Seminar on Arithmetic Geometry and Algebraic Groups
2023 Session
2023 Fall Session
Date: 24.11.2023 Time: 13:30–14:30 (Beijing Time)
or
Date: 24.11.2023 Time: 14:30–15:30 (Tokyo Time)
Zoom Meeting ID: available upon demand
Speaker: Takao Yamazaki (Chuo University)
Title: Torsion birational motives of surfaces and unramified cohomology
Abstract: Kahn-Sujatha’s birational motive is a variant of Chow motive that synthesis the ideas of birational geometry and motives. We explain our result saying that the unramified cohomology is a universal invariant for torsion motives of surfaces. We also exhibit examples of complex varieties violating the integral Hodge conjecture. If time permits, we also discuss a pathology in positive characteristic. (Joint work with Kanetomo Sato.)
Video recording: available upon request
Ref: K. Sato and T. Yamazaki, Torsion birational motives of surfaces and unramified cohomology
Date: 08.11.2023 Time: 12:00–13:00 (Beijing Time)
or
Date: 08.11.2023 Time: 17:00–18:00 (New Zealand Standard Time)
Zoom Meeting ID: available upon demand
Speaker: Brendan Creutz (University of Canterbury)
Title: The Brauer-Manin obstruction for curves over global function fields
Abstract: For a curve C over a global field k it is expected that the Brauer-Manin obstruction is the only obstruction to the existence of rational points and to weak approximation.
Over number fields this is wide open, but over global function fields there has been significant progress. Poonen and Voloch proved this for ‘most’ non-isotrivial curves over global function fields over a decade ago.
I will discuss more recent works on this topic, joint with Voloch, in which we prove the result for all non-isotrivial curves and also give some partial results in the isotrivial case.
Video recording: available upon request
Ref: B. Creutz and J. Voloch, The Brauer-Manin obstruction for curves over global function fields.
Date: 26.10.2023 Time: 16:00–17:00 (Beijing Time)
or
Date: 26.10.2023 Time: 10:00–11:00 (Central European Summer time)
Zoom Meeting ID: available upon demand
Speaker: Vitezslav Kala (Charles University)
Title: Universal quadratic forms and Northcott property of infinite number fields
Abstract: Universal quadratic forms generalize the sum of four squares about which it is well known that it represents all positive rational integers. In the talk, I’ll start by discussing some results on universal quadratic forms over totally real number fields. Then I’ll move on to the - markedly different! - situation over infinite degree extensions K of Q. In particular, I’ll show that if K doesn’t have many small elements (i.e., “K has the Northcott property”), then it admits no universal form. The talk should be broadly accessible, and is based on a very recent joint work with Nicolas Daans and Siu Hang Man.
Video recording: available upon request
Date: 18.10.2023 Time: 16:00–17:00 (Beijing Time)
or
Date: 18.10.2023 Time: 10:00–11:00 (CEST)
Zoom Meeting ID: available upon demand
Speaker: Victor Y. Wang (IST Austria)
Title: Harmonic analysis towards Manin–Peyre
Abstract: I will outline the shape of algebro-geometric harmonic analysis in my work on Manin’s conjecture for cubic fourfolds and/or ax+b compactifications, as time permits
Video recording: available upon request
2023 Spring Session
Date: 12.07.2023 Time: 16:00–17:00 (Beijing Time)
or
Date: 12.07.2023 Time: 10:00–11:00 (Central European Summer time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Eva Bayer-Fluckiger (École Polytechnique Fédérale de Lausanne)
Title: Automorphisms of K3 surfaces and cyclotomic polynomials
Abstract: Let X be a complex projective K3 surface, and let T be its transcendental lattice; the characteristic polynomials of the isometries of T induced by automorphisms of X are powers of cyclotomic polynomials. Which powers of cyclotomic polynomials occur ? The aim of this note is to answer this question, as well as related ones, and give an alternative approach to some results of Kondo, Machida, Oguiso, Vorontsov, Xiao and Zhang.
Ref: Eva Bayer-Fluckiger, Automorphisms of K3 surfaces and cyclotomic polynomials
Date: 10.07.2023 Time: 15:30–16:30 (Beijing Time)
or
Date: 10.07.2023 Time: 13:00–14:00 (Indian Standard Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Amalendu Krishna (Indian Institute of Science)
Title: Ramified class field theory of curves over local fields
Abstract: In this talk, I will present some results on the class field theory of smooth projective curves over a local field where one allows arbitrary ramification along a proper closed subset. We shall derive these results using some new results on the class field theory of 2-local fields and a duality theorem. This is based on a joint work with Subhadip Majumder.
Ref:
Date: 05.07.2023 Time: 15:00–16:00 (Beijing Time)
or
Date: 05.07.2023 Time: 10:00–11:00 (Jerusalem time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Boris Kunyavskii (Bar-Ilan University)
Title: Violation of local-to-global principles for rationality and linearizability
Abstract: Click here
Ref: B.Kunyavskii, Tori and surfaces violating a local-to-global principle for rationality
Date: 21.06.2023 Time: 16:00–17:00 (Beijing Time)
or
Date: 21.06.2023 Time: 10:00–11:00 (Paris time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: David Harari (Université Paris-Saclay)
Title: Finiteness of certain Tate–Shafarevich sets
Abstract: Let G be a smooth algebraic group defined over a field K. The Tate–Shafarevich set of G is related to the failure of the local-global principle for principal homogeneous spaces of G. We discuss the finiteness of this set over various fields (number fields, p-adic function fields, function fields of a curve over a number field).
