Pith Number

pith:LHJYC4BF

pith:2026:LHJYC4BF4BFC6JRHCZRJOI2SAZ

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Factorization of Additive Polynomials and van der Geer--van der Vlugt curves in characteristic 2

Daichi Takeuchi, Takahiro Tsushima, Tetsushi Ito

Factorization of additive polynomials yields a simpler formula for the Frobenius eigenvalues of van der Geer--van der Vlugt curves in characteristic 2.

arxiv:2605.16729 v1 · 2026-05-16 · math.NT

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Claims

C1strongest claim

We prove a new formula for the Frobenius eigenvalues using factorization of additive polynomials. The resulting formula is simpler and is useful for explicit computations. As applications, we provide a method for constructing maximal and minimal van der Geer--van der Vlugt curves, and show that every such curve arises from this construction.

C2weakest assumption

Factorization of the relevant additive polynomials yields a uniform description of the Frobenius eigenvalues that does not reintroduce the many auxiliary choices present in the quotient approach.

C3one line summary

New simpler formula for Frobenius eigenvalues of vdGV curves in char 2 via additive polynomial factorization, enabling construction of all maximal and minimal curves.

References

13 extracted · 13 resolved · 0 Pith anchors

[1] R. Blache and T. Pierre,Zeta functions of quadratic Artin–Schreier curves in characteristic two, Acta Arith.207(2023), No. 1, 39–56 2023

[2] D. W. Bump,Automorphic forms and representations, Cambridge Studies in Advanced Mathe- matics, 55, Cambridge Univ. Press, Cambridge, 1997 1997

[3] R. S. Coulter,The number of rational points of a class of Artin–Schreier curves, Finite Fields Appl.8(2002), No. 4, 397–413 2002

[4] Goss,Basic Structures of Function Field Arithmetic, Springer, 1996 1996

[5] van der Geer and M 1992

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:02:38.772400Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

59d3817025e04a2f26271662972352064e900bef92ec3367e2b73459e7be3b6b

Aliases

arxiv: 2605.16729 · arxiv_version: 2605.16729v1 · doi: 10.48550/arxiv.2605.16729 · pith_short_12: LHJYC4BF4BFC · pith_short_16: LHJYC4BF4BFC6JRH · pith_short_8: LHJYC4BF

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/LHJYC4BF4BFC6JRHCZRJOI2SAZ \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 59d3817025e04a2f26271662972352064e900bef92ec3367e2b73459e7be3b6b

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "bd1ec37e4eab8b802846fac6d96bc7c7896fe3ef89922eeacf2ed40c5fefa36f",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-16T00:42:22Z",
    "title_canon_sha256": "54968d40271ee25357ad6617fed3027c2ba1a9034237f3a6349c8819a72f9241"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16729",
    "kind": "arxiv",
    "version": 1
  }
}