Chris Xu
You can find me at HSS 3073.
My email is chx007(at)(four letter school name)(dot)edu.
Here is a photo of me at the Mordell conference (center-left, with the UCSD shirt). Here is another photo of me at the ANTS XVI conference the following week (center-right).
About me
As of Winter 2025, I am a 4th year graduate student at University of California San Diego. I work in algebraic number theory. Currently I have Kiran Kedlaya and Aaron Pollack as advisors. I expect to graduate in 2027.
Here is my CV.
Papers
- Cohomology classes on G2 and their relations, in preparation
- Equationless linear Chabauty for modular curves (with Y. Huang and I. Rendell), in preparation
- Rigorous expansions of modular forms at CM points, II (with Y. Huang and I. Rendell), in preparation
- Rigorous expansions of modular forms at CM points, I: Denominators, submitted to LuCaNT
- Skelet #17 and the fifth Busy Beaver number, preprint
I have other "research papers" from my undergrad but they're not good. If you are interested you can find all of them on MIT's math website.
What I'm thinking about
Rational points on curves
- Say you have a modular curve. You want to work with the modular curve without ever needing an equation for it. It turns out you can use the moduli interpretation (elliptic curves with some level structure) and things will work out just as well. You also have a good supply of points that are defined over number fields that aren't too complicated: those are the CM points and cusps. They are dense in the modular curve and in particular cover every residue disk.
- Develop an explicit theory for Noam Kantor's thesis (see also a recent paper of Corwin-Zehavi). I like to call this the CKLV method, meaning "Chabauty-Coleman-Kantor-Kim-Lawrence-Venkatesh", where it is understood that the "C" and "K" are each referring to two people. It is basically what you would think of if you tried to combine the methods of Chabauty-Kim and Lawrence-Venkatesh. Hopefully an explicit theory will give stronger dimension bounds than simply motivic (unipotent) Chabauty, so you won't have to go to as high of a depth, and ergo save on computation effort.
- Say you have a modular curve. You want to find all of its rational points. You would like to automate this process. Here is a brief strategy for such an algorithm. Concretely, input a modular curve of arbitrary level H (call this X_H) and a number field K; output X_H(K). Step 1: find a semistable model of X_H (you can probably do this using Jared Weinstein's paper and then quotienting out by H). Step 2: perform quadratic or motivic Chabauty on X_H using Jennifer Balakrishnan's papers or David Corwin's papers. (Remark 1: you need a semistable model for both cases; also, the motives associated to X_H are all related to Res_{F_v/Q_p}GL_2, [F := End^0(J_H)], because everything is of GL_2-type, so hopefully things do not get too bad here.) (Remark 2: you might be tempted to look at the line bundle \omega of modular forms and then perform CKLV using the Tannakian category generated by \omega. But this will only work if X_H is representable i.e. there is a universal elliptic curve E -> X_H.) Filling in the blanks of this strategy is one of the ultimate end goals for explicit nonabelian Chabauty.
- Serre's uniformity conjecture for non-split Cartan modular curves (the remaining case). Recall that Bilu-Parent prove Serre uniformity for a split Cartan modular curve of prime level p by (1) bounding the j-invariant (of any point) below by something exponential in p, and (2) bounding the j-invariant (of a non-CM rational point) above by something subexponential in p. The way they prove (2) is by performing an analysis near cusps, in a way that only works in the split Cartan case. Is there a way to perform an analysis near CM points, rather than cusps? Might this be controlled by a Gross-Zagier formula, which tells you what the height pairing is on CM points?
- Michael Stoll has found a genus two curve X over \Q with at least 642 rational points. Its Mordell-Weil rank is at least 22. Can one use motivic Chabauty (involving the representation theory of GSp_4) to prove that #X(\Q) is exactly 642? This would be a good test case for Corwin's work on motivic Chabauty.
Automorphic forms
- Special cycles on G2. You can embed (the split form of) G_2 into O(4,3). From this you can perform a theta lift to the two-fold cover of SL_2. On cohomology classes of G_2, this means that you get half-integral weight modular forms whose coefficients are given by integrals of "special cycles" on G_2. Note that Kudla-Millson have already developed such a theory for orthogonal groups, and that our "special cycle" is just pulled back from the ones that they consider. But the construction that I hope to make will hopefully be simpler than their paper.
- There are three classes of discrete series representations of G2. One of them, quarternionic discrete series, has been investigated by Pollack, and there is a nice theory of Fourier coefficients. But there is a more mysterious class that Pollack, in conversations with me, has deemed "middle discrete series". What can we learn about them? For instance, can they be detected by theta lifts from simpler groups?
What I'm learning about
Some of us number theorists at UCSD have made a Discord server for the purposes of organizing seminars each quarter. In particular to every bullet point below there is a corresponding seminar ongoing right now. Please email Shubhankar Sahai if you are UCSD affiliated and interested in joining the Discord.
- Analytic stacks. Attempting to understand Camargo Rodriguez's paper on the analytic de Rham stack. Also, hopefully the real local Langlands manuscript.
- Modular curves. Katz-Mazur. Trying to learn this thoroughly for obvious purposes (see above).
- Representations of p-adic groups. For the purposes of Arizona Winter School 2025.
Teaching
- Winter 2025 (current): Math 20E
- Fall 2024: Math 104/105
- Summer 2024: Math 20E
- Spring 2024: Math 20E
- Winter 2024: Math 20D
- Fall 2023: Math 20E
- Spring 2023: Math 20E
- Winter 2023: Math 20C
- Fall 2022: Math 20B
- Spring 2022: Math 106
- Winter 2022: Math 20C
- Fall 2021: Math 20B
Federico Ardila's axioms
- Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
- Everyone can have joyful, meaningful, and empowering mathematical experiences.
- Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
- Every student deserves to be treated with dignity and respect.
Last updated: February 4th, 2025