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This year the seminar will be organised at the IISER, Pune on 23-24th September, 2022.
Venue-Lecture hall complex (LHC 304)
23rd September
Chair: Dr. Chandrasheel Bhagwat
Supriya Pisolkar (IISERP): 3 pm-4pm.
Anand Chitrao (TIFR): 4.30-5.30 PM
Chair: Dr. Kaneenika Sinha
K. Mallesham (IITB): 5.45-6.45 PM.
Saturday 24th September
Chair: Prof. Eknath Ghate
Sandeep Varma (TIFR): 9-10 AM (Zoom talk)
Ankit Rai (IITB): 10.30-11.30 AM
Chair: Prof. U. K. Anandavardhanan
Prashant Arote (IISERP): 12 noon -1 PM (zoom .
Title and abstracts:-
Supriya Pisolkar (IISERP)
Title: Towards Fontaine-Mazur Conjecture for bi-quadratic extensions - an example.
Abstract :
We prove that the Galois group of the maximal everywhere unramified pro-3-extension of the biquadratic field Q( \sqrt{−26}, \sqrr{229}) has no infinite p-adic analytic pro-3 quotient. This answers negatively a question asked by Nigel Boston in his fundamental 1992 paper in his search for some counter-example to Fontaine-Mazur conjecture. This is a joint work with Ramla Abdellatif.
Anand Chitrao (TIFR)
Title : Blow-ups in rigid analysis.
Abstract:
In this talk, we will construct the blow-up of the 2-dimensional unit ball at the origin as a rigid analytic variety. Using some facts from p-adic analysis, we will check that this rigid analytic variety satisfies the universal property for blow-ups.
K. Mallesham (IITB)
Title: Sub-Weyl strength bound for short character sums of twist of $GL(2)$ Fourier coefficients.
Abstract:
Let $$S(N) = \sum_{n \sim N}^{\text{smooth}} \, \lambda_{f}(n) \, \chi(n),$$
where $\lambda_{f}(n)$'s are Fourier coefficients of a Hecke eigenform $f$, and $\chi$ is a primitive character of conductor $p^{r}$. In this talk I will discuss about a sub-Weyl strength bound for $S(N)$ with $r$ varying and $p$ being fixed. This talk is based on a joint work with Aritra Ghosh.
Sandeep Varma (TIFR)
Title: Some comments on the stable Bernstein center of a reductive $p$-adic group
Abstract:-
This will be a quasi-expository talk, in which I will report on joint work in progress, partially with Yeansu Kim, concerning the stable Bernstein center of a quasi-split reductive $p$-adic group $G$. Literature suggests at least two candidates for what deserves to be called the stable Bernstein center of $G$ --- the vector subspace $\mathcal{Z}_1(G)$ ofthe Bernstein center $\mathcal{Z}(G)$ consisting of stable distributions, and the subring $\mathcal{Z}_2(G) \subset \mathcal{Z}(G)$ of elements convolution with which preserves the space of stable distributions. According to (some version of) the stablecenter conjecture, the easy inclusion $\mathcal{Z}_2(G)\subset \mathcal{Z}_1(G)$ is expected to be an equality. Haines has also formulated a so called $\mathcal{Z}$-transfer conjecture relating the behavior of the stable Bernstein center with respect to endoscopic transfer. I will discuss these and related statements, and explain why some standard expectations regarding tempered $L$-packets and
endoscopic transfer imply the truth of these conjectures. Though these results are tentative (this being work in progress), it is likely that many of these results are known to experts, at least in principle.
Ankit Rai (IITB)
Title : Lefschetz properties for orthogonal Shimura varieties.
Abstract :
Let $G$ be a semisimple algebraic group over the rationals with real points $SO(2,n)$. Let $H$ be a subgroup of $G$ with real points $SO(2,n-1)$. For arithmetic subgroups $\Gamma \subset G(\mathbb{Q})$ and $\Gamma_H = \Gamma \cap H(\mathbb{Q})$, let $X_{\Gamma}$ and $X_{H,\Gamma_H$ be the associated locally symmetric varieties. We will prove a Lefschetz property for restriction of cohomology from $X_{\Gamma}$ to finitely many subvarieties of the form $X_{H,\Gamma_H}$ for different subgroups $H$. The proofs combine geometric techniques (mixed Hodge theory, compactifications of locally symmetric varieties) and techniques from representation theory and automorphic forms.
Prashant Arote
Title:-Harish-Chandra Induction and Jordan Decomposition of Characters}
Abstract :
Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_{q}$ and let
$F:G\rightarrow G$ be the corresponding Frobenius morphism. Let $L$ be a Levi factor of a $F$-stable parabolic subgroup of $G$. Endomorphism algebras of representations parabolically induced from cuspidal representations of $L^F$ were described by Howlett and Lehrer. In this talk, we show that any such algebra is isomorphic to the twisted endomorphism algebra of a representation which is parabolically induced from a unipotent cuspidal representation of Levi subgroup of a finite reductive group. As an application, we show that for any finite connected reductive group, a Jordan decomposition can be chosen such that it commutes with Harish-Chandra induction. This is a joint work with Manish Mishra.
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