Lattices and MordellWeil groups, Francesco Pappalardi and Peter Stevenhagen
1. Elliptic curves over number fields:

Projective coordinates and projective space, homogeneous equations, singular points;
 Weierstrass equation, sum of points, duplication of points, points at infinity, explicit formulas for operations;
 The group of rational points over a field, points of order two and points of order three;
 Points of finite order and endomorphisms of elliptic curves.
2. Three proofs of the associativity of the group law:

Computer assisted proof;
 Proof via the Pappus theorem
 Proof via Picard group
3. MordellWeil lattices
 The weak MordellWeil theorem $+$ descent $\Rightarrow$ the MordellWeil theorem
 heights on projective spaces
 NeronTate height
 MordellWeil lattices

Lattices and Lie algebras, Laura Geatti
 Abstract root systems;
 Existence of a base;
 The lattice associated to an abstract root system (root lattice);
 The classification of root systems;
 Examples of root lattices.
Lecture 2.
 Complex semisimple Lie algebras;
 Cartan subalgebras and root decomposition;
 The Lie algebraa $\mathfrak{sl}_2$ and its representations.
Lecture 3.
 The root system of a semisimple Lie algebra
 Cartan matrix and Dynkin diagram.
 The weight lattice.
Notes
References
 J. E. Humphreys, Introduction to Lie algebras and Representation theory, Graduate Texts in Mathematics, Springer Verlag, New York, 1972
 J.P. Serre, Complex semisimple Lie algebras, Springer Verlag, New York, 1966
 Course notes.

Lattices and modular forms, Valerio Talamanca
Lecture 1
Lattices in the complex plane and elliptic curves;
 Action of $\operatorname{SL}_2(\mathbb{Z})$ on the set of lattices in the complex plane;
 Moduli space for complex elliptic curves.
Lecture 2
Classsical theta functions and their properties;
 Modular forms for $\operatorname{SL}_2(\mathbb{Z})$, cusp forms. The space $M_k$ of weight 2k modular forms;
 Invariants of elliptic curves as modular functions;
 Eisenstein series and the explicit determination of a basis of $M_k$.
Lecture 3
Theta series of lattices;
 Even Unimodular lattices;
 The theta series of even unimodular lattices is a modular form;
 Explicit examples.
References
J.P. Serre, A Course in Arithmetic, Springer Verlag, New York, 1973 (Chapter VII)
