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September 29 |
Organizational meeting (APM 7321) |
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October 6 |
Christian Klevdal (UCSD)
(Joint with Stefan Patrikis.) In this talk, we discuss a strong form of independence of $\ell$ for canonical $\ell$-adic local systems on Shimura varieties, and sketch a proof of this for Shimura varieties arising from adjoint groups whose simple factors have real rank $\geq 2$. Notably, this includes all adjoint Shimura varieties which are not of abelian type. The key tools used are the existence of companions for $\ell$-adic local systems and the superrigidity theorem of Margulis for lattices in Lie groups of real rank $\geq 2$. The independence of $\ell$ is motivated by a conjectural description of Shimura varieties as moduli spaces of motives. For certain Shimura varieties that arise as a moduli space of abelian varieties, the strong independence of $\ell$ is proven (at the level of Galois representations) by recent work of Kisin and Zhou, refining the independence of $\ell$ on the Tate module given by Deligne's work on the Weil conjectures. |
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October 13 |
Shishir Agrawal (UCSD)
Let $G$ be a $p$-adic reductive group with $\mathfrak{g} = \mathrm{Lie}(G)$. I will summarize work with Matthias Strauch in which we construct an exact functor from category $\mathcal{O}^\infty$, the extension closure of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ inside the category of $U(\mathfrak{g})$-modules, into the category of admissible locally analytic representations of $G$. This expands on an earlier construction by Sascha Orlik and Matthias Strauch. A key role in our new construction is played by $p$-adic logarithms on tori, and representations in the image of this functor are related to some that are known to arise in the context of the $p$-adic Langlands program. |
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October 20 |
Jon Aycock (UCSD)
In 1978, Katz gave a construction of the $p$-adic $L$-function of a CM field by using a $p$-adic analog of the Maass--Shimura operators acting on $p$-adic Hilbert modular forms. However, this $p$-adic Maass--Shimura operator is only defined over the ordinary locus, which restricted Katz's choice of $p$ to one that splits in the CM field. In 2021, Andreatta and Iovita extended Katz's construction to all $p$ for quadratic imaginary fields using overconvergent differential operators constructed by Harron--Xiao and Urban, which act on elliptic modular forms. Here we give a construction of such overconvergent differential operators which act on Hilbert modular forms. |
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October 27 |
Rusiru Gambheera Arachchige (UCSD)
In 2015 Greither and Popescu constructed a new class of Iwasawa modules, which are the number field analogues of $p-$adic realizations of Picard 1- motives constructed by Deligne. They proved an equivariant main conjecture by computing the Fitting ideal of these new modules over the appropriate profinite group ring. This is an integral, equivariant refinement of Wiles' classical main conjecture. As a consequence they proved a refinement of the Brumer-Stark conjecture away from 2. All of the above was proved under the assumption that the relevant prime $p$ is odd and that the appropriate classical Iwasawa $\mu$–invariants vanish. Recently, Dasgupta and Kakde proved the Brumer-Stark conjecture, away from 2, unconditionally, using a generalization of Ribet's method. We use the Dasgupta-Kakde results to prove an unconditional equivariant main conjecture, which is a generalization of that of Greither and Popescu. As applications of our main theorem we prove a generalization of a certain case of the main result of Dasgupta-Kakde and we compute the Fitting ideal of a certain naturally arising Iwasawa module. This is joint work with Cristian Popescu. |
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November 3 |
Finn McGlade (UCSD)
Let $E/\mathbb{Q}$ be an imaginary quadratic extension and suppose $G$ is the unitary group attached to hermitian space over $E$ of signature $(2,n)$. The symmetric domain $X$ attached to $G$ is a quaternionic Kahler manifold. In the late nineties N. Wallach studied harmonic analysis on $X$ in the context of this quaternionic structure. He established a multiplicity one theorem for spaces of generalized Whittaker periods appearing in the cohomology of certain $G$-bundles on $X$. We prove an analogous multiplicity one statement for some degenerate generalized Whittaker periods and give explicit formulas for these periods in terms of modified K-Bessel functions. Our results give a refinement of the Fourier expansion of certain automorphic forms on $G$ which are quaternionic discrete series at infinity. As an application, we study the Fourier expansion of cusp forms on $G$ which arise as theta lifts of holomorphic modular forms on quasi-split $\mathrm{U}(1,1)$. We show that these cusp forms can be normalized so that all their Fourier coefficients are algebraic numbers. (joint with Anton Hilado and Pan Yan) |
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November 10 |
Kalyani Kansal (Johns Hopkins) (paper)
The Emerton-Gee stack for $\mathrm{GL}_2$ is a stack of $(\varphi, \Gamma)$-modules whose reduced part $\mathcal{X}_{2, \mathrm{red}}$ can be viewed as a moduli stack of mod $p$ representations of a $p$-adic Galois group. We compute criteria for codimension one intersections of the irreducible components of $\mathcal{X}_{2, \mathrm{red}}$, and interpret them in sheaf-theoretic terms. We also give a cohomological criterion for the number of top-dimensional components in a codimension one intersection. |
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November 17 |
Romyar Sharifi (UCLA) (paper)
We will discuss Galois cohomology groups of “intermediate” quotients of an induced module, which sit between Iwasawa cohomology up a tower and cohomology over the ground field. Special elements in Iwasawa cohomology that arise from Euler systems become divisible by a certain Euler factor upon norming down to the ground field. In certain instances, there are reasons to wonder whether this divisibility can also hold for the image in intermediate cohomology. Using “intermediate” Coleman maps, we shall see that the situation locally at $p$ is as nice as one could imagine. |
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December 1 |
Christopher Keyes (Emory) (paper)
If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$, what is the probability that the superelliptic curve with equation $C \colon: y^m = f(x, z)$ has a $p$-adic point, or better, points everywhere locally? In joint work with Lea Beneish, we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94%, with the exact value given by an explicit infinite product of rational function expressions. |