Abelian Varieties: geometry and arithmetic

This Ph.D. course aims at discussing the basic theory of abelian varieties. Abelian Varieties represent a class of algebraic varieties of fundamental importance in mathematics from many different points of view, e.g. complex analytic, algebro-geometric, hyperbolic, arithmetic, ... The course will give the basic definitions and properties of abelian varieties, both over the complex numbers and over arbitrary fields. It will then deepend specific aspects depending on the interests of the audience.

Organization. The course will start with a couple of introductory lectures and then will continue with seminars prepared and presented by the participants. These seminars will be shared among the participants taking into account backgrounds and expertise).

#	Content	Speaker	Date and Time
1	Introduction 1	Amos Turchet	Jan 23, h 14:00, Aula 108
2	Introduction 2	Amos Turchet	Jan 30, h 16:00, Aula B
3	Basic definitions and properties 1	Danilo Avaro, Wei Chen, Ilaria Cruciani	Feb 6, h 14:00, Aula 108
4	Basic definitions and properties 2	Danilo Avaro, Wei Chen, Ilaria Cruciani	Feb 13, h 10:00, Aula C
5	Group Schemes	Leonardo Carofiglio, Francesco Tropeano	Feb 20, h10:00, Aula 108
6	Abelian varieties over \(\mathbb{C}\)	Nicola Ottolini	Feb 27, h11:00, Aula 108
7	Jacobians	Luigi De Filpo	Mar 20, h11:00, Aula 108

• C. Birkenhake and H. Lange. Complex Abelian Varieties. Springer, 2004;
• S. Bosch, W. Lütkebohmert, and M. Raynaud. Néron models, vol. 21 of Ergebnisse der Math. Springer-Verlag, 1990;
• B. Conrad, Abelian Varieties;
• O. Debarre, Complex tori and abelian varieties. SMF/AMS Texts Monogr., 11, 2005;
• B. Edixhoven, B. Moonen, and B. van Geemen. Abelian varieties;
• S. Grushevsky's Lectures on Abelian Varieties;
• S. Lang, Abelian varieties. Interscience Tracts in Pure and Applied Mathematics. No. 7, 1959;
• H. Lange, Abelian varieties over the complex numbers, a graduate course, Grundlehren Text Editions, Springer, Cham, 2023;
• D. Lombardo: Abelian Varieties;
• D. Mumford. Abelian varieties. In Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, 1970;
• D. Mumford, Appendix: Curves and Their Jacobians. In: The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol 1358, Springer, 1999;
• V. M. Murty, Introduction to abelian varieties. CRM Monogr. Ser., 3 AMS, 1993;
• J.S. Milne: Abelian Varieties;
• A. Polishchuk, Abelian varieties, theta functions and the Fourier transform. Cambridge Tracts in Math., 153, 2003;

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Lecturer: Amos Turchet