On Furusho's analytic continuation of Drinfeld logarithms (Q6562900)

In this article, the author studies the analytic continuation of Drinfeld logarithms by using \(t\)-frames of Drinfeld modules and the techniques of \textit{H. Furusho} [Tunis. J. Math. 4, No. 3, 559--586 (2022; Zbl 1498.11226)].\N\NFor elliptic curves, the Weierstraß \(\wp\)-function and its derivative define the curve, while the exponential function \(\operatorname{exp}_{\Lambda}\) associated to a lattice \(\Lambda\) provides an isomorphism between \(\mathbb{C}/\Lambda\) and \(E_{\Lambda}(\mathbb{C})\). Moreover, one can construct the inverse of the complex analytic group homomorphism \(\operatorname{exp}_{\Lambda_E}^{-1}: E(\mathbb{C}) \to \mathbb{C}/\Lambda\) by using the analytic continuation of an elliptic integral of the first kind. The author addresses a similar problem in the context of Drinfeld modules, which serve as analogues of elliptic curves in the function field setting.\N\NNotation: Let \(\mathbb{P}^1_{\mathbb{F}_q}\) be the projective line over \(\mathbb{F}_q\), a finite field with \(q\) elements of characteristic \(p\). Fix a closed point \(\infty\) on \(\mathbb{P}^1_{\mathbb{F}q}\). Let \(A\) be the ring of regular functions outside \(\infty\), and let \(k\) be its fraction field. Denote by \(k_{\infty}\) the completion of \(k\) with respect to the normalized norm associated to \(\infty\). Let \(\mathbb{C}_{\infty}\) denote the completion of an algebraic closure of \(k_{\infty}\).\N\NFor a given \(A\)-lattice \(\Lambda\) in, one can define an exponential morphism \(\operatorname{exp}_{\Lambda}\) and a Drinfeld module \(E\). Moreover, we have an isomorphism \(\operatorname{exp}_{\Lambda}: \mathbb{C}_{\infty}/\Lambda \xrightarrow{\sim} E^{\Lambda}(\mathbb{C}_{\infty})\). In fact, the converse is also true. Given a Drinfeld module \(E\), there is a unique formal power series \(\operatorname{exp}_E\) that gives an entire, surjective, \(\mathbb{F}_q\)-linear function on \(\mathbb{C}_{\infty}\), from which one obtains the period lattice \(\Lambda_E = \ker(\operatorname{exp}_E)\). Furthermore, one can define a formal inverse \(\operatorname{log}_E\) of the formal power series \(\operatorname{exp}_E\), which is called the Drinfeld logarithm. However, the formal power series \(\operatorname{log}_E\) is not defined on all of \(\mathbb{C}_{\infty}\). Inspired by the classical setting, this paper finds a solution to this problem and defines an analytic continuation of \(\operatorname{log}_E\) so that one can define the inverse isomorphism \(\operatorname{exp}^{-1}: E(\mathbb{C}_{\infty}) \xrightarrow{\sim} \mathbb{C}_{\infty}/\Lambda\).\N\NThe author first uses the \(t\)-frame of Anderson's dual \(t\)-motives, which is also recalled in the present paper with examples in Section 2, to construct the deformation series of the Drinfeld logarithm and provide explicit formulas. Then, the author uses the constructed deformation series together with an adaptation of Furusho's ideas to construct the analytic continuation of \(\operatorname{log}\).