Seminar on Motives, Standard Conjectures and Weil Cohomology Theories, Spring 2022

    Description of the seminar

The primary goal of the seminar would be to get acquainted with the language of motives and the questions surrounding them.

The first few talks aim at defining the category of correspondences and the symmetrical monoidal category of Chow Motives M(k), where k is an algebraically closed field. After that we will discuss classical Weil cohomology theories H*, tying them in with motives due to the property of factoring through the category M(k) and their realization functors. (We will follow the chapter of stacks project [0FFG]).

Later we will study the interaction between standard conjectures and motives: in particular we will focus on semisimplicity (and how it doesn't need standard conjectures in order to be proved [J]), on the existence of graded decomposition on the motive h(X), on the property of being Tannakian and the Motivic Galois Group. (We will use Milne's paper [M1] in order to navigate the topics, and as references André's book [A] and papers [J] & [S]).

Depending on the audience's preferences (as well as how much time we have left), we can either dig in some concrete computations from Scholl's paper [S] (graded decomposition for motives h(A) of abelian varieties, and computation of motives h1(X) and h2d−1(X), with X variety of dimension d [S]) or discuss Motives over finite fields (following [M2]).

    Logistics Info

The seminar will be held on Monday afternoons from 1:00pm to 2:20pm in Room 528 in the Math Department at Columbia.

Please email me at mp3947 at columbia dot edu if you are interested in giving a talk and/or you want to be added to the mailing list. Below there is a tentative list of topics: but please feel free to make suggestions about any related topic which you may be interested in.


    List of Talks

  1. Feb 21
            Speaker: Morena Porzio
            Title: Motivations to Motives and Seminar Roadmap
            Abstract: I will talk about why we could be interested in the category of motives, giving some motivations and the roadmap of the seminar.
            We will also review the bare minimum needed to talk about intersection theory and adequate equivalence relations.
  2. Feb 28
            Speaker: Morena Porzio
            Title: Category of correspondences and the one of rational motives
            Abstract: The program for today is to see the construction of the category of correspondences and the one of rational motives M_rat(k).
            Then we will discuss some of the properties of M_rat(k), in particular its symmetric monoidal structure and additive structure. The existence of (left) duals is postponed to next week.
  3. Mar 07
            Speaker: Morena Porzio
            Title: Classical Weil cohomology theories and their factorization through the category of Chow motives
            Abstract: We will resume the proof that Mrat(k) is Karoubian and has left duals. Then we will focus on Classical Weil cohomology theories, in particular on their factorization through
            the category of rational motives Mrat(k). In order to do so, some formal properties of Weil cohomology theories will be recalled (but without proof).
  4. Mar 21
            Speaker: Caleb Ji
            Title: The Standard Conjectures on Algebraic Cycles
            Abstract: In his work on motives and the Weil conjectures, Grothendieck formulated a set of conjectures on algebraic cycles which he called the standard conjectures.
            We explain what they are and the known implications between them. Then we will explain their application to the Weil conjectures, and in particular to the proof of the Riemann hypothesis over function fields.
            Caleb Ji's Notes: Standard Conjectures
  5. Mar 28
            Speaker: Morena Porzio
            Title: Manin's identity principle and abelianity and semisimplicity of M_~(k)
            Abstract: Today we will compute the motives of some concrete examples using Manin's identity principle, and looking at the relation between the motive of a curve and its Jacobian. Then we will see why
            the category of motives fails to be abelian and why the category of motives M_~(k) is semisimple iff ~ is the numerical equivalence. At the end, we will discuss which direction to focus on
            in the upcoming meetings (if you have preferences, feel free to share them!).
  6. Apr 04
            Speaker: Avi Zeff
            Title: The Tannakian formalism and the motivic Galois group
            Abstract: We'll discuss rigid symmetric monoidal categories, their properties, and some examples, and then see how this structure gives rise to the Tannakian formalism. We'll then try to apply
            this formalism to the category of motives, using the standard conjectures, and try to describe the motivic Galos group.
  7. Apr 11
            Speaker: Morena Porzio
            Title: The Tannakian formalism and the motivic Galois group (continuation)
            Abstract: We'll resume discussing Tannakian categories from where we left: in particular we'll see the general notion of Tannakian category via fibre functors, how to associate an affine gerbe to any Tannakian
            category C, and then how to realize C as a category of representations. After that we will focus on particular fibre functors, namely the realization functors associated with specific weil cohomology theories.
            We'll see an overview of what is known/folklore about the motivic galois groups for Hodge structures of weight k and for l-adic cohomology theory.
  8. Apr 18
            Speaker: Morena Porzio
            Title: Numerical Motives over finite fields
            Abstract: Following Milne's Motives over Finite Fields, we will focus on the category of numerical motives over finite fields F_q assuming the truth of Tate's conjecture. In particular we will study
            the fiber functors w_l for l-adic cohomology theories, with the aim of describing the motivic galois group GMot_{num}(F_q).
  9. Apr 25
            Speaker: Morena Porzio
            Title: Motivic Galois Group of M_num(F_q) w.r.t. l-adic cohomology and over the algebraic closure of Q
            Abstract: We will finish the discussion about the essential image of the realization functor w_l : M_num(F_q) \otimes Q_l ---> V_l(F_q) to then pass to the Motivic Galois
            group of it, giving a description of it as a proreductive affine group scheme. Then we will see that for any fiber functor w defined over the algebraic closure of Q the automorphisms group
            Aut^{\otimes}(w) is a group scheme P(q) whose characters coincide with the Weil q-numbers.

    Tentative list of Topics