Grant: Discovery Launch Supplement

We focus on problems in prime number theory, arithmetic statistics, and diophantine equations. Prime number theory is one of the oldest and most distinguished subjects in mathematics, dat- ing back to the work of Euclid in ancient Greece. Recent spectacular progress has been achieved by a plethora of mathematicians including Green, Tao, Maynard, and Zhang. Diophantine problems is an equally old subject, dating back to its namesake, Diophantus. It is now recognized that diophantine equations are deeply connected to arithmetic and algebraic geometry, most famously demonstrated by Sir Andrew Wiles' solution of Fermat's Last Theorem. Arithmetic statistics is a relatively new area, dating back to the work of Gauss and invigorated by Cohen and Lenstra in the 1970's. Recent spectacular work by Bhargava and his coauthors, ultimately leading to Bhargava receiving the Fields Medal in 2014, has cemented the status of arithmetic statistics as a centrepiece of modern number theory. Our goal is to investigate the following problems over the next five years: (1) Counting elliptic curves with a rational 2, 3, 5, 7-isogeny whose conductors have few prime factors; (2) Poonen's question (joint work with S. Yamagishi); (3) Constructing Sn and An-fields ramified at very few primes (joint work with A. Entein and J. Konig); and (4) Counting monic abelian quartics. In the realm of elliptic curves, our work will provide exact asymptotic formulae for certain families of elliptic curves when ordered by conductor. Except in very artificial examples, this has not been achieved. This work will involve sophisticated methods from arithmetic geometry as well as prime number theory, both new and classical. Poonen's question asks whether there exists a bivariate polynomial in two integer variables which rep- resents all non-negative integers, and does not represent any negative integer. It is expected that no such polynomial exists. Our work with S. Yamagishi establishes a negative answer for degrees 4 and 6. In arithmetic statistics a basic problem is to count number fields by the size of their discriminant. De- spite much progress in the last twenty years, we are still not done even with the case of quartic number fields. The last hold-out is the set of A4-quartic fields, and our work will make further progress on estimat- ing the density of A4-quartic fields among all quartic number fields. Addressing these problems will propel number theory forward overall, with deeper understanding of the connection between the distribution of prime numbers and the distributions of algebraic and arithmetic objects of interest. This also contributes to my long term goal of building a large research community in arithmetic statistics and prime number theory in Western Canada and beyond.