2021 봄학기 정수론 세미나

• 6월 11일 (금) 14:30~15:30, Zoom 838 1595 9158

Speaker : 주장원 (울산대학교)

Title : The pentagonal theorem of sixty-three and generalizations of Cauchy's lemma

Abstract : In this talk, we study the representability of integers as sums of pentagonal numbers. In particular, we prove the ``pentagonal theorem of $63$", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers

1, 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 17, 18, 19, 23, 28, 31, 33, 34, 39, 42, and 63.

We also introduce a method to obtain a generalized version of Cauchy's lemma using representations of binary integral quadratic forms by quaternary quadratic forms, which plays a crucial role in proving the results. This is a joint work with Daejun Kim.

• 5월 14일 (금) 15:30~16:30, Zoom 836 3968 3307

Speaker : 진석호(중앙대학교)

Title : On some results about non-holomorphic Jacobi forms

Abstract : In this talk, we want to consider some class of non-holomorphic Jacobi forms and its relationship to modular forms. This work was motivated by the paper “Taylor coefficients of non-holomorphic Jacobi forms and applications”, by Kathrin Bringmann, which had been motivated by Zwegers’ work on so-called Appell’s functions. We study Appell's function more, and applying the ideas arising from harmonic Maass forms, we relate some series coming from combinatorics.

• 5월 6일 (목) 14:30~15:30, Zoom 854 1988 1532

Speaker : 강순이 (강원대학교)

Title : Arithmetic properties of weakly holomorphic modular functions of arbitrary level

Abstract : The canonical basis of the space of modular functions on the modular group of genus zero form a Hecke system. From this fact, many important properties of modular functions were derived. Recently, we have proved that the Niebur-Poincare basis of the space of Harmonic Maass functions also forms a Hecke system. In this talk, we show its applications, including divisibility of Fourier coefficients of modular functions of arbitrary level, higher genus replicability, and values of modular functions on divisors of modular forms.

This is a joint work with Daeyeol Jeon and Chang Heon Kim.

• 4월 2일 (금) 15:30~16:30, Zoom 865 0413 2358

Speaker : Ming-Lun Hsieh (Academia Sinica Institute of Mathematics)

Title : CM congruence and the derivatives of p-adic L-functions for imaginary quadratic fields

Abstract : Darmon, Dasgupta and Pollack in 2011 applied the Eisenstein congruence for Hilbert modular forms to prove the rank one Gross conjecture for Deligne-Ribet p-adic L-functions under some technical assumptions. These assumptions were later lifted by Ventullo. In this talk, we will apply their ideas in the setting of CM congruence to compute the first derivative of the Katz p-adic L-functions associated with ring class characters of imaginary quadratic fields at the exceptional zero. We will present a precise first derivate formula of the Katz p-adic L-functions in terms of certain Gross regulator and p-adic logarithms of elliptic units. This proves a formula proposed in a recent work of Betina and Dimitrov. This talk is based on a joint work with Masataka Chida.