Weil Operators and Remainders of Anderson Generating Functions over a Punctured Projective Line
Abstract
Let Aρ be the coordinate ring of the projective line over Fq with a closed point of degree N removed. We give finite formulas for higher Weil operators in a basis of Riemann--Roch spaces, which recover the formulas of Hu--Ou--Yang when N=1. We identifythese operators with representatives of the residue Casimir tensors and obtain a determinant formula for the Weil pairing, relative to a chosen determinant realization. For a normalized rank-one Drinfeld module, we compute the remainder of its Anderson generating function modulo any nonconstant p∈ Aρ. The formula includes multiple zeros and a zero at the ordinary point at infinity. Its last coefficient is a nonzero scalar multiple of an exponential division value. When (p) is a finite principal prime, a value of the associated special function at one zero of p generates the torsion field after adjoining that zero. We give an explicit q-linear formula recovering a torsion generator from this value. We also specialize the evaluation theorem of Gazda--Maurischat and compute the different as the principal ideal (∂xp).
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BibTeX
@misc{hu2026weil,
title = {{Weil Operators and Remainders of Anderson Generating Functions over a Punctured Projective Line}},
author = {Chuangqiang Hu and Xiao-Min Huang and Guorui Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41131},
eprint = {2610.41131},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41131},
note = {CC0},
}
RIS
TY - UNPB
TI - Weil Operators and Remainders of Anderson Generating Functions over a Punctured Projective Line
AU - Chuangqiang Hu
AU - Xiao-Min Huang
AU - Guorui Ma
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41131
UR - https://arxiv.org/abs/2610.41131
AB - Let A_ρ be the coordinate ring of the projective line over Fq with a closed point of degree N removed. We give finite formulas for higher Weil operators in a basis of Riemann--Roch spaces, which recover the formulas of Hu--Ou--Yang when N=1. We identifythese operators with representatives of the residue Casimir tensors and obtain a determinant formula for the Weil pairing, relative to a chosen determinant realization. For a normalized rank-one Drinfeld module, we compute the remainder of its Anderson generating function modulo any nonconstant p∈ A_ρ. The formula includes multiple zeros and a zero at the ordinary point at infinity. Its last coefficient is a nonzero scalar multiple of an exponential division value. When (p) is a finite principal prime, a value of the associated special function at one zero of p generates the torsion field after adjoining that zero. We give an explicit q-linear formula recovering a torsion generator from this value. We also specialize the evaluation theorem of Gazda--Maurischat and compute the different as the principal ideal (∂_xp).
ER -