Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 58 min ago
PreprintCC BY 4.0arXiv

Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms

  • Nicolas Swanson
Open at sourcePDFDOICite

Abstract

For a fixed characteristic p, let δ(p) denote the smallest degree needed to guarantee the existence of an isogeny from any supersingular curve E to its Frobenius conjugate E(p). We translate existence questions concerning δ(p) into questions about positive definite integral ternary quadratic forms and use Voronoi's theory of perfect forms in dimension three to characterize when δ(p) is nearly maximal. For sufficiently large primes, we show that δ(p) attains its upper bound precisely when p is represented by one of nine explicit cubic polynomials, reducing the infinitude of such primes to whether one of these cubics takes prime values infinitely often. We also prove that, for almost all primes, δ(p) lies at least p1/6−o(1) below its upper bound.

Related

  1. Iwasawa theory of CM elliptic curves at potentially supersingular primes
    Ben Forrás, Antonio Lei
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. On Sun's Conjectures for Truncated Jacobi-Symbol Determinants via Supersingular Elliptic Curves
    Guo Li, Xiaojun Yan
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. An Explicit Sextic Twist of a Supersingular Elliptic Curve in Characteristic Two
    Guillermo Sosa-Gómez
    MathematicsPeer-reviewedCC BY 4.0Geometry & topologyCrossref
  4. Higher rank elliptic sieve via CM endomorphisms
    Hang Lok Edison Au-Yeung
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Torsion points in families of elliptic curves
    Ester Sgallová
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{swanson2026supersingular,
  title = {{Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms}},
  author = {Nicolas Swanson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03839},
  eprint = {2609.03839},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03839},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Supersingular Elliptic Curves Without Inseparable Small Degree Endomorphisms
AU  - Nicolas Swanson
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03839
UR  - https://arxiv.org/abs/2609.03839
AB  - For a fixed characteristic p, let δ(p) denote the smallest degree needed to guarantee the existence of an isogeny from any supersingular curve E to its Frobenius conjugate E^(p). We translate existence questions concerning δ(p) into questions about positive definite integral ternary quadratic forms and use Voronoi's theory of perfect forms in dimension three to characterize when δ(p) is nearly maximal. For sufficiently large primes, we show that δ(p) attains its upper bound precisely when p is represented by one of nine explicit cubic polynomials, reducing the infinitude of such primes to whether one of these cubics takes prime values infinitely often. We also prove that, for almost all primes, δ(p) lies at least p^1/6−o(1) below its upper bound.
ER  -