An introduction to elliptic fibrations

Cecília Salgado
Universidade Federal do Rio de Janeiro
Abstract:
We will present a brief introduction to the theory of elliptic fibrations in 3 lectures. We will assume that the students are not familiar with elliptic curves, but we will take into consideration that a course treating such objects will happen in parallel.
 
The classes will be divided as follows1:
- Day 1: Elliptic curves, Weierstrass equations, reduction modulo a prime. Elliptic surfaces, first examples, singular fibers.
- Day 2: Mordell-Weil and Néron-Severi groups (lattices), torsion sections.
- Day 3: More geometry, some arithmetic and open problems involving elliptic fibrations.
 
The material is all contained in [2] and [3] and it would be good if the student has a first look at these references before the classes. They cover way more than we will be able to do in 3 hours. I also recommend the previous reading of the first chapter of [1], of [4] until III.3, and a browsing of the book [5].
References:
[1] R. Hartshorne, Algebraic Geometry, Springer-Verlag New York, 1977.
[2] R. Miranda, The basic theory of elliptic surfaces, http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf
[3] M.Schuett, T. Shioda, M.Schuett, T. Shioda, https://arxiv.org/pdf/0907.0298.pdf
[4] J. Silverman, The arithmetic of elliptic curves, Springer-Verlag New York, 2009.
[5] J. Silverman, J. TateRational points on elliptic curves, Springer International Publishing, 2015.
 
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1This  is a first idea of the themes we will threat but might suffer alterations.