Irrationality of Periods and Arithmetic of Abelian Varieties

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This research project concerns work in Diophantine geometry and arithmetic geometry, which are essentially ways to understand solutions of families of polynomial equations. The first part of the project studies irrationality in Diophantine geometry. A classical way to prove a number is irrational is by showing that there exists a sequence of rational numbers that approximate this number very well. There are many interesting (conjecturally) irrational numbers in the literature with approximations by rational numbers that are not good enough to apply the classical methods. The principal investigator and collaborators will develop a new framework to explore the properties of certain power series constructed from these approximations in order to prove the conjectured irrationality in some important cases. The second part of the project studies the arithmetic of abelian varieties, which are higher dimensional analogues of elliptic curves. These geometric objects can be defined by polynomial equations over the integers. The principal investigator and collaborators will study the behavior of certain abelian varieties modulo different prime numbers. The proposed work includes the training of undergraduate and graduate students. For the first part, the classical way of proving irrationality can be formulated as studying the convergence radii of the power series associated to the rational approximations and comparing them to the denominator type of the power series. In earlier studies of rationality and algebraicity criterion of power series, the convergence radii have been replaced by many variants, which are numerically larger; therefore, there are rational approximations whose convergence radii are too small compared to the denominator type while these variants are large enough. The PI and collaborators expect to explore these larger radii variants to solve some irrationality questions. For the second part, the PI and collaborators expect to generalize Elkies’s theorem on infinitude of supersingular reductions of elliptic curves to certain abelian varieties parametrized by genus 0 Shimura curves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.