Mini-workshop

Quadratic Forms and Related Topics

Time and Place

July 10th, 2023, 1 pm,  Building no. 3, Room no. 207, Academic College of Tel-Aviv-Yaffo.

Organizer

Adam Chapman.

Speakers

Ahmed Laghribi, Diksha Mukhija, Marco Zaninelli and Uriya First.

Participants

Uzi Vishne, Louis Rowen, Eliyahu Matzri, Ido Efrat, Alexander Guterman, Solomon Vishkautsan, Ilan Levin, Shira Gilat, Elad Paran, Pavel Shteyner, Boris Bychkov, Saurabh Gosavi.

Titles and Abstracts

Marco Zaninelli

Title: Upper bounds for the Pythagoras number of fields

Abstract: The Pythagoras number of a field K is the minimum number n such that any sum of squares in K can be written as a sum of n squares in K, or ∞ if such a number does not exist. Despite its elementary definition, computing the Pythagoras number of a field can be very complicated. In this talk, based on a joint work with Karim Johannes Becher, we describe a method to prove, in presence of a certain local-global principle for n-fold Pfister forms for n ∈ N,  that the Pythagoras number of a field is at most 2n+1. This method provides original upper bounds for the Pythagoras number of certain fields; in addition, it allows to retrieve several known upper bounds using more elementary techniques.

Uriya First

Title: An exact octagon of Witt groups of Azumaya algebras and applications

Abstract: Let L/K be a quadratic Galois extension of fields. It is a classical result of Pfister that the kernel of W(K)-->W(L) is generated by the norm form of L/K. This can be restated as saying that there is an exact sequence of Witt groups W(L,s)-->W(K)-->W(L), where s:L-->L is the nontrivial K-automorphism of L and W(L,s) is the Witt group of s-hermitian forms over L.

Over the years, many more examples of exact sequences relating Witt groups of different central simple algebras (CSAs) with involution have emerged. Eventually, almost all of these sequences turned out to be special cases of an 8-periodic exact sequence ('octagon') of Witt groups of CSAs introduced by Grenier-Boley and Mahmoudi. A celebrated application of (a piece) of this sequence is the proof of Serre's Conjecture II for classical groups by Bayer-Fluckiger and Parimala.

I will talk about a recent work where we show that the octagon is still exact if we replace CSAs with involution by Azumaya algebras with involution over a semilocal ring. I will also discuss various applications of this result, e.g., a generalization of the aforementioned theorem of Pfister to quadratic Galois extensions of 2-dimensional regular rings.

Ahmed Laghribi

Title: On the descent for quadratic and bilinear forms

Abstract: Let K/F be a field extension. The aim of this talk is to discuss conditions under which a K-form (quadratic or bilinear) Q is defined over the base field F when K is the function field of a projective quadric and F is of characteristic 2. This will be done in the spirit of the descent conjectures proposed by Kahn in characteristic not 2. We will give a complete answer for forms Q of dimension less than or equal to 4. (This a joint work with Diksha Mukhija.)

Diksha Mukhija

Title: The behavior of singular quadratic forms under purely inseparable extensions

Abstract: Working over a field $F$ of characteristic $2$, we distinguish between three types of quadratic forms: nonsingular forms, semisingular forms and totally singular forms. One of the central problems in the algebraic study of quadratic forms deals with giving conditions under which a given quadratic form becomes isotropic or hyperbolic after extending scalars to a field extension of $F$. A recent result of Sobiech gives a complete answer to the hyperbolicity of nonsingular $F$-quadratic forms over purely inseparable extensions of $F$. In this talk, we will give a complete classification of semisingular $F$-quadratic forms that have maximal Witt index and defect index over a purely inseparable modular field extension of $F$. To achieve this, we'll first study totally singular forms under modular extensions, for which the notion of hyperbolicity is generalized to quasi-hyperbolicity. We'll then introduce the notion of strictly quasi-hyperbolicity for semisingular quadratic forms. We'll also apply this result to give necessary and sufficient conditions under which an anisotropic semisingular $F$-quadratic form has a given Witt index over a purely inseparable extension with the modularity condition.

https://www.youtube.com/watch?v=6KdqQLUUMLg&list=PLnr2SPGV58Ko9HngCepqSVCoCkWhQWKEV&pp=gAQBiAQB