Sam Mundy: Vanishing of Selmer Groups for Siegel Modular Forms

21st March, 2024

Let π be a cuspidal automorphic representation of Sp2n over ℚ which is holomorphic discrete series at infinity, and χ a Dirichlet character. Then one can attach to π an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is irreducible, that π is ordinary at p, and that p does not divide the conductor of χ. I will describe work in progress which aims to prove that the Bloch-Kato Selmer group attached to the twist of ρ by χ vanishes, under some mild ramification assumptions on π; this is what is predicted by the Bloch-Kato conjectures.

The proof uses "ramified Eisenstein congruences" by constructing p-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of non-classical weight, and using these families to construct ramified Galois cohomology classes for the Tate dual of the twist of ρ by χ.

Pierre Charollois: On Eisenstein’s Jugendtraum for Complex Cubic Fields

22nd February, 2024

In the early 2000s Ruijsenaars and Felder-Varchenko have introduced the elliptic gamma function, a remarkable multivariable meromorphic q-series that comes from mathematical physics. It satisfies modular functional equations under the group SL3(ℤ) which make it a higher-dimensional analogue of the Jacobi theta function. In this work, we unveil the place that this function and its avatars play in number theory. Our main thesis is that these functions play the role of modular units in extending the theory of complex multiplication to complex cubic fields. In other words we propose a conjectural solution to Hilbert’s 12th problem for complex cubic fields, following a line of research actually initiated by G. Eisenstein. We give a lot of numerical evidence that support this conjecture, and relate it to the Stark conjecture by proving an analogue of the Kronecker limit formula in this cubic setting.

Michael Robert Magee: Convergence of Unitary Representations and Spectral Gaps of Manifolds

26th January, 2024

Let G be an infinite discrete group. Finite dimensional unitary representations of G are usually quite hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps.

Tony Feng: Mirror Symmetry and the Breuil-Mezard Conjecture

4th December, 2023

The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group G. Most of the progress thus far has been concentrated on the case G = GL2, which has several special features. I will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). It is based on the intuition that the Breuil-Mezard conjecture is analogous to homological mirror symmetry.

Gene Kopp: The Shintani-Faddeev Modular Cocycle

9th November, 2023

We ask the question, "how does the infinite q-Pochhammer symbol transform under modular transformations?" and connect the answer to that question to the Stark conjectures. The infinite q-Pochhammer symbol transforms by a generalized factor of automorphy, or modular 1-cocycle, that is analytic on a cut complex plane. This "Shintani–Faddeev modular cocycle" is an SL2(ℤ)-parametrized family of functions generalizing Shintani’s double sine function and Faddeev’s non-compact quantum dilogarithm. We relate real multiplication values of the Shintani-Faddeev modular cocycle to exponentials of certain derivative L-values, conjectured by Stark to be algebraic units generating abelian extensions of real quadratic fields.