Title of Seminar: Number Theory Seminar
Title of Talk: Periodicity in filtrations of mod p representations of $GL_2(F_q)$
Speaker: Arindam Jana, TIFR
Date: April 26, 2023
Time: 14:30:00 Hours
Venue: AG-77
Abstract: The irreducible mod $p$ representations of ${GL}_2(\mathbb{F}_p)$ are exactly the twists of $V_r,$ the $r$-th symmetric power of the standard representations of ${GL}_2(\mathbb{F}_p)$ for small values of $r.$ In this talk, for sufficiently large $r,$ we investigate the periodicity in a filtration of $V_r$ defined by the powers of the polynomial $\theta:=X^pY-XY^p,$ motivated by a classical result of Glover. Ghate and Vangala studied the periodicity of the higher quotients in the filtration of $V_r$ using generalized dual numbers. We strengthen their result by defining an explicit isomorphism between these quotients of $V_r$ and generalized mod $p$ principal series representations using differential operators, and extend it to ${GL}_2(\mathbb{F}_q)$ for $q=p^f, f\geq 1.$ In search of a similar periodicity result in case of cuspidal representations, Reduzzi proved that the reduction mod $p$ of a cuspidal representation of ${GL}_2(\mathbb{F}_q)$ is isomorphic to the cokernel of a differential operator on $V_r$ defined by Serre. This isomorphism is proved using crystalline cohomology and is not explicit. We define this isomorphism explicitly after tensoring with $V_{q-1.}$ This work is joint with Eknath Ghate.
Title of Seminar: Algebraic Geometry Preprint Seminar 2022
Title of Talk: Existence of moduli spaces for algebraic stacks - II
Speaker: Yogish Holla, TIFR, Mumbai
Date: April 26, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this lecture we will describe the proof the existence of good/adequate moduli spaces for algebraic stacks which are locally reductive. The paper under consideration is by Jarod Alper, Daniel Halpern-Leister, and Jochen Heinloth (arXiv:1812.01128v4 [math.AG] 27 Jun 2022). This theorem is a generalisation of a well known theorem of Keel and Mori. This can be applied for the construction of coarse moduli spaces for many interesting moduli spaces like principal bundles and Bridgeland semistable objects in the derived category of coherent sheaves.
Title of Talk: Periodicity in filtrations of mod p representations of $GL_2(F_q)$
Speaker: Arindam Jana, TIFR
Date: April 26, 2023
Time: 14:30:00 Hours
Venue: AG-77
Abstract: The irreducible mod $p$ representations of ${GL}_2(\mathbb{F}_p)$ are exactly the twists of $V_r,$ the $r$-th symmetric power of the standard representations of ${GL}_2(\mathbb{F}_p)$ for small values of $r.$ In this talk, for sufficiently large $r,$ we investigate the periodicity in a filtration of $V_r$ defined by the powers of the polynomial $\theta:=X^pY-XY^p,$ motivated by a classical result of Glover. Ghate and Vangala studied the periodicity of the higher quotients in the filtration of $V_r$ using generalized dual numbers. We strengthen their result by defining an explicit isomorphism between these quotients of $V_r$ and generalized mod $p$ principal series representations using differential operators, and extend it to ${GL}_2(\mathbb{F}_q)$ for $q=p^f, f\geq 1.$ In search of a similar periodicity result in case of cuspidal representations, Reduzzi proved that the reduction mod $p$ of a cuspidal representation of ${GL}_2(\mathbb{F}_q)$ is isomorphic to the cokernel of a differential operator on $V_r$ defined by Serre. This isomorphism is proved using crystalline cohomology and is not explicit. We define this isomorphism explicitly after tensoring with $V_{q-1.}$ This work is joint with Eknath Ghate.
Title of Seminar: Algebraic Geometry Preprint Seminar 2022
Title of Talk: Existence of moduli spaces for algebraic stacks - II
Speaker: Yogish Holla, TIFR, Mumbai
Date: April 26, 2023
Time: 16:00:00 Hours
Venue: AG-77
Abstract: In this lecture we will describe the proof the existence of good/adequate moduli spaces for algebraic stacks which are locally reductive. The paper under consideration is by Jarod Alper, Daniel Halpern-Leister, and Jochen Heinloth (arXiv:1812.01128v4 [math.AG] 27 Jun 2022). This theorem is a generalisation of a well known theorem of Keel and Mori. This can be applied for the construction of coarse moduli spaces for many interesting moduli spaces like principal bundles and Bridgeland semistable objects in the derived category of coherent sheaves.