Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Calabi-Yau Manifolds & Mirror Symmetry Seminar

CYMS Seminar - Noriko Yui (Queen's University)

Thursday, October 18th, 2018

Time: 11:30 a.m - 12:20 p.m Place: Jeffery Hall 422

Speaker: Noriko Yui (Queen's University)

Title: Four-dimensional Galois representations arising from certain Calabi--Yau threefolds Part II

Abstract: We consider the (irreducible) four-dimensional Galois representations arising from certain Calabi--Yau threefolds over ${\bf{Q}}$ with all the Hodge numbers of the third cohomology groups equal to $1$. There are many examples of (families) of such Calabi--Yau threefolds. The modularity/automorphy of such Calabi--Yau threefolds will be the main topic of discussion. There are two venues to be considered. In one venue, we ought to count the number of rational points over finite fields of these Calabi--Yau threefolds to concoct their L-series. In the other venue, we ought to construct some modular varieties, in this case, conjecturally, Siegel modular forms of weight $3$ and genus $2$ on some paramodular subgroups of $Sp(4,{\bf{Z}})$, and then compute their L-series. Such modular forms may be constructed using Borcherds forms. The ultimate aim is to establish a Langlands correspondence between the two L-series, thereby establishing the modularity/automorphy of such Calabi--Yau threefolds.

This is a joint work with Yifan Yang (National Taiwan University).

CYMS Seminar - Noriko Yui (Queen's University)

Thursday, September 27th, 2018

Time: 11:30 a.m - 12:20 p.m Place: Jeffery Hall 422

Speaker: Noriko Yui (Queen's University)

Title: Four-dimensional Galois representations arising from certain Calabi--Yau threefolds

Abstract: We consider the (irreducible) four-dimensional Galois representations arising from certain Calabi--Yau threefolds over ${\bf{Q}}$ with all the Hodge numbers of the third cohomology groups equal to $1$. There are many examples of (families) of such Calabi--Yau threefolds. The modularity/automorphy of such Calabi--Yau threefolds will be the main topic of discussion. There are two venues to be considered. In one venue, we ought to count the number of rational points over finite fields of these Calabi--Yau threefolds to concoct their L-series. In the other venue, we ought to construct some modular varieties, in this case, conjecturally, Siegel modular forms of weight $3$ and genus $2$ on some paramodular subgroups of $Sp(4,{\bf{Z}})$, and then compute their L-series. Such modular forms may be constructed using Borcherds forms. The ultimate aim is to establish a Langlands correspondence between the two L-series, thereby establishing the modularity/automorphy of such Calabi--Yau threefolds.

This is a joint work with Yifan Yang (National Taiwan University).

CYMS Seminar - Richard Gottesman (Queen's University)

Thursday, September 20th, 2018

Time: 11:30 a.m - 12:20 p.m Place: Jeffery Hall 422

Speaker: Richard Gottesman (Queen's University)

Title: Vector-Valued Modular Forms on $\Gamma_{0}(2)$C

Abstract: The collection of vector-valued modular forms form a graded module over the graded ring of modular forms. I will explain how understanding the structure of this module allows one to show that the component functions of vector-valued modular forms satisfy an ordinary differential equation whose coefficients are modular forms. In certain cases, we can use a Hauptmodul to transform such a differential equation into a Fuchsian differential equation on the projective line minus three points. We then are able to use the Gaussian hypergeometric series to explicitly solve this differential equation.
Finally, we make use of these ideas together with some algebraic number theory to study the prime numbers that divide the denominators of the Fourier coefficients of the component functions of vector-valued modular forms.

CYMS Seminar - Yasuhiro Goto (Hokkaido University)

Thursday, September 13th, 2018

Time: 11:30 a.m - 12:20 p.m Place: Jeffery Hall 422

Speaker: Professor Yasuhiro Goto (Hokkaido University of Education, Hakodate)

Title: Formal groups of low dimensional Calabi-Yau varieties

Abstract: Calabi-Yau varieties are associated with formal groups of dimension one. When they are defined over an algebraically closed field of positive characteristic, the formal groups are classified by the height. Using weighted Delsarte varieties of low dimensions (say, $2$, $3$ or $4$), we describe how to compute the height of their formal groups and show various numerical data for them.