Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 41 min ago
PreprintarXiv licencearXiv

Improved bounds for Szpiro's conjecture

  • Hector Pasten
Open at sourcePDFDOICite

Abstract

In the direction of Szpiro's conjecture for all elliptic curves E over ℚ, the current best bound is log Δ≪ N log N where Δ is the absolute value of the minimal discriminant of E and N is the conductor of E (the implicit constant is absolute). This bound dates back to 2013. We obtain the stronger bound log Δ≪ Nlog log N which was previously known only under GRH. In addition, for semistable elliptic curves we prove log Δ≪ N. Our methods build on the theory developed by the author for approaching the abc conjecture via Shimura curves.

Related

  1. Improved bounds for the two-point logarithmic Chowla conjecture
    Cédric Pilatte
    Journal of the American Mathematical SocietyPeer-reviewedLicence not statedFull text22 citedAlgebra & number theoryOpenAlex
  2. Improved Bounds on the Szeged-Wiener Gap and the BKLPS Conjecture
    Lily Zhang, Evan Li
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. Counterexamples to a conjecture and improved bounds for bipanchromatic colorings
    Stefan Novaković, Duško Jojić
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. Improved Bounds for the Bilu-Linial Conjecture via Spectral Recovery from Mixed Determinantal Polynomials
    Fangfang Lin, Hong Zhou
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Improved bounds in Birch's theorem for forms in many variables
    Yijie Diao, Lena Wurzinger
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{pasten2026improved,
  title = {{Improved bounds for Szpiro's conjecture}},
  author = {Hector Pasten},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17390},
  eprint = {2609.17390},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17390},
}
RIS
TY  - UNPB
TI  - Improved bounds for Szpiro's conjecture
AU  - Hector Pasten
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17390
UR  - https://arxiv.org/abs/2609.17390
AB  - In the direction of Szpiro's conjecture for all elliptic curves E over ℚ, the current best bound is log Δ≪ N log N where Δ is the absolute value of the minimal discriminant of E and N is the conductor of E (the implicit constant is absolute). This bound dates back to 2013. We obtain the stronger bound log Δ≪ Nlog log N which was previously known only under GRH. In addition, for semistable elliptic curves we prove log Δ≪ N. Our methods build on the theory developed by the author for approaching the abc conjecture via Shimura curves.
ER  -