Titles and Abstracts
Speaker: Francesc, Bars(Universitat Autònoma de Barcelona)
Speaker: Francesc, Bars(Universitat Autònoma de Barcelona)
Title: On determining bielliptic quotient modular curves I,II
Title: On determining bielliptic quotient modular curves I,II
Abstract: In the lectures we will explain the different ideas and tools to determine bielliptic quotient modular curves. In particular, we will present different programmes in Magma and Mathematica which allow to us to determine when a quotient modular curve is bielliptic or not.
Abstract: In the lectures we will explain the different ideas and tools to determine bielliptic quotient modular curves. In particular, we will present different programmes in Magma and Mathematica which allow to us to determine when a quotient modular curve is bielliptic or not.
Speaker: SoYoung, Choi(Gyeongsang National University)
Speaker: SoYoung, Choi(Gyeongsang National University)
Title: On the zeros of weakly holomorphic modular forms in the level two
Title: On the zeros of weakly holomorphic modular forms in the level two
Abstract: We study the properties related to the zeros of weakly holomorphic modular forms in the level two. In particular, we investigate the location of zeros of natural basis elements and their transcendence. We also present bounds of the number of zeros certain combinations of the Eisenstein series in the level two.
Abstract: We study the properties related to the zeros of weakly holomorphic modular forms in the level two. In particular, we investigate the location of zeros of natural basis elements and their transcendence. We also present bounds of the number of zeros certain combinations of the Eisenstein series in the level two.
Speaker: WonTae Hwang(Jeonbuk National University)
Speaker: WonTae Hwang(Jeonbuk National University)
Title: Automorphism groups of polarized abelian varieties over finite fields - summary of current status, and one of its applications
Title: Automorphism groups of polarized abelian varieties over finite fields - summary of current status, and one of its applications
Abstract: As one of my main research interests, we briefly summarize the current status on the subject of classifying the finite groups that can be realized as the full automorphism group of some polarized abelian variety over a finite field, and as a potentially interesting application of our previous results to a group theoretic problem, we briefly describe the Jordan constants of simple abelian surfaces over fields of positive characteristic, with the aid of a similar computation on the Jordan constants of some arithmetic objects.
Abstract: As one of my main research interests, we briefly summarize the current status on the subject of classifying the finite groups that can be realized as the full automorphism group of some polarized abelian variety over a finite field, and as a potentially interesting application of our previous results to a group theoretic problem, we briefly describe the Jordan constants of simple abelian surfaces over fields of positive characteristic, with the aid of a similar computation on the Jordan constants of some arithmetic objects.
Speaker: Daeyeol, Jeon(Kongju National University)
Speaker: Daeyeol, Jeon(Kongju National University)
Title: Modular curves with infinitely many rational points
Title: Modular curves with infinitely many rational points
Abstract: Modular curves are moduli spaces parametrizing elliptic curves, and they play a central role in studying elliptic curves. In this talk, we consider the problem to determine the modular curves with infinitely many rational points over the number fields for a fixed degree. Faltings proved that a curve of genus greater than 1 has only finitely many rational points over a fixed number field. Thus we allow the number fields to vary for a fixed degree. First, we present some results on the modular curves with infinitely many rational points over the number fields with a fixed degree, and then we consider their application to elliptic curves.
Abstract: Modular curves are moduli spaces parametrizing elliptic curves, and they play a central role in studying elliptic curves. In this talk, we consider the problem to determine the modular curves with infinitely many rational points over the number fields for a fixed degree. Faltings proved that a curve of genus greater than 1 has only finitely many rational points over a fixed number field. Thus we allow the number fields to vary for a fixed degree. First, we present some results on the modular curves with infinitely many rational points over the number fields with a fixed degree, and then we consider their application to elliptic curves.
