Mar 13, 2023
Monday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
 
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09:30 AM  10:30 AM


An AFL Conjecture for the Whole Hecke Algebra
Michael Rapoport (Universität Bonn)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will present an extension of Wei Zhang's Arithmetic Fundamental Lemma conjecture to the whole Hecke algebra, and present some evidence for it. Joint work with Chao Li and Wei Zhang.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Diagonal Classes and the Birch and SwinnertonDyer Conjecture
Massimo Bertolini (Universität DuisburgEssen)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
This talk considers tripleproduct padic Lfunctions corresponding to various regions of classical interpolation. It describes their relations with rational points and Selmer classes on modular elliptic curves, obtained from explicit reciprocity laws for appropriate diagonal classes.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 
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 Abstract
 
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02:00 PM  03:00 PM


Arithmetic Theta Kernel and Liftings
Tonghai Yang (University of WisconsinMadison)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Let $K=\Q(\sqrt d)$ be an imaginary quadratic field. Let $L$ be a unimodular $O_K$lattice of signature $(n1, 1)$, and let $\mathcal X$ be the compactified Shimura variety over $O_K$ associated at $L$. In this talk, we describe roughly how to construct an arithmetic theta series (arithmetic theta kernel), and prove that it is a modular form of level $d$, weight $n$, character $\chi_d^n$ with values in $\widehat{CH}^1(\mathcal X^*)$. Using this arithmetic theta kernel, we can produce classical modular forms from arithmetic $1$cycles (Faltings’ height pairing) and produce arithmetic divisors from classical cusp forms. This is joint work with Bruinier, Howard, Kudla, and Rapoport for $n \ge 3$. The case $2$ is a joint work in progress with Qiao He and Yousheng Shi.
 Supplements


03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


KudlaRapoport Conjecture for Krämer Models
Chao Li (Columbia University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The KudlaRapoport conjecture, proved jointly with Wei Zhang, is a precise identity relating arithmetic intersection numbers of special cycles on unitary Shimura varieties with good reduction and central derivatives of Siegel Eisenstein series. We discuss how to formulate and prove an analogous identity for certain unitary Shimura varieties with bad reduction (Krämer models at ramified places). We will motivate these conjectures and highlight interesting new phenomena in the presence of bad reduction. This is joint work with Qiao He, Yousheng Shi and Tonghai Yang.
 Supplements



Mar 14, 2023
Tuesday

09:30 AM  10:30 AM


On the Modularity of Elliptic Curves Over Imaginary Quadratic Fields
Ana Caraiani (Imperial College, London)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
I will discuss recent joint work with James Newton where we establish the modularity of elliptic curves over a large class of imaginary quadratic fields.
 Supplements


03:30 PM  04:30 PM


Level Raising via Unitary Shimura Varieties with Good Reduction and an Ihara Lemma
Yifeng Liu (Zhejiang University)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
Recall a classical theorem of Ribet: Fix a prime l; consider a weight2 levelN newform f satisfying the mod l levelraising condition at a prime p coprime to Nl. Then Ribet shows that the first Galois cohomology of the mod l Galois representation of Q_p associated with f can be realized as the AbelJacobi image of the supersingular locus of the levelN modular curve over F_p. In this talk, we will discuss how one can generalize this phenomenon to higherdimensional unitary Shimura varieties at inert places (which remains a conjecture in general), and its relation with a certain Ihara type lemma for such varieties. We will explain cases for which we have confirmed such conjecture; and if time permits, we will mention its numbertheoretical implications. This is a joint work in progress with Yichao Tian and Liang Xiao.
 Supplements



Mar 15, 2023
Wednesday

09:00 AM  10:00 AM


On Quadratic Twists of Elliptic Curves
Ye Tian (Academy of Mathematics and Systems Science)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
We discuss 2Selmer groups and Lvalues of elliptic curves under quadratic twists. This is based on joint works with Pan and with HeXiong.
 Supplements



10:00 AM  10:30 AM


Break + Conference Photo

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Irrationality of 2Adic Zeta 5
Yunqing Tang (University of California, Berkeley)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
In this talk, we will discuss the proof of irrationality of 2adic zeta value at 5. It uses an arithmetic holonomicity theorem, which can be viewed as a refinement of André’s algebraicity criterion and has also been studied in the recent work of Bost and Charles. We will also discuss some other linear independence problems.
This is joint work with Frank Calegari and Vesselin Dimitrov.
 Supplements



11:30 AM  11:45 AM


Break

 Location
 
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11:45 AM  12:45 PM


Modularity of Higher Theta Series for Function Fields
Zhiwei Yun (Massachusetts Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In earlier work we defined a theta series of algebraic cycles on the moduli space of unitary Shtukas with an arbitrary number of legs. This can be viewed as a generalization of theta series and arithmetic theta series for function fields. A main open problem after this construction is the modularity of the higher theta series. In this talk, I will explain a proof of the modularity of higher theta series when restricted to the generic fiber. This is joint work with Tony Feng and Wei Zhang.
 Supplements



12:45 PM  02:00 PM


Lunch

 Location
 
 Video


 Abstract
 
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03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Mar 16, 2023
Thursday

