# Philippe Michaud-Rodgers

I am a second-year PhD student in number theory under the supervision of Professor Samir Siksek and Dr Damiano Testa. I am especially interested in Diophantine equations, modular curves, and how the two are related.

I am currently studying ternary Diophantine equations using modularity. I have most recently worked on certain generalized Fermat equations, using modularity results over totally real fields. I have also recently worked on Fermat's Last Theorem over real quadratic fields, with a focus on quadratic points on modular curves. Please see below for papers on these topics.

During the first year of my PhD I studied quadratic points on some non-split Cartan modular curves. This led to proving that certain classes of elliptic curves over quadratic fields are modular. See below for a link to the project, as well the paper I wrote based on the findings.

Previously, I completed my MMath degree at the University of Warwick.

Email: p dot rodgers at warwick dot ac dot uk

**Papers and Preprints:**

- On some Generalized Fermat Equations of the form $x^2 + y^{2n} = z^p$ ( PDF
- A Unique Perfect Power Decagonal Number ( PDF ) ( arXiv ), June 2021, to appear in the Bulletin of the Australian Mathematical Society;
- Fermat's Last Theorem and Modular Curves over Real Quadratic Fields ( PDF ) ( arXiv ), accompanying code, February 2021;
- Quadratic Points on Non-Split Cartan Modular Curves ( published version ) ( arXiv ), accompanying code, International Journal of Number Theory, published online, July 2021.

**Talks Given:**

- Fermat's Last Theorem - Not Enough Margin! Young Researchers in Mathematics, 10th edition, University of Bristol (Online), 8th June 2021, slides and compressed slides;
- Fermat's Last Theorem and Modular Curves over Real Quadratic Fields, Seminar on Number Theory and Algebra, University of Zagreb (Online), 17th May 2021, slides and compressed slides;
- Modular Curves as Moduli Spaces, Modular Forms Study Group, University of Warwick (Online), 5th March 2021, handwritten notes;
- Fermat's Last Theorem Over Totally Real Fields, London Junior Number Theory Seminar (Online), 16th February 2021, slides and compressed slides;
- Who Wants to Be a Millionaire? (The Hard Way), Warwick Maths Society Talks Series, University of Warwick (Online), 1st December 2020, slides and compressed slides, Video;
- Global Class Field Theory: Classical Approach, Class Field Theory Study Group, University of Warwick (Online), 20th November 2020, handwritten notes;
- Fermat's Last Theorem and the Modular Method, Warwick Postgraduate Seminar (Online), 4th November 2020, slides and compressed slides;
- Congruent Numbers and Elliptic Curves, WIMP (Warwick-Imperial) Autumn 2019 Conference, University of Warwick, 30th November 2019, handwritten notes;
- Formalism of Period Rings and $B_{HT}$, $p$-adic Hodge Theory study group, University of Warwick, 15th November 2019, handwritten notes;
- Sums of Three Squares, WIMP (Warwick-Imperial) Spring 2019 Conference, Imperial College London, 9th March 2019, slides;
- Magic Squares of Squares, Warwick Maths Society Talks Series, University of Warwick, 26th February 2019, slides.

**Teaching and Other Responsibilities:**

- Referee for Journal of Number Theory, 2021;
- Exam marking (online) for the third-year module Galois Theory (MA3D5), approx. 160 questions, Term 3, 2021;
- Video transcript editing for the first-year module Linear Algebra (MA106), approx. 8 hours of video, April 2021;
- Teaching assistant (online) for the third-year module Galois Theory (MA3D5), Term 2, 2021;
- Supervisor (online) to first-year maths and philosophy undergraduates, Terms 1 and 2, 2020-2021;
- Teaching assistant (online) for the third-year module Algebraic Number Theory (MA3A6), Term 1, 2020;
- Supervisor to first-year maths undergraduates, Terms 1 and 2, 2018-2019;
- Analysis I class helper, Term 1 2016 and Term 1, 2017.

**Other Mathematical Writing:**

- Quadratic Points on Non-Split Cartan Modular Curves ( PDF ), first-year PhD project supervised by Professor Samir Siksek and Dr Damiano Testa, September 2020;
- Magic Squares of Squares ( PDF ), fourth-year undergraduate project supervised by Dr Damiano Testa, April 2019;
- Quadratic Forms and the Three-Square Theorem ( PDF ), third-year undergraduate essay supervised by Professor John Cremona, April 2018;
- Oddly Perfect ( PDF ), second-year undergraduate essay, March 2017.

**Conferences, Seminars, and Summer Schools Attended:**

- Young Researchers in Mathematics, 10th edition, University of Bristol (Online), 7th - 9th June 2021;
- Modern Breakthroughs in Diophantine Problems, BIRS Online, 31st August - 4th September 2020;
- Warwick Number Theory Seminar, University of Warwick / Online, 2019-2021;
- ROW (Reading-Oxford-Warwick) Diophantine Geometry Seminar, Online, 2020-2021;
- WIMP (Warwick-Imperial) Autumn 2019 Conference, University of Warwick, 30th November 2019;
- WIMP (Warwick-Imperial) Spring 2019 Conference, Imperial College London, 9th March 2019;
- London Mathematical Society Undergraduate Research School, University of Glasgow, 16th - 27th July, 2018.