Frontiers in Mathematical Sciences

Mathematical Logic and Number Theory

Jamshid Derakhshan
University of Oxford


While the model theory of locally compact fields are well understood (starting from pioneering decidability works of Tarski and Ax-Kochen), very little is known regarding the model theory of number fields due to their Godelian undecidability where a direct approach is not possible. In this talk, I will give some model-theoretic results for number fields that are established from local theories. Such a local to global transition is made via motivic and $p$-adic integration to establish a result on meromorphic continuation of Dirichlet series that are Euler products of local integrals defined by means of model-theoretic data, and to understand model theory and measure theory of adeles. This result will then give solution to a number of open problems in number theory and algebra.