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Special values of Goss $L$-series attached to Drinfeld modules of rank 2
Number Theory · Submitted 28 May 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, for a fixed choice of a power $q$ of a prime number and a given Drinfeld module $\phi$ of rank 2 with a certain condition on its coefficients, we give explicit formulas for the values of Goss $L$-series attached to $\phi$ at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$.
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The abstract, sentence by sentence
Inspired by the classical setting, Goss defined $L$-series attached to Drinfeld modules. In this paper, for a fixed choice of a power $q$ of a prime number and a given Drinfeld module $\phi$ of rank 2 with a certain condition on its coefficients, we give explicit formulas for the values of Goss $L$-series attached to $\phi$ at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Inspired by the classical setting, Goss defined $L$-series attached to Drinfeld modules. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | In this paper, for a fixed choice of a power $q$ of a prime number and a given Drinfeld module $\phi$ of rank 2 with a certain condition on its coefficients, we give explicit formulas for the values of Goss $L$-series attached to $\phi$ at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$. | result 0.51 | 0.97 | 0.95 | asserted 1.00 |
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Original abstract
Inspired by the classical setting, Goss defined $L$-series attached to Drinfeld modules. In this paper, for a fixed choice of a power $q$ of a prime number and a given Drinfeld module $\phi$ of rank 2 with a certain condition on its coefficients, we give explicit formulas for the values of Goss $L$-series attached to $\phi$ at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$.
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Using [ref] and a similar argument as in [ref] , we have [equation] Note that $N^m=0$ when $m≥n+1$ and $N^j-m=0$ if $j-m≥n+1$.
Deformation of Multiple Zeta Values and Their Logarithmic Interpretation in Positive Characteristic · The Analysis on the Logarithm series $_G_n$ · 0.81
Cites as background 1 paper
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The logarithm series $_G∈_d(L)[[τ]]$ of $G$ which is the formal inverse of the exponential series is given by [equation] with respect to the condition [equation] Similar to the exponential series, the logarithm series $_G$ induces the logarithm function which is an $𝔽_q$-linear homomorphism $_G:𝔻→(G)(_∞)$ defined by [equation] where $𝔻$ is a subset of $G(_∞)$ consisting elements in the domain of the convergence of $_G$ (see [ref] for details).
Pink's theory of Hodge structures and the Hodge conjecture over function fields · The $t$-module $G_n$ · 0.93
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