Deformation Rings and Group Schemes

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The goal of the proposal is to better understand the structure of (restricted) deformation rings arising from Galois representations. For example, it is a fundamental problem to relate algebraic properties of deformation rings to properties of the representations being classified. This is, at present, very poorly understood. Out of analogy with algebraic geometry, can one define a suitable cohomology group whose vanishing corresponds to formal smoothness of the deformation ring? A representative non-trivial example which we will study is the case of representations arising from finite, flat group schemes over highly ramified bases. This example should shed some light on more general cases, and it should also have applications to modularity questions for Galois representations and elliptic curves. This research is in the area of number theory, which is the branchof mathematics that is concerned with questions about the integers.Number theory is a very old subject, but is full of difficult problemsand significant conjectures. Many of the fundamental problems in the subject concern either the behavior of prime numbers or properties of the integer solutions to systems of polynomial equations. These kinds of problems can be studied with a vast range ofanalytic, algebraic, and geometric tools. Such number-theoretic ideas also have important real-world applications in the development of secure and accurate electronic communications systems. The elliptic curve factorization algorithm is one example of such an application.