Ref: D.Harari and T. Szamuely, On Tate-Shafarevich groups of one-dimensional families of commutative group schemes over number fields, Math. Z.302(2022), no.2, 935–948.
Date: 29.05.2023 Time: 16:30–17:30 (Beijing Time)
or
Date: 29.05.2023 Time: 09:30–10:30 (London Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Damián Gvirtz-Chen (University of Glasgow)
Title: Surfaces defined by pairs of polynomials
Abstract: Surfaces in P3 defined by a pair of polynomials over a finitely generated field of characteristic zero are relevant to the study of rational points because their Brauer group is known to be finite up to constants. Via a topological deformation to the diagonal case we determine these Brauer groups provided our surface is, in an appropriate sense, sufficiently general. This generalises previous results obtained by Colliot-Thélène–Kanevsky– Sansuc, Bright, Uematsu and Santens. (Joint work with A. N. Skorobogatov.)
Ref: D. Gvirtz-Chen, A. Skorobogatov, Surfaces defined by pairs of polynomials
Date: 24.04.2023 Time: 15:30–16:30 (Beijing Time)
or
Date: 24.04.2023 Time: 09:30–10:30 (Prague/CEST Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Nicolas Daans (Charles University)
Title: The Pythagoras number of function fields
Abstract: The Pythagoras number of a field K is the smallest natural number n such that every sum of squares of elements of K is a sum of n squares of elements of K, or infinity, if such a natural number does not exist. Let us denote the Pythagoras number of K by p(K). Any non-zero natural number (and infinity) is the Pythagoras number of some field. Very little is known about the behaviour of the Pythagoras number under field extensions, in particular how quickly and freely the Pythagoras number can grow. For example, when L/K is a finite field extension, we only in general know that p(L) is bounded by [L : K]p(K), but in practice, we do not know of any example where p(L) > p(K) + 2 when L/K is a finite field extension. A related open question is whether, when p(K) is finite, then also p(K(X)) is finite, where K(X) is a rational function field over K. We also do not know of any example where p(K(X)) > p(K) + 2. In this talk, I discuss joint work with Karim Johannes Becher, David Grimm, Gonzalo Manzano-Flores, and Marco Zaninelli, in which we prove for an arbitrary field K that, if p(K(X)) = 2 (the lowest possible value), then p(L) is at most 5 for any finite field extension L of K(X), thereby providing a step in the direction of this open question.
Date: 17.04.2023 Time: 16:00–17:00 (Beijing Time)
or
Date: 17.04.2023 Time: 10:00–11:00 (Paris Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Julian Lawrence Demeio (University of Basel)
Title: Weak weak approximation on Del Pezzo surfaces of low degree
Abstract: In recent joint work with Sam Streeter and Rosa Winter, we show that weak weak approximation holds for Del Pezzo surfaces of degree 2 (over a number field) with a rational point not lying on the ramification curve or on the intersection of 4 exceptional curves. To prove this, we use two geometric “procedures” to produce rational points on the surface. The points obtained by a certain iteration of these two procedures are parametrized by a rational higher-dimensional cover of the surface, and we deduce our result by proving the arithmetic surjectivity of the morphism defining the cover.
Date: 23.03.2023 Time: 08:00–09:00 (Beijing Time)
or
Date: 22.03.2023 Time: 20:00–21:00 (Atlanta Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Raman Parimala (Emory University)
Title: Hasse principle over semiglobal fields
Abstract: We review results on the Hasse principle for homogeneous spaces under connected linear algebraic groups over function fields of curves over complete discrete valued fields. We present a recent result on the Hasse principle for projective homogeneous spaces under connected reductive groups over such fields. (Joint work with Philippe Gille.)
Video recording: please contact the organizers
Ref: P. Gille and R. Parimala, A local-global principle for twisted flag varieties
Date: 15.03.2023 Time: 16:00–17:00 (Beijing Time)
or
Date: 15.03.2023 Time: 10:00–11:00 (Israel Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Mikhail Borovoi (Tel Aviv University)
Title: Galois cohomology of reductive groups over global fields
Abstract: Let G be a connected reductive group over a global field F (a number field or a global function field). Let M=\pi_1(G) denote the “algebraic fundamental group” of G, which is a certain finitely generated abelian group endowed with an action of the absolute Galois group Gal(F^s/F). Using and generalizing a result of Tate for tori, we give a closed formula for the Galois cohomology set H^1(F,G) in terms of the Galois module M and the Galois cohomology sets H^1(F_v,G) for the “real” places v of F.
Moreover, let T be a torus over a global field F and let M=\pi_1(T)=X_{\star}(T) denote the cocharacter group of T. We give a closed formula for H^2(F,T) in terms of the Galois module M.
This is a joint work with Tasho Kaletha.
Ref: M. Borovoi and T. Kaletha, Galois cohomology of reductive groups over global fields
Date: 05.01.2023 Time: 10:00–11:00 (Beijing Time)
or
Date: 05.01.2023 Time: 11:00–12:00 (Tokyo Time)
Zoom Meeting ID: 967 248 8008
Passcode: available upon demand
Speaker: Thomas H. Geisser (Rikkyo University)
Title: Brauer groups and Neron-Severi groups of surfaces over finite fields
Abstract: For a smooth and proper surface over a finite field, the formula of Artin and Tate relates the behaviour of the zeta-function at $1$ to other invariants of the surface. We give a version of the formula which equates invariants related to the Brauer group to invariants to the Neron-Severi group. To illustrate our results we give some applications for abelian surfaces.
Ref: