An infinite family of counterexamples to Goss's conjecture on L-functions of cyclotomic function fields
Abstract
Let p be a prime, q = p, A = F_p[T], and let P be a monic irreducible of degree d with cyclotomic function field K_P and mod-P Teichmuller character omega_P. For a character chi = omega_P^i, Goss defined g(X, chi), the "congruent to one modulo p" part of the Artin L-function L(X, chi), and conjectured that deg_X g(X, omega_P^i) <= 1 for every q-magic index i; this is an analogue for function fields of Vandiver's conjecture, and was raised as an open problem by Angles. We disprove it. For every a in F_p^*, put P_a = T^p - T - a, irreducible of degree p, and let i = p^n - 1 with 0 <= n <= p-1, a q-magic index. We prove an exact congruence showing that the L-polynomial reduces to (1-X)^n modulo P_a, whence deg_X g = n-1. For p >= 5 and 3 <= n <= p-1 this gives deg_X g >= 2, contradicting Goss's conjecture; moreover deg_X g = n-1 is unbounded, attaining p-2.
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BibTeX
@misc{giraudin2026infinite,
title = {{An infinite family of counterexamples to Goss's conjecture on L-functions of cyclotomic function fields}},
author = {David Niedbala Giraudin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37466},
eprint = {2609.37466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37466},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An infinite family of counterexamples to Goss's conjecture on L-functions of cyclotomic function fields
AU - David Niedbala Giraudin
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.37466
UR - https://arxiv.org/abs/2609.37466
AB - Let p be a prime, q = p, A = F_p[T], and let P be a monic irreducible of degree d with cyclotomic function field K_P and mod-P Teichmuller character omega_P. For a character chi = omega_P^i, Goss defined g(X, chi), the "congruent to one modulo p" part of the Artin L-function L(X, chi), and conjectured that deg_X g(X, omega_P^i) <= 1 for every q-magic index i; this is an analogue for function fields of Vandiver's conjecture, and was raised as an open problem by Angles. We disprove it. For every a in F_p^*, put P_a = T^p - T - a, irreducible of degree p, and let i = p^n - 1 with 0 <= n <= p-1, a q-magic index. We prove an exact congruence showing that the L-polynomial reduces to (1-X)^n modulo P_a, whence deg_X g = n-1. For p >= 5 and 3 <= n <= p-1 this gives deg_X g >= 2, contradicting Goss's conjecture; moreover deg_X g = n-1 is unbounded, attaining p-2.
ER -