Class groups, Elliptic curves and Diophantine Equations

Ziel

This project will develop new techniques in arithmetic statistics, in particular concerning the distribution of class groups, the rank distribution of elliptic curves and solvability statistics of Diophantine equations. Breakthrough work of Alexander Smith from 2017 introduced important new techniques in arithmetic statistics. Since then, there has been a surge of activity in this area, including several works by myself. In particular, I have already resolved several notable long-standing problems in arithmetic statistics, including Nagell's conjecture from the 1930s (later refined by Stevenhagen) on the solvability of the negative Pell equation, Hilbert's tenth problem for all finitely generated rings (posed in the 1970s) and improved bounds for the proportion of integers that can be written as sums of two rational cubes.

The underlying techniques of these results are underexplored and are ripe for further development. Emphasis will lie on enhancing the currently available techniques in two key aspects: improved uniformity and error terms in the analytic techniques, and greater applicability of the techniques when the relevant class groups and Selmer groups have additional symmetries. I have extensive expertise in these areas, and am therefore ideally situated for this task. The new techniques will greatly expand the scope of the current techniques, and will be a critical ingredient in unlocking new applications.

Natural applications of these new techniques will include:
1. finding the distribution of 2-Sylows of class groups of quadratic extensions of Q(i),
2. a proof (conditional on a converse theorem) that 50% of integers are sums of two rational cubes, and 50% are not,
3. an improved bound for the number of elliptic curves having rank greater or equal than 2 in quadratic twist families.

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