Algebraic curves | Abelian varieties | Algebraic surfaces | Geometry of divisors

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally from abelian varieties defined over number fields to ones defined over finite fields and various local fields. Since a number field is the fraction field of a Dedekind domain, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain by the prime, which is a finite field for all finite primes. This induces a map from the fraction field to any such finite field. Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field. Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1. Abelian varieties have Kodaira dimension 0. (Wikipedia).

Kazuya Kato - Logarithmic abelian varieties

Correction: The affiliation of Lei Fu is Tsinghua University. This is a joint work with T. Kajiwara and C. Nakayama. Logarithmic abelian varieties are degenerate abelian varieties which live in the world of log geometry of Fontaine-Illusie. They have group structures which do not exist in

From playlist Conférence « Géométrie arithmétique en l’honneur de Luc Illusie » - 5 mai 2021

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

Jacob explains the fundamental concepts in group theory of what groups and subgroups are, and highlights a few examples of groups you may already know. Abelian groups are named in honor of Niels Henrik Abel (https://en.wikipedia.org/wiki/Niels_Henrik_Abel), who pioneered the subject of

From playlist Basics: Group Theory

We give a buttload of definitions for morphisms on various categories of complexes. The derived category of an abelian category is a category whose objects are cochain complexes and whose morphisms I describe in this video.

From playlist Derived Categories

Every Group of Order Five or Smaller is Abelian Proof

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Every Group of Order Five or Smaller is Abelian Proof. In this video we prove that if G is a group whose order is five or smaller, then G must be abelian.

From playlist Abstract Algebra

Abelian varieties not isogenous to Jacobians - Jacob Tsimerman

Joint IAS/Princeton University Number Theory Seminar Topic: Abelian varieties not isogenous to Jacobians Speaker: Jacob Tsimerman Affiliation: University of Toronto Date: December 01, 2021 Katz and Oort raised the following question: Given an algebraically closed field k, and a positive

From playlist Mathematics

David Masser: Avoiding Jacobians

Abstract: It is classical that, for example, there is a simple abelian variety of dimension 4 which is not the jacobian of any curve of genus 4, and it is not hard to see that there is one defined over the field of all algebraic numbers \overline{\bf Q}. In 2012 Chai and Oort asked if ther

From playlist Algebraic and Complex Geometry

Before we carry on with our coset journey, we need to discover when the left- and right cosets are equal to each other. The obvious situation is when our group is Abelian. The other situation is when the subgroup is a normal subgroup. In this video I show you what a normal subgroup is a

From playlist Abstract algebra

Stefano Marseglia, Computing isomorphism classes of abelian varieties over finite fields

VaNTAGe Seminar, February 1, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in this talk: Honda: https://doi.org/10.2969/jmsj/02010083 Tate: https://link.springer.com/article/10.1007/BF01404549 Deligne: https://eudml.org/doc/141987 Hofmann, Sircana: https://arxiv.org/ab

From playlist Curves and abelian varieties over finite fields

Group theory 17: Finite abelian groups

This lecture is part of a mathematics course on group theory. It shows that every finitely generated abelian group is a sum of cyclic groups. Correction: At 9:22 the generators should be g, h+ng not g, g+nh

From playlist Group theory

Ananth Shankar, Picard ranks of K3 surfaces and the Hecke orbit conjecture

VaNTAGe Seminar, November 23, 2021

From playlist Complex multiplication and reduction of curves and abelian varieties

Francesc Fité, Sato-Tate groups of abelian varieties of dimension up to 3

VaNTAGe seminar on April 7, 2020 License: CC-BY-NC-SA Closed captions provided by Jun Bo Lau.

From playlist The Sato-Tate conjecture for abelian varieties

Lucia Mocz: A new Northcott property for Faltings height

Abstract: The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness stat

From playlist Algebraic and Complex Geometry

What is the Mordell-Lang problem?

It is my intention to eventually explain some things about the Mordell-Lang problem and the higher dimensional versions of these. The presentation in this video is due to Mazur and can be found in an MSRI article he wrote that introduces these things.

From playlist Mordell-Lang

Christelle Vincent, Exploring angle rank using the LMFDB

VaNTAGe Seminar, February 15, 2022 License: CC-NC-BY-SA Links to some of the papers mentioned in the talk: Dupuy, Kedlaya, Roe, Vincent: https://arxiv.org/abs/2003.05380 Dupuy, Kedlaya, Zureick-Brown: https://arxiv.org/abs/2112.02455 Zarhin 1979: https://link.springer.com/article/10.100

From playlist Curves and abelian varieties over finite fields

Rachel Pries - The geometry of p-torsion stratifications of the moduli space of curve

The geometry of p-torsion stratifications of the moduli space of curve

From playlist 28ème Journées Arithmétiques 2013

Valentijn Karemaker, Mass formulae for supersingular abelian varieties

VaNTAGe seminar, Jan 18, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in this talk: Oort: https://link.springer.com/chapter/10.1007/978-3-0348-8303-0_13 Honda: https://doi.org/10.2969/jmsj/02010083 Tate: https://link.springer.com/article/10.1007/BF01404549 Tate: https

From playlist Curves and abelian varieties over finite fields

Claire Voisin: Gonality and zero-cycles of abelian varieties

Abstract: The gonality of a variety is defined as the minimal gonality of curve sitting in the variety. We prove that the gonality of a very general abelian variety of dimension g goes to infinity with g. We use for this a (straightforward) generalization of a method due to Pirola that we

From playlist Algebraic and Complex Geometry