July 21, Monday

9:00 – 9:30

Registration Session

9:30 – 10:30

Max Xu 徐文强

Arithmetic aspects of random multiplicative functions

10:30 – 10:45 

Break

10:45 – 11:45

Ryan Chen

Near-center derivatives and arithmetic 1-cycles

11:45 – 13:30

Lunch Break (Lunch boxes will be provided)

13:30- 14:30

Hao Peng 彭淏

On the Beilinson–Bloch–Kato conjecture for polarized motives

14:30 - 15:00

Tea Break

15:00 - 16:00

Zhiyu Zhang 张志宇

Twisted triple products and arithmetic heights

16:15 - 17:15

Kai-Wen Lan 蓝凯文

Some vanishing results for the rational completed cohomology of Shimura varieties

Max Xu 徐文强

Title: Arithmetic aspects of random multiplicative functions

Abstract: I will give an overview of recent progress in random multiplicative functions (random models for multiplicative functions), an active area in probabilistic number theory, and their connection to the study of the Fyodorov–Hiary–Keating conjecture (motivated from random matrix theory and mathematical physics), and Polya's question on nonnegative character sums; and applications to character sums estimates and Mobius cancellations.

Ryan Chen

Title: Near-center derivatives and arithmetic 1-cycles

Abstract: Theta series for lattices count lattice vectors of fixed norm. Such theta series give some of the first examples of automorphic forms. It is possible to form theta series in other geometric contexts, e.g., for counting problems involving abelian varieties. It is expected that these theta series again have additional automorphic symmetry. I will explain some “near-central” instances of an arithmetic Siegel–Weil formula from Kudla’s program. These geometrize the classical Siegel–Weil formulas, on lattice and lattice vector counting via Eisenstein series. At these near-central points of functional symmetry, we observe that both the leading special value (complex volumes) and the sub-leading first derivative (arithmetic volume) simultaneously have geometric meaning. The key input is a new limit phenomenon relating positive characteristic intersection numbers and heights in mixed characteristics, as well as its automorphic counterpart.

Hao Peng 彭淏

Title: On the Beilinson–Bloch–Kato conjecture for polarized motives

Abstract: We show that the rank-zero case of the Beilinson–Bloch–Kato conjecture for unitary Rankin–Selberg motives, as proved by Liu, Tian, Xiao, Zhang, and Zhu, implies the rank-zero case of the same conjecture for standard unitary motives. The argument is based on the use of theta correspondence. An analogous implication also holds in the orthogonal situation. If time permits, I will also discuss my ongoing work on proving the Beilinson–Bloch–Kato conjecture for orthogonal Rankin–Selberg motives. 

Zhiyu Zhang 张志宇

Title: Twisted triple products and arithmetic heights

Abstract: We are interested in explicit arithmetic of elliptic curves and their products, using cycles on compactified Shimura varieties. Liu’s fundamental work (2016) proves rank 0 case of Bloch–Kato conjecture on twisted triple products over Q under assumptions (E1–E3), using Hirzebruch–Zagier cycles. For rank 1 case, we still need to prove (conjectural) height formulas on cycles and L-functions. In this talk, I will focus on another version of twisted triple products over totally real fields, using non-reductive cycles, mostly on results towards height formulas. Unlike (E1–E3), this approach might allow quadratic twists and CM elliptic curves. I will give some evidence, e.g., the twisted GGP conjecture, and the twisted arithmetic fundamental lemma.

Kai-Wen Lan 蓝凯文

Title: Some vanishing results for the rational completed cohomology of Shimura varieties

Abstract: I will start with some introduction to Shimura varieties and their completed cohomology, and report on my joint work in progress with Lue Pan which shows that, in the rational p-adic completed cohomology of a general Shimura variety, sufficiently regular infinitesimal weights (whose meaning will be explained) can only show up in the middle degree. I will give some examples and explain the main ingredients in our work if time permits.