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Number Theory Seminar

Counting central simple algebras over number fields

Benjamin Linowitz

Location:  Room 425
Date & time: Tuesday, 21 January 2020 at 2:00PM - 3:00PM

ABSTRACT: This talk will begin with a friendly introduction to central simple algebras over number fields. After reviewing some non-commutative algebra we will discuss the elements of class field theory which are needed to classify central simple algebras in terms of their ramification data. We will then use a Tauberian theorem in order to count the number of central simple algebras with discriminant less than X. In the case that the central simple algebras we are counting have dimension 4 (i.e., are quaternion algebras), we can do much better and count the number of quaternion algebras over a number field k which admit embeddings of any finite number of fixed quadratic field extensions of k and which have discriminant at most X. We will conclude the talk by discussing some applications of this to hyperbolic geometry. In particular we will use the aforementioned results to count the number of arithmetic hyperbolic manifolds containing closed geodesics with prescribed lengths.