Abelian Varieties: Winter 2014-2015

This is the website for the seminar Abelian varieties  held at the Johannes-Gutenberg University of Mainz during the Wintersemester of 2014-2015



Practical information

Organizers: Ariyan Javanpeykar, Ronan Terpereau

Schedule:  Tuesday 16h00-17h30 in Room 04-422



  1. David Mumford. Abelian varieties, pages viii+242. Tata Institute of Fundamental. Research Studies in Mathematics, No. 5. Published for the Tata. Institute of Fundamental Research, Bombay, 1970.
  2. Christina Birkenhake and Herbert Lange. Complex abelian varieties. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 302. Springer-Verlag, Berlin, 2004.
  3. Alexander Polishchuk. Abelian varieties, theta functions and the Fourier transform. Cambridge Tracts in Mathematics, 153. Cambridge University Press, Cambridge, 2003.
  4. Gerard van der Geer and Ben Moonen. Abelian varieties. Book available on-line.
  5. Arnaud Beauville. Classical theta functions and their generalization. Notes on-line.
  6. Greg H. Anderson. Logarithmic derivatives of Dirichlet L-functions and the periods of abelian varieties. Compositio Mathematica 1982, tome 45, no.3, p. 315-332.
  7. Oleksandr Iena. Vector bundles on elliptic curves and factors of automorphy. Rend. Istit. Mat. Univ. Trieste 43 (2011), 61–94.
  8. Frans Oort. http://www.math.nyu.edu/~tschinke/books/finite-fields/final/05_oort.pdf
  9. Janos Kollar. https://web.math.princeton.edu/~kollar/book/chap1.pdf
  1. Gary Cornell, Joseph Silverman. Arithmetic geometry. Springer-Verlag New-York, 1986.
  1. Michel Brion. Compactification de l'espace des modules des variétés abéliennes principalement polarisées. Séminaire Bourbaki, 48 (p1-32), 2005-2006.
  2. Yung-sheng Tai. On the Kodaira dimension of the moduli space of abelian varieties.  Inventiones mathematicae 1982, Volume 68, Issue 3, pp 425-439.
  3. Carel Faber, Gerard van der Geer, Frans Oort. Moduli of Abelian Varieties. Progress in Mathematics Volume 195, 2001, pp 345-416.
  4. Iku Nakamura. Compactification by GIT-stability of the moduli space of abelian varieties. arXiv:1406.0174
  5. Samuel Grushevsky. Geometry of A_g and its compactifications. arXiv:0711.0094
  6. Martin Olsson. https://math.berkeley.edu/~molsson/Overview3.pdf
  7. Martin Olsson. https://math.berkeley.edu/~molsson/mono020807.pdf




  1. Holomorphic line bundles on complex tori (Laura Biroth), 28th of October, notes
    content: Appel-Humbert theorem, dual complex torus, Poincaré line bundle.
    references: [Polishchuk,I.1], [Mumford, I.1 and I.2], [Birkenhake-Lange, 2.1-2.5]
  2. Theta functions I (Xin Lu), 4th of November, notes
    content: isogenies, Riemann forms, theta functions, algebraizability of complex tori.
    references: [Mumford, I.3],  [Cornell-Silverman, IV.1-IV.3]
  3. Theta functions II (Helge Ruddat), 11th of November, notes
    content: isogenies, Riemann forms, theta functions, algebraizability of complex tori.
    references: [Mumford, I.3],  [Cornell-Silverman, IV.1-IV.3]
  4. Cohomology of line bundles (Markus Pauly), 18th of November, notes
    content: vanishing theorem of Mumford and Kempf, dimension of the cohomology group of a line bundle, Riemann-Roch theorem.
    references: [Polishchuk,I.7], [Mumford III.16],  [Birkenhake-Lange, 3.1-3.6]
  5. Abelian varieties and theorem of the cube (Carolin Peternell), 25th of November, notes
    content: algebraic group point of view, rigidity lemma, theorem of the cube, ampleness criterion.
    references: [Polishchuk,II.8], [Mumford, II.4-II.6]
  6. 2nd of December: NO TALK!
  7. The dual abelian variety (Levente Nagy), 9th of December, notes
    content: construction of the dual abelian variety, functorial point of view, (principal) polarization, construction of the quotient of abelian varieties.
    references: [Polishchuk,II.9], [Mumford, II.7-II.9], [Birkenhake-Lange, 4.1]
  8. The Jacobian  (Adam Czaplinski), 16th of December, notes
    content: symmetric power of a curve, construction of the jacobian of a curve, (statement of) the Torelli theorem.
    references: [Polishchuk,III.16 and III.21], [Birkenhake-Lange, 11.1], [Cornell-Silverman, VII.1-VII.4 and VII.12]
  9. Singular cohomology of complex tori vs étale cohomology of abelian varieties (Thomas Weissschuh), 6th of January, notes
    content: étale cohomology groups of abelian varieties, singular cohomology groups of complex tori.
    references: [Birkenhake-Lange, 1.3 and 1.4], [Cornell-Silverman, V.15]
  10. Abelian surfaces (Maksim Zhykhovich), 13th of January, notes
    content: Kummer surfaces and other examples, Reider's theorem, product of elliptic curves.
    references: [Birkenhake-Lange, 10.1-10.6]
  11. Period distribution of Abelian varieties (Maximilian Preisinger), 20th of January, notes
    content: periods of Abelian varieties, endomorphism ring of Abelian varieties,  CM fields
    references: [Anderson]
  12. 27th of January: NO TALK!
  13. Moduli theory of abelian varieties (Ariyan Javanpeykar),  2nd of February (at 16.00 in room 04-422), notes
    content: representable functors, examples of moduli functors, moduli  of abelian varieties and its basic properties (smoothness, separatedness, non-properness, etc)
  14. Abelian varieties over finite fields and Weil numbers (Axel Stäbler), 3rd of February, notes
    content: abelian varieties over finite fields, Weil numbers, Honda-Tate theory
    references: [Oort]
  15. Compactification of the moduli space of abelian varieties (Ronan Terpereau), 10th of February, notes
    content : case of elliptic curves , minimal compactification, modular compactification of Alexeev, semi-abelian varieties
    references: [Brion], [Olsson]