Finite fields | Algebraic geometry | Algebraic number theory

Drinfeld module

In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. Drinfeld modules were introduced by Drinfeld, who used them to prove the Langlands conjectures for GL2 of an algebraic function field in some special cases. He later invented shtukas and used shtukas of rank 2 to provethe remaining cases of the Langlands conjectures for GL2. Laurent Lafforgue proved the Langlands conjectures for GLn of a function field by studying the moduli stack of shtukas of rank n. "Shtuka" is a Russian word штука meaning "a single copy", which comes from the German noun “Stück”, meaning “piece, item, or unit". In Russian, the word "shtuka" is also used in slang for a thing with known properties, but having no name in a speaker's mind. (Wikipedia).

Video thumbnail

Terrorists Using Drinfeld Modules

This comes from Twenty Four, Season 04, Episode 11

From playlist Mathematical Shenanigans

Video thumbnail

Cell Programming Kit - Elowitz Lab

Researchers at Caltech have developed a kind of biological toolkit of parts that can be assembled to create custom circuits for cells.

From playlist Our Research

Video thumbnail

C73 Introducing the theorem of Frobenius

The theorem of Frobenius allows us to calculate a solution around a regular singular point.

From playlist Differential Equations

Video thumbnail

Abel formula

This is one of my all-time favorite differential equation videos!!! :D Here I'm actually using the Wronskian to actually find a nontrivial solution to a second-order differential equation. This is amazing because it brings the concept of the Wronskian back to life! And as they say, you won

From playlist Differential equations

Video thumbnail

Linear Transformations: Onto

Linear Algebra: Continuing with function properties of linear transformations, we recall the definition of an onto function and give a rule for onto linear transformations.

From playlist MathDoctorBob: Linear Algebra I: From Linear Equations to Eigenspaces | CosmoLearning.org Mathematics

Video thumbnail

Building Drone Rotors - PART 1

We present a multi-part series covering the construction and testing of large multi-rotor propellers.

From playlist Drones

Video thumbnail

Integration 1 Riemann Sums Part 1 - YouTube sharing.mov

Introduction to Riemann Sums

From playlist Integration

Video thumbnail

Drinfeld Module Basics - part 01

This is a very elementary introduction to Drinfeld Modules. We just give the definitions. My wife helped me with this. Any mistakes I make are my fault.

From playlist Drinfeld Modules

Video thumbnail

New developments in the theory of modular forms... - 9 November 2018

http://crm.sns.it/event/416/ New developments in the theory of modular forms over function fields The theory of modular forms goes back to the 19th century, and has since become one of the cornerstones of modern number theory. Historically, modular forms were first defined and studied ov

From playlist Centro di Ricerca Matematica Ennio De Giorgi

Video thumbnail

Build a Computer Part IV: Processor and RAM

This video shows you how to add your processor and RAM to the motherboard. Credits: , HowStuffWorks

From playlist Classic HowStuffWorks

Video thumbnail

New developments in the theory of modular forms... - 6 November 2018

http://crm.sns.it/event/416/ New developments in the theory of modular forms over function fields The theory of modular forms goes back to the 19th century, and has since become one of the cornerstones of modern number theory. Historically, modular forms were first defined and studied ov

From playlist Centro di Ricerca Matematica Ennio De Giorgi

Video thumbnail

New developments in the theory of modular forms... - 8 November 2018

http://crm.sns.it/event/416/ New developments in the theory of modular forms over function fields The theory of modular forms goes back to the 19th century, and has since become one of the cornerstones of modern number theory. Historically, modular forms were first defined and studied ov

From playlist Centro di Ricerca Matematica Ennio De Giorgi

Video thumbnail

New developments in the theory of modular forms... - 7 November 2018

http://crm.sns.it/event/416/ New developments in the theory of modular forms over function fields The theory of modular forms goes back to the 19th century, and has since become one of the cornerstones of modern number theory. Historically, modular forms were first defined and studied ov

From playlist Centro di Ricerca Matematica Ennio De Giorgi

Video thumbnail

Drinfeld's lemma for schemes - Kiran Kedlaya

Joint IAS/Princeton University Algebraic Geometry Seminar Topic: Drinfeld's lemma for schemes Speaker: Kiran Kedlaya Affiliation: University of California, San Diego; Visiting Professor, School of Mathematics Date: February 4, 2019 For more video please visit http://video.ias.edu

From playlist Joint IAS/PU Algebraic Geometry

Video thumbnail

Early Stomper Tests

Enjoy a small collection of explosions from our preliminary tests of the prototype of what is to become Project Stomper.

From playlist Project Stomper Videos

Video thumbnail

K. Kato - Log Drinfeld modules and moduli spaces

We construct toroidal compactifications of the moduli space of Drinfeld modules of rank d with N-level structure. We obtain them as the moduli spaces of log Drinfeld modules of rank d with N-level structure. The theory of toroidal compactifications was announced by Pink long ago (using the

From playlist Arithmetic and Algebraic Geometry: A conference in honor of Ofer Gabber on the occasion of his 60th birthday

Video thumbnail

Drinfeld Module Basics - part 03

We talk about periods of Drinfeld modules in this video. We show how to use a rigid analytic version of Weierstrass factorization for the Carlitz exponential to show that the period lattice for the Carlitz module is generated by the Carlitz period,

From playlist Drinfeld Modules

Video thumbnail

Geordie Williamson: Langlands and Bezrukavnikov II Lecture 18

SMRI Seminar Series: 'Langlands correspondence and Bezrukavnikov’s equivalence' Geordie Williamson (University of Sydney) Abstract: The second part of the course focuses on affine Hecke algebras and their categorifications. Last year I discussed the local Langlands correspondence in bro

From playlist Geordie Williamson: Langlands correspondence and Bezrukavnikov’s equivalence

Video thumbnail

Manipulating Functions Algebraically and Evaluating Composite Functions

Now that we know what functions are, what can we do with them? Can we do operations with them? Yes! Can we plug functions into other functions? Yes! Oh, what won't these functions do? Let's play with a few of them now. Watch the whole Mathematics playlist: http://bit.ly/ProfDaveMath Clas

From playlist Algebra 1 & 2

Video thumbnail

Drinfeld Module Basics - part 02

Definition of exponentials. We use the Carlitz exponential as an example and indicate how rigid analysis is used to prove these things. We don't prove the full existence of the exponential for Drinfeld Modules.

From playlist Drinfeld Modules

Related pages

Dedekind domain | Level structure (algebraic geometry) | Scheme (mathematics) | Algebraic stack | Twisted polynomial ring | Carlitz exponential | Mathematics | Finite field | Moduli stack of elliptic curves | Complex multiplication | Algebraic function field | Vector bundle | Module (mathematics) | Additive polynomial