{
  "abstract": "We introduce refinements of the theory of modular symbols for a subgroup Γ of SL 2 (Z). The group H\n\n] by the Manin relations. Our more general refinement consists in replacing the Manin relations by what we call the Lewis relation. We apply this first to Γ = Γ 0 (N ), for N prime. The Hecke algebra acting on refined modular symbols has an Eisenstein ideal I L , which refines Mazur's classical situation. If N ̸ ≡ 1 (mod 8), we have\n\nThe second application concerns Γ = Γ 1 (N ), for any integer N . Sharifi has constructed a map Π\n\n, where ζ N is a primitive N -th root of unity. Sharifi and Venkatesh have established that such a map is Eisenstein. Relying on their work, we construct a lift of Π • N on refined modular symbols to K 2 (Z[ζ N , 1/N ]) that retains the Eisenstein property.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Emmanuel Lecouturier"
    },
    {
      "name": "Loïc Merel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b4nm0bevcjpq6.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "b4nm0bevcjpq6",
  "issns": [],
  "landing_url": "https://hal.science/hal-05752071v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05752071/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b4nm0bevcjpq6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:35Z",
      "native_id": "hal-05752071",
      "source": "hal",
      "url": "https://hal.science/hal-05752071v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "REFINED MODULAR SYMBOLS AND REFINED EISENSTEIN IDEALS",
  "title_text": "REFINED MODULAR SYMBOLS AND REFINED EISENSTEIN IDEALS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:35Z",
  "url": "https://research.draconitic.com/paper/b4nm0bevcjpq6",
  "venue": "HAL",
  "versions": []
}