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Chieh-Yu Chang

Publications and source records attributed to Chieh-Yu Chang.

At least 19 recordsLinked to original sources
arXiv2026

On Multiple Eisenstein Series in Positive Characteristic: Direct Sum Result

In this paper, we study multiple Eisenstein series (MES) in positive characteristic. By computing and analyzing the $t$-expansions of MES, we determine the precise "weights" of their coefficients. This framework enables us to establish a graded algebra structure for MES, extending the direct sum result for Thakur's multiple zeta values proved in [Cha14]. Our result may also be viewed as a function field analogue of the corresponding direct sum result of Bachmann and Kanno [BK26].

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arXiv2025

$v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited

Let $v$ be a finite place of $\mathbb{F}_q(θ)$. In this paper, we interpret $v$-adic arithmetic gamma values in terms of the $v$-adic crystalline-de Rham periods of Carlitz motives with Complex Multiplication, and establish an Ogus-type Chowla-Selberg formula. Furthermore, we prove the algebraic independence of these $v$-adic periods by employing the technique of switching "$v$ and $\infty$", and determining the dimension of relevant motivic Galois groups on the "$\infty$-adic" side through an adaptation and refinement of existing methods. As a consequence, all algebraic relations among $v$-adic arithmetic gamma values over $\mathbb{F}_q(θ)$ can be derived from standard functional equations together with Thakur's analogue of the Gross-Koblitz formula.

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arXiv2022

Function Field Analogue of Shimura's Conjecture on Period Symbols

In this paper we introduce the notion of Shimura's period symbols over function fields in positive characteristic and establish their fundamental properties. We further formulate and prove a function field analogue of Shimura's conjecture on the algebraic independence of period symbols. Our results enable us to verify the algebraic independence of the coordinates of any nonzero period vector of an abelian t-module with complex multiplication whose CM type is non-degenerate and defined over an algebraic function field. This is an extension of Yu's work on Hilbert-Blumenthal t-modules.

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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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arXiv2020

Taylor coefficients of Anderson-Thakur series and explicit formulae

For each positive characteristic multiple zeta value (defined by Thakur), the first and third authors constructed a $t$-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson-Thakur series and $t$-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.

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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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arXiv2017

Log-algebraic identities on Drinfeld modules and special L-values

We formulate and prove a log-algebraicity theorem for arbitrary rank Drinfeld modules defined over the polynomial ring F_q[theta]. This generalizes results of Anderson for the rank one case. As an application we show that certain special values of Goss L-functions are linear forms in Drinfeld logarithms and are transcendental.

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arXiv2017

On multiple polylogarithms in characteristic $p$: $v$-adic vanishing versus $\infty$-adic Eulerianness

In this paper, we give a simultaneous vanishing principle for the $v$-adic Carlitz multiple polylogarithms (abbreviated as CMPLs) at algebraic points, where $v$ is a finite place of the rational function field over a finite field. This principle establishes the fact that the $v$-adic vanishing of CMPLs at algebraic points is equivalent to its $\infty$-adic counterpart being Eulerian. This reveals a nontrivial connection between the $v$-adic and $\infty$-adic worlds in positive characteristic.

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arXiv2016

On finite Carlitz multiple polylogarithms

In this paper, we define finite Carlitz multiple polylogarithms and show that every finite multiple zeta value over the rational function field $\mathbb{F}_{q}(θ)$ is an $\mathbb{F}_{q}(θ)$-linear combination of finite Carlitz multiple polylogarithms at integral points. It is completely compatible with the formula for Thakur MZV's established in [C14].

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arXiv2016

Linear relations among double zeta values in positive characteristic

The study of this paper is inspired by the conjecture of Zagier on the explicit dimension formula for the space of the same weight double zeta values in terms of the dimension of cusp forms for SL_{2}(Z). Our main result is to devise an effective criterion for computing the dimension of the same weight double zeta values over a rational function field F_{q}(theta) in positive characteristic. Contrary to the Zagier's conjecture, the analogue of Zagier's conjectural dimension provides a lower bound for the dimension of the double zeta values when the weight is $A$-even.

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arXiv2015

An effective criterion for Eulerian multizeta values in positive characteristic

Characteristic p multizeta values were initially studied by Thakur, who defined them as analogues of classical multiple zeta values of Euler. In the present paper we establish an effective criterion for Eulerian multizeta values, which characterizes when a multizeta value is a rational multiple of a power of the Carlitz period. The resulting "t-motivic" algorithm can tell whether any given multizeta value is Eulerian or not. We also prove that if zeta_A(s_1,...,s_r) is Eulerian, then zeta_A(s_2,...,s_r) has to be Eulerian. When r=2, this was conjectured (and later on conjectured for arbitrary r) by Lara Rodriguez and Thakur for the zeta-like case from numerical data. Our methods apply equally well to values of Carlitz multiple polylogarithms at algebraic points and zeta-like multizeta values.

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arXiv2009

Geometric Gamma values and zeta values in positive characteristic

In analogy with values of the classical Euler Gamma-function at rational numbers and the Riemann zeta-function at positive integers, we consider Thakur's geometric Gamma-function evaluated at rational arguments and Carlitz zeta-values at positive integers. We prove that, when considered together, all of the algebraic relations among these special values arise from the standard functional equations of the Gamma-function and from the Euler-Carlitz relations and Frobenius p-th power relations of the zeta-function.

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arXiv2009

Periods of third kind for rank 2 Drinfeld modules and algebraic independence of logarithms

In analogy with the periods of abelian integrals of differentials of third kind for an elliptic curve defined over a number field, we introduce a notion of periods of third kind for a rank 2 Drinfeld Fq[t]-module rho defined over an algebraic function field and derive explicit formulae for them. When rho has complex multiplication by a separable extension, we prove the algebraic independence of rho-logarithms of algebraic points that are linearly independent over the CM field of rho. Together with the main result in [CP08], we completely determine all the algebraic relations among the periods of first, second and third kinds for rank 2 Drinfeld Fq[t]-modules in odd characteristic.

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arXiv2009

Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic

In analogy with the Riemann zeta function at positive integers, for each finite field F_p^r with fixed characteristic p we consider Carlitz zeta values zeta_r(n) at positive integers n. Our theorem asserts that among the zeta values in {zeta_r(1), zeta_r(2), zeta_r(3), ... | r = 1, 2, 3, ...}, all the algebraic relations are those algebraic relations within each individual family {zeta_r(1), zeta_r(2), zeta_r(3), ...}. These are the algebraic relations coming from the Euler-Carlitz relations and the Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.

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