Online Junior Number Theory Seminar

- Motivation -

A few years ago, a rather naive idea came to my mind: As graduate students, we spend a lot of time working on mini-projects, Master’s and PhD theses, or simply following our noses and exploring whatever catches our interest – so why not share these with each other?

Motivated by this, I started a small and somewhat irregular colloquium with friends, where we took turns talking about what we had been thinking about. It turned out to be a surpisingly nice and efficient way to get exposed to exciting bits of mathematics.

In this sense, the present seminar is a continuation of those earlier, informal activities. As I’ve grown (hopefully) and changed over time, the focus of the seminar has naturally shifted along with me. Now, the idea is to invite junior researchers in number theory and arithmetic geometry to introduce their research projects. There is absolutely no need to present new results – the goal is simply to keep things clear, accessible, and enjoyable.

More than anything, it is a chance for us to get together, chat a bit, and enjoy some mathematics.

Logistics

  • Time: Fridays 9am EST, 2pm GMT, 3pm CET, and 9pm CST
  • Zoom id: 949 924 7490
  • Recording: available on request

We are quite flexible when it comes to scheduling, and speakers are free to design their talks as they wish :)


- Upcoming Talks -


Confirmed speakers Autumn 2026:

  • Ignacio Muñoz Jiménez (Genova)
  • Jeremy Feusi (ETH Zürich)
  • Zhenghang Du (Regensburg)
  • Julia Meng (MIT)
  • Jiahao Niu (Stanford)


Title: TBC

26 June 2026, Felix Sefzig (University of Zurich)

Abstract:


Anti-Cyclotomic p-adic L-functions

04 June 2026, Frederick Thøgersen (Warwick)

Abstract: p-adic L-functions represent an essential part of the study of critical values of L-functions by creating devices that p-adically interpolate them. Furthermore, they prominently feature in the Iwasawa main conjectures where they control elements of the structure certain ideal class groups and related. A recent project of mine focuses on constructing p-adic L-functions for imaginary quadratic number fields where there are two flavours: the cyclotomic and anti-cyclotomic “directions”. In my talk, I will attempt to discuss (i) The motivation, (ii) My recent results for anti-cyclotomic definite unitary U_{2n} and (iii) a bit about the (“overconvergent”) method.


- Past Talks -


Arithmetic theta lifts and the Arithmetic Gan–Gross–Prasad conjecture for orthogonal groups

29 May 2026, Xinran Qian (Arizona)

Abstract: Gross and Prasad formulated a conjecture relating the central value of certain Rankin–Selberg L-function to SO(n) periods of automorphic forms on SO(n)×SO(n + 1), generalising the Waldspurger formula in the case n=2. Their conjecture was further generalized to include all classical groups in the book of Gan–Gross–Prasad. Parallel to the periods of automorphic forms, there is a conjectural generalization of the Gross–Zagier formula to higher-dimensional Shimura varieties, known as the arithmetic GGP conjecture. In my talk, I will present an equational refinement of the arithmetic Gan–Gross–Prasad conjecture for SO(3)×SO(4) in some endoscopic cases using the theory of arithmetic theta lifting, following the work of Xue in the unitary case.


Newton stratifications of integral local Shimura varieties and mod-p representations of p-adic groups

23 May 2026, Xinyu Zhou (Boston)

9:00am Chicago, 10:00am EST, 3:00pm GMT, 4:00pm CET, and 10:00pm CST

Abstract: The mod-p fiber of a moduli space of abelian varieties admits the Newton stratification based on the isocrystals of abelian varieties. This stratification is also defined for local Shimura varieties (e.g. Rapoport–Zink spaces). However, in the local case, another stratification, the generic Newton stratification, can be defined via the Fargues–Fontaine curve. I will discuss the interplay between the two stratifications and how to understand the strata via Banach–Colmez spaces. I then show some applications to mod-p representations of p-adic groups. It time permits, I will also discuss applications to chromatic homotopy theory. This is partially based on the joint work with Tobias Barthel, Lucas Mann, Rin Ray, Andrew Senger, Tomer Schlank, Jared Weinstein.


