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

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Title:
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Potentially crystalline deformation ring and Serre weight conjectures for GSp4
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Speaker:
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Heejong Lee (University of Toronto)

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Abstract:
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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 GSp4. This talk will be mostly expository.

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

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Title:
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Horizontal non-vanishing mod $p$ of $L$-values
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Speaker:
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Hae-Sang Sun (UNIST)

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Abstract:
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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

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Title:
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Several questions on p-adic slopes of modular forms
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Speaker:
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Liang Xiao (Peking University)

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Abstract:
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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

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Title:
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Congruences for weakly holomorphic modular forms of half-integral weight
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Speaker:
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Subong Lim (Sungkyunkwan University)

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Abstract:
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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

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Title:
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Robba-valued cohomology
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Speaker:
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Koji Shimizu (UC Berkeley)

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Abstract:
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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

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Title:
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On Euler systems for adjoint modular Galois representations
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Speaker:
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Eric Urban (Columbia University)

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Abstract:
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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

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Title:
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Some analytic quantities having arithmetic information on elliptic curves
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Speaker:
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Masato Kurihara (Keio University)

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Abstract:
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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

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Title:
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Slopes of modular forms and the ghost conjecture
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Speaker:
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Robert Pollack (Boston University)

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Abstract:
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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 pth 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.