Schedule:

08:20-09:00 Registration

Chair: Choie

09:00-09:45 Kohnen

10:00-10:45 Sun

11:00-11:45 Pohl

11:45-14:00 Lunch

Chair: Oh

14:00-14:45 Kim

15:00-15:45 Ehlen

16:00-16:45 Ju

Organizing Committee:

• YoungJu Choie (POSTECH)
• Bo-Hae Im (KAIST)
• Byeong-Kweon Oh (Seoul National University)

Abstracts

• Stephan Ehlen (University of Koeln) - New variants of the Doi- Naganuma lift and applications

I will report on joint work with Yingkun Li in which we construct new analogues of the Doi and Doi-Naganuma lifts, which lift elliptic modular forms of integral weight k to Hilbert modular forms of parallel weight k on the full Hilbert modular group over a real quadratic field of discriminant D. Namely, we extend the lifts in two directions: 1) we extend them to harmonic Maass forms (building on work of Borcherds who extended it to weakly holomorphic forms) and 2) we allow forms of level d for any d dividing D. The functions we obtain are analogues of polar harmonic Maass forms on Hilbert modular surfaces. Moreover, as one of our applications, we find that the Eisenstein series appearing in Gross's and Zagier's paper on the factorization of singular moduli is a lift of an incoherent Eisenstein series of weight one attached to an imaginary quadratic field. This connects the proof by Gross and Zagier of the factorization formula with the one given by Schofer using regularized theta lifts.

• Jangwon Ju (Seoul National University) - Universal sums of generalized m-gonal numbers

For an integer $m\geq3$, a generalized polygonal number of oder $m$ (or a generalized $m$-gonal number) is defined by $P_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$. For integers $\alpha_i>0$ and $m_j\geq3$ $(1\leq i,j\leq k)$, a sum $\Phi_{\alpha_1,\alpha_2,\dots,\alpha_k} ^{m_1,m_2,\dots,m_k}(x_1,x_2,\cdots,x_k)=\alpha_1P_{m_1}(x_1)+\alpha_2P_{m_2}(x_2)+\cdots+\alpha_kP_{m_k}(x_k)$ of generalized polygonal numbers is called universal if $\Phi_{\alpha_1,\alpha_2,\dots,\alpha_k}^{m_1,m_2,\dots,m_k}(x_1,x_2,\cdots,x_k)=N$ has an integer solution for any nonnegative integer $N$. In this talk, under the some conditions on $m_i$, we find all universal sums $\Phi_{\alpha_1,\alpha_2,\dots,\alpha_k} ^{m_1,m_2,\dots,m_k}$ of generalized polygonal numbers.

Furthermore, in each case, under the same conditions on $m_i$, we provide an effective criterion on the universality of an arbitrary sum $\Phi_{\alpha_1,\alpha_2,\dots,\alpha_k} ^{m_1,m_2,\dots,m_k}$ of generalized polygonal numbers. This might be considered as a generalization of the "15-theorem" of Conway and Schneeberger. This is a joint work with Byeong-Kweon Oh.

• Wansu Kim (KIAS) - Equivariant BSD conjecture over global function fields

Under a certain finiteness assumption of Tate-Shafarevich groups, Kato and Trihan showed the BSD conjecture for abelian varieties over global function fields of positive characteristic. We explain how to generalise this to semi-stable abelian varieties ``twisted by Artin character" over global function field (under some technical assumptions). This is a joint work with David Burns and Mahesh Kakde.

• Winfried Kohnen (University of Heidelberg) - Bounds for Fourier-Jacobi coefficients of Siegel cusp forms of degree two

We discudss and prove several estimates involving Petersson norms of Fourier-Jacobi coefficients of Siegel cusp forms of degree two.

• Anke Pohl (University of Bremen) - Dynamical characterization of Maass forms

We discuss how certain discretizations of the geodesic flow on hyperbolic surfaces allow to provide a dynamical characterization of Maass forms, and to develop a natural notion of period functions for Maass forms.

• Hae-Sang Sun (UNIST) - Continued fractions and Mazur-Rubin conjecture on modular symbols

Based on numerical evidences, Mazur-Rubin proposed several conjectures on the distribution of modular symbols. In 2017, Petridis and Risager successfully gave a proof for an average version using a sophisticated theory on Eisenstein series twisted by moments of modular symbols. In the talk, we present a proof of the average version using the dynamics of continued fractions. This is joint work with Jungwon Lee.