Number Theory Seminar

Some geometric counting problems have very pretty exact answers. For instance, if K is a finite field with q elements, the number of elliptic curves over K (counted with a certain natural weight factor) is exactly q, and the number of genus-2 curves over K (counted with the same weight factor) is exactly q^3. These results come from the fact that elliptic curves and genus-2 curves have nice moduli spaces. Jeff Achter (Colorado State University) and I consider the problem of counting genus-2 curves over K whose Jacobian varieties are not simple --- that is to say, genus-2 curves that have non-trivial maps to elliptic curves. These curves correspond to a subset of the moduli space of genus-2 curves that is a countably infinite union of surfaces, so we do not expect a nice clean answer. We show that up to explicit logarithmic factors, the number of such curves is bounded above and below by q^{5/2}.