KIAS Number Theory Seminar



Upcoming seminars

June 9 (Thu), 2022, 10:00--11:00, TBA

Title: On the probability distribution of $2$-Selmer groups of quadratic twist families of elliptic curves over global function fields

Speaker: Sun Woo Park (University of Wisconsin-Madison / NIMS)

Abstract: We present recent progress on computing the probability distribution of $2$-Selmer groups of quadratic twist families of a non-isotrivial elliptic curve $E$ over global function fields $\mathbb{F}_q(t)$ of characteristic coprime to $2$ and $3$. The elliptic curves we consider satisfy the condition that the Galois groups of the field extensions generated by their $2$-torsion points are isomorphic to $S_3$. The key component of this talk will focus on how the Riemann hypothesis over $\mathbb{F}_q(t)$ allows one to show that the probability distribution conforms to the Poonen-Rains heuristics with explicit error bounds, a simplification of previous well-studied approaches on computing the desired distribution over the rational numbers $\mathbb{Q}$.


July 21 (Thu), 2022, 10:00--11:00, online

Title: TBA

Speaker: Gilyoung Cheong (UC Irvine)

Abstract: TBA





Past seminars

March 31 (Thu), 2022, 10:00--11:00, online

Title: Continued fraction expansions in real quadratic field and an invariant measure of Gauss-Kuzmin type in dimension two

Speaker: Junyeong Park (Seoul National University)

Abstract: We investigate a two-dimensional dynamical system that models a Euclidean algorithm for algebraic integers in the real quadratic field of discriminant five. It represents certain non-simple continued fraction expansions. A Markov partition is exhibited, which we use to prove that the transfer operator has a dominant eigenvalue. As a consequence, we show that it has a unique invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. The system is expanding in a non-uniform way. This is joint work with Dohyeong Kim.


April 21 (Thu), 2022, 10:00--11:00, online

Title: On Stevenhagen's conjecture

Speaker: Peter Koymans (University of Michigan)

Abstract: In this talk we will study the negative Pell equation, which is the conic $C_D : x^2 - D y^2 = -1$ to be solved in integers $x, y \in \mathbb{Z}$. We shall be concerned with the following question: as we vary over squarefree integers $D$, how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic formula for such $D$. Fouvry and Klüners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen's conjecture, and potential applications of the new proof techniques. This is joint work with Carlo Pagano.


May 12 (Thu), 2022, 17:00--18:00, online

Title: Polarizations of abelian varieties over finite fields via canonical liftings

Speaker: Stefano Marseglia (Utrecht University)

Abstract: We describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical liftings to characteristic zero, i.e., a lifting for which the reduction morphism induces an isomorphism of endomorphism rings. This is joint work with Jonas Bergström and Valentijn Karemaker.


May 19 (Thu), 2022, 17:00--18:00, online

Title: Arithmetic of $\theta$-critical $p$-adic $L$-functions

Speaker: Kazim Büyükboduk (University College Dublin)

Abstract: In joint work with Denis Benois, we give an étale construction of Bellaïche's $p$-adic $L$-functions about $\theta$-critical points on the Coleman–Mazur eigencurve. I will discuss applications of this construction towards leading term formulae in terms of $p$-adic regulators on what we call the thick Selmer groups, which come attached to the infinitesimal deformation at the said $\theta$-critical point along the eigencurve, and an exotic ($\Lambda$-adic) $\mathcal{L}$-invariant. Besides our interpolation of the Beilinson–Kato elements about this point, the key input to prove the interpolative properties of this $p$-adic $L$-function is a new $p$-adic Hodge-theoretic "eigenspace-transition via differentiation" principle.





Please send one of the organizers an email if you are interested in giving a talk. If you want to receive regular email announcement, please send an email with your name/affiliation, in title of "subscribe", to jilee.math@gmail.com.

The first number theory seminar started March 10th, 2005 and it became a regular seminar since then. For more information on the early history of the number theory seminar, see the article (by Soon-Yi Kang in KIAS News Letter, Fall 2005) and the following archive files.

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Number Theory Group at KIAS

Choi, Youn-Seo
Choi, Junhwa (jhchoi.math@gmail.com)
Kim, Chan-Ho
Kim, Daejun
Lee, Chulhee
Lee, Jungin
Lee, Youngmin