Curves, Jacobians, and Abelian Varieties


Workshop in Algebraic Geometry

Fakulteti i Shkencave të Natyrës
Universiteti i Tiranës
Tirana, Albania
December 15-22, 2019.


Dorisa Tabaku
Depart. of Mathematics
University of Tirana
Tirana, Albania
Tony Shaska
Depart. of Mathematics
Oakland University


Algebraic curves are one of the most classical objects of mathematics, Their study led to the concept of Jacobian and more generally that of an Abelian variety. The goal of this session is to focus on the arithmetic aspects of theory. We will explore minimal models of curves, rational points on curves, Abelian varieties, isogenies, Honda-Tate theory, Weil descent, applications to isogeny based cryptography, etc. The area is a very active area of research and we expect that the session will be well attended. We intend to invite many younger mathematicians, graduate students, and recent PhD’s.


  • Equations of curves over their minimal field of definition
  • Field of moduli versus the field of definition
  • Models of curves with minimal height
  • Moduli height of curves
  • Rational points on curves
  • Rational points in the moduli space of curves
  • Jacobians of curves and their decompositions
  • Neron-Tate models of algebraic curves
  • Neron-Tate heights on Jacobians
  • Minimal discriminants and conductors
  • Selmer groups in Jacobians
  • Arithmetic invariant theory
  • Pairings and Weil descent
  • Mordell-Weil group
  • Abelian varieties with complex multiplication
  • Lectures

    Lectures will be all in English. There will be 5 days of lectures, Monday through Friday. This is equivalent to an 8-weeks course. This will be a crash course on algebraic curves. Lectures will be based on the book "From hyperelliptic to superelliptic curves", to be published by the American Math Society.
    1. Function fields and coverings of the projective line
    2. Riemann Existence Theroem
    3. Automorphisms of Curves
    4. Hurwitz spaces and the braid action
    5. Moduli space of curves (Hurwitz, Severi, Grothendick, Fulton, Fried, Mumford, Deligne)
    6. Equations of curves
    7. Invariant theory
    8. Field of moduli versus the minimal fields of definition
    9. Weighted moduli space and weighted heights
    10. Minimal models, Neron heights


    If you are interested to attend, please contact Dorisa Tabaku at

    Organizing Institutions

    Akademia e Shkencave të Shqipërisë


    Universiteti i Tiranës


    Oakland University


    Albanian Algebraic Geometry Group