AGNT - Ankara-Istanbul Algebraic Geometry and Number Theory Meetings
09 Mayıs 2015
Ankara - Bilkent University
Arithmetic Statistics - Kazim Buyukboduk, Ilhan Ikeda, Ekin Ozman
Arithmetic Statistics: The question of ﬁnding all rational solutions to Diophantine equations is one of the oldest and most central questions in number theory. Such solutions correspond to solutions of polynomials in rational numbers or, geometrically speaking,Q-rational points on the curve described by that polynomial. It has proved highly fruitful to view such problems not just from an algebraic perspective but also a geometric one. A recent approach is to view a family of equations(or curves) all together instead of studying them individually and asking questions about average number of rational points of a family or distribution of such points among the family. This new approach, sometimes called Arithmetic Statistics, has been studied from different perspectives by many mathematicians including Bhargava, Mazur, Poonen and Rubin. In this series of talks, we will start with defining main objects of this study and then focus on introducing this new approach following the approach of Mazur, Rubin and their collaborator Klagsbrun.
The theme of "variation of arithmetic invariants in families" goes all the way back to Gauss, who then inquired about the (*horizontal*) variation of class numbers of quadratic fields. Although there are heuristics about the statistics governing that, we still fall short of a definitive answer. In contrast with this rather erratic behaviour in *horizontal* families, Iwasawa proved a neat variational formula for the class numbers along (*vertical*) Z_p towers, in effect abusing the "p-adic analytic variation" of class groups. This lead Greenberg and Mazur (and many others) to a much broader study of the p-adic analytic variation in various arithmetic families, which was emphasized in the last lecture of the Fall Semester (on Rigid Geometry and Langlands' program). The perspective we shall take this semester is akin to the *horizontal* approach (which looks much harder, from historical perspective), following the lead of Bhargava, Mazur, Poonen and Rubin as indicated above, investigating the statistics of the Mordell-Weil ranks of elliptic curves ranging in horizontal families.
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