**Special Day on Number Theory**

23 December 2011, Sabancı University Karaköy Communication Center

Baran, Büyükboduk, Özman

**Program:**

11:30-12:30 Burcu Baran (University of Michigan, Ann Arbor)

*Serre’s uniformity problem via modular curves*

Lunch Break

15:00-16:00 Ekin Özman (University of Texas, Austin)

*Twisting Modular Curves*

17:00-18:00 Kâzım Büyükboduk (Koç University, Istanbul)

*Deformations of Kolyvagin systems*

This workshop is supported by TUBITAK ISBAP project 107T897 and Sabancı University.

**Abstracts:**

**Burcu Baran - Serre’s uniformity problem via modular curves**

I will discuss Serre’s uniformity problem over Q, an open problem in the theory of Galois representations of elliptic curves. Past work by Serre, Mazur and Bilu-Parent has led to important progress but has not solved the problem. The remaining and most diﬃcult part amounts to a problem concerning rational points of modular curves associated to normalizers of non-split Cartan subgroups. I will discuss this case and also introduce my work on these modular curves. This includes an exceptional isomorphism which can be constructed in two diﬀerent ways.

**Kâzım Büyükboduk - Deformations of Kolyvagin systems**

Mazur's theory of Galois deformations, inspired by Hida's earlier work on families of modular forms, has led to the resolution of many important problems in Number Theory: Wiles and Taylor/Wiles proved Taniyama-Shimura conjecture (to conclude with the proof of FLT), Buzzard/Taylor and Taylor used it to prove many cases of Artin's conjecture. In this talk, I will first give a general outline of Mazur's abstract theory and explain how it is used to attack concrete arithmetic problems. At the end, I will talk about a recent result that Kolyvagin systems (which Mazur and Rubin prove to exist for mod p Galois representations) do often deform to a big Kolyvagin system for the "Universal Galois deformation" representation. I will touch upon important applications of this result in arithmetic.

**Ekin Özman - Twisting Modular Curves**

In this talk, we will give results on the poly-quadratic twists of the modular curve X_0(N) through the Atkin-Lehner involution w_p where p is a prime divisor of N and a poly-quadratic extension K/Q. For the case of quadratic twists, we give necessary and sufficient conditions for the existence of a Q_p-rational point on the twisted curve whenever p is not simultaneously ramified in K and Q(\sqrt{-N}). For other kind of extensions, we give an algorithm that produces several families which has local points everywhere. If time permits, we will give a population of curves which have local points everywhere but no points over Q; in several cases we show that this obstruction to the Hasse Principle is explained by the Brauer-Manin obstruction.