Number Theory Seminars 2021

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December 2 (Thu), 2021, 15:00--16:30, online

Title: Factorization of Beilinson-Kato's element

Speaker: Shanwen Wang (Renmin University of China)

Abstract: In Kato's seminal work on his Euler system, he relates his Euler system of modular form to special value of $L$-functions via the Rankin-Selberg method. We will explain an algebraic version of this fact via the factorization of Beilinson-Kato's element. This talk is based on the joint work with Pierre Colmez.

November 11 (Thu), 2021, 16:00--17:00, online

Title: A hyperelliptic curve mapping to specified elliptic curves

Speaker: Bo-Hae Im (KAIST)

Abstract: We consider the non-existence of rational curves on certain Kummer varieties. In particular, we prove that if the $j$-invariants of a $21$-tuple of elliptic curves $E_1,\ldots,E_{21}$ over $\mathbb{C}$ are algebraically independent over $\mathbb{Q}$, there is no hyperelliptic curve which admits a non-trivial morphism to each of the $E_i$. This is a joint work with Michael Larsen and Sailun Zhan.

October 14 (Thu), 2021, 16:00--17:00, online

Title: The a-values of the Riemann zeta function near the critical line

Speaker: Yoonbok Lee (Incheon National University)

Abstract: We study the value distribution of the Riemann zeta function near the line $Re(s) =1/2$. We find an asymptotic formula for the number of a-values in the rectangle $ 1/2 + h_1 / (logT)^\theta \leq Re(s) \leq 1/2 + h_2 / (logT)^\theta$, $T \leq Im(s) \leq 2T $ for fixed $h_1 , h_2>0$ and $0 < \theta <1/13$. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwill's recent results on the discrepancy between the distribution of the Riemann zeta function and its random model. We also propose the secondary main term for the Selberg's central limit theorem by providing sharper estimates on the line $ Re(s) = 1/2 +1/(logT)^\theta$.

September 9 (Thu), 2021, 14:00--15:00, online

Title: Diophantine study of moduli spaces with nonlinear symmetry

Speaker: Junho Peter Whang (Seoul National University)

Abstract: We discuss two families of Diophantine equations with nonlinear automorphism groups, describing moduli spaces of different geometric objects. The first are the moduli spaces of special linear rank two local systems on surfaces, endowed with mapping class group dynamics. Building on an earlier work of ours, we prove that the set of integral points on these varieties is decidable and finitely generated in a precise sense. The second family are the moduli spaces of Stokes matrices, admitting a braid group action via mutations. We discuss the finite generation problem for integral points on these varieties, describing joint work with Fan establishing connections to the former moduli spaces, and formulating new results and questions.

August 3, 5, 10, 12 (Tue/Thu), 2021, 16:00--17:30, online

Title: Geometric Eisenstein series and the Fargues-Fontaine curve

Speaker: David Hansen (Max Planck Institute)

Abstract: The cohomology of local Shimura varieties, and of more general spaces of local shtukas, is of fundamental interest in the Langlands program. The Harris-Viehmann conjecture, roughly speaking, describes how the cohomology of local Shimura varieties interacts with parabolic induction. I will formulate a generalization of the HV conjecture, and then lay out a proof strategy, based on studying geometric Eisenstein series functors for stacks of bundles on the Fargues-Fontaine curve. Joint work in progress with Peter Scholze.

June 24 (Thu), 2021, 11:00--12:00, online

Title: Potentially crystalline deformation ring and Serre weight conjectures for $\text{GSp}_4$

Speaker: Heejong Lee (University of Toronto)

Abstract: The theory of Galois deformation has played crucial role in the Langlands program, especially to understand modularity/automorphy of Galois representation starting from the work of Wiles, Taylor--Wiles. It is expected that to understand the $p$-adic / mod $p$ Galois representation of global field, it is crucial to understand what happens at the places dividing $p$. In this talk, I'll explain the role of potentially crystalline deformation rings in the context of modularity problem and how to study them. At the end, I'll comment on my work on the case of $\text{GSp}_4$. This talk will be mostly expository.

