I propose to work on p-adic interpolation of modular eigenforms, and on statistical issues in arithmetic arising from, for example, the recent advances in the conjecture of Sato-Tate; I will continue a long-range project, joint with Karl Rubin, to study Selmer groups; recently we have been applying our work in various directions. For example, to obtain some results in mathematical logic (we show that a piece of the classical Shafarevich-Tate Conjecture implies that Hilbert's Tenth Problem phrased for any commutative ring of infinite cardinality that is finitely generated over the ring of rational integers has a negative answer) and to resolve a conjecture of Darmon regarding a refined form of the classical ''analytic formula'' for the first derivative at zero of a Dirichlet L-function, and to study the proportions, in a family of quadratic twists, of the elliptic curves with 2-Selmer rank equal to a given number. Our goal is to make advances in each of these directions.
The broader issues that are the focus of the aspects of my proposal described above are the classical ones: determining the structure of solutions to polynomial equations; of delineating arithmetic problems that admit of no algorithmic solution; and studying the statistical features of data important to the study of arithmetic.
