Pellarin's Identity, Carlitz Period, and Anderson Generating Functions over Arbitrary Curves

Source abstract

Let $A$ be the coefficient ring of a smooth projective curve over $\mathbb{F}_q$ with a closed point at infinity of arbitrary degree $\mathrm{N}=°(\infty)\ge 1$, and let $\varphi$ be a rank-one Drinfeld $A$-module. In this paper, we prove three main results concerning the module of special functions and its relation to Pellarin's $\mathcal L(1)$-series. First, the module $\mathrm{sf}(\varphi)$ of special functions contains a Tate-algebra unit exactly when it is free of rank one, equivalently when its period lattice is isomorphic to the module of regular differentials. This settles the Gazda--Maurischat conjecture. The proof evaluates Cauchy kernels; on the period lattice all such evaluations agree, and their common characteristic residue recovers the period. Second, for suitable isogeny data to a principal-period target $ψ$, a strictly normalized shtuka product, together with one characteristic residue, gives an explicit formula for the fundamental period $\tildeπ$ (the generator of the free lattice $Λ_ψ$). The same defect equation also yields the residue formula for the Drinfeld logarithm. Third, the first Frobenius twist of the normalized shtuka differential is identified with a pairing built from finite torsion traces, whose coefficients are finite étale traces over the field of definition. As an application, Pellarin's $\mathcal L(1)$-series is expressed exactly as the first Frobenius twist of the generating differential at a fundamental period. A notable feature of these results is that they hold for arbitrary $\mathrm{N} \geq 1$, and the source period lattice need not be free.