arXiv subjects

Yen-Tsung Chen

Publications and source records attributed to Yen-Tsung Chen.

16 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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arXiv2026

Linear equations on $t$-modules

Let $F$ be a number field. Given finitely many $F$-valued points on a commutative algebraic group defined over $F$, a question of interest to number theorists is the determination of the group of their linear relations. In this article, we investigate an analogous problem in the $t$-module setting. Let $L$ be a global function field, and $E$ be a $d$-dimensional $t$-module defined over $L$. Given finitely many points on $E$ with entries in $L$, we establish the connection between their $\mathbb{F}_q[t]$-linear relations and polynomial solutions of Frobenius difference equations. Consequently, we deduce an algorithm to compute the module of their $\mathbb{F}_q[t]$-linear relations.

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arXiv2026

On the partial derivatives of Drinfeld modular forms of arbitrary rank

In this paper, we obtain an analogue of the Serre derivation acting on the product of spaces of Drinfeld modular forms which generalizes the differential operator introduced by Gekeler in the rank two case. We further introduce a finitely generated algebra $\mathcal{M}_r$ containing all the Drinfeld modular forms for the full modular group and show its stability under the partial derivatives.

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arXiv2025

On special values of meromorphic Drinfeld modular forms of arbitrary rank at CM points

In the present paper, we introduce meromorphic Drinfeld modular forms of arbitrary rank equipped with a particular arithmeticity property. We also study their special values at CM points and show the algebraic independence of these values under some conditions. Our results may be seen as a generalization of Chang's results on the special values of arithmetic Drinfeld modular forms in the rank two setting.

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arXiv2025

Linear relations among algebraic points on tensor powers of the Carlitz module

In the present paper, we study linear equations on tensor powers of the Carlitz module using the theory of Anderson dual $t$-motives and a detailed analysis of a specific Frobenius difference equation. As an application, we derive some explicit sufficient conditions for the linear independence for Carlitz polylogarithms at algebraic points in both $\infty$-adic and $v$-adic settings.

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arXiv2025

Analytic continuation of Kochubei multiple polylogarithms and its applications

In the present paper, we propose an analytic continuation of Kochubei multiple polylogarithms using the techniques developed by Furusho. Moreover, we produce a family of linear relations and a linear independence result for values of our analytically continued Kochubei polylogarithms at algebraic elements from a cohomological aspect.

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arXiv2024

On the third kind periods for abelian $t$-modules

Inspired by the relations between periods of elliptic integrals of the third kind and the periods of the extensions of the corresponding elliptic curves by the multiplicative group, we introduce the notion of the third kind periods for abelian $t$-modules and establish an evaluation for these periods that is parallel to the classical setting. When we specialize our result to the case of Drinfeld modules, an explicit formula for these third kind periods is established. We also prove the algebraic independence of periods of the first, the second, and the third kind for Drinfeld modules of arbitrary rank. This generalizes prior results of Chang for rank $2$ Drinfeld modules.

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arXiv2023

Nearly holomorphic Drinfeld modular forms and their special values at CM points

In the present paper, we introduce the notion of nearly holomorphic Drinfeld modular forms and study an analogue of Maass-Shimura operators in this context. Furthermore, for a given nearly holomorphic Drinfeld modular form, we show that its special values at CM points are algebraically independent whenever the associated endomorphism algebras are distinct. As an application of our results on nearly holomorphic Drinfeld modular forms, we study Drinfeld quasi-modular forms for arbitrary congruence subgroups and investigate the structure of the vector spaces and the algebras generated by them.

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arXiv2023

A $v$-adic variant of Anderson-Brownawell-Papanikolas linear independence criterion and its application

Let $\overline{k}$ be a fixed algebraic closure of $k$. When the finite place $v$ is of degree one, we show that all $\overline{k}$-linear relations among $v$-adic Carlitz multiple polylogarithms at algebraic points arise from $k$-linear relations among these values of the same weight. As an application, we establish a function field analogue of Furusho-Yamashita's conjecture for $v$-adic multiple zeta values whenever the degree of the place $v$ is one.

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arXiv2022

On Furusho's analytic continuation of Drinfeld logarithms

In the present paper, we establish an analytic continuation of Drinfeld logarithms by using the techniques introduced in [Fur20]. This result can be seen as an analogue of the analytic continuation of the elliptic integrals of the first kind for Drinfeld modules.

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arXiv2022

Linear equations on Drinfeld modules

Let $L$ be a finite extension of the rational function field over a finite field $\mathbb{F}_q$ and $E$ be a Drinfeld module defined over $L$. Given finitely many elements in $E(L)$, this paper aims to prove that linear relations among these points can be characterized by solutions of an explicitly constructed system of homogeneous linear equations over $\mathbb{F}_q[t]$. As a consequence, we show that there is an explicit upper bound for the size of the generators of linear relations among these points. This result can be regarded as an analogue of a theorem of Masser for finitely many $K$-rational points on an elliptic curve defined over a number field $K$.

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arXiv2022

On Thakur's basis conjecture for multiple zeta values in positive characteristic

In this paper, we study multiple zeta values (abbreviated as MZV's) over function fields in positive characteristic. Our main result is to prove Thakur's basis conjecture, which plays the analogue of Hoffman's basis conjecture for real MZV's. As a consequence, we derive Todd's dimension conjecture, which is the analogue of Zagier's dimension conjecture for classical real MZV's.

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arXiv2021

On Drinfeld modular forms of higher rank and quasi-periodic functions

In the present paper, we introduce a special function on the Drinfeld period domain $Ω^{r}$ for $r\geq 2$ which gives the false Eisenstein series of Gekeler when $r=2$. We also study its functional equation and relation with quasi-periodic functions of a Drinfeld module as well as transcendence of its values at CM points.

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arXiv2020

On lower bounds of the dimensions of multizeta values in positive characteristic

In this paper, we study the linear independence of special values, including the positive characteristic analogue of multizeta values, alternating multizeta values and multiple polylogarithms, at algebraic points. Consequently, we establish linearly independent sets of these values with the same weight indices and a lower bound on the dimension of the space generated by depth r > 2 multizeta values of the same weight in positive characteristic.

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arXiv2020

Algebra structure of multiple zeta values in positive characteristic

This paper is a culmination of [CM20] on the study of multiple zeta values (MZV's) over function fields in positive characteristic. For any finite place $v$ of the rational function field $k$ over a finite field, we prove that the $v$-adic MZV's satisfy the same $\bar{k}$-algebraic relations that their corresponding $\infty$-adic MZV's satisfy. Equivalently, we show that the $v$-adic MZV's form an algebra with multiplication law given by the $q$-shuffle product which comes from the $\infty$-adic MZV's, and there is a well-defined $\bar{k}$-algebra homomorphism from the $\infty$-adic MZV's to the $v$-adic MZV's.

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arXiv2020

Integrality of $v$-adic multiple zeta values

In this article, we prove the integrality of $v$-adic multiple zeta values (MZVs). For any index $\mathfrak{s}\in\mathbb{N}^r$ and finite place $v\in A:=\mathbb{F}_q[θ]$, Chang and Mishiba introduced the notion of the $v$-adic MZVs $ζ_A(\mathfrak{s})_v$, which is a function field analogue of Furusho's $p$-adic MZVs. By estimating the $v$-adic valuation of $ζ_A(\mathfrak{s})_v$, we show that $ζ_A(\mathfrak{s})_v$ is a $v$-adic integer for almost all $v$. This result can be viewed as a function field analogue of the integrality of $p$-adic MZVs, which was proved by Akagi-Hirose-Yasuda and Chatzistamatiou.

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