Programme for Active Period in Arithmetic Geometry
Week 2: Monday 21 April – Friday 25 April

• Monday 21 April, Seminar
  Time: 11 – 12
  Room: B3.02
  Speaker: Evis Ieronymou (Imperial),
  Title: Brauer group of diagonal quartic surfaces, and elements of order 2
  Abstract: We work over the complex numbers, and consider an elliptic fibration of a diaqonal quartic surface. We construct elements of the Brauer group of the generic fibre, as biquaternion algebras over its function field. This is done by considering torsors under Z/2. A key fact of the process is that the generic fibre can be given the structure of a 2-covering of its jacobian, which can be lifted to a 4-covering. We then check which of the elements thus constructed belong to the Brauer group of the original surface. Having constructed one such element we then explore its arithmetic applications. We show that it provides an obstruction to weak approximation for a specific diagonal quartic, over a degree 8 extension of the rational numbers. We note that over this extension the algebraic part of the Brauer group is trivial.
  
  
• Monday 21 April, Seminar
  Time: 2 – 3
  Room: MS02
  Speaker: Sir Peter Swinnerton-Dyer (Cambridge),
  Title: Rational points on certain pencils of curves
  
  
• Tuesday 22 April – Wednesday 23 April, Short Course
  Times and Rooms: Tuesday 10 – 11, 12 – 1, B3.02
  Wednesday 10 – 11, 12 – 1, B3.02
  Speaker: Johan Bosman (Leiden)
  Title: Galois Representations of Modular Forms
  Abstract: In this short course we will start by giving an introduction to classical modular forms and Galois representations associated to them. In the end of the course we will discuss some results in Bas Edixhoven's project on the computation of coefficients of modular forms. Included topics are: Hecke operators, modular curves and Serre's conjecture. Knowlegde of modular forms is not required to understand the lectures.
  
  
• Friday 25 April, Mathematics Institute Colloquium
  Time: 4 – 5
  Room: B3.02
  Speaker: Tom Fisher (Cambridge)
  Title: The Arithmetic of Plane Cubics
  Abstract: In this talk I will describe the process of 3-descent on elliptic curves over the rationals, as has recently been made more explicit in joint work with Cremona, O'Neil, Simon and Stoll. I will begin by reviewing some of the classical geometry related to the Hesse pencil of plane cubics. I will then define the group of rational points (or Mordell-Weil group) of an elliptic curve, and explain how computing its rank is related to searching for rational points on plane cubics. The aim of a 3-descent calculation is then, starting from an elliptic curve, to find the relevant plane cubics.
  
  

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