Michigan Number Theory Day

The second Michigan Number Theory Day will be held at the University of Michigan on Saturday, April 11, 2026.

Speakers

• Patrick Allen (McGill University)

• Petar Bakic (University of Utah)

• Ananth Shankar (Northwestern University)

• Mingjia Zhang (Princeton University)

Registration

Please register here by March 20. There is no fee and your registration will help us with planning (especially with ordering enough food for lunch).

Schedule

All talks will be held in East Hall Room 1360. The breaks and lunch will be in the East Hall Upper Atrium.

• 9:30am-10am: Registration and light breakfast

• 10-11am: Petar Bakic - Fourier coefficients for modular forms on G2

  • In his classic work on the Shimura correspondence, Waldspurger used the theta correspondence to relate the Fourier coefficients of certain modular forms of half-integral weight to special values of the corresponding L-functions. In this talk, we investigate an analogous formula for Fourier coefficients of modular forms on the group G2, using the exceptional theta lift from PU(3). This is the subject of a long-standing conjecture of Gross. This is joint work with A. Horawa, S.D. Li-Huerta, and N. Sweeting.

• 11-11:30am: Coffee break

• 11:30am-12:30pm: Mingjia Zhang - Categorical local Langlands and cohomology of Shimura varieties

  • In recent years, categorical versions of the local Langlands conjectures have been formulated and significant progresses towards the conjecture have been made, due to the work of Fargues—Scholze, Zhu, Hansen—Mann. These have been applied to the study of cohomology of Shimura varieties, reaping rich rewards. I will discuss some methods, results, and if time permits, conjectures in this framework.

• 12:30pm-1:30pm: Lunch break

• 1:30pm-2:15pm: Career panel

• 2:30-3:30pm: Ananth Shankar - Geometric Manin-Mumford in positive characteristic

  • The Manin-Mumford conjecture (now a theorem) posits that a subvariety of an abelian variety (or torus) in characteristic zero that contains a Zariski-dense set of torsion points must be the translate of an abelian sub-variety (or subtorus) by a torsion point. In positive characteristic, this conjecture is necessarily false as every point defined over a finite field is torsion. I will discuss a geometric version of this conjecture in positive characteristic, and if time permits, also other results having a similar flavour. This is based on work in progress with Anup Dixit, Philip Engel, and Ruofan Jiang.

• 3:30-3:45pm: Break

•  3:45-4:45pm: Patrick Allen - Modular orbifolds and derived R = T

  • An example of Serre shows that in the strong form of his modularity conjecture, one can't also fix the nebentypus. Serre and Carayol independently explained that this obstruction is due to nontrivial isotropy groups on certain modular orbifolds, hence only occurs for the primes 2 and 3 and certain Galois representations that we'll call badly dihedral. Curiously, when studying the deformation theory of a mod p modular Galois representation for an odd prime p, the same badly dihedral representations for p = 3 arise: it is for these that the minimal deformation ring does not appear to be a flat local complete intersections over the ring of Witt vectors. We explain this link via a derived version of a minimal R = T theorem. As a corollary, we can characterize when these badly dihedral representations admit lifts with minimal weight, level, and nebentypus. This is joint work in progress with Preston Wake.

Location

East Hall Room 1360

530 Church St

Ann Arbor, MI 48109

To find the Upper Atrium, enter East Hall into the open atrium area and go up the stairs to the mezzanine.

Organizers

Organized by Preston Wake of MSU and Charlotte Chan and Kartik Prasanna of University of Michigan.