Enumerating odd-degree hyperelliptic curves and abelian surfaces over P1

Abstract

As explained therein by Venkatesh, in many interesting number theory problems (e.g., counting number fields, arithmetic curves or abelian varieties over a number field) one has not only a main term in the asymptotic count, but a secondary term or more. We have very little understanding of these lower order terms. They are not just of theoretical interest: when one tries to verify the conjectures numerically, one finds that the secondary terms are dominant in the computational range. For example, the number of cubic number fields of height ≤B for certain constants a,b>0 is aB+bB5/6+o(B56).