Seminar on Arithmetic Geometry and Algebraic Groups
2022 Session
2022 Fall Session
Date: 19.12.2022 Time: 15:00–16:00 (Beijing Time)
or
Date: 19.12.2022 Time: 16:00–17:00 (Tokyo Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Takashi Suzuki (Chuo University)
Title: Class field theory, Hasse principles and Picard-Brauer duality for two-dimensional local rings
Abstract: Saito gives class field theory and arithmetic duality for two-dimensional normal local rings with finite residue field. On the other hand, Serre and Hazewinkel give a geometric class field theory for local fields with general perfect residue field. In this talk, I will explain my work on a geometric refinement of Saito’s two-dimensional theory in Serre-Hazewinkel’s style. It treats two-dimensional normal local rings of mixed characteristic with general perfect residue field and gives the relevant arithmetic invariants certain structures as ind-pro-algebraic groups. We construct class field theory, Hasse principles and Picard-Brauer duality taking these structures into account.
Date: 14.12.2022 Time: 16:30–17:30 (Beijing Time)
or
Date: 14.12.2022 Time: 09:30–10:30 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Nguyen Manh Linh (Université Paris-Saclay)
Title: Arithmetics of homogeneous spaces over p-adic function fields
Abstract: Let K be the function field of a p-adic curve, a field of cohomological dimension 3. If X is a smooth geometrically integral K-variety, we are interested in the following arithmetic questions for X:
- Local-global principle (LGP): If X has K_v-points for all closed points on a smooth projective model of K, does X have K-points?
- Weak approximation (WA): If X has K-points, is X(K) dense in the topological product of the X(K_v)’s?
Generalizing the Brauer-Manin obstruction over number fields, we may use the group H^3_nr(X, Q/Z(2)) of unramified degree 3 cohomology to detect the failure of LGP and WA (“reciprocity obstruction”). It is natural to ask if this obstruction is the only one. Using global duality Poitou-Tate style duality theorem and parts of Poitou-Tate sequences, Harari, Scheiderer, Szamuely, and Izquierdo provided the positive answer for tori. Tian established the same result for certain reductive groups. In my talk, I shall present similar results for homogeneous spaces of SLn with geometric stabilizers of type umult (extension of a group of multiplicative type by a unipotent group), obtained by the same techniques. This is my latest preprint https://arxiv.org/abs/2211.08986.
Ref: Nguyen Manh Linh, Arithmetics of homogeneous spaces over p-adic function fields
Date: 07.12.2022 Time: 16:00–17:00 (Beijing Time)
or
Date: 07.12.2022 Time: 09:00–10:00 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Jean-Louis Colliot-Thélène (Université Paris-Saclay)
Title: On the arithmetic of intersections of two quadrics
Abstract: Lichtenbaum proved that index and period coincide for a curve of genus one over a $p$-adic field. Salberger proved that the Hasse principle holds for a smooth complete intersection of two quadrics $X \subset P^n$ over a number field, if it contains a conic and if $n\geq 5$. Building upon these two results, we extend recent results of Creutz and Viray (2021) on the existence of a quadratic point on intersections of two quadrics over $p$-adic fields and number fields. We then recover Heath-Brown’s theorem (2018) that the Hasse principle holds for smooth complete intersections of two quadrics in $P^7$. We also give an alternate proof of a theorem of Iyer and Parimala (2022) on the local-global principle in the case $n=5$.
Date: 24.11.2022 Time: 16:00–17:00 (Beijing Time)
or
Date: 24.11.2022 Time: 09:00–10:00 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: François Charles (École normale supérieure Paris - Université Paris-Saclay)
Title: Some uniform bounds for Chow groups and Brauer groups
Abstract: Let $X\rightarrow S$ be a smooth projective family defined over a number field. Various conjectures on algebraic cycles predict that some invariants of the closed fibers of this family, be they Mordell-Weil ranks, Chow groups or Brauer groups, satisfy some finiteness conditions. The goal of this talk is to investigate what sort of uniformity can be expected from these finiteness conditions. We will focus on the case of Brauer groups (joint work with Cadoret) and Chow groups in codimension 2 (joint work with Pirutka).
