# Christopher Birkbeck

I am a final year PhD student, working in Number Theory under the supervision of Lassina Dembele and John Cremona. I am working on cases of Langlands p-adic functoriality over totally real fields. In particular, I have extended Chenevier's overconvergent Jacquet-Langlands correspondence to totally real fields and then used this to compute slopes of overconvergent Hilbert modular forms near the centre and the boundary of the weight space.

Email: c (dot) d (dot) birkbeck @warwick.ac.uk

**Publications/Preprints:**

-Extensions of vector bundles on the Fargues-Fontaine curve (joint with Tony Feng, David Hansen, Serin Hong, Qirui Li, Anthony Wang, Lynnelle Ye) (arxiv, pdf)

**-**Slopes of overconvergent Hilbert modular forms via the Jacquet-Langlands correspondence (arxiv , pdf)

**Code: **

Here is some code to compute the slopes in the inert case. This is the case done in my thesis and in the preprint above.

**Talks:**

-Number Theory Seminar, University of Warwick, 09/01/2017.** **

-Seminaire d'arithmetique, ENS de Lyon, 12/05/2016.

**Old stuff**

**PhD year 1 Report:** Here is my first year report which is to do with the Jacquet--Langlands correspondence. It first defines automorphic forms and then uses them to define Hilbert modular forms. It then goes on to talk about quarnionic modular forms and the (classical) Jacquet--Langlands correspondence. This is very rough, so please read at your own risk!

**Msci Project**: This is my fourth year project that I did at Imperial College under the supervion of Prof. Kevin Buzzard. It is based on a paper by R. Langlands called Representations of Abelian Algebraic Groups. Basically it is setting the ground work for what is known as the abelian or GL1 case of the Langlands program. In my project I generalize his results to work for the GL1 case of the p-adic Langlands program. It starts with a short exposition on Group Cohomology and a very brief introduction to Class Field Theory, and it ends with a short explanation of how Langlands ideas can be used to generalize Class Field Theory. There are still some typos so read at your on risk!

Langlands Correspondence for Algebraic Tori.

I once computed a table of congruences for primes which guarantee that a prime p can be written in the form x^2+ny^2, for the 65 known values of n for which it is known to be possible to find such a condition. The table is here.

Also in my first year as an undergrad student I found myself needing to factor the following 150 digit number. It took my laptop 1 month of running at 100% to complete and it died shortly after, so in its memory I have the number and it factors:

9209927592397944934920379845651267773097680921924769291116603583273617641456927248867317822193720416353231538795723470673266303828 99130751718966390493

Factors: 979863517748281070738228384371465939352816287338433450546979231984403057791 and 939919430163324037588145448416699718267248619754167542463564282295611164323