Special values of Goss \(L\)-series attached to Drinfeld modules of rank 2 (Q2240427)

Let \({\mathbb F}_q\) be the finite field of \(q\)-elements and let \(A={\mathbb F}_q [\theta]\) be the polynomial ring over \({\mathbb F}_q\). The goal of this paper is, for a Drinfeld \(A\)-module \(\phi\) of rank \(2\) with certain conditions on its coefficients, to give explicit formulas for the values of the 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 \(\phi\). Let \(\phi_{\theta}=\theta +a \tau+b\tau^2\) be a Drinfeld \(A\)-module of rank \(2\), \(a\in {\mathbb F}_q\), \(b\in {\mathbb F}_q^*\). Then \(L(M_{\phi},n+1)\) is given explicitly for any positive integer \(n\) such that \(2n+1\leq q\), where \(M_{\phi}\) is a certain effective \(t\)-motive and \(L\) is Taelman's \(L\)-series. This is Theorem 5.9, the main result of this paper. The paper runs as follows. The author defines the \(t\)-module \(G_n\) given by the tensor product of a Drinfeld \(A\)-module of rank \(2\) and the \(n\)-th tensor power of the Carlitz module. Next, certain entries of the coefficients of the logarithm series \(\mathrm{Log}_{G_n}\) of \(G_n\) are analyzed. In Section 4, the unit module \(U(G_n/A)\) of \(G_n\) is introduced and the generators of \(U(G_n/A)\) as \(A\)-module are given in terms of the values of \(\mathrm{Log}_{G_n}\) at some algebraic points. In Section 5, the results of \textit{B. Anglès} et al. [Adv. Math. 372, Article ID 107313, 32 p. (2020; Zbl 1458.11095)] are applied to the construction given in this paper.