A. Cadoret and F. Charles, A remark on uniform boundedness for Brauer groups, Algebraic Geometry 7 (2020), pp. 512-522.
Date: 19.10.2022 Time: 15:30–16:30 (Beijing Time)
or
Date: 19.10.2022 Time: 09:30–10:30 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Diego Izquierdo (École Polytechnique)
Title: Milnor K-theory and zero-cycles over p-adic function fields
Abstract: In 1986, Kato and Kuzumaki introduced a set of conjectures in order to characterize the cohomological dimension of fields in diophantine terms. The conjectures are known to be wrong in full generality, but they provide interesting arithmetical problems over various usual fields in arithmetic geometry. The goal of this talk is to discuss the case of function fields of p-adic curves. This is joint work with G. Lucchini Arteche.
Date: 12.10.2022 Time: 16:00–17:00 (Beijing Time)
or
Date: 12.10.2022 Time: 10:00–11:00 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Olivier Wittenberg (Université Sorbonne Paris Nord)
Title: Rational points and fibrations of small rank
Abstract: Given a family of rationally connected varieties over the projective line, the fibration method aims at constructing rational points on the total space. We revisit this method and make it work fully when the locus of non-split fibres has degree at most 2, as well as, under Schinzel’s hypothesis, when the bad fibres are split by a cyclic extension. We do not make any arithmetic assumption on the smooth fibres, thus solving a problem left open since the 1990’s. This is a joint work with Yonatan Harpaz and Dasheng Wei.
Ref: Y. Harpaz, D. Wei and O. Wittenberg, Rational points on fibrations with few non-split fibres
Date: 28.09.2022 Time: 15:30–16:30 (Beijing Time)
or
Date: 28.09.2022 Time: 09:30–10:30 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Vlerë Mehmeti (Sorbonne Université)
Title: Patching on analytic fibers and local-global principles
Abstract: Patching techniques have become an important tool for proving local-global principles over function fields of curves. These principles provide a way to study the existence of rational points on varieties. I will explain how using non-archimedean analytic spaces and patching techniques on them one obtains an approach to local-global principles in higher dimensional settings. The talk will begin with a brief introduction of the objects that will be used.
Ref:
V. Mehmeti, Patching over analytic fibers and the local-global principle. Math. Ann. 383, No. 3-4, 1825-1904 (2022).
or
2022 Spring Session
Date: 11.08.2022 Time: 09:00–10:00 (Beijing Time)
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Date: 10.08.2022 Time: 21:00–22:00 (New York Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Wai Kiu Chan (Wesleyan University)
Title: On the exceptional sets of integral quadratic forms
Abstract: A collection S of equivalence classes of positive definite integral quadratic forms in n variables is called an n-exceptional set if there exists a positive definite integral quadratic form which represents all equivalent classes of positive definite integral quadratic forms in n variables except those in S. In this talk, I will describe a recent joint work with Byeong-Kewon Oh which shows that for any given positive integers m and n, there is always an n-exceptional set of size m and there are only finitely many of them.
Ref: W. K. Chan and B.-K Oh, On the exceptional sets of integral quadratic forms
Date : 29.06.2022 Time: 16:00–17:00 (Beijing Time)
or
Date: 29.06.2022 Time: 09:00–10:00 (London Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Alexei N. Skorobogatov (Imperial College London)
Title: Reduction of Kummer surfaces
Abstract: If a Kummer surface over a discretely valued field of characteristic zero has good reduction, then it comes from an abelian surface with good reduction.The converse holds if the residual characteristic is not 2. In a joint work with Chris Lazda we obtain a necessary and sufficient condition for good reduction of Kummer surfaces attached to abelian surfaces with good, non-supersingular reduction in residual characteristic 2. We also give a similar criterion for ‘twisted’ Kummer surfaces attached to 2-coverings of abelian surfaces. The supersingular case is very interesting but seems to be completely open.
