{
  "abstract": "Arithmetic terms are fixed finite compositions of addition, truncated subtraction, multiplication, integer division and exponentiation on natural numbers. We construct such terms for the number of affine solutions of $ y^2 = x^3 + Ax + B $. For arbitrary moduli $ n \\geq 1 $, a specialization of Prunescu's general construction reduces the number of monomial contributions from fourty-nine to fifteen and the size of the packed integer from approximately $ 2n^{11} $ to approximately $ 2n^5 $ binary digits. For prime moduli $ p \\geq 17 $, the Hasse invariant and the trace of Frobenius give a term of about thirty operations. For curves defined over $ \\mathbb F_p $, a further term counts the points over $ \\mathbb F_{p^k} $ with $ k $ as a variable. For primes $ p \\geq 5 $, we also obtain arithmetic terms for the counts over $ \\mathbb Z / p^k \\mathbb Z $ with $ k $ variable, covering good reduction, nodes and cusps, including non-minimal equations and zero discriminant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05371",
  "author_total": 2,
  "authors": [
    {
      "name": "Bogdan Dumitru"
    },
    {
      "name": "Mihai Prunescu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cfk2fmr8rq55q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05371",
  "fulltext_url": null,
  "id": "cfk2fmr8rq55q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05371",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7211905508",
  "pdf_url": "https://arxiv.org/pdf/2609.05371",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cfk2fmr8rq55q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.05371",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05371"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields",
  "title_text": "Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/cfk2fmr8rq55q",
  "venue": "arXiv",
  "versions": []
}