Adelic Problems in Algebraic Number Theory

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Larsen 9701410 This award funds the research of Professor Michael Larsen, who will investigate two problems in algebraic number theory. The first is a project undertaken jointly with Richard Pink to analyze in a qualitative sense the full image of a compatible system of Galois representations. The underlying classical problem, finding the Galois group of the points of finite order on an abelian variety, goes back at least to Kronecker in special cases but remains unsolved in general. The second project is more narrowly focused and concerns the search for a generalization to n-dimensional varieties of Hasse's reciprocity law for central simple algebras over number fields. This research falls into the general mathematical field of Number Theory. Number Theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.