2021 가을학기 정수론 세미나

• 1월 20일 (목) 11:00 AM, 27동 116호

Speaker : Gyujin Oh (Princeton University)

Title : Local cohomology of Hilbert modular varieties and harmonic Hilbert Maass forms

Abstract : Guided by our algebro-geometric interpretation of harmonic Maass forms, we study the local cohomology of open and closed Hilbert modular varieties. In this talk, we will focus on the case of Hilbert modular surfaces; we will review the geometry of cusp singularities of Hilbert modular surfaces, compute the local cohomology, and interpret it as the "Hilbert modular analogue" of the space of harmonic Maass forms.

• 1월 20일 (목) 10:00 AM, 27동 116호

Speaker : Dong Gyu Lim (UC Berkeley)

Title : Nonemptiness of affine Deligne-Lusztig varieties

Abstract : In the study Shimura varieties, it is an important question to count the points reduction modulo p (Langlands-Rapoport conjecture) as it provides a way to compute the Hasse-Weil zeta function. The most interesting piece showing up in the point counting is affine Deligne-Lusztig variety (ADLV) and it has been studied in various level structures including the hyperspecial level and the Iwahori level. In this talk, we will see explicit examples of ADLV described as a set of certain lattices and flags. Moreover, we will think about the nonemptiness criterion of ADLV along with the results already known and newly discovered. If time permits, the dimension formula will be discussed shortly.

• 12월 21일 (화) 11:00 AM, 27동 116호

Speaker : Gyujin Oh (Princeton University)

Title : Arithmetic geometry and representation theory of harmonic Maass forms

Abstract : Harmonic Maass forms are exotic generalizations of the usual modular forms where functions are required to be harmonic (as opposed to being holomorphic). These have been concretely studied, which led to many discoveries in number theory and combinatorics. We explain the algebro-geometric interpretation of the notion of harmonic Maass forms, which is the first of its kind. From this, harmonic Maass forms make sense over more general base rings, such as Q, Z, Q_p, and F_p. This interpretation is also amenable to generalizations to other Shimura varieties, and we explain its Hilbert modular analog as an example. Finally, we discuss how this fits into the "categorical" generalization of the Langlands program.

• 12월 3일 (금) 10:30 AM, Zoom 896 5654 6548 / 157067

Speaker : Abhishek Oswal (Caltech)

Title : Algebraization theorems in complex and non-archimedean geometry

Abstract : Algebraization theorems originating from o-minimality have found striking applications in recent years to Hodge theory and Diophantine geometry. The utility of o-minimality originates from the 'tame' topological properties that sets definable in such structures satisfy. O-minimal geometry thus provides a way to interpolate between the algebraic and analytic worlds. One such algebraization theorem that has been particularly useful is the definable Chow theorem of Peterzil and Starchenko which states that a closed analytic subset of a complex algebraic variety that is simultaneously definable in an o-minimal structure is an algebraic subset. In this talk, I shall discuss a non-archimedean version of this result and some recent applications of these algebraization theorems.

• 11월 5일 (금) 4:00 PM, Zoom 847 0852 7472 / 056920

Speaker : Jaesung Kwon (UNIST)

Title : Bianchi modular symbols and p-adic L-functions

Abstract : In this talk, we will discuss the integral L-values and p-adic L-functions of Bianchi modular forms. Also I will give the brief proof of the generation of the first homology groups by Bianchi modular symbols. From this, we obtain the result toward mu=0 conjecture.

• 10월 15일 (금) 10:30 AM, Zoom 819 9587 9639 / 378135

Speaker : Ananth Shankar (University of Wisconsin, Madison)

Title : Canonical heights on Shimura varieties and the Andre-Oort conjecture

Abstract : Let S be a Shimura variety. The Andre-Oort conjecture posits that the Zariski closure of special points must be a sub Shimura subvariety of S. The Andre-Oort conjecture for A_g (the moduli space of principally polarized Abelian varieties) — and therefore its sub Shimura varieties — was proved by Jacob Tsimerman.
However, this conjecture was unknown for Shimura varieties without a moduli interpretation. I will describe joint work with Jonathan Pila and Jacob Tsimerman where we prove the Andre Oort conjecture in full generality.