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October 1 |
Organizational meeting (UCSD)
This is an organizational meeting for the remainder of the term. The seminar itself will begin one week later. |
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October 8 |
Aaron Pollack (UCSD)
The exceptional group $E_{7,3}$ has a symmetric space with Hermitian tube structure. On it, Henry Kim wrote down low weight holomorphic modular forms that are "singular" in the sense that their Fourier expansion has many terms equal to zero. The symmetric space associated to the exceptional group $E_{8,4}$ does not have a Hermitian structure, but it has what might be the next best thing: a quaternionic structure and associated "modular forms". I will explain the construction of singular modular forms on $E_{8,4}$, and the proof that these special modular forms have rational Fourier expansions, in a precise sense. This builds off of work of Wee Teck Gan and uses key input from Gordan Savin. |
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October 15 |
Cristian Popescu (UCSD) (paper)
I will present a vast generalization of Taelman's 2012 celebrated class-number formula for Drinfeld modules to the setting of (rigid analytic) L-functions of Drinfeld module motives with Galois equivariant coefficients. I will discuss applications and potential extensions of this formula to the category of t-modules and t-motives. This is based on joint work with Ferrara, Green and Higgins, and a result of meetings in the UCSD Drinfeld Module Seminar. |
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October 22 |
Paul Van Koughnett (Purdue) (paper)
This will be a mainly expository talk about some recent applications of number theory to topology. The crux of these applications is the construction of a cohomology theory called topological modular forms (TMF) out of the moduli of elliptic curves. I'll explain what TMF is, what we have been doing with it, and what we'd still like to know; I'll also discuss more recent attempts to extend the theory using level structures, higher-dimensional abelian varieties, and K3 surfaces. Time permitting, I'll talk about my work with Dominic Culver on some partial number-theoretic interpretations of TMF co-operations. |
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October 29 |
Brandon Alberts (UCSD)
We construct a random group with a local structure that models the behavior of the absolute Galois group ${\rm Gal}(\overline{K}/K)$, and prove that this random group satisfies Malle's conjecture for counting number fields ordered by discriminant with probability 1. This work is motivated by the use of random groups to model class group statistics in families of number fields (and generalizations). We take care to address the known counter-examples to Malle's conjecture and how these may be incorporated into the random group. |
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November 5 |
Samuel Mundy (Columbia)
I will talk about some recent work determining the archimedean components of certain Eisenstein series and CAP forms induced from the long root parabolic of $G_2$. I will also discuss how these results are being used in some work in progress on producing nonzero classes in symmetric cube Selmer groups. |
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November 12 |
James Upton (UCSD)
Let $X/\mathbb{F}_q$ be a smooth affine curve over a finite field of characteristic $p > 2$. In this talk we discuss the $p$-adic variation of zeta functions $Z(X_n,s)$ in a pro-covering $X_\infty:\cdots \to X_1 \to X_0 = X$ with total Galois group $\mathbb{Z}_p$. For certain ``monodromy stable'' coverings over an ordinary curve $X$, we prove that the $q$-adic Newton slopes of $Z(X_n,s)/Z(X,s)$ approach a uniform distribution in the interval $[0,1]$, confirming a conjecture of Daqing Wan. We also prove a ``Riemann hypothesis'' for a family of Galois representations associated to $X_\infty/X$, analogous to the Riemann hypothesis for equicharacteristic $L$-series as posed by David Goss. This is joint work with Joe Kramer-Miller. |
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November 19, 4:00PM |
Yifeng Liu (Yale University)
In this talk, we study the Chow group of the motive associated to a tempered global L-packet \pi of unitary groups of even rank with respect to a CM extension, whose global root number is -1. We show that, under some restrictions on the ramification of \pi, if the central derivative L'(1/2,\pi) is nonvanishing, then the \pi-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and L-functions. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain \pi-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative L'(1/2,\pi) and local doubling zeta integrals. This is a joint work with Chao Li. |
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December 3, 10:00AM |
Maxim Mornev (ETHZ)
The theory of Drinfeld modules is remarkably similar to the theory of abelian varieties, but their local monodromy behaves differently and is poorly understood. In this talk I will present a research program which aims to fully describe this monodromy. The cornerstone of this program is a "z-adic" variant of Grothendieck's l-adic monodromy theorem. The talk is aimed at a general audience of number theorists and arithmetic geometers. No special knowledge of monodromy theory or Drinfeld modules is assumed. |
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December 10 |
Bao Le Hung (Northwestern University)
The mod p cohomology of locally symmetric spaces for definite unitary groups at infinite level is expected to realize the mod p local Langlands correspondence for GL_n. In particular, one expects the (component at p) of the associated Galois representation to be determined by cohomology as a smooth representation. I will describe how one can establish this expectation in many cases when the local Galois representation is Fontaine-Laffaille. This is joint work with D. Le, S. Morra, C. Park and Z. Qian. |