June 17 (Thu), 2021, 16:00--17:00, online

Title: Horizontal non-vanishing mod $p$ of $L$-values

Speaker: Hae-Sang Sun (UNIST)

Abstract: Most results on the non-vanishing mod $p$ of $L$-values are concerned with Dirichlet or ray class characters of prime power conductors, or with quadratic characters. Burungale-Sun obtained the horizontal non-vanishing results for the Dirichlet $L$-values, i.e., results for the Dirichlet characters of general moduli. Even though the results are the first of their kind, they are not optimal and there is much room for improvement. In the talk, I will survey the result and explain briefly a possible method to obtain an improvement. This is a research in progress.

June 10 (Thu), 2021, 11:00--12:00, online

Title: Several questions on p-adic slopes of modular forms

Speaker: Liang Xiao (Peking University)

Abstract: The $p$-adic slope is the $p$-adic valuation of the $U_p$-eigenvalue on the space of modular forms. The study of $p$-adic slopes goes back to pioneer works of Gouvea and Mazur in 1990s, as well as later conjectures by Coleman, Mazur, Buzzard, Kilford, .... In this talk, I will discuss two parallel but different phenomena of slopes of modular forms, and hopefully explain how such questions can be understood in terms of $p$-adic local Langlands correspondence. This includes several joint works with Ruochuan Liu, Nha Truong, Daqing Wan, and Bin Zhao.

May 20 (Thu), 2021, 15:30--16:30, online

Title: Congruences for weakly holomorphic modular forms of half-integral weight

Speaker: Subong Lim (Sungkyunkwan University)

Abstract: In this talk, we discuss congruences involving $U_l$ operator for weakly holomorphic modular forms of half-integral weight. By using this, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singuli moduli.

April 15 (Thu), 2021, 10:00--11:00, online

Title: Robba-valued cohomology

Speaker: Koji Shimizu (UC Berkeley)

Abstract: Rigid cohomology is a good $p$-adic cohomology theory for algebraic varieties in characteristic $p$. In this talk, I will explain a relevant idea by Monsky and Washnitzer and then discuss my ongoing attempt to define a $p$-adic cohomology theory for rigid analytic varieties over a local field of characteristic $p$.

April 01 (Thu), 2021, 17:00--18:00, online

Title: On Euler systems for adjoint modular Galois representations

Speaker: Eric Urban (Columbia University)

Abstract: The purpose of this talk is to illustrate a new method to construct Euler systems based on the study of congruences between modular forms of arbitrary weights and level and their corresponding deformations of Galois representations. We will focus on the case of adjoint modular Galois representations attached to an ordinary eigenform and connect our construction to a conjecture for the Fitting ideal of the equivariant congruence module attached to the abelian base changes of that moduar form.

March 05 (Fri), 2021, 10:30--11:30, online

Title: Some analytic quantities having arithmetic information on elliptic curves

Speaker: Masato Kurihara (Keio University)

Abstract: I will discuss some analytic quantities constructed from modular symbols for a rational elliptic curve, which have some interesting arithmetic information on the Mordell-Weil group and related subjects.

February 04 (Thu), 2021, 10:00--11:00, online

Title: Slopes of modular forms and the ghost conjecture

Speaker: Robert Pollack (Boston University)

Abstract: Modular forms are mathematical objects born in complex analysis but in fact have become central objects in number theory. In particular, special modular forms, called eigenforms, have Fourier coefficients which are algebraic integers and these integers contain loads of number theoretic data. In this talk, we will focus on one slice of this data, namely, the highest power of $p$ dividing the $p$th Fourier coefficient (called the slope of the form). Through many numerical examples, we will discuss the properties of these slopes and ultimately state the so-called "ghost conjecture" which predicts them in a combinatorial way.