09:00 AM  10:00 AM


Zeta Morphisms for Rank Two Universal Deformations
Kentaro Nakamura (Saga University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
In his work on the Iwasawa main conjecture for elliptic Hecke eigen cusp newforms, Kazuya Kato constructed a map which we call the zeta morphism whose target is the Iwasawa cohomology of the associated padic Galois representations. Combining FukayaKato’s idea for constructing the zeta morphisms for Hida families with many deep results in the padic (local and global) Langlands correspondence for GL_2/Q, we extend this map for the universal deformations of odd absolutely irreducible mod p Galois representations of rank two. As an application, we prove a theorem, which roughly says that, under some mu= 0 assumption, the Iwasawa main conjecture (without padic L function) for one modular form implies the same conjecture for arbitrary congruent modular forms.
 Supplements


10:00 AM  10:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:45 AM


Mazur's Main Conjecture at Eisenstein Primes
Giada Grossi (Université de Paris XIII (ParisNord))

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
Let E be an elliptic curve over the rationals and p an odd prime of good reduction. In 1972, Mazur formulated a conjecture describing points on the elliptic curve defined over the padic tower of cyclotomic extensions of Q in terms of a padic Lfunction. I will report on a joint work with F.Castella and C.Skinner, where we prove the conjecture in cases where E admits a rational pisogeny. Our proof is based on a congruence argument exploiting the cyclotomic Euler system of BeilinsonFlach classes, combined with a strengthening of our previous results on the anticyclotomic Iwasawa theory of E over an imaginary quadratic field. If time permits I will also discuss about how this refined anticyclotomic result yields new cases of Kolyvagin's conjecture on indivisibility of Heegner points.
 Supplements



11:45 AM  11:50 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:50 AM  12:50 PM


Zeta Functions of Shimura Varieties: Past, Present, and the Near Future
Yihang Zhu (University of Maryland)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will first recall the general expectations of Shimura, Langlands, and Kottwtiz on the shape of the zeta function of a Shimura variety, or more generally its etale cohomology. I will then report on some recent progress which partially fulfills these expectations, for Shimura varieties of unitary groups and special orthogonal groups. Finally, I will give a preview of some foreseeable developments in the near future.
 Supplements



12:50 PM  02:00 PM


Lunch

 Location
 
 Video


 Abstract
 
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02:00 PM  03:00 PM


Semisimplicity of Etale Cohomology of Some Shimura Varieties
Si Ying Lee (MaxPlanckInstitut für Mathematik)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
The LanglandsKottwitz method allows us to understand the zeta function of the Shimura variety, which in turn allows us to understand the Galois representations appearing in the etale cohomology of the Shimura variety, but only up to semisimplification. I will talk about a result that shows that a certain part of the etale cohomology of various abeliantype Shimura varieties is semisimple.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Exceptional Theta Functions
Aaron Pollack (University of California, San Diego)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Classical pluriharmonic theta functions give one a way to explicitly write down cuspidal holomorphic (Siegel) modular forms, together with their exact Fourier expansion. These classical theta functions are associated to algebraic modular forms on definite orthogonal groups. I will describe an "exceptional" analogue of these classical theta functions, which are associated to algebraic modular forms on definite forms of the exceptional groups G_2 and F_4. I will also describe some applications.
 Supplements


04:30 PM  06:20 PM


Reception

 Location
 SLMath: Front Courtyard
 Video


 Abstract
 
 Supplements




Mar 17, 2023
Friday

09:30 AM  10:30 AM


Algebraic and PAdic Aspects of LFunctions, with a View toward Spin LFunctions for GSp_6
Ellen Eischen (University of Oregon)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
I will discuss developments for algebraic and padic aspects of Lfunctions, with a view toward my ongoing joint work with G. Rosso and S. Shah on spin Lfunctions for GSp_6. I will emphasize how these developments fit into a broader context, highlighting common ingredients among constructions during the past several decades, while also indicating where new technical challenges arise. All are welcome, and expertise in spin Lfunctions is unnecessary to follow this lecture.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


PAdic LFunctions for FiniteSlope Families on Symplectic and Unitary Groups
Zheng Liu (University of California, Santa Barbara)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
We use the doubling method to construct padic Lfunctions for finiteslope families of automorphic forms on symplectic and unitary groups. A key construction is the padic deformation of iterations of MaassShimura differential operators on nearly overconvergent families. We will explain how to generalize the construction for GL(2) by Andreatta and Iovita, and how to bypass explicit computations of formulas of iterations of differential operators on qexpansions by using dynamics of U_p operators.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 
 Video


 Abstract
 
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02:00 PM  03:00 PM


Period Relations for Arithmetic Automorphic Periods on Unitary Groups
Jie Lin (Universität DuisburgEssen)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
Given an automorphic representation of a unitary group, one can define an arithmetic automorphic period as the Petersson inner product of a deRham rational form. Here the deRham rational structure comes from the cohomology of Shimura varieties. When the form is holomorphic, the period can be related to special values of Lfunctions and is better understood. In this talk, we formulate a conjecture on relations among general arithmetic periods of representations in the same Lpacket and explain a conditional proof.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


GanGrossPrasad Cycles and Derivatives of PAdic LFunctions
Daniel Disegni (Ben Gurion University of the Negev)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Certain RankinSelberg motives of rank n(n+1) are endowed with algebraic cycles arising from maps of unitary Shimura varieties. GanGrossPrasad conjectured that these cycles are analogous to Heegner points, in the sense that their nontriviality should be detected by derivatives of Lfunctions.
I will propose another nontriviality criterion, based on padic Lfunctions. Under some assumptions, this variant can be established in a refined quantitative form, via a comparison of padic relativetrace formulas. Together with the recent LTXZZ Euler system, this gives cases of the padic BeilinsonBlochKato conjecture of PerrinRiou. (Joint work with Wei Zhang.)
 Supplements



