Arithmetic & Algebraic Geometry Seminar at HRI (archive)

September 20, 2023: Charanya Ravi (Indian Statistical Institute, Bangalore) 16:00–17:00 (IST) #HRI Auditorium

Title: Equivariant localization theorem

Abstract: The Atiyah-Bott localization theorem says that the equivariant cohomology of a space can be recovered, up to inverting some elements, from the equivariant cohomology of the fixed point subspace. We discuss a categorified version of this result which allows us to deduce the theorem for all oriented theories (cohomology and Borel-Moore homology). This is based on a joint work with Adeel Khan.

August 9, 2023: Anand Sawant (TIFR) 16:30–17:30 (IST) #Higgs Lecture Hall

Title: Obstructions to the Rost nilpotence principle

Abstract: The Rost nilpotence principle is an important tool in the study of motivic decompositions of smooth projective varieties over a field. I will introduce Chow motives, the Rost nilpotence principle and briefly survey the cases in which it is known to hold. Voevodsky's proof of the Bloch-Kato conjecture yields powerful motivic techniques, which enable one to precisely write down the obstruction for the Rost nilpotence principle to hold. I will outline some recent work on the determination of these obstructions.

July 26, 2023: Kim Tuan Do (UCLA) 09:00–10:00 (IST) #Zoom

Title: Construction of an anticyclotomic Euler System

Abstract: We construct a new anticyclotomic Euler system for the Galois representation attached to a modular form of weight $2k$, twisted by an anticyclotomic Hecke character $\chi$ of infinity type $(m,-m)$, denoted $V_{f,\chi}$, when the Heegner hypothesis is not satisfied and $m\ge k$. If time permits, we will then show some arithmetic applications (in a joint work with F. Castella) of the constructed Euler system, including evidence of the Bloch-Kato Conjecture and the Iwasawa Main Conjecture for $V_{f,\chi}$.

July 19, 2023: Annette Huber-Klawitter (University of Freiburg) 16:00–17:00 (IST) #Zoom

Title: Species and dimension formulas for period spaces

Abstract: (joint work with Martin Kalck) Periods are complex numbers obtained by integrating algebraic differential forms over $\mathbf{Q}$ over a domain of algebraic nature. This includes numbers like $\pi$, $\log(2)$ or the values of the Riemann zeta function at integers. More conceptually, periods are the entries of the period pairing between the singular and de Rham realisation of a mixed motive over $\mathbf{Q}$. The period conjecture makes a sweeping qualitative prediction on the relations between period numbers, e.g., that $\pi$ is transcendental. We develop and apply the abstract theory of a very special class of Noetherian rings, namely finite dimensional $\mathbf{Q}$-algebras, in order to deduce formulas for the expected dimension of the space of periods of a single motive. This is of particular interest in the case of $1$-motives (or periods of curves), where the linear version of the period conjecture is a theorem due to Huber and Wüstholz.

June 21, 2023: Antonio Cauchi (Concordia University, Montreal) 17:30–18:30 (IST) #Zoom

Title: Towards new Euler systems for automorphic Galois representations

Abstract: The construction of Euler systems for Galois representations associated to automorphic forms often relies on the existence of Rankin-Selberg integrals which calculate the corresponding $\mathrm{L}$-function. I will discuss a new Rankin-Selberg integral, which represents a twist of the degree $5$ $\mathrm{L}$-function of cusp forms on $GSp_4$, and its potential application to the study of the arithmetic of the standard Galois representation associated to cusp forms on $GSp_4$. This is joint work with Armando Gutierrez.

June 07, 2023: Joaquín Rodrigues Jacinto (Université Sorbonne Paris Nord) 16:30–17:30 (IST) # Zoom

Title: Solid locally analytic representations

Abstract: I will explain a joint work with Juan Esteban Rodríguez Camargo, where we develop new foundations for the theory of locally analytic representations of a p-adic Lie group through the use of condensed mathematics. As an application of this new formalism, I will explain a geometric interpretation of the category of locally analytic representations and, if time permits, some comparison results between different cohomology theories for solid representations.

