Talks

A bit of representation theory of groups over local fields, parabolic induction, cuspidal representations. Review of the Local Langlands correspondence, L-functions and epsilon factors. L-packets, the Jacquet-Langlands correspondence, The GGP conjectures: both local and global conjectures.

We first prove, for a prime p>3 unramified in a CM quadratic extension of a totally real field F, h(M/F)L(\chi)|H(\psi)|h(M/F)F(\chi) (h(M/F)=h(M)/h(F)) in \Lambda for the congruence power serie H(\psi) of \psi lifting a fixed anti-cyclotomic character \chi and anticyclotomic Katz p-adic L-function L(\chi) of branch character \chi, built on the lectures by Tilouine proving this over \Lambda[1/p]. Here \Lambda is the many variable Iwasawa algebra of M. In the second lecture, we give a sketch of the proof of the reverse divisibility: H(\psi)|h(M/F)L(\chi) resulting in the main conjecture, as H(\psi)=h(M/F)F(\chi) for the anticyclotomic Iwasawa power series F(\chi) by the “R=T”-theorem.

Abstract

How do chemical reactions change when they’re run at temperatures a billion times colder than a Canadian winter? What can we learn when we have perfect quantum control of the reactants? Before answering these questions, we’ll discuss the fascinating techniques of laser cooling that allow us to cool atoms and molecules to within a few billionths of a degree above absolute zero. We’ll then look at how molecules prepared at such temperatures allow us to control chemical reactions at the quantum level, beginning to open a new understanding of chemistry and new possibilities for technologies of the future.

About the Speaker

Dr. Alan Jamison is an Assistant Professor at the University of Waterloo, jointly appointed to the Department of Physics and Astronomy and the Institute for Quantum Computing (IQC). He leads the Jamison Lab, which investigates ultracold atoms and molecules to explore quantum many-body physics, quantum chemistry, and quantum information science. Dr. Jamison earned his B.S. in Mathematics from the University of Central Florida in 2007, followed by an M.S. and Ph.D. in Physics from the University of Washington in 2008 and 2014, respectively.

After completing his Ph.D., he joined the group of Nobel Laureate Wolfgang Ketterle at the Massachusetts Institute of Technology (MIT) as a postdoctoral researcher. At the University of Waterloo, Dr. Jamison's research centers on using ultracold atoms and molecules to investigate complex quantum systems. His lab aims to achieve precise control over chemical reactions at ultracold temperatures, providing insights into quantum chemistry and enabling advancements in quantum computing and simulation.

The arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, expresses the degrees of certain 0-cycles on integral models of unitary Shimura varieties in terms of the nondegenerate Fourier coefficients of the central derivative of an Eisenstein series.
Feng-Yun-Zhang proved a higher derivative version of this arithmetic Siegel-Weil formula in the function field setting, now expressing degrees of 0-cycles on moduli spaces of unitary shtukas to the nondegenerate Fourier coefficients of higher central derivatives of an Eisenstein series.
The goal of my lecture series is (1) to explain all of this background, (2) extend the results of Feng-Yun-Zhang to include some degenerate coefficients, and (3) deduce from this extension an arithmetic application: the nonvanishing of higher central derivatives of certain Langlands L-functions implies the nonvanishing of classes in the Chow groups of moduli spaces of shtukas.
All of the new results are joint work with Tony Feng and Mikayel Mkrtchyan.

Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.

Helen Hargreaves, MSW, RSW will present a workshop for PI Residents, providing information on the basics of Emotion Theory, how to assess ones own needs and communicate them. This presentation will particularly focus on Autistic and other neurodivergent ways of experiencing emotions and stress and how to better support neurodivergent team members in the workplace. Helen Hargreaves is a Neurodivergent Therapist with over 15 years experience workings with Neurodivergent clients. She is the Director of Rainbow Brain, a social work group practice that focuses on providing queer, trans and neurodivergent affirming therapy. Please note that this will be a 1.5 hour session with presentation and experiential components.

The local epsilon conjecture is one of a series of Kato's conjectures on a generalization of the Iwasawa main conjecture to general families of p-adic Galois representations. It gives a precise description of a p-adic variation of the p-adic Hodge theoretic invariants, like local (L-, and epsilon) factors, Bloch-Kato's cohomologies, and Hodge-Tate weights which are only defined for de Rham representations, in p-adic families of local p-adic Galois representations. In my lectures, I will explain the formulation of this conjecture, the proof of the conjecture for the rank two case using the p-adic Langlands for GL_2(Q_p), and it's application to a generalization of Rubin's local sign decomposition conjecure.

Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.

We first prove, for a prime p>3 unramified in a CM quadratic extension of a totally real field F, h(M/F)L(\chi)|H(\psi)|h(M/F)F(\chi) (h(M/F)=h(M)/h(F)) in \Lambda for the congruence power serie H(\psi) of \psi lifting a fixed anti-cyclotomic character \chi and anticyclotomic Katz p-adic L-function L(\chi) of branch character \chi, built on the lectures by Tilouine proving this over \Lambda[1/p]. Here \Lambda is the many variable Iwasawa algebra of M. In the second lecture, we give a sketch of the proof of the reverse divisibility: H(\psi)|h(M/F)L(\chi) resulting in the main conjecture, as H(\psi)=h(M/F)F(\chi) for the anticyclotomic Iwasawa power series F(\chi) by the “R=T”-theorem.

The arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, expresses the degrees of certain 0-cycles on integral models of unitary Shimura varieties in terms of the nondegenerate Fourier coefficients of the central derivative of an Eisenstein series.
Feng-Yun-Zhang proved a higher derivative version of this arithmetic Siegel-Weil formula in the function field setting, now expressing degrees of 0-cycles on moduli spaces of unitary shtukas to the nondegenerate Fourier coefficients of higher central derivatives of an Eisenstein series.
The goal of my lecture series is (1) to explain all of this background, (2) extend the results of Feng-Yun-Zhang to include some degenerate coefficients, and (3) deduce from this extension an arithmetic application: the nonvanishing of higher central derivatives of certain Langlands L-functions implies the nonvanishing of classes in the Chow groups of moduli spaces of shtukas.
All of the new results are joint work with Tony Feng and Mikayel Mkrtchyan.