Number Theory Seminars 2020

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December 09, 10 (Wed, Thu), 2020, 10:00--11:00, online

Title: Iwasawa theory and rational points on elliptic curves

Speaker: Francesc Castella (UCSB)

Abstract: In the early 2000s, Greenberg-Vatsal introduced a method for studying the cyclotomic Iwasawa theory of elliptic curves $E$ over $\mathbb{Q}$ at Eisenstein primes (i.e., primes $p$ for which $E$ admits a rational $p$-isogeny). Combined with Kato's work, their method yields important results towards the Birch and Swinnerton-Dyer conjecture in rank $0$ ($p$-converse, $p$-part of BSD formula). In these talks, I will explain recent joint work with G. Grossi, J. Lee, and C. Skinner in which we develop the method of Greenberg-Vatsal in the anticyclotomic setting, leading to new applications towards the Birch and Swinnerton-Dyer conjecture in rank $1$. If time permits, I will also discuss progress towards extending some of these results to small primes.

December 09, 10 (Wed, Thu), 2020, 11:00--12:00, online

Title: A conjecture of Perrin-Riou

Speaker: Ashay A. Burungale (Caltech)

Abstract: Let $E$ be an elliptic curve over the rationals with conductor $N$. A conjecture due to Perrin-Riou describes a rank one facet of the associated $p$-adic Beilinson--Kato elements for a prime $p$. In the case $(p, 6N)=1$, the talks plan to describe - the set-up and strategy of - our proof of the conjecture (joint with Christopher Skinner and Ye Tian).

December 07 (Mon), 2020, 10:00--11:00, online

Title: On the local conditions describing quartic fields

Speaker: Robert Hough (Stony Brook University)

Abstract: In his thesis, Bhargava described local conditions describing the maximality of quartic and quintic rings. I will discuss recent work which gives an orbit description of these local conditions in the quartic case and their Fourier transform, and discuss applications of these methods to the study of the distribution of low dimensional number fields. Part of this work is joint with Eun Hye Lee.

November 23, 25, 26, 27 (Mon, Wed, Thu, Fri), 2020, 14:00--15:30, online

Title: Congruences of zeta elements and its application to Iwasawa main conjecture

Speaker: Kentaro Nakamura (Saga University)

Abstract: In his work on Iwasawa main conjecture for Hecke eigen cusp newforms, Kazuya Kato constructed zeta elements, which are elements in Iwasawa cohomology of p-adic Galois representations associated to Hecke eigen cusp newforms. He also predicted that such elements exist for arbitrary families of $p$-adic global Galois representations (generalized Iwasawa main conjecture).
In this series of lectures, I explain the results of my recent work ( on this conjecture, precisely, the construction of zeta elements for universal deformations of absolutely irreducible mod $p$ odd Galois representations of rank two, and its application to Iwasawa main conjecture (without $p$-adic L-function).
I (plan to) first recall Kato's results on zeta elements for eigen cusp newforms and his generalized Iwasawa main conjecture. Then, I explain my main result on zeta elements for universal deformations. As a corollary of this result, we can obtain congruences for zeta elements between congruent eigen cusp newforms. As an application of these congruences, we can show that the validity of Iwasawa main conjecture (without $p$-adic L-function) for one eigen cusp new forms are equivalent to that for arbitrary congruent eigen cusp new forms. I explain the outline of the proof of this application. After that, I'll explain the idea of the construction of zeta elements for universal deformations. In our construction, many deep results in $p$-adic (global and local) Langlands correspondence for $\text{GL}_{2, \mathbb{Q}}$ are crucial. I'll recall these results, and explain how to use these results to construct our zeta elements.

November 17 (Tue), 2020, 15:00--16:00, Room 1424

Title: The embedding property for profinite groups

Speaker: Junguk Lee (KAIST)

Abstract: We aim to give an alternative proof of the existence of a universal embedding cover of a given profinite group. This was first proved by D. Haran and A. Lubotzky for the finitely generated case, and proved by M. D. Fried and M. Jarden for the general case. Also we will talk about model theoretic proof of the uniqueness of a universal embedding cover of a profinite group, proved by Z. Chatzidakis. In my knowledge, this is the only known proof for the uniqueness of a universal embedding cover working for all profinite groups. This approach for the existence and uniqueness of universal embedding covers also works for sorted profinite groups.

November 17 (Tue), 2020, 16:00--17:00, Room 1424

Title: Semi-stable deformation rings in even Hodge-Tate weights

Speaker: Wan Lee (UNIST)

Abstract: Integral lattices of semi-stable $p$-adic Galois representations of $\mathbb{Q}_p$ with small Hodge-Tate weights can be represented by strongly divisible lattices. In this talk, we explicitly find strongly divisible lattices of two dimensional semi-stable representations with Hodge-Tate weights $(0,2m)$ where $0<2m$.

