernoulli-Carlitz numbers and the Fitting ideal of class modules is a special zeta value. We also express the regulator of $\underline{A}(n)$ for positive $n$ in terms of generalized Carlitz polylogarithms; after establishing their algebraic relations using difference Galois theory together with the Anderson-Brownawell-Papanikolas criterion, we prove that the regulator is an isomorphism if, and only if, $n$ is prime to the characteristic.

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arXiv2022

Residue of special functions of Anderson $A$-modules at the characteristic graph

Let $E$ be an Anderson $A$-module over $\mathbb{C}_{\infty}$. The period lattice of $E$ is related to its module of special functions by means of a non-canonical isomorphism introduced by the authors in [GM21]. In this paper, we explain how a modification of the inverse map is canonical by interpreting it as a residue morphism along the characteristic graph. This phenomenon has already been observed in various situations. The main innovation of this text is that of costability (costable admissible opens, costable site, etc.) which provides a convenient framework to develop the notion of sheaves of $E(\mathbb{C}_{\infty})$-valued meromorphic functions on the rigid analytic plane.

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arXiv2021

Abelian equals A-finite for Anderson A-modules

Anderson introduced t-modules as higher dimensional analogs of Drinfeld modules. Attached to such a t-module, there are its t-motive and its dual t-motive. The t-module gets the attribute "abelian" when the t-motive is a finitely generated module, and the attribute "t-finite" when the dual t-motive is a finitely generated module. The main theorem of this article is the affirmative answer to the long standing question whether these two attributes are equivalent. The proof relies on an invariant of the t-module and a condition for that invariant which is necessary and sufficient for both being abelian and being t-finite. We further show that this invariant also provides the information whether the t-module is pure or not. Moreover, we conclude that also over general coefficient rings A, i.e. for Anderson A-modules, the attributes of being abelian and being A-finite are equivalent.

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arXiv2021

Algebraic independence of the Carlitz period and its hyperderivatives

This paper deals with the fundamental period $\tilde{\pi}$ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field $\mathbb{F}_q(\theta)$. Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in $\tilde{\pi}$ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here.

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arXiv2020

Taylor coefficients of Anderson generating functions and Drinfeld torsion extensions

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the $\mathfrak{p}$-adic Tate module lies in the $\mathfrak{p}$-adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the $t$-adic case.

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arXiv2019

Anderson t-modules with thin t-adic Galois representations

Pink has given a qualitative answer to the Mumford-Tate conjecture for Drinfeld modules in the 90s. He showed that the image of the v-adic Galois representation is v-adically open in the motivic Galois group for any prime v. In contrast to this result, we provide a family of uniformizable Anderson t-modules for which the Galois representations of their t-adic Tate-modules are "far from" having t-adically open image in their motivic Galois groups. Nevertheless, the image is still Zariski-dense in the motivic Galois group which is in accordance to the Mumford-Tate conjecture. For the proof, we explicitly determine the motivic Galois group as well as the Galois representation for these t-modules.

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arXiv2019

Reduced group schemes as iterative differential Galois groups

This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which is finitely generated over its algebraically closed field of constants. We also introduce the notion of equivalence of iterative derivations on a given field - a condition which implies that the inverse Galois problem over equivalent iterative derivations are equivalent.

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arXiv2019

Special Functions and Gauss-Thakur Sums in Higher Rank and Dimension

Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 80s where they were at the heart of comparison isomorphisms, further important applications e.g. to transcendence theory have only been discovered recently. The Anderson-Thakur special function interpolates L-values via Pellarin-type identities, and its values at algebraic elements recover Gauss-Thakur sums, as shown by Angl\`es and Pellarin. For Drinfeld-Hayes modules, generalizations of Anderson generating functions have been introduced by Green-Papanikolas and -- under the name of `special functions' -- by Angl\`es-Ngo Dac-Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A-module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A-motive. We also undertake the study of Gauss-Thakur sums for Anderson A-modules, and show that the result of Angl\`es-Pellarin relating values of the special functions to Gauss-Thakur sums holds in this generality.

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arXiv2019

On field extensions given by periods of Drinfeld modules

In this short note, we answer a question raised by M. Papikian on a universal upper bound for the degree of the extension of $K_\infty$ given by adjoining the periods of a Drinfeld module of rank 2. We show that contrary to the rank 1 case such a universal upper bound does not exist, and the proof generalises to higher rank. Moreover, we give an upper and lower bound for the extension degree depending on the valuations of the defining coefficients of the Drinfeld module. In particular, the lower bound shows the non-existence of a universal upper bound.

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arXiv2018

Periods of $t$-modules as special values

In this article we show that all periods of uniformizable $t$-modules (resp. their coordinates) can be obtained via specializing a rigid analytic trivialization of a related dual $t$-motive at $t=\theta$. The proof is even constructive. The central object in the construction is a subset $H$ of the Tate algebra points of $E$ which turns out to be isomorphic to the period lattice of $E$ via kind of generating series in one direction and residues in the other. This isomorphism even holds for arbitrary $t$-modules $E$, even non-abelian ones.

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arXiv2017

Prolongations of t-motives and algebraic independence of periods

In this article we show that the coordinates of a period lattice generator of the $n$-th tensor power of the Carlitz module are algebraically independent, if $n$ is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for $t$-motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another ingredient is a theorem which shows hypertranscendence for the Anderson-Thakur function $\omega(t)$, i.e. that $\omega(t)$ and all its hyperderivatives with respect to $t$ are algebraically independent.

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arXiv2016

An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions

Let $\mathfrak{p}$ be a monic irreducible polynomial in $A:=\mathbb{F}_q[\theta]$, the ring of polynomials in the indeterminate $\theta$ over the finite field $\mathbb{F}_q$, and let $\zeta$ be a root of $\mathfrak{p}$ in an algebraic closure of $\mathbb{F}_q(\theta)$. For each positive integer $n$, let $\lambda_n$ be a generator of the $A$-module of Carlitz $\mathfrak{p}^n$-torsion. We give a basis for the ring of integers $A[\zeta,\lambda_n] \subset K(\zeta, \lambda_n)$ over $A[\zeta] \subset K(\zeta)$ which consists of monomials in the hyperderivatives of the Anderson-Thakur function $\omega$ evaluated at the roots of $\mathfrak{p}$. We also give an explicit field normal basis for these extensions. This builds on (and in some places, simplifies) the work of Angl\`es-Pellarin.

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arXiv2016

A chain rule formula for higher derivations and inverses of polynomial maps

The multidimensional chain rule formula for analytic functions and its generalisation to higher derivatives perfectly work in the algebraic setting in characteristic zero. In positive characteristic one runs into problems due to denominators in these formulas. In this article we show a direct analog of these formulas using higher derivations which are defined in any characteristic. We also use these formulas to show how higher derivations to different coordinate systems are related to each other. Finally, we apply this to polynomial automorphisms in arbitrary characteristic and obtain a formula for the inverse of such a polynomial automorphism.

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arXiv2015

Non-free iterative differential modules

In the article "Picard-Vessiot theory of differentially simple rings" we established a Picard-Vessiot theory over differentially simple rings which may not be fields. Differential modules over such rings were proven to be locally free but do not have to be free as modules. In this article, we give a family of examples of non-free differential modules, and compute Picard-Vessiot rings as well as Galois groups for them.

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