Michigan Number Theory Day

The inaugural Michigan Number Theory Day will be held at MSU on Saturday, November 1, 2025.

Speakers

• Vesselin Dimitrov (Caltech)

• Sean Howe (University of Utah)

• Gilbert Moss (University of Maine)

• Naomi Sweeting (Princeton University)

Registration

There is no fee to register, but to help our planning, please register by October 18.

Please click here to register.

Schedule

All talks in C304 Wells Hall. 

• 9:30am-10am: Registration

• 10-11am: Gilbert Moss - The universal Harish-Chandra j-function for p-adic groups

  • Abstract: Harish-Chandra envisioned generalizations of the Plancherel formula in the representation theory of p-adic reductive groups, involving integration with the Plancherel measure, a function constructed using intertwining operators between parabolically induced representations. In the last several decades, it has become apparent that many of the analytic tools used to build this theory can be replaced with purely algebraic constructions. In this talk we describe recent work where we push the algebraic framework further to construct intertwining operators and the Plancherel measure for representations with coefficients in arbitrary commutative Noetherian Z[1/sqrt{p}]-algebras. This generalizes constructions of Waldspurger, Dat, and Girsch. We prove a generic Schur's lemma result, which circumvents the need for generic irreducibility in defining the j-function. Finally, we will describe an application using the j-function to characterize a putative local Langlands correspondence in families for classical groups.

• 11-11:30am: Coffee break

• 11:30am-12:30pm: Vesselin Dimitrov - On Diophantine effectivization

  • Abstract: Two of Carl Ludwig Siegel’s timeless contributions to number theory were separation inequalities containing theoretically ineffective constants due to using some kind of lever in loosely similar ways. These are, of course, the Thue—Siegel inequality (which contains as a consequence the finiteness of the integral points of a non-rational affine algebraic curve), and the separation from s = 1 of a putative Landau—Siegel zero. In my talk, I will outline two new ways to partially bypass the levers and replace them with a concept of multivalent holonomy bounds which arose in an ongoing collaboration with Frank Calegari and Yunqing Tang. In the former case, we are led into an effectivization of the binomial case of Siegel’s theorem, and a new effective resolution of the Thue and the two-variable S-unit equations, by means of a dihedral algebraic construction which replaces the anchoring lever in Thue’s hypergeometric method. In the latter case, we may outline a path that leads to a new resolution of the classical class number one problem (for quadratic imaginary fields), using a method of square roots and ultimately the exact same Diophantine analysis core. 

• 12:30-2:30pm: Lunch break

• 2:30-3:30pm: Naomi Sweeting - On the Bloch–Kato Conjecture for some four-dimensional symplectic Galois representations

  • Abstract: The Bloch–Kato Conjecture predicts a relation between Selmer ranks and orders of vanishing of L-functions for Galois representations arising from etale cohomology of algebraic varieties. In this talk, I’ll describe results towards this conjecture in ranks 0 and 1 for the self-dual Galois representations that come from Siegel modular forms on GSp(4) with parallel weight (3, 3); these contribute to cohomology of classical Siegel threefolds. The key step in the proof is a construction of auxiliary ramified Galois cohomology classes, which then give bounds on Selmer groups. The ramified classes come from level-raising congruences and the geometry of special cycles on Shimura varieties.

• 3:30-3:45pm: Break

•  3:45-4:45pm: Sean Howe - The infinitesimal structure of moduli spaces in p-adic Hodge theory

  • Abstract: The modern theory of p-adic geometry via v-sheaves is built up from perfectoid rings, which are very non-Noetherian Banach algebras over the p-adic numbers characterized by the existence of approximate p-power roots. These perfectoid rings are well-adapted to the study of p-adic cohomology (p-adic Hodge theory), but the existence of approximate p-power roots forces them to behave poorly from a differential perspective.  In this talk, I will explain some aspects of a differential theory of p-adic geometry (inscribed v-sheaves) that is built on top of the “topological” theory of perfectoid spaces and v-sheaves. This differential theory has applications to p-adic Hodge theory, p-adic automorphic forms, and unlikely intersections, but I will mostly focus on explaining the infinitesimal structures arising in simple examples. No prior knowledge of perfectoid rings, etc. will be assumed!

Location

Make sure to enter in the C or D wing. Wells Hall C Wing doors in Google Maps. 

619 Red Cedar Road

C304 Wells Hall (C Wing)

East Lansing, MI 48824

Organizers

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