Speaker: Soon-Yi Kang(Kangwon National University)
Speaker: Soon-Yi Kang(Kangwon National University)
Title: Divisibility of Fourier coefficients of weakly holomorphic modular functions
Title: Divisibility of Fourier coefficients of weakly holomorphic modular functions
Abstract: We give a thorough survey of the various congruence and divisibility properties that have been proven for bases of weakly holomorphic modular functions. In particular, we focus on the fact that the Fourier coefficients of those bases elements are highly divisible by small primes and we present the results on the prime number 11 that have been omitted from the literature.
Abstract: We give a thorough survey of the various congruence and divisibility properties that have been proven for bases of weakly holomorphic modular functions. In particular, we focus on the fact that the Fourier coefficients of those bases elements are highly divisible by small primes and we present the results on the prime number 11 that have been omitted from the literature.
Speaker: Byoung Du Kim(Victoria University of Wellington)
Speaker: Byoung Du Kim(Victoria University of Wellington)
Title: Construction of anti-cyclotomic Euler systems of modular abelian varieties, and the ranks of their Mordell-Weil groups
Title: Construction of anti-cyclotomic Euler systems of modular abelian varieties, and the ranks of their Mordell-Weil groups
Abstract: In this presentation, I will present me, Daeyeol Jeon, and Chang Heon Kim's construction of certain points on $X_1(N)$ over ring class fields (and therefore construction of points on the abelian varieties associated to newforms of level $\Gamma_1(N)$). Our work generalizes Bryan Birch's Heegner points on $X_0(N)$. Then, we show that these points form Euler systems (like the Heegner points), and we improve Kolyvagin's Euler system techniques to show that for our point $P_{\tau_K/c}$ and any ring class character $\chi$ of the extended ring class field of conductor $c$ satisfying $\chi=\overline{\chi}$, if $P_{\tau_K/c}^\chi$ is non-torsion and $G_K \to \operatorname{Aut} A_f[\pi]$ is surjective, then the corank of ${\rm Sel}(A_\chi/K)$ is 1, which implies the rank of $A_f(K)^\chi$ is 1.
Abstract: In this presentation, I will present me, Daeyeol Jeon, and Chang Heon Kim's construction of certain points on $X_1(N)$ over ring class fields (and therefore construction of points on the abelian varieties associated to newforms of level $\Gamma_1(N)$). Our work generalizes Bryan Birch's Heegner points on $X_0(N)$. Then, we show that these points form Euler systems (like the Heegner points), and we improve Kolyvagin's Euler system techniques to show that for our point $P_{\tau_K/c}$ and any ring class character $\chi$ of the extended ring class field of conductor $c$ satisfying $\chi=\overline{\chi}$, if $P_{\tau_K/c}^\chi$ is non-torsion and $G_K \to \operatorname{Aut} A_f[\pi]$ is surjective, then the corank of ${\rm Sel}(A_\chi/K)$ is 1, which implies the rank of $A_f(K)^\chi$ is 1.
Speaker: Chang Heon Kim(Sungkyunkwan University)
Speaker: Chang Heon Kim(Sungkyunkwan University)
Title: Weakly holomorphic Hecke eigenforms
Title: Weakly holomorphic Hecke eigenforms
Abstract: It is well-known that the space of holomorphic cusp forms has a basis consisting of eigenforms for the Hecke operators. It does not hold for the space of weakly holomorphic modular forms, but if we give an appropriate equivalence relation on the space, we can find Hecke eigenforms. In this talk, I will show how to find a basis consisting of weakly holomorphic Hecke eigenforms for some congruence subgroups of which the genera are zero or one. Further, I will investigate the relations between such eigenforms and Poincare series. (This is a joint work with Jihyun Hwang.)
Abstract: It is well-known that the space of holomorphic cusp forms has a basis consisting of eigenforms for the Hecke operators. It does not hold for the space of weakly holomorphic modular forms, but if we give an appropriate equivalence relation on the space, we can find Hecke eigenforms. In this talk, I will show how to find a basis consisting of weakly holomorphic Hecke eigenforms for some congruence subgroups of which the genera are zero or one. Further, I will investigate the relations between such eigenforms and Poincare series. (This is a joint work with Jihyun Hwang.)