2023 봄학기 정수론 세미나

• 3월  31일 (금) 16:00 - 17:00  27동 325호

    Speaker  : Homin Lee (Northwestern University)

    Title         : Height gap theorem and almost law

    Abstract : E. Breuillard showed that finite subsets $F$ of matrices in $GL_{d}(\overline{Q})$ generating non-virtually solvable groups have normalized height $\widehat{h}(F) \ge \epsilon$, for some positive $\epsilon>0$. This can be thought of as a non-abelian analog of Lehmer’s Mahler measure problem and has a nice application such as uniform Tits alternative. Recently, it also leads to the arithmetic Margulis lemma by M.Fraczyk, S.Hurtado, and J. Raimbault. In this talk, we will discuss a relatively elementary proof of E.Breuillard's height gap theorem which avloids Bruhat-Tits geometry, and deep results on algebraic tori that are used in the original E.Breuillard's proof. The key idea is a usage of a mysterious word map so-called "almost law". This is joint work with Lvzhou Chen (Joe Chen) and Sebastian Hurtado.  

• April  14 (Fri) 16:00 - 17:00  27-325

    Speaker  : Jaesung Kwon (UNIST)

    Title         : Homological interpretation of Hecke $L$-values over real quadratic fields.

    Abstract :  In this talk, I will demonstrate how to represent the Hecke $L$-values as the integral on the homology classes, which lives in the 2-dimensional complex manifolds with boundary. Also, I will construct a cyclotomic $p$-adic $L$-function, which interpolates the Hecke $L$-values, and give a criterion for the $\mu$-invariant of the $p$-adic $L$-function to be zero. This is a joint work with Jungyun Lee and Hae-Sang Sun.

• April  21 (Fri) 14:00 - 15:00 27-317

    Speaker  : Yoonbok Lee (Incheon National University)

    Title         : Selberg's central limit theorem of L -functions near the critical line

    Abstract :  Selberg's central limit theorem says that the logarithm of the Riemann zeta function has a Gaussian distribution in the complex plane on and near the critical line. We find an asymptotic expansion of a multi-dimensional version of Selberg's central limit theorem for L-functions near the critical line

• June 2 (Fri) 16:00 - 17:00 27-325

    Speaker  : Chanho Kim (KIAS)

    Title         :  A refined Tamagawa number conjecture for modular forms

    Abstract :  We discuss an explicit formula for the structure of Bloch–Kato Selmer groups of the central critical twist of modular forms if the analytic rank is ≤ 1 or the Iwasawa main conjecture localized at the augmentation ideal holds. This formula reveals more refined arithmetic information than the p-part of the Tamagawa number conjecture for motives of modular forms and reduces the corresponding Beilinson–Bloch–Kato conjecture to a purely analytic statement. We do not impose any good ordinary or Fontaine–Laffaille assumption.

• June  30 (Fri) 16:00 - 18:00  27-325

    Speaker  : Seung uk Jang (University of Chicago)

    Title         : Do Tropical Markov Cubics dream of Hyperbolic Origami?

    Abstract :  Non-archimedean fields and varieties over them admit the operation of tropicalizations, which provides a piecewise-linear approximate sketch of varieties that encapsulates many key aspects. For Markov surfaces $x^2+y^2+z^2+xyz=D$, this viewpoint was initiated by works of Spalding and Veselov, who focused on its tropical and dynamical aspects.

          In this talk, we will be working on a more general family of Markov surfaces and discover that, for any parameters, we have a copy or a shadow of the hyperbolic plane with the $(\infty,\infty,\infty)$-triangle reflection group action. Such a viewpoint easily yields corollaries in Fatou domains (dynamical side) or the finiteness of orbits of rational points with prime power denominators (number theory side). Some interesting number-theoretic aspects of this system may be introduced, such as Farey triples.