CAREER: Genus one curves: rational points and arithmetic statistics

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A central topic in number theory is the study of diophantine equations, that is, the search for integer solutions to polynomial equations with integer coefficients. It has proved highly fruitful to view such equations not only from an algebraic perspective but also from a geometric one; the field of arithmetic geometry provides a language to do so. Elliptic curves (and the closely related genus one curves) are a fascinating class of diophantine equations. The proposed research bears on basic questions of solvability of algebraic equations that define genus one curves, as well as other arithmetic-geometric aspects of such equations. The PI plans to organize a collaborative workshop with specialists from three different areas of number theory that impact the proposed research; and a lecture series on recent developments in number theory which will expose faculty, graduate students, and undergraduates to new and exciting mathematics in this area. The basic objects of study of the proposed research are genus one curves and their rational points.The PI will investigate them from three distinct perspectives: proving, in joint work with A. Wiles, the existence of solvable points on genus one curves defined over the rationals; analyzing the structure of local points on supersingular elliptic curves over Z_p extensions; and studying the arithmetic statistics of the existence of points on quadratic twists of elliptic curves viewed as genus one curves.