Ref: C. Lazda and A. Skorobogatov, Reduction of Kummer surfaces modulo 2 in the non-supersingular case
Date: 15.06.2022 Time: 15:30–16:30 (Beijing Time)
or
Date: 15.06.2022 Time: 09:30–10:30 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Kestutis Cesnavicius (Université Paris-Saclay)
Title: Reductive group torsors on a complement of a smooth divisor
Abstract: A conjecture of Nisnevich predicts that for a smooth variety X over a field, a smooth divisor D in X, and a totally isotropic reductive X-group scheme G, every generically trivial G-torsor on X \ D trivializes Zariski locally on X. I will discuss this conjecture and related questions about torsors under reductive groups over regular rings.
Ref: K.Cesnavivius, The Bass-Quillen phenomenon for reductive group torsors
Date: 10.06.2022 Time: 09:00–10:00 (Beijing Time)
or
Date: 09.06.2022 Time: 21:00–22:00 (New York Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Brendan Hassett (Brown University)
Title: Equivariant rationality questions
Abstract: Fix a finite group G. We seek to classify varieties with G-action equivariantly birational to a representation of G on affine or projective space. The equivariant rationality problem is analogous to Diophantine questions over nonclosed fields. We explore how invariants – both classical cohomological invariants and recent symbol constructions – control rationality in some cases. Universal torsors are a powerful geometric tool for analyzing equivariant stable birational equivalence. (joint with Tschinkel)
Ref: B. Hassett and Y. Tschinkel, Torsors and stable equivariant birational geometry
Date: 16.05.2022 Time: 15:30–16:30 (Beijing Time)
or
Date: 16.05.2022 Time: 09:30–10:30 (Rome/Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Claudio Quadrelli (Insubria University)
Title: Nets for fishing absolute Galois pro-p groups
Abstract: Click here
Date: 15.04.2022 Time: 09:00–10:00 (Beijing Time)
or
Date: 14.04.2022 Time: 21:00–22:00 (New York Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: David Harbater (University of Pennsylvania)
Title: Bounding cohomology classes over semi-global fields
Abstract: It is a classical problem to relate the period of a Brauer class to its index. This has been carried out over certain ground fields, including by the speaker and collaborators in the case of semi-global fields; i.e. function fields of curves over a complete discretely valued field. This talk concerns an analog of this question, in which one considers higher cohomology classes rather than Brauer classes. Motivated by work of Saurabh Gosavi on Brauer groups, we also consider the simultaneous index of a finite set of cohomology classes in this context. This is joint work with Julia Hartmann and Daniel Krashen.
Ref: D. Harbater, J. Hartmann and D. Krashen, Bounding cohomology classes over semiglobal fields
Date: 12.04.2022 Time: 19:30–20:30 (Beijing Time)
or
Date: 12.04.2022 Time: 13:30–14:30 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Haowen Zhang (Sorbonne Université)
Title: Weak approximation for homogeneous spaces over C((X,Y)) or C((X))(Y)
Abstract: We study obstructions to weak approximation for connected linear groups and homogeneous spaces with connected or abelian stabilizers over finite extensions of C((X, Y)) or function fields of curves over C((X)). We show that for connected linear groups, the usual Brauer–Manin obstruction works as in the case of tori, using the same dévissage argument. However, this Brauer–Manin obstruction is not enough for homogeneous spaces, so we should somehow combine the Brauer–Manin obstruction with the descent obstruction using torsors under quasi-trivial tori, another natural tool used in the study of such questions, as done by Izquierdo and Lucchini Arteche for the study of obstruction to rational points.
Date: 28.03.2022 Time: 15:30–16:30 (Beijing Time)
or
Date: 28.03.2022 Time: 09:30–10:30 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Philippe Gille (Université de Lyon 1)
Title: R-equivalence for group schemes
Abstract: This is a report on a joint work with Anastasia Stavrova (St Petersburg). For a group scheme G over a ring A, we define the R-equivalence on G(A) in a compatible way with the case of algebraic groups. We compute the invariant G(A)/R in the case of a local ring when G is a torus or G is isotropic semisimple simply connected.
Ref: P. Gille and A. Stavrova, R-Equivalence on Group Schemes