Sam Frengley

Email: stf32 (at) cam (dot) ac (dot) uk
https://orcid.org/0000-0002-8904-6253/
https://github.com/SamFrengley/

About me

Hi, I am a PhD student at the University of Cambridge. My supervisor is Tom Fisher.
I am originally from New Zealand and am currently funded by a Woolf Fisher Scholarship.

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

Publications and preprints

• Generic models for genus 2 curves with real multiplication
  • joint with Alex Cowan and Kimball Martin,
  • preprint. [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]
• 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, LNCS, vol. 14603. Springer, Cham. Best paper award. [Abstract] [ePrint] [BibTex] [Code]
  • An excellent accessible blog post by Maria Corte-Real Santos.
• On 12-congruences of elliptic curves,
  • Int. J. Number Theory 20 (2024), no. 02, 565-601. [Abstract] [arXiv] [BibTex] [Code]
  • MR 4709643, 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

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

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.

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.