November 13 (Fri), 2020, 14:00--16:00, online

Title: On the Gan-Gross-Prasad conjecture for skew-hermitian unitary group

Speaker: Jae-ho Haan (KIAS)

Abstract: One of the most fundamental problem in the relative Langlands program is to find the relation between period integrals and some automorphic L-function. Among them, the Gan-Gross-Prasad(GGP) conjecture is the most famous problem in this context. Building upon the huge development of the theory of endoscopy for the past 10 years (for example, Ngo's Fundamental lemma, Arthur's endoscopic classification of automorphic representations), there have been remarkable progresses toward the GGP conjecture. In the first hour of this talk, I will introduce the GGP conjecture and report some celebrated achievements have been made so far. In the second hour, we will discuss the proof of the one direction of the GGP conjecture for skew-hermitian unitary groups. If time permits, we propose several attractive and challenging problem related to GGP the conjecture.

November 13 (Fri), 2020, 11:00--12:00, Room 1424

Title: Combinatorial Aspects of Multiple Zeta Values

Speaker: Kyunghwan Song (Sun Moon University)

Abstract: We introduce (1) The definition and a sum formula of the Multiple Zeta Values (2) Stuffle and Shuffle product of Multiple Zeta Values (3) Alternative proofs of some elementary theorems related to combinatorics using the properties of Multiple Zeta Values (4) Finite Multiple Zeta Values.

November 05 (Thu), 2020, 10:30--12:00, online

Title: Parametrizing Rings of Rank 6

Speaker: Seok Hyeong Lee (Princeton University)

Abstract: Since Manjul Bhargava's Higher Composition Laws (2008), parametrizing rings of rank n has been discussed in various perspectives. Most ring parametrization models relate rank n rings to equations for n points in projective space, and Melanie Wood (2010) described an explicit procedure of constructing rings from those equations using hypercohomology of sheaves. In this talk, we will apply this procedure to some classical determinantal varieties of degree 6 to obtain rank 6 rings, and discuss their properties.

October 22 (Thu), 2020, 10:00--11:30, online

Title: Quotients, subsheaves and extensions of vector bundles on the Fargues-Fontaine curve

Speaker: Serin Hong (University of Michigan)

Abstract: Since its construction in early 2010s, the Fargues-Fontaine curve has played a pivotal role in p-adic Hodte theory and related fields. Vector bundles on this ``curve" are of particular interest, as they provide a geometric interpretation of numerous constructions from those fields. Most notably, by the upcoming work of Fargues-Scholze, the local Langlands correspondence can be realized in terms of certain sheaves on the stack of vector bundles on the Fargues-Fontaine curve. In this talk, we discuss several classification theorems about vector bundles on the curve, including a complete classification of quotients and subsheaves of a given vector bundle. Our proof crucially relies on Scholze's theory of diamonds, which provides a correct framework for dimension counting of various moduli spaces of bundle maps.

October 15 (Thu), 2020, 16:00--17:30, online

Title: Higher codimension behavior in equivariant Iwasawa theory for CM-fields

Speaker: Takenori Kataoka (Keio University)

Abstract: In classical Iwasawa theory, we mainly study codimension one behavior of Iwasawa modules. Against this background, Bleher-Chinburg-Greenberg-Kakde-Pappas-Sharifi-Taylor started studying codimension two behavior of unramified Iwasawa modules which are assumed to be pseudo-null. In this talk, we propose a new perspective of their works by using the determinant modules of perfect complexes. This enables us to describe the higher codimension behavior, and moreover to extend the results to equivariant settings.

October 08 (Thu), 2020, 16:00--17:30, online

Title: The arithmetic of a classical family of elliptic curves

Speaker: Yukako Kezuka (University of Regensburg)

Abstract: The Birch-Swinnerton-Dyer conjecture is one of the most celebrated open problems in number theory. In this talk, I will explain some recent progress on the study of this conjecture for elliptic curves $E$ of the form $x^3+y^3=N$ for a positive integer $N$ prime to $3$. They are cubic twists of the Fermat elliptic curve $x^3+y^3=1$, and admit complex multiplication by the ring of integers of $\mathbb{Q}(\sqrt{-3})$. First, I will explain the Tamagawa number divisibility satisfied by the central $L$-values, and exhibit a curious relation between the $3$-part of the Tate-Shafarevich group of $E$ and the number of prime divisors of $N$ which are inert in $\mathbb{Q}(\sqrt{-3})$. I will then explain my joint work with Yongxiong Li where we study in more detail the case when $N=2p$ or $2p^2$ for an odd prime number $p$ congruent to $2$ or $5$ modulo $9$. For these curves, we establish the $3$-part of the Birch-Swinnerton-Dyer conjecture and a relation between the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ and the $2$-Selmer group of $E$, which can be used to study non-triviality of the $2$-part of the Tate-Shafarevich group.

