2022 가을학기 정수론 세미나

• 2023년 2월  6일 (월) / 7일 (화)  11:00 AM, 27동 325호 (2 talks)

    Speaker  : Yong Suk Moon (BIMSA) 

    Title         : Prismatic F-crystals and applications to p-adic Galois representations

    Abstract :  Prismatic cohomology, which is recently developed by Bhatt and Scholze, is a p-adic cohomology theory unifying etale, de Rham, and crystalline cohomology. In this series of two talks, we will discuss its central object of study called prismatic F-crystals, and some applications to studying p-adic Galois representations. The first part will be mainly devoted to explaining motivational background on the topic. Then we will discuss the relation between prismatic F-crystals and crystalline local systems on p-adic formal scheme, and talk about applications on purity of crystalline local system and on crystalline deformation ring. If time permits, we will also discuss recent work in progress on log prismatic F-crystals and semistable local systems. A part of the results is based on joint work with Du, Liu, Shimizu.

• 12월  20일 (화)  14:00 PM, 129동 301호

    Speaker  : Dong Gyu Lim (UC Berkeley) 

    Title         : Connected components of affine Deligne-Lusztig varieties

    Abstract :  Among various questions on ADLV, the question on the connected components turns out to be a fairly important problem. For example, Kisin, in his proof of the Langlands-Rapoport conjecture (in a weak sense) for abelian type Shimura variety with the hyperspecial level structure, crucially used the description of the set of connected components. Since then, many authors have answered this question in various restricted cases. I will discuss these previous works and my new result (joint work with Ian Gleason and Yujie Xu) which finishes the question in the mixed characteristic case.

• 12월  20일 (화)  11:00 AM, 129동 301호

    Speaker  : Dong Gyu Lim (UC Berkeley)

    Title         : Some problems on affine Deligne-Lusztig varieties

    Abstract :  Deligne and Lusztig constructed a geometric object called Deligne-Lusztig varieties whose certain cohomology groups contain all irreducible representations of the corresponding finite group of Lie type. Affine Deligne-Lusztig varieties can be analogously defined in the context of a local field. As it turns out, they are related to the special fibers of Shimura varieties. I will explain some known results and open problems on ADLV.

• 12월  2일 (금)  10:00 AM, Zoom 910 1345 711, Passcode 331303

    Speaker  : Julie Desjardins (University of Toronto)

    Title         : Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one

    Abstract :  The blow up of the anticanonical base point on X, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface E with only irreducible fibers. The sections of minimal height of E are in correspondence with the 240 exceptional curves on X. A natural question arises when studying the configuration of those curves : 

If a point of X is contained in « many » exceptional curves, it is torsion on its fiber on E?

In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there is 56 exceptional curves) that if « many » equals 4 or more, then yes. With Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if « many » equals 9 or more, then yes. Additionnally we find counterexamples where a torsion point lies at the intersection lies at the intersection of 7 exceptional curves.

• 11월  11일 (금)  16:00 - 17:00 27동 325호

    Speaker  : Jungin Lee (KIAS)

    Title         : Mixed moments and the joint distribution of random groups

    Abstract : The moment problem is to determine whether a probability distribution is uniquely determined by its moments. Recently, the moment problem for random groups has been applied to the distribution of random groups, in particular the cokernels of random p-adic matrices. In this talk, we introduce the mixed moments of random groups and apply this to the joint distribution of random abelian and non-abelian groups. In the abelian case, we provide three universality results for the joint distribution of the multiple cokernels for random p-adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations. We also explain the universality of the cokernel of random Hermitian matrices over the ring of integers of a quadratic extension of Q_p, which is an analogue of the universality of random symmetric matrices over Z_p proved by Wood.

• 11월  4일 (금)  16:00 - 17:00 27동 325호

    Speaker  : Lee, Seul Bee / 이슬비 (IBS-CGP)

    Title         : Approximations with mod 2 congruence conditions

    Abstract : In the study of Diophantine approximation, a natural question is which rationals p/q minimize |qx-p| with a bounded condition over q. We call such rationals the best approximations. The regular continued fraction gives an algorithm generating the best approximations. From a general perspective, we are interested in the best approximations with congruence conditions on their numerators and denominators. It is known that the continued fraction allowing only even integer partial quotients generates the best approximations whose numerator and denominator have different parity. In this talk, we explain the connection between the best approximations and the Ford circles. Then we explain how we can induce continued fraction algorithms that give the best approximating rationals with congruence conditions of modulo 2. This is joint work with Dong Han Kim and Lingmin Liao.

• 9월  16일 (금)  16:00 - 17:00  27동 325호

    Speaker  : Jun-Yong Park / 박준용 (Max Planck Institute for Mathematics)

    Title         : Height-moduli and lower order terms 

    Abstract : We introduce the `Height-moduli' which allows us to differentiate between integral points and rational points on proper algebraic stacks with respect to stacky height functions formulated by Ellenberg, Zureick-Brown, and Satriano. Focusing upon Weierstrass equations and their moduli counterparts of cyclotomic stacks, the distinction naturally accounts for the origin of second and third main terms in the sharp enumerations of elliptic curves over global function fields ordered by discriminant. This is joint work with Dori Bejleri (Harvard) and Matthew Satriano (Waterloo).