Chris Xu

You can find me at HSS 3073.
My email is chx007(at)(four letter school name)(dot)edu.

About me

As of Spring 2026, I am a 5th year graduate student at University of California San Diego.
I work in algebraic number theory, under Kiran Kedlaya and Aaron Pollack.
This fall, I will be at Tsinghua University working as a postdoc under Koji Shimizu.
Here is my CV.

Papers

1. Algorithmic modular curve Chabauty-Coleman without equations, PhD thesis, in preparation
2. Rigorous expansions of modular forms at CM points, I: Denominators, submitted to LuCaNT
3. Skelet #17 and the fifth Busy Beaver number, preprint

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. (For Shimura curves things are even easier because you already have a p-adic uniformization with which you can then apply Chabauty to, modular curves only enjoy this near the cusps.) Also, low weight Eisenstein series give you all the cusp forms, have nice expressions for Hecke action, and have a nice moduli interpretation thanks to work of Khuri-Makdisi and Mascot.
• 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.
• You want to prove Serre uniformity in very general settings. (The conjecture gives bounds for how large a prime p you can go before the mod p image of Galois for any elliptic curve has to surject.) Does a version of the Eisenstein ideal exist for Edixhoven-Lido construction? Perhaps the most interesting curves to consider are X_0^*(N) for N squarefree and X_ns^+(N) for N prime, where quadratic Chabauty methods are required.
• 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.
• Develop a Mazur's Program B for Shimura varieties of higher dimension. Develop generalizations of Chabauty for higher-dimensional varieties; currently, most of the work here is done for symmetric powers of curves, for which Siksek and co. have pioneered.

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?
• Replicate Khuri-Makdisi's moduli interpretation of Eisenstein series for Hilbert modular forms.

What I'm learning about

• (Spring 2026, current) Alterations. See here.
• (Winter 2025) Analytic stacks. Attempting to understand Camargo Rodriguez's paper on the analytic de Rham stack. Also, hopefully the real local Langlands manuscript.
• (Winter 2025) Modular curves. Katz-Mazur. Trying to learn this thoroughly for obvious purposes (see above).
• (Winter 2025) Representations of p-adic groups. For the purposes of Arizona Winter School 2025.

Teaching

• Spring 2026 (current): Math 20E
• Winter 2026: Math 20D
• Fall 2025: Math 120A
• Spring 2026: Math 20E
• Winter 2025: 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

1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
4. Every student deserves to be treated with dignity and respect.