Number Theory Seminars 2022

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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.

June 9 (Thu), 2022, 10:00--11:00, Room 1424

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 27 (Wed), 2022, 15:00--16:00, online

Title: Cyclotomic units and the scarcity of Euler systems

Speaker: Dominik Bullach (King's College London)

Abstract: Ever since their introduction, Euler systems have played an important role in spectacular advances in arithmetic geometry. In this talk, I will discuss Coleman's Conjecture (proved in recent joint work with Burns, Daoud, and Seo), which gives a precise description of the set of all classical Euler systems over the rationals. If time permits, I will also indicate striking consequences of these ideas towards the conjecture of Bloch and Kato over general number fields.

July 28 (Thu), 2022, 10:00--11:00, online

Title: Jordan-Landau theorem for matrices over finite fields

Speaker: Gilyoung Cheong (UC Irvine)

Abstract: We asymptotically compute the probability that the characteristic polynomial of a random $n \times n$ matrix over a fixed finite field is square-free with specified number of irreducible factors for large $n$. This is a matrix version of the following two theorems: Landau's theorem that computes the probability that a positive integer is square-free with specified number of prime factors and Jordan's theorem that computes the probability that a random permutation of $n$ letters have specified number of cycle factors. This is joint work with Hayan Nam and Myungjun Yu.

July 28 (Thu), 2022, 15:00--16:00, online

Title: Distribution of Hecke eigenvalues for holomorphic Siegel modular forms

Speaker: Henry Kim (University of Toronto)

Abstract: We study distribution of Hecke eigenvalues for holomorphic Siegel cusp forms for $\text{Sp}_{2n}$ in level aspect. We give several applications including the vertical Sato-Tate theorem and low-lying zeros for degree $2n+1$ standard $L$-functions of holomorphic Siegel cusp forms. It is a joint work with S. Wakatsuki and T. Yamauchi.

August 4 (Thu), 2022, 10:00--11:00, online

Title: Arithmetic dynamics and rational maps with nontrivial automorphisms

Speaker: Minsik Han (University of Rochester)

Abstract: Formulated in the 1990s, arithmetic dynamics studies number theoretic aspects of discrete dynamical systems, with inspirations from other fields including complex dynamics and arithmetic geometry. In this talk, we will specifically consider rational maps with nontrivial automorphisms and introduce recent results regarding two topics in arithmetic dynamics, Misiurewicz polynomials and dynamical uniform boundedness principle.

October 25 (Tue), 2022, 15:00--16:00, Room 1424

Title: On the supersingular locus of fake Hilbert-Siegel modular varieties

Speaker: Chia-Fu Yu (Academia Sinica)

Abstract: Shimura curves associated to an indefinite quaternion $\mathbb{Q}$-algbera are also called fake modular curves. Similarly, the moduli spaces associated to an indefinite quaternion algebra $B$ over a totally real field $F$ of genus $m$ are called fake Hilbert-Siegel modular varieties $M$. In this talk we shall discuss the non-emptiness of the superspecial locus of $M$, the mass formula, and recent progress on the geometry of supersingular locus of $M$. This is joint work in progress with Yasuhiro Terakado.

October 25 (Tue), 2022, 16:00--17:00, Room 1424

Title: Mass formulas and the basic locus of unitary Shimura varieties

Speaker: Yasuhiro Terakado (NCTS)

Abstract: A Shimura variety of PEL type is a moduli space of abelian varieties with additional structure. The mod-$p$ reduction naturally decomposes into finitely many "Newton strata'', which are given by the isogeny classes of the $p$-divisible groups of abelian varieties. There is a unique closed Newton stratum, called the basic locus. In this talk we study the geometry and arithmetic of the basic locus of the $\text{GU}(r, s)$-Shimura variety associated with an imaginary quadratic field. We discuss mass formulas for abelian varieties on the basic locus, and compute the number of the irreducible components. This is joint work with Chia-Fu Yu.

October 27 (Thu), 2022, 10:00--11:00, online

Title: Bounds on the Torsion Subgroups of Néron-Severi Group Schemes

Speaker: Hyuk Jun Kweon (University of Georgia)

Abstract: Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$, and $\mathbf{Pic}^0 X$ be the identity component of $\mathbf{Pic}\, X$. The Néron-Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}^0 X)_{\mathrm{red}}$, and the Néron-Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$. We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_{\mathrm{et}}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements in various situations.