a-Numbers, higher Cartier ranks and maximality of van der Geer-van der Vlugt curves
Abstract
Let p be an odd prime and let CR be the van der Geer-van der Vlugt curve yp−y=xR(x) over an algebraically closed field of characteristic p, where R(x)=∑i=0haixpi is an additive polynomial of degree ph. We determine the image and the rank of every power of the Cartier operator Cart on H0(CR,Omg). For 1≤ r≤ h−1 the image of Cartr is spanned by an explicit set of monomial differentials that does not depend on R, and rank(Cartr)=ph−r−1binomp+rr+2. The last iterate detects the linear coefficient: rank(Carth)=0 if a0=0 and rank(Carth)=binom(p−1)/2+hh+1 if a0≠0, so the nilpotency index of Cart is h or h+1. Hence, for h≥2, the loci a0=0 and a0≠0 have the same a-number, which is the largest a-number of a one-point Artin-Schreier curve with pole order ph+1, but they share no Ekedahl-Oort type. Over finite fields we deduce that a curve over Fp2n that is geometrically isomorphic to CR can be Fp2n-maximal or Fp2n-minimal only if n≥ h, and only if n≥ h+1 when a0≠0. Explicit examples show that the second bound is attained for (p,h)=(3,1), (3,2) and (5,2).
Related
- Maximality and configuration spaces on real algebraic curves
- Ranks and integer points on elliptic curves induced by Fibonacci triples
- Tamagawa numbers and torsion of elliptic curves over function fields
- Graded Betti numbers of general curves of large degree
- Higher order unknotting numbers for knots and links
BibTeX
@misc{filho2026anumbers,
title = {{$a$-Numbers, higher Cartier ranks and maximality of van der Geer-van der Vlugt curves}},
author = {Gilberto B. Almeida Filho and Saeed Tafazolian and Stéfani C. Vieira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17476},
eprint = {2610.17476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17476},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - a-Numbers, higher Cartier ranks and maximality of van der Geer-van der Vlugt curves
AU - Gilberto B. Almeida Filho
AU - Saeed Tafazolian
AU - Stéfani C. Vieira
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17476
UR - https://arxiv.org/abs/2610.17476
AB - Let p be an odd prime and let C_R be the van der Geer-van der Vlugt curve y^p−y=xR(x) over an algebraically closed field of characteristic p, where R(x)=∑_i=0^ha_ix^p^i is an additive polynomial of degree p^h. We determine the image and the rank of every power of the Cartier operator Cart on H^0(C_R,Omg). For 1≤ r≤ h−1 the image of Cart^r is spanned by an explicit set of monomial differentials that does not depend on R, and rank(Cart^r)=p^h−r−1binomp+rr+2. The last iterate detects the linear coefficient: rank(Cart^h)=0 if a_0=0 and rank(Cart^h)=binom(p−1)/2+hh+1 if a_0≠0, so the nilpotency index of Cart is h or h+1. Hence, for h≥2, the loci a_0=0 and a_0≠0 have the same a-number, which is the largest a-number of a one-point Artin-Schreier curve with pole order p^h+1, but they share no Ekedahl-Oort type. Over finite fields we deduce that a curve over F_p^2n that is geometrically isomorphic to C_R can be F_p^2n-maximal or F_p^2n-minimal only if n≥ h, and only if n≥ h+1 when a_0≠0. Explicit examples show that the second bound is attained for (p,h)=(3,1), (3,2) and (5,2).
ER -