Mazur proposes to work on the statistical and algorithmic aspects of number theory---specifically as related to elliptic curves, abelian varieties, modular curves, and automorphic forms. For example, he is engaged in a research program with Karl Rubin to investigate the Markovian statistics of Selmer groups of quadratic twist families of elliptic curves. Selmer groups provide a delicate---and computable---gauge for the arithmetic of elliptic curves and abelian varieties. A close understanding of Selmer groups can be applied to questions as far-ranging as issues of solvability and unsolvability (Hilbert?s Tenth Problem) or refined analytic formulas (where Rubin and the PI have solved a good part of a conjecture of Darmon). Mazur also proposes to continue the study of p-adic interpolation of modular eigenforms and of their L-functions.
In broad terms, Mazur wants to understand polynomial equations and the algorithms that might lead us to the solutions of such equations. It is vastly instructive to study such solutions, both specifically and statistically, bringing the relevant analytic and arithmetic tools to bear on the problem. Such investigations have had, in the past, surprisingly broader application---both practical and theoretical---as the PI hopes will be the case in this instance.
