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PreprintCC BY 4.0arXiv

Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$

  • Saeed Tafazolian
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Abstract

Let k be an algebraically closed field of characteristic $2$ and let q=2h, h≥3. We construct ordinary bielliptic curves X of genus q+1 for which Aut(X)≅ Dih(Cq)× C2, |Aut(X)|=4q=4(g(X)−1). These curves realize case \textup{(ib)} in the classification of Giulietti--Korchmáros and give an infinite family answering a problem posed by Korchmáros. More generally, we prove that every curve in case \textup{(ib)} arises from the same construction. Case \textup{(ib)} occurs exactly in genera g=2h+1 with h≥2. For g≥9 we determine the full automorphism group; in genus $5$ we determine its Sylow $2$-subgroup but do not claim the full automorphism group. For every fixed q≥8, the isomorphism classes in case \textup{(ib)} of genus q+1 are parametrized bijectively by (k×)2. The construction is described in terms of an ordinary elliptic curve and an invariant differential. We determine the short orbits and ramification, the unramified cyclic quotients, and the quotients by the central involutions. For q=8 an explicit plane model over F2 is given.

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Cite

BibTeXRIS
BibTeX
@misc{tafazolian2026nakajimaextremal,
  title = {{Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$}},
  author = {Saeed Tafazolian},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14830},
  eprint = {2609.14830},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14830},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$
AU  - Saeed Tafazolian
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14830
UR  - https://arxiv.org/abs/2609.14830
AB  - Let k be an algebraically closed field of characteristic $2$ and let q=2^h, h≥3. We construct ordinary bielliptic curves X of genus q+1 for which Aut(X)≅ Dih(C_q)× C_2, |Aut(X)|=4q=4(g(X)−1). These curves realize case \textup{(ib)} in the classification of Giulietti--Korchmáros and give an infinite family answering a problem posed by Korchmáros. More generally, we prove that every curve in case \textup{(ib)} arises from the same construction. Case \textup{(ib)} occurs exactly in genera g=2^h+1 with h≥2. For g≥9 we determine the full automorphism group; in genus $5$ we determine its Sylow $2$-subgroup but do not claim the full automorphism group. For every fixed q≥8, the isomorphism classes in case \textup{(ib)} of genus q+1 are parametrized bijectively by (k^×)^2. The construction is described in terms of an ordinary elliptic curve and an invariant differential. We determine the short orbits and ramification, the unramified cyclic quotients, and the quotients by the central involutions. For q=8 an explicit plane model over F_2 is given.
ER  -