Elliptic Curves and Their Statistics

It is curious how aggregates rather than single instances creep into our subject even when we aren't looking for statistical trouble. Here is an example. In the Erdös spirit, I'll offer a $5 prize for anyone who can manage to provide a proof of the fact that • every linear form aX+b with a, b ∈ Z relatively prime represents at least one prime number; and yet • the proof doesn't actually show that it represents infinitely many primes. I think my $5 is safe, but the point I want to make is that a certain amount of our work is—whether we want it or not—inescapably about " aggregates. " The statement in the first bullet above is true 1 , and Dirichlet proved it in 1837 by showing, more precisely, that there is a positive density of primes in any arithmetic progression with g.c.d.(a, b) = 1. Specifically, 1 An analogous statement is conjectured to be true for any polynomial with integer coefficients that satisfies some natural requirements (as would follow from Schinzel's Conjecture).