2019 가을학기 정수론 세미나
12월 13일 (금) 16:00~18:00, 27동 325호
Speaker : Soon-Yi Kang (Kangwon National University)
Title : Divisibility property of the Fourier coefficients of (mock) modular functions
Abstract : Ramanujan's conjecture on the tau function, the Fourier coefficients of the discriminant function, led to the development of Hecke Theory. Many divisibility property of Fourier coefficients of modular functions were proved using the theory. Like the canonical basis of the space of modular functions form a Hecke system, we show that the Niebur-Poincare basis of the space of Harmonic Maass forms also form a Hecke system. As consequences, several arithmetic properties of Fourier coefficients of modular functions on the higher genus modular curves and mock modular functions are established.
11월 15일 (금) 16:00~18:00, 27동 325호
Speaker : Jaehyeok Lee (POSTECH)
Title : Tannakian reconstruction of representable presheaves of groups on the category of cocommutative differential graded coalgebras
Abstract : The reconstruction part of the Tannaka-Krein duality states that a compact group X can be recovered from the category of its finite dimensional unitary representations. Grothendieck suggested and Saavedra showed in his thesis that an analogous reconstruction holds true for affine group schemes. This result lead to the notion of Tannakian categories, which was further studied by Deligne.
In this talk, we study representable presheaves of groups on the category of cocommutative differential graded coalgebras, motivated by rational homotopy theory. Such presheaves can be considered as a dual notion to differential graded affine group schemes. We introduce an analogous reconstruction result for these presheaves. More precisely, we reconstruct such presheaf from the category of its (not necessarily finite dimensional) representations. As a consequence, we give an alternative reconstruction for (differential graded) affine group schemes.
This is a joint work with Jae-Suk Park.
11월 1일 (금) 16:00~18:00, 27동 325호
Speaker : Jaehoon Lee (KAIST)
Title : Structure of the Mordell--Weil Group over Z_p-extensions
Abstract : In this talk, we study the Lambda-module structure of the Mordell--Weil, Selmer, and Tate--Shafarevich groups of an abelian variety over Z_p-extensions.
10월 4일 (금) 16:00~18:00, 27동 325호
Speaker : Dohoon Choi (Korea University)
Title : Application of Langlands program to the Linnik problem for automorphic representations
Abstract : The Dirichlet prime number theorem states that if two integers a and b are relatively prime, then there are infinitely many primes of the form an+b. It is a natural question after the Dirichlet theorem to find an upper bound for the smallest prime having the form an+b. Linnik answered the question, which is called 'Linnik problem' now. This problem can be stated in terms of a certain character, and so the Linnik problem is extended to a question on automorphic representations. In this talk, I will talk about application of Langalnds program to the Linnik problem for automorphic representations.
9월 20일 (금) 16:00~18:00, 27동 325호
Speaker : Sangtae Jeong
Title : Ergodic functions over the p-adic integers
Abstract : In this talk, we present an ergodicity criterion of a certain class of 1-Lipschitz functions on Z_p for arbitrary primes p, known as B-functions. These functions are locally analytic functions of order 1 (and therefore contain polynomials). For arbitrary primes p>3, this erodicity criterion leads to an efficient and practical method of constructing ergodic polynomials on Z_p that realize a given unicyclic permutation modulo p. In particular, for polynomials over Z_3, we provide a complete ergodicity criterion in terms of its coefficients. This method can be applied to a Z_p for general primes p.