A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular. (Wikipedia).
Modular Forms | Modular Forms; Section 1 2
We define modular forms, and borrow an idea from representation theory to construct some examples. My Twitter: https://twitter.com/KristapsBalodi3 Fourier Theory (0:00) Definition of Modular Forms (8:02) In Search of Modularity (11:38) The Eisenstein Series (18:25)
From playlist Modular Forms
This lecture is part of an online graduate course on modular forms. We introduce modular forms, and give several examples of how they were used to solve problems in apparently unrelated areas of mathematics. I will not be following any particular book, but if anyone wants a suggestion
From playlist Modular forms
Modular forms: Eisenstein series
This lecture is part of an online graduate course on modular forms. We give two ways of looking at modular forms: as functions of lattices in C, or as invariant forms. We use this to give two different ways of constructing Eisenstein series. For the other lectures in the course see http
From playlist Modular forms
In this talk, we will define elliptic curves and, more importantly, we will try to motivate why they are central to modern number theory. Elliptic curves are ubiquitous not only in number theory, but also in algebraic geometry, complex analysis, cryptography, physics, and beyond. They were
From playlist An Introduction to the Arithmetic of Elliptic Curves
Modular Functions | Modular Forms; Section 1.1
In this video we introduce the notion of modular functions. My Twitter: https://twitter.com/KristapsBalodi3 Intro (0:00) Weakly Modular Functions (2:10) Factor of Automorphy (8:58) Checking the Generators (15:04) The Nome Map (16:35) Modular Functions (22:10)
From playlist Modular Forms
Modular forms: Modular functions
This lecture is part of an online graduate course on modular forms. We classify all meromorphic modular functions, showing that they are all rational functions of the elliptic modular function j. As an application of j we use it to prove Picard's theorem that a non-constant meromorphic
From playlist Modular forms
Genus of abstract modular curves with level ℓℓ structure - Ana Cadoret
Ana Cadoret Ecole Polytechnique; Member, School of Mathematics November 21, 2013 To any bounded family of 𝔽ℓFℓ-linear representations of the etale fundamental of a curve XX one can associate families of abstract modular curves which, in this setting, generalize the `usual' modular curves w
From playlist Mathematics
Modular forms: Theta functions
This lecture is part of an online graduate course on modular forms. We show that the theta function of a 1-dimensional lattice is a modular form using the Poisson summation formula, and use this to prove the functional equation of the Riemann zeta function. For the other lectures in th
From playlist Modular forms
Complex analysis: Classification of elliptic functions
This lecture is part of an online undergraduate course on complex analysis. We give 3 description of elliptic functions: as rational functions of P and its derivative, or in terms of their zeros and poles, or in terms of their singularities. We end by giving a brief description of the a
From playlist Complex analysis
8ECM EMS Prize Lecture: Jack Thorne
From playlist 8ECM EMS Prize Lectures
Birch Swinnerton-Dyer conjecture: Introduction
This talk is an graduate-level introduction to the Birch Swinnerton-Dyer conjecture in number theory, relating the rank of the Mordell group of a rational elliptic curve to the order of the zero of its L series at s=1. We explain the meaning of these terms, describe the motivation for the
From playlist Math talks
Ana Caraiani, Modularity over CM fields
VaNTAGe Seminar, May 24, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in the talk: Freitas-Le Hung-Siksek: https://arxiv.org/abs/1310.7088 Poonen-Schaefer-Stoll: https://arxiv.org/abs/math/0508174 Harris-Lan-Taylor-Thorne: https://link.springer.com/article/10.1186/s406
From playlist Modularity and Serre's conjecture (in memory of Bas Edixhoven)
CTNT 2020 - 3-adic images of Galois for elliptic curves over Q - Jeremy Rouse
The Connecticut Summer School in Number Theory (CTNT) is a summer school in number theory for advanced undergraduate and beginning graduate students, to be followed by a research conference. For more information and resources please visit: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2020 - Conference Videos
CTNT 2020 - A virtual tour of the LMFDB: the L-functions and Modular Forms DataBase
This video is part of a series of videos on "Computations in Number Theory Research" that are offered as a mini-course during CTNT 2020. In this video, we take a virtual tour of the LMFDB - the L-functions and modular forms database. Please click on "show more" to see the links below. Abo
From playlist CTNT 2020 - Computations in Number Theory Research
Ana Caraiani - Modularity over CM fields
I will discuss joint work in progress with James Newton, where we prove a local-global compatibility result in the crystalline case for Galois representations attached to torsion classes occurring in the cohomology of locally symmetric spaces. I will then explain an application to the modu
From playlist Franco-Asian Summer School on Arithmetic Geometry (CIRM)
Elliptic Curves and Modular Forms | The Proof of Fermat’s Last Theorem
Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem. This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't hav
From playlist Famous Unsolved Problems
"A Brief History of Fermat's Last Theorem" by Prof. Kenneth Ribet
The speaker discussed work on Fermat's Last Theorem over the last 350+ years. The theorem was proved in the mid-1990s using tools from contemporary arithmetic algebraic geometry. The speaker focused on such objects as elliptic curves, Galois representations and modular forms that are cen
From playlist Number Theory Research Unit at CAMS - AUB
Jeremy Rouse, l-adic images of Galois for elliptic curves over Q
VaNTAGe seminar, June 22, 2021 License: CC-BY-NC-SA
From playlist Modular curves and Galois representations
Modular forms: Theta functions in higher dimensions
This lecture is part of an online graduate course on modular forms. We study theta functions of even unimodular lattices, such as the root lattice of the E8 exceptional Lie algebra. As examples we show that one cannot "her the shape of a drum", and calculate the number of minimal vectors
From playlist Modular forms
Moduli spaces of shtukas over function fields - Jared Weinstein
Joint IAS/Princeton University Number Theory Seminar Topic: Moduli spaces of shtukas over function fields Speaker: Jared Weinstein Affiliation: Boston University Date: February 13, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics