Variation of Selmer Groups

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Abstract for award DMS-0400232 of Ramakrishna: The main purpose of this project is to continue aninvestigation of the PI of mod p Galois representations,p-adic Galois representations and their relations withproblems in number theory and arithmetic algebraic geometry.A particular emphasis will be the understanding of how Selmergroups of various sorts change as more primes are addedto the ramification set. In one context this variationof Selmer groups relates to how many `new at q' newforms there arecongruent to a particular oldform. The PI's research is in Algebraic Number Theory.This field has its roots in the study of whole numbers and theirvarious properties. A key tool in the PI's researchis Galois theory. Galois theory is the study of symmetries of solutionsof equations. The understanding of these symmetriesshed slight on the the solutions themselves, abasic mathematical question. An example of such a symmetry is thefact that complex solutions to polynomial equations occur incomplex conjugate pairs.