Sam Frengley

Email: sam (dot) frengley (at) bristol (dot) ac (dot) uk
https://orcid.org/0000-0002-8904-6253/
https://arxiv.org/a/0000-0002-8904-6253/
https://github.com/SamFrengley/

About me

Hi, I am a postdoc at the University of Bristol. Previously I was a PhD student at the University of Cambridge where my supervisor was Tom Fisher. I am originally from New Zealand and previously held a Woolf Fisher Scholarship.

My research interests are in arithmetic geometry and number theory, including computational aspects.

Publications and preprints

• Galois groups of low dimensional abelian varieties over finite fields,
  • joint with Santiago Arango-Piñeros and Sameera Vemulapalli,
  • preprint, arXiv:2412.03358. [Abstract] [arXiv] [BibTex] [Code]
• On the geometry of the Humbert surface of square discriminant,
  • preprint, arXiv:2408.09830. [Abstract] [arXiv] [BibTex] [Code]
• Explicit 7-torsion in the Tate-Shafarevich groups of genus 2 Jacobians,
  • to appear in J. Théor. Nombres Bordeaux,
  • arXiv:2405.11693. [Abstract] [arXiv] [BibTex] [Code]
• Generic models for genus 2 curves with real multiplication,
  • joint with Alex Cowan and Kimball Martin,
  • preprint, arXiv:2403.03191. [Abstract] [arXiv] [BibTex] [Code]
• Appendix A: An example with 7-torsion in Ш,
  • joint with Timo Keller and Michael Stoll,
  • appendix to Timo Keller and Michael Stoll, Complete verification of strong BSD for many modular abelian surfaces over Q,
  • preprint, arXiv:2312.07307. [arXiv]
• An algorithm for efficient detection of (N,N)-splittings and its application to the isogeny problem in dimension 2,
  • joint with Maria Corte-Real Santos and Craig Costello,
  • Public-Key Cryptography (PKC) 2024 (Best paper award), LNCS, vol. 14603. Springer, Cham. [Abstract] [ePrint] [BibTex] [Code]
  • An excellent accessible blog post by Maria Corte-Real Santos.
  • MR 4763489, DOI 10.1007/978-3-031-57725-3_6
• On 12-congruences of elliptic curves,
  • Int. J. Number Theory 20 (2024), no. 02, 565-601. [Abstract] [arXiv] [BibTex] [Code]
  • MR 4709643, Zbl 07819768, DOI 10.1142/S1793042124500301
• Congruences of elliptic curves arising from non-surjective mod N Galois representations,
  • Mathematics of Computation 92 (2023), no. 339, 409-450. [Abstract] [arXiv] [BibTex] [Code]
  • MR 4496970, Zbl 07603636, DOI 10.1090/mcom/3770
• Explicit moduli spaces for curves of genus 1 and 2,
  • PhD Thesis, University of Cambridge (2023). [Permalink] [BibTex] [Code] [Code permalink]
  • DOI 10.17863/CAM.109127

BibTex for the above papers (or as a .bib file).

Software

• humbert
  • This package is attached to my paper "On the geometry of the Humbert surface of square discriminant" but provides more functionality than is required there. It provides some functionality to depict intersections of divisors on Hilbert modular surfaces of square discriminant.
  • 

Slides

• On the geometry of the Humbert surface of square discriminant,
  • Joint Meeting of the NZMS, AustMS and AMS, December 2024. [Slides]
• An algorithm for efficient detection of (N,N)-splittings and its application to the isogeny problem in dimension 2,
  • PKC, April 2024. [Slides]
• On the geometry of the Humbert surface of discriminant N²,
  • YRANT, September 2023. [Slides]
• N-congruences between quadratic twists of elliptic curves,
  • YRANT, August 2021. [Slides]
• ℓ-adic Galois representations and Weil-Deligne representations,
  • my talk from this reading group on automorphy lifting theorems. [Slides]

If you've seen me talk and there were no slides, send me an email and I can give you the notes I was working off.

Teaching

• Algebraic surfaces,
  • TCC, Autumn 2024,
  • Course notes. [pdf] [html]

Reading groups

• Bhargava's proof of van der Waerden's conjecture,
  • Michaelmas 2022. [Schedule]

Misc

• Some Magma I find useful sometimes:
  • Some code which relates LMFDB and Cremona labels in Magma.
  • It can be nice to automatically attach your personal intrinsics. This can be done with user startup specification files. You can e.g., create a ~/.global_spec spec file, followed by adding the command export MAGMA_USER_SPEC="~/.global_spec" to your .bashrc file.
• Check out Maths Craft, if you're based in New Zealand they run events, and in any case they have some fun resources.
• A record which remains standing from the first test match.