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 follows^{1}: 
 Day 1: Elliptic curves, Weierstrass equations, reduction modulo a prime. Elliptic surfaces, first examples, singular fibers. 
 Day 2: MordellWeil and NéronSeveri 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, SpringerVerlag New York, 1977. 
[2] R. Miranda, The basic theory of elliptic surfaces, http://www.math.colostate.edu/~miranda/BTESMiranda.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, SpringerVerlag New York, 2009. 
[5] J. Silverman, J. Tate, Rational points on elliptic curves, Springer International Publishing, 2015. 

 
^{1}This is a first idea of the themes we will threat but might suffer alterations. 