Reading course on the Langlands Program and the Modularity Lifting Theorem

First part. Organisers: Dario Antolini, Pietro Mercuri.

  • Title: Introduction,
    Speaker: Pietro Mercuri,
    When: Monday 24/09/2018, hours 09:00-11:00.
    Where: Room "D'Antoni", "Tor Vergata" University, Rome.
    Abstract: In this lecture I introduce the Langlands Program and Modularity Lifting Theorem and I give an overview about the background, the ideas, the tools and the strategies used to approach to these problems.
    Main references: "Automorphic Forms and Representations", Bump D.; "Automorphic Forms on Adele Groups", Gelbart S.S.; "The Local Langlands Conjecture for GL(2)", Bushnell C.J., Henniart G.; "Modular Forms and Fermat's Last Theorem", Cornell G., Silverman J.H., Stevens G.; Toby Gee notes (Arizona Winter School 2013); Peter Bruin, Arno Kret notes (Galois Representations Course).

  • Title: Locally profinite groups and local fields,
    Speaker: Lorenzo Pagani,
    When: Monday 08/10/2018, hours 09:00-11:00.
    Where: Room "D'Antoni", "Tor Vergata" University, Rome.
    Abstract: In this talk we will recall the basic properties of profinite groups and infinite Galois theory. We will study valuations over fields in order to deduce the inertia-decomposition exact sequence first in the local case then in the global one. Finally, we will introduce the notion of adeles.
    Main references: "Local fields", Cassels J.W.S.; P. Stevenhagen notes.

  • Title: Elliptic curves and abelian varieties,
    Speaker: Claudio Fabroni,
    When: Monday 15/10/2018, hours 09:00-11:00.
    Where: Room "D'Antoni", "Tor Vergata" University, Rome.
    Abstract: In this talk we introduce the notion of a C-group or a group object in a category, then i will treat the special case where C is the category of S-scheme. At this point we define an abelian varieties X over a field K and we investigate X(K) in the case K is the complex field. Finally we focus on elliptic curves, i.e. the case of abelian varieties of dimension 1.
    Main references: "Geometric modular forms and elliptic curves", Hida H.; "The arithmetic of elliptic curves", Silverman J.; J. Milne notes.

  • Title: Modular forms: the classical setting,
    Speaker: Ankan Pal,
    When: Monday 22/10/2018, hours 09:00-11:00.
    Where: Room "D'Antoni", "Tor Vergata" University, Rome.
    Abstract: Modular Forms are ubiquitous because of their rich mathematical structure and have become an indispensable tool. They were a key ingredient in the much coveted proof of 'Fermat's Last Theorem'. In this talk, Modular Forms will be introduced with some motivating examples. The concept of Congruence Subgroups will be defined and how the Fourier coefficients are related to Modular Forms will be discussed. Riemann-Roch theorem will be introduced and 'Dimension Formulas for even k' will be derived. Hecke Algebra will be discussed at length and the connection of Modular forms with L-functions would be elaborated. Eigenforms and Abelian varieties associated to a Modular Form will be presented. As a concluding example, a cusp form with rational coefficients will be taken and the L-function and elliptic modular curve associated with it will be shown.
    Main reference: "A First Course in Modular Forms", F. Diamond, J. Shurman.
    Notes: pdf.

  • Title: Representations associated to modular forms,
    Speaker: Pietro Mercuri,
    When: Monday 29/10/2018, hours 09:00-11:00.
    Where: Room "D'Antoni", "Tor Vergata" University, Rome.
    Abstract: In this talk I explain how to associate representations of GL(2) and Galois representations to a cusp form.
    Main reference: "A First Course in Modular Forms", F. Diamond, J. Shurman; "Automorphic Forms and Representations", Bump D.; "Automorphic Forms on Adele Groups", Gelbart S.S.; W.W. Li notes.

Second part. Organiser: Dario Antolini.

See Antolini's webpage.


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