Mar 13, 2023
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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An AFL Conjecture for the Whole Hecke Algebra
Michael Rapoport (Universität Bonn)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- 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
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Diagonal Classes and the Birch and Swinnerton-Dyer Conjecture
Massimo Bertolini (Universität Duisburg-Essen)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
This talk considers triple-product p-adic L-functions 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
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12:00 PM - 02:00 PM
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Lunch
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- Location
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- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Arithmetic Theta Kernel and Liftings
Tonghai Yang (University of Wisconsin-Madison)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Let $K=\Q(\sqrt d)$ be an imaginary quadratic field. Let $L$ be a unimodular $O_K$-lattice of signature $(n-1, 1)$, 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
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Kudla-Rapoport Conjecture for Krämer Models
Chao Li (Columbia University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
The Kudla-Rapoport 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
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Mar 14, 2023
Tuesday
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09:30 AM - 10:30 AM
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On the Modularity of Elliptic Curves Over Imaginary Quadratic Fields
Ana Caraiani (Imperial College, London)
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- Location
- SLMath: Online/Virtual
- Video
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- 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
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03:30 PM - 04:30 PM
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Level Raising via Unitary Shimura Varieties with Good Reduction and an Ihara Lemma
Yifeng Liu (Zhejiang University)
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- Location
- SLMath: Online/Virtual
- Video
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- Abstract
Recall a classical theorem of Ribet: Fix a prime l; consider a weight-2 level-N newform f satisfying the mod l level-raising 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 Abel-Jacobi image of the supersingular locus of the level-N modular curve over F_p. In this talk, we will discuss how one can generalize this phenomenon to higher-dimensional 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 number-theoretical implications. This is a joint work in progress with Yichao Tian and Liang Xiao.
- Supplements
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Mar 15, 2023
Wednesday
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09:00 AM - 10:00 AM
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On Quadratic Twists of Elliptic Curves
Ye Tian (Academy of Mathematics and Systems Science)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
We discuss 2-Selmer groups and L-values of elliptic curves under quadratic twists. This is based on joint works with Pan and with He-Xiong.
- Supplements
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10:00 AM - 10:30 AM
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Break + Conference Photo
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:30 AM
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Irrationality of 2-Adic Zeta 5
Yunqing Tang (University of California, Berkeley)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
In this talk, we will discuss the proof of irrationality of 2-adic 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
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11:30 AM - 11:45 AM
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Break
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- Location
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- Video
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- Abstract
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- Supplements
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11:45 AM - 12:45 PM
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Modularity of Higher Theta Series for Function Fields
Zhiwei Yun (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- 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
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12:45 PM - 02:00 PM
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Lunch
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- Location
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- Video
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- Abstract
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- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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Mar 16, 2023
Thursday
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09:00 AM - 10:00 AM
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Zeta Morphisms for Rank Two Universal Deformations
Kentaro Nakamura (Saga University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- 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 p-adic Galois representations. Combining Fukaya-Kato’s idea for constructing the zeta morphisms for Hida families with many deep results in the p-adic (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 p-adic L- function) for one modular form implies the same conjecture for arbitrary congruent modular forms.
- Supplements
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10:00 AM - 10:30 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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10:30 AM - 11:45 AM
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Mazur's Main Conjecture at Eisenstein Primes
Giada Grossi (Université de Paris XIII (Paris-Nord))
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- 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 p-adic tower of cyclotomic extensions of Q in terms of a p-adic L-function. 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 p-isogeny. Our proof is based on a congruence argument exploiting the cyclotomic Euler system of Beilinson-Flach 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
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11:45 AM - 11:50 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:50 AM - 12:50 PM
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Zeta Functions of Shimura Varieties: Past, Present, and the Near Future
Yihang Zhu (University of Maryland)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- 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
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12:50 PM - 02:00 PM
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Lunch
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- Location
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- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Semisimplicity of Etale Cohomology of Some Shimura Varieties
Si Ying Lee (Max-Planck-Institut für Mathematik)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
The Langlands-Kottwitz 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 abelian-type Shimura varieties is semisimple.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Exceptional Theta Functions
Aaron Pollack (University of California, San Diego)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- 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
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Front Courtyard
- Video
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- Supplements
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Mar 17, 2023
Friday
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09:30 AM - 10:30 AM
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Algebraic and P-Adic Aspects of L-Functions, with a View toward Spin L-Functions for GSp_6
Ellen Eischen (University of Oregon)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will discuss developments for algebraic and p-adic aspects of L-functions, with a view toward my ongoing joint work with G. Rosso and S. Shah on spin L-functions 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 L-functions is unnecessary to follow this lecture.
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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P-Adic L-Functions for Finite-Slope Families on Symplectic and Unitary Groups
Zheng Liu (University of California, Santa Barbara)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
We use the doubling method to construct p-adic L-functions for finite-slope families of automorphic forms on symplectic and unitary groups. A key construction is the p-adic deformation of iterations of Maass-Shimura 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 q-expansions by using dynamics of U_p operators.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
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- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Period Relations for Arithmetic Automorphic Periods on Unitary Groups
Jie Lin (Universität Duisburg-Essen)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- 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 L-functions and is better understood. In this talk, we formulate a conjecture on relations among general arithmetic periods of representations in the same L-packet and explain a conditional proof.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Gan-Gross-Prasad Cycles and Derivatives of P-Adic L-Functions
Daniel Disegni (Ben Gurion University of the Negev)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Certain Rankin-Selberg motives of rank n(n+1) are endowed with algebraic cycles arising from maps of unitary Shimura varieties. Gan-Gross-Prasad conjectured that these cycles are analogous to Heegner points, in the sense that their nontriviality should be detected by derivatives of L-functions.
I will propose another nontriviality criterion, based on p-adic L-functions. Under some assumptions, this variant can be established in a refined quantitative form, via a comparison of p-adic relative-trace formulas. Together with the recent LTXZZ Euler system, this gives cases of the p-adic Beilinson-Bloch-Kato conjecture of Perrin-Riou. (Joint work with Wei Zhang.)
- Supplements
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