May 24, 2023: Kuntal Chakraborty (HRI) 16:00–17:00 (IST) # HRI Auditorium

A criterion of completability of unimodular rows over non-singular affine domain of dimension $d$

In this talk we first prove that for regular local ring $R$ containing an infinite field, the natural map $SL_r(R[X], (X^2-X))/E_r(R[X], (X^2-X))\rightarrow SK_1(R[X], (X^2-X))$ is an isomorphism for $r\geq3$. Then we prove that for a smooth affine domain $R$ over an infinite field of dimension $d$, a unimodular row of length $d+1$ over $R$ is completable, if $${\rm Um}_{d+1}(Q(R)[X_1,X_2,\dots,X_d],(X_1^2-X_1)(X_2^2-X_2)\dots (X_d^2-X_d))= e_1 SL_{d+1}(Q(R)[X_1,X_2,\dots,X_d],(X_1^2-X_1)(X_2^2-X_2)\dots(X_d^2-X_d)),$$ where $Q(R)$ is the quotient field of $R$. This is a joint work with Prof. Ravi A. Rao.

April 26, 2023: K. V. Shuddhodan (IHÉS) 16:00–17:00 (IST) # Zoom

The (non-uniform) Hrushovski-Lang-Weil estimates

In 1996 using techniques from model theory and intersection theory, Hrushovski obtained a generalization of the Lang-Weil estimates. Subsequently, the estimate has found applications in group theory, algebraic dynamics, and algebraic geometry. We shall discuss an $l$-adic proof of these estimates' non-uniform version and the rationality of the associated generating function.

April 12, 2023: Lucas Mann (University of Münster, Münster) 16:00–17:00 (IST) # Zoom

A p-Adic 6-Functor Formalism on Rigid Varieties.

We explain how Clausen-Scholze's condensed mathematics can be used to construct a 6-functor formalism for p-adic sheaves on rigid varieties (i.e. "p-adic manifolds"), which in particular implies p-adic Poincaré duality in this setting. When applied to classifying stacks of p-adic groups, this 6-functor formalism provides new insights into the p-adic Langlands program.

March 29, 2023: Jitendra Rathore (TIFR, Mumbai) 16:00–17:00 (IST) # Higgs Lecture Hall (HRI)

Class field theory for schemes over local fields

The (tame) class field theory for a smooth variety $X$ is the study of describing the abelianized (tame) {\'e}tale fundamental group of $X$ in terms of some groups which are defined using algebraic cycles of $X$.

In this talk, we study the tame class field theory for smooth varieties over local fields. We will begin with defining few notions and recalling various results from the past to overview the historical background of the subject. We will then study abelianized tame fundamental group denoted as $\pi^{ab,t}_{1}(X)$, with the help of reciprocity map $\rho^{t}_{X} : C^{t}(X) \rightarrow \pi^{ab,t}_{1}(X)$ and will describe the kernel and topological cokernel of this map. This talk is based on a joint work with Prof. Amalendu Krishna and Dr. Rahul Gupta.

March 15, 2023: Shubhodip Mondal (MPIM, Bonn) 16:00–17:00 (IST) # Zoom

Unipotent homotopy theory of schemes

Building on Toen's work on higher (affine) stacks, I will discuss a notion of homotopy theory for schemes, which we call ``unipotent homotopy theory". Over a field of char. $p>0$, I will explain how our unipotent homotopy group schemes recover (1) unipotent completion of the Nori fundamental group scheme, (2) p-adic étale homotopy groups, and (3) certain formal group laws arising from algebraic varieties constructed by Artin--Mazur. Joint work with Emanuel Reinecke.

February 22, 2023: Amalendu Krishna (IISC, Bengaluru) 16:45–17:45 (IST) # Higgs Lecture Hall (HRI)

Ramified class field theory of curves over local fields

The class field theory for ramified coverings of smooth projective curves over finite fields is a classical result in arithmetic geometry. However, the same is not yet understood over local fields. We shall review the known results on this question and present some new results.