October 08 (Thu), 2020, 10:00--11:30, online

Title: From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture

Speaker: Seoyoung Kim (Queen's University)

Abstract: Let $E$ be an elliptic curve over $\mathbb{Q}$ with discriminant $\Delta_E$. For primes $p$ of good reduction, let $N_p$ be the number of points modulo $p$ and write $N_p=p+1-a_p$. In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies $$\lim_{x\to\infty}\frac{1}{\log x}\sum_{\substack{p\leq x\\ p\nmid \Delta_{E}}}\frac{a_p\log p}{p}=-r+\frac{1}{2},$$ where $r$ is the order of the zero of the $L$-function $L_{E}(s)$ of $E$ at $s=1$, which is predicted to be the Mordell-Weil rank of $E(\mathbb{Q})$. We show that if the above limit exits, then the limit equals $-r+1/2$. We also relate this to Nagao's conjecture. This is a recent joint work with M. Ram Murty.

August 21 (Fri), 2020, 16:00--17:00, Online seminar

Title: A twisted Ichino formula

Speaker: Shunsuke Yamana (Osaka City University)

Abstract: Let $\pi_i$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A})$ with central character $\omega_i$, where $\mathbb{A}$ is an adele ring of a number field. When the product $\omega_1\omega_2\omega_3$ is the trivial character of $\mathbb{A}^\times$, Atsushi Ichino proved a formula for the central value $L\bigl(\frac{1}{2},\pi_1\times\pi_2\times\pi_3\bigl)$ of the triple product $L$-series in terms of global trilinear forms that appear in Jacquet's conjecture. I will extend this formula to the case when $\omega_1\omega_2\omega_3$ is a quadratic character. This is a joint work with Ming-Lun Hsieh.

August 13 (Thu), 2020, 15:00--16:30, Room 8309

Title: Siegel modular forms and Gross-Keating invariants

Speaker: Lee, Chul-hee (KIAS)

Abstract: The Siegel series, attached to a quadratic form over the p-adic integers, is an important object in the study of Siegel modular forms. Its fundamental role can be seen in an explicit formula for the Fourier coefficients of the Siegel-Eisenstein series and some lifting methods to construct Siegel modular forms such as the Duke-Imamoglu-Ikeda lift. In this talk, I will review some classical applications of the Siegel series to the theory of Siegel modular forms, and explain the role of the Gross-Keating invariant of a quadratic form in its determination.

July 24 (Fri), 2020, 14:00--15:00, Room 1424

Title: On certain multiple Dirichlet series

Speaker: Lee, Eun Hye (Stony Brook University)

Abstract: In this talk, I will be talking about the analytic properties of multiple Dirichlet series defined using the space of binary cubic forms. First I will construct the double zeta function from the 2 (out of 4) relative invariants of the binary cubic forms, and then I will prove its meromorphic continuation to the whole $\mathbb{C}^2$ via a functional equation and as time permits, show where the poles are. This work is joint work with Ramin Takloo-Bighash.

July 06 (Mon), 2020, 14:00--15:00, Room 8309

Title: On the number of abelian varieties over finite fields

Speaker: Jungin Lee (POSTECH)

Abstract: Let $B(q, g)$ be the number of isomorphism classes of abelian varieties of dimension $g$ over a finite field $\mathbb{F}_q$. Since the number $B(q, g)$ is known to be finite, it is natural to consider its asymptotic behavior. In this talk, we will provide a lower bound and an upper bound of $B(p, g)$ as $g \rightarrow \infty$ for a fixed prime $p$. We will also investigate the number of simple isogeny classes of abelian varieties over $\mathbb{F}_q$ of dimension $g$.

February 20 (Thu), 2020, 16:00--17:00, Room 1424

Title: On Galois covers of the projective line and their specializations

Speaker: Francois Legrand (TU Dresden)

Abstract: Studying Galois covers of the projective line and their specializations is a central topic in inverse Galois theory, due to their connection with the inverse Galois problem. In this talk, we shall present two applications to diophantine geometry and the theory of modular forms. Firstly, we shall explain how deriving many curves over $\mathbb{Q}$ failing the Hasse principle, under the abc-conjecture. Secondly, we shall construct infinitely many non-liftable Hecke eigenforms of weight one over $\overline{\mathbb{F}_p}$ with pairwise non-isomorphic projective Galois representations, for $p \in \{3,5,7,11\}$. This talk is based on joint works with Joachim König, and with Sara Arias-de-Reyna and Gabor Wiese.