Michael Lipnowski University of Toronto
Topological Explosion and Supercancellation in Arithmetic Statistics
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Abstract:
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The fundamental question (*) How many solutions does a system of polynomial equations have modulo a fixed prime number p?听 drives the burgeoning field of arithmetic statistics.听
Understanding the statistical behavior of the number of solutions within families of algebraic varieties of increasing complexity is subtle due to "topological explosion".听 We will describe several interesting families exhibiting topological explosion and the "supercancellation" necessary for proving that听the statistics of (*) are captured by reasonable heuristic models. 听
We will also discuss joint work with Jacob Tsimerman on the failure of supercancellation for听abelian varieties of large dimension over finite fields.听 This failure results in some statistically counterintuitive behavior.听 For example, most principally polarized abelian varieties of large听dimension over a fixed finite field are essentially powers of elliptic curves and do not obey听Cohen-Lenstra heuristics.听