Saturday, November 7, 2020

(Click on the titles for the recordings.)

### 10:30am-11:00am

Virtual welcome gathering.

### 11:00am-12:00: Irina Bobkova (Texas A&M)

**Title:** Elliptic curves and chromatic homotopy theory

**Abstract: **Computation of the stable homotopy groups of spheres is a long-standing open problem in algebraic topology, which has deep connections to number theory and derived algebraic geometry. I will introduce chromatic homotopy theory and explain how it splits this problem into simpler chromatic pieces, which can be understood using the theory of formal group laws and their deformations. Then I will talk about recent results and work in progress at the second chromatic level. In particular, I will talk about a self-dual decomposition of the sphere at the second chromatic level, where the decomposition pieces are constructed using the supersingular elliptic curves. This talk is based on joint work with M. Behrens, D. Culver and P. VanKoughnett.

### 12:15pm-1:15pm: Will Sawin (Columbia)

**Title:** The Shafarevich conjecture for hypersurfaces in abelian varieties. (Slides.)

**Abstract: **Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension n, defined over the rational numbers, with good reduction outside a fixed finite set of primes, up to isomorphism. In joint work with Brian Lawrence, we prove an analogous finiteness statement for hypersurfaces in a fixed abelian variety with good reduction outside a finite set of primes. I will broadly discuss the strategy of the proof, which combines ideas from multiple fields of algebraic geometry and number theory.

### 1:15pm-2:00pm

Break

### 2:00pm-3:00pm: Chao Li (Columbia)

**Title:** On the Beilinson-Bloch conjecture for unitary Shimura varieties. (Slides.)

**Abstract: **For certain automorphic representations π on unitary groups, we show that if L(s, π) vanishes to order one at the center s=1/2, then the associated π-localized Chow group of a unitary Shimura variety is nontrivial. This proves part of the Beilinson-Bloch conjecture for unitary Shimura varieties, which generalizes the BSD conjecture. Assuming the modularity of Kudla’s generating series of special cycles, we further prove a precise height formula for L'(1/2, π). This proves the conjectural arithmetic inner product formula, which generalizes the Gross-Zagier formula to Shimura varieties of higher dimension. We will motivate these conjectures and discuss some aspects of the proof. This is joint work with Yifeng Liu.

### 3:15pm-4:15pm: Gordan Savin (Utah)

**Title:** Bernstein projectors for positive depth Moy Prasad types.

**Abstract: **Let G be a p-adic reductive group. For a fixed Moy-Prasad type, every smooth representation of G can be written as a direct sum of two summands where one summand is generated by vectors in the type. We use the refined building of G to write down an explicit idempotent in

the Bernstein’s center that gives this decomposition for every smooth

representation. (Joint with A. Moy)

### 4:15pm-4:45pm

Coffee break

### 4:45pm-5:45pm: Sam Raskin (UT Austin)

**Title:** Geometric Langlands for *l*-adic sheaves. (Slides.)

**Abstract: **In celebrated work, Beilinson-Drinfeld formulated a categorical analogue of the Langlands program for unramified automorphic forms. Their conjecture has appeared specialized to the setting of algebraic D-modules: non-holonomic D-modules play a prominent role in known constructions

In this talk, we will discuss a categorical conjecture suitable in other geometric settings, including l-adic sheaves. One of the main constructions is a suitable moduli space of local systems. We will also discuss applications to unramified automorphic forms for function fields. This is joint work with Arinkin, Gaitsgory, Kazhdan, Rozenblyum, and Varshavsky.