Pellarin's Identity, Carlitz Period, and Anderson Generating Functions over Arbitrary Curves
Abstract
Let A be the coefficient ring of a smooth projective curve over 𝔽q with a closed point at infinity of arbitrary degree N=°(∞)≥ 1, and let φ 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 ℒ(1)-series. First, the module sf(φ) 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 π̃ (the generator of the free lattice Λψ). 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 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 N ≥ 1, and the source period lattice need not be free.
Related
- Weil Operators and Remainders of Anderson Generating Functions over a Punctured Projective Line
- Pellarin Values from Cauchy Coordinates on Arbitrary Curves
- A generating function identity for finite and symmetric multiple zeta values via Dougall's bilateral 2H2 summation
- Bilinear generating functions of the multivariable Al-Salam-Carlitz polynomials and applications
- Generating Functions for Casimir Invariants of Simple Lie Algebras
BibTeX
@misc{hu2026pellarins,
title = {{Pellarin's Identity, Carlitz Period, and Anderson Generating Functions over Arbitrary Curves}},
author = {Chuangqiang Hu and Stephen S. -T. Yau and Lishan Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03134},
eprint = {2610.03134},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03134},
}
RIS
TY - UNPB
TI - Pellarin's Identity, Carlitz Period, and Anderson Generating Functions over Arbitrary Curves
AU - Chuangqiang Hu
AU - Stephen S. -T. Yau
AU - Lishan Yu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03134
UR - https://arxiv.org/abs/2610.03134
AB - Let A be the coefficient ring of a smooth projective curve over 𝔽_q with a closed point at infinity of arbitrary degree N=°(∞)≥ 1, and let φ 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 ℒ(1)-series. First, the module sf(φ) 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 π̃ (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 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 N ≥ 1, and the source period lattice need not be free.
ER -