Tag - Number theory

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.

Craig Westerland: The Stable Homology of the Braid Group with Coefficients Arising from the Hyperelliptic Representation

2nd November, 2023

The braid group B2g+1 has a description in terms of the hyperelliptic mapping class group of a curve X of genus g. This equips it with an action on V = H1(X), and we may produce a wealth of new representations Sλ(V) by applying Schur functors to V. The goal of this talk is to describe the stable (in g) group homology of these representations. Following an idea of Randal-Williams in the setting of the full mapping class group, one may extract these homology groups as Taylor coefficients of the functor given by the stable homology of the space of maps from the universal hyperelliptic curve to a varying target space. We compute that stable homology by way of a scanning argument, much as in Segal’s original computation of the stable homology of configuration spaces. This is joint work with Bergström, Diaconu, and Petersen. Dan will speak afterwards on the application of these results to the conjecture of Andrade-Keating on moments of quadratic L-functions in the function field setting.

Dan Petersen: Moments of Families of Quadratic L-Functions Over Function Fields Via Homotopy Theory

2nd November, 2023

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. (This will be explained in Craig Westerland's lecture on Nov 2.) We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. (This will be explained in my lecture on Nov 3.) Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.

Jordan Ellenberg: Stable Homology and the BKPLR Heuristics Over Function Fields

1st November, 2023

A basic question in arithmetic statistics is:  what does the Selmer group of a random abelian variety look like?  This question is governed by the Poonen-Rains heuristics, later generalized by Bhargava-Kane-Lenstra-Poonen-Rains, which predict, for instance, that the mod p Selmer group of an elliptic curve has size p+1 on average.  Results towards these heuristics have been very partial but have nonetheless enabled major progress in studying the distribution of ranks of abelian varieties.

Rebecca Bellovin: Modularity of Trianguline Galois Representations

12th October, 2023

The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation ρ : Galℚ → GL2(ℚ̅p) is modular if it is unramified outside finitely many places and de Rham at p. I will talk about what this means, and I will discuss an analogous modularity result for Galois representations ρ : Galℚ → GL2(L) when L is instead a non-archimedean local field of characteristic p.

Vadim Vologodsky: Prismatic F-gauges and Fontaine-Laffaille Modules

11th October, 2023

With every bounded prism Bhatt and Scholze associated a cohomology theory of formal p-adic schemes. The prismatic cohomology comes equipped with the Nygaard filtration and the Frobenius endomorphism. The Bhatt-Scholze construction has been advanced further by Drinfeld and Bhatt-Lurie who constructed a cohomology theory with values in a stable ∞-category of prismatic F-gauges. The new cohomology theory is universal, meaning that, for every bounded prism, the associated prismatic cohomology theory factors through the category of prismatic F-gauges.

In this talk, I will explain how a full subcategory of the category of prismatic F-gauges formed by objects whose Hodge-Tate weights lie in the interval [0,p-2] is equivalent to the derived category of Fontaine-Laffaille modules with a similar weight constraint. In the geometric context, this means that the prismatic F-gauge associated with a formally smooth scheme over p-adic integers of dimension less than p-1 can be recovered from its Hodge filtered de Rham cohomology equipped with the Nygaard refined crystalline Frobenius endomorphism.

If time permits, I will explain a generalization of the above statement to the case of prismatic F-gauges over a smooth p-adic formal scheme.

Kalyani Kansal: Irregular Loci in the Emerton-Gee Stack for GL2

9th October, 2023

Let K be a finite extension of ℚp. The Emerton-Gee stack for GL2 is a stack of etale (φ, Γ)-modules of rank two. Its reduced part, X, is an algebraic stack of finite type over a finite field, and can be viewed as a moduli stack of two-dimensional mod p representations of the absolute Galois group of K. By the work of Caraiani, Emerton, Gee and Savitt, it is known that in most cases, the locus of mod p representations admitting crystalline lifts with specified regular Hodge-Tate weights is an irreducible component of X. Their work relied on a detailed study of a closely related stack of etale phi-modules which admits a map from a stack of Breuil-Kisin modules with descent data. In our work, we assume K is unramfied and further study this map with a view to studying the loci of mod p representations admitting crystalline lifts with small, irregular Hodge-Tate weights. We identify these loci as images of certain irreducible components of the stack of Breuil-Kisin modules and obtain several inclusions of the non-regular loci into the irreducible components of X.

Jacob Lurie: What is… p-adic geometry?

9th October, 2023

Let p be a prime number. Roughly speaking, rigid analytic geometry is a counterpart of complex analysis where one replaces the field ℂ of complex numbers by the field ℚp of p-adic rational numbers (or some extension thereof).

In this talk, I'll try to explain some of the motivations which led to the development of this theory, and to give some flavour of the recent progress which make it a timely subject for our Special Year. No prior knowledge of p-adic geometry will be assumed.