Special Day on Number Theory
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Special Day on Number Theory
23 December 2011, Sabancı University Karaköy Communication Center
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
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.