► Lecture Series : 조재현 (UNIST)

► Problem Session : 이석형 (Seoul National University)

Lecture 1. Introduction to low-lying zeros of L-functions

Abstract. We introduce Katz and Sarnak’s philosophy on low-lying zeros of L-functions. We review some known results and applications.

Lecture 2. The average analytic rank of elliptic curves

Abstract. BSD conjecture claims that the analytic rank of an elliptic curve coincides with its algebraic rank. Using the n-level density of elliptic curve L-functions, we study the distribution of analytic rank of elliptic curves. This talk is based on a joint work with K. Jeong.

Lecture 3. Average of the smallest prime in a conjugacy class

Abstract. A number field K of degree n is a G-field if its Galois closure over the rational numbers is a G Galois extension. For a conjugacy class C of the group G, we define N_C as the smallest prime p whose Frobenius automorphism Frob_p belongs to C. For S_n-fields, n=3,4, and 5, we show that the average of N_C exists under GRH.

- Lecture Note 1: https://drive.google.com/file/d/1yfqLFBiuvaKAEGT7a_4noHgxqWAWfuzX/view

- Lecture Note 2: https://drive.google.com/file/d/1OCiaKMSzkut-2tmWQE_kLa1rAATvk-J1/view

- Lecture Note 3: https://drive.google.com/file/d/1NALslad9978uJcWaVY221-QXf9cIQ4Nk/view

- Problem Set: https://drive.google.com/file/d/1nUo_uZZhkc7zxIbFvV5v6Km2KMVs803i/view

► Lecture Series : 박선우 (University of Wisconsin-Madison)

► Problem Session : 강태엽 (POSTECH)

Title: Arithmetic statistics over global function fields

Abstract. This series of lecture focuses on giving an overview of obtaining asymptotic estimates for counting families of arithmetic objects indexed by square-free polynomials of degree n over global function fields. As an example, we will focus on obtaining different types of asymptotic statistical estimates for the sizes of prime Selmer groups of quadratic twist families of non-isotrivial elliptic curves. 

- Lecture Note: https://drive.google.com/file/d/1qKf3RkyKZR7LRvRMdcmQXNc-glyvOfDQ/view 

- Problem Set: https://drive.google.com/file/d/1qVSl34pJpzWZagCy4zycf5XNwpGIrUzd/view 

[Slide 1] https://drive.google.com/file/d/16ZXNe2lcIZJ2rHkia0bz1_OkDPaPC3bW/view

[Slide 2] https://drive.google.com/file/d/1d7r14b6BtwMgEA1XA4VkQGXpk5bm6uCr/view

[Slide 3] https://drive.google.com/file/d/1ewoHuOrdJzw9bBFMiYefxeWiVoxIgodm/view

► Lecture Series : 이정인 (KIAS)

► Problem Session : 정길용 (University of California-Irvine)

Title: Distribution of random p-adic matrices and random groups 

Abstract. In this lecture series, we will study the distribution of random p-adic matrices and random profinite groups. We will cover the following topics in the lecture.

(1) The work of Friedman-Washington and the Cohen-Lenstra heuristics

(2) Joint distribution of the multiple cokernels for random p-adic matrices

(3) Universality for random p-adic matrices and the moments of random groups

(4) Distribution of random profinite groups and its applications 

- Lecture Note: https://drive.google.com/file/d/1y7WvJ_MtBLGKMWoxmgm2rVMcgJgs7WvL/view 

- Problem Set : https://drive.google.com/file/d/1r15NhMQBAKUcJDRA8s87s15JAh62Fxu8/view