Arithmetic Statistics: Groups of Elliptic Curves and Abelian Varieties, and Zeroes of Families of Curves over Finite Fields.

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This research proposal is concerned with Arithmetic Statistics, a new branch of numbertheory which emerged from the spectacular new developments of the last decadesabout distribution questions related to number theoritic objects. The two maindistribution questions that we are investigating are:(1)If we look at a random sequence of groups,such as class groups of quadratic imaginary extensions, or groups ofpoints of elliptic curves over finite fields, how are those groups distributed?A naive probabilistic model would be that groups of the same size are uniformlydistributed, but strong evidence indicates that the probability is inversely proportional to the size of the automorphism group, as predicted by the Cohen-Lenstra heuristics.(2) What are the statistics for the distribution of the number of points in families of curves over finite fields? When studying this variation in families where the size of the finite field gets large, the distribution is coming from random matrix theory as proven by Katz and Sarnak. By contrast, there is no general conjectural picture of what one should expect for the study of families at the large genus limit when the size of the finite field.