On Galois Representations associated with mod p Hilbert eigenforms

16 May 2026, Deding Yang (Chicago)

9:30am Chicago, 10:30am EST, 3:30pm GMT, 4:30pm CET, and 10:30pm CST

Abstract: Given a modular eigenform of weight k, it is well known that there exists an associated l-adic Galois representation satisfying certain compatibility conditions away from l and the level. It is then natural to ask the converse of this problem. In the mod p world, the desired weight k of which \rho is modular is encoded in the weight part of Serre’s conjecture (For the Hilbert case, this is the Buzzard-Diamond-Jarvis conjecture), also referred to as “algebraic modularity” by Diamond and Sasaki. They also defined “geometric modularity” for mod p Hilbert eigenforms, and conjectured that the two notions of “modularity” are equivalent when the weight k satisfies certain conditions. In this talk, we prove this conjecture for all quaternionic Shimura varieties. This is a joint work in progress with Siqi Yang.


Companion Forms of Picard Modular Forms and Split Local Galois Representations

08 May 2026, Haocheng Fan (Peking University)

Abstract: We show that for a Picard modular form, the existence of the companion form is equivalent to a splitting property of the associated local Galois representation. This result is obtained by computing the monodromy group of the closure of the non-ordinary Newton stratum in the special fiber of the Picard modular surface at a split prime, and then using this computation in a cohomological argument. It can be used to detect the ramification of the associated eigenvariety.


Explicit p-adic Hodge theory and p-adic monodromy group

01 May 2026, Moqing Chen (Strasbourg)

Abstract: Over a p-adic field, the p-adic Tate module of an abelian variety carries a natural Galois representation. The associated p-adic monodromy group, defined as the Zariski closure of its image, provides an important invariant along Hecke orbits on Shimura varieties of Hodge type.

In this talk, I will give several families of Galois representations arising from abelian surfaces with supersingular good reduction over Q_p, then classify the p-adic monodromy groups associated with them. I will then introduce a coarse moduli space parameterizing the corresponding p-adic Galois representations and describe how the p-adic monodromy groups distribute on this space.


Mumford–Tate Conjecture for Hyper-Kähler Varieties

24 April 2026, Haitao Zou (Bielefeld)

Abstract: The Mumford–Tate conjecture serves as a bridge between the analytic world of Hodge theory and the arithmetic world of Galois representations. While the conjecture is difficult even for abelian varieties, hyper-Kähler varieties offer a promising testing ground due to their similarity to K3 surfaces.

In this talk, I will introduce the Mumford–Tate conjecture and the geometry of hyper-Kähler varieties. I will then present recent results (joint with Zhichao Tang) showing that the conjecture holds after taking semisimplification. This generalizes previous results known only for specific deformation types.


Some Recent Progress of the Arithmetic Inner Product Formula

17 April 2026, Zhuoni Chi (Zhejiang University)

Abstract: This talk briefly introduces the arithmetic inner product formula for unitary groups and explains how it connects theta lifting, automorphic L-functions, and arithmetic cycles on Shimura varieties. After reviewing the classical picture and method, I will focus on the new difficulties that arise at ramified places with local root number -1.


Previous Edition

  • Minhua Cheng (Utah), Introduction to p-adic Hodge theory
  • Zipei Nie (Huawei - Centre de recherche Lagrange), Card Guessing Game with Partial Feedback
  • Huatao Gui (ETH Zürich), An introduction to right-angled Artin groups
  • Markus Schwagenscheidt (ETH Zürich), From generating functions to modular forms
  • Matthias Gröbner (ETH Zürich), The Riemann zeta function from an adelic perspective
  • Raphael Appenzeller (ETH Zürich), Generalized metric spaces and the Lean theorem prover
  • Francesco Naccarato (Scuola Normale Superiore Pisa), Tunnell’s Theorem and the analytic rank of elliptic curves
  • Linpu Gao (Tsinghua) Kac’s theorem and quiver representations over finite fields
  • Feusi Jeremy (ETH Zürich), Galois groups and fundamental groups, interesting properties and similarities
  • Yixuan Li (UC Berkeley), Geometric representation theory
  • Jiahao Niu (UCAS/Stanford University), Six functor formalism
  • Cunyuan Zhao (ETH Zürich), Bounded cohomology and actions on the circle
  • Zhongkai Tao (UC Berkeley), An introduction to Selberg trace formula
  • Xiangyu Pan (Peking University), Introduction to étale cohomology