Explicit Galois Deformation Theory, Modular Forms, and Iwasawa Theory

Michigan State University, East Lansing MI

Investigators

Abstract

Any scientific effort at classification proceeds in two steps: first, classify the most basic objects, and, second, classify the different ways the basic objects can be combined to form more complex objects. For example, in the classifications of molecules in chemistry, the most basic objects are the chemical elements appearing on the periodic table, and the combinations are the various bonds that can form between them. In algebraic number theory, the subject of this project, the goal is to classify algebraic structures called Galois representations, which capture the symmetries of solutions to "Diophantine equations," polynomial equations where we seek solutions among the integers. The basic objects are called "irreducible," and the ways they can be combined are called "extensions." There is a very rich theory of extensions even between relatively simple irreducible pieces. In this project, the PI aims to establish instances of conjectures that parameterize the structure of these extensions by objects coming from other areas of mathematics, including analysis and geometry. The guiding principle comes from deformation theory: starting with the simplest possible extension between two irreducibles, if there are many ways to deform it, then there must be many other extensions, whereas if it is more rigid, the structure of extensions is simpler. When explicit deformations can be found, this gives information about extensions. Deformation theory is crucial in modern number theory and played a central role in Andrew Wiles's proof of Fermat's Last Theorem. In this project, the extension groups to be analyzed are Selmer groups. The PI will develop techniques for studying the deformation theory of reducible representations by systematically using pseudorepresentations. In this case, explicit deformations come from congruences between Eisenstein series and cuspidal modular forms. Analysis of deformations leads to results relating the size of Selmer groups to special values of L-functions, as predicted by the Bloch-Kato conjectures. Of particular interest are refined conjectures, such as Sharifi's conjecture, which extract finer information about Selmer groups from more delicate properties of congruences. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.