Abstract:
Siegel's theorem on integer points on curves was originally
nonexplicit; Faltings' theorem still is. For many applications,
what is needed is quantitative versions of Siegel's and Faltings'
theorems; certain simplifying assumptions may be justified, but
the quantitative results must be as strong as possible. We will see
how to bound the number of integer points on elliptic curves and
the number of rational points of bounded height on curves of higher genus
by means of sphere packings. The said bounds allow us to strengthen
the existing results on the squarefree sieve. Joint work by
the speaker and A. Venkatesh has brought about two further applications:
the number of elliptic curves of conductor N is shown to be at most
O(N^{0.2239}), whereas the number of 3torsion elements in the class
group of Q(\/N) is at most O(N^{0.442}). (In both cases, the best
previous bounds were of the order of O(N^{1/2+epsilon}).) We obtain
these bounds by a refinement of the simplest approach based on sphere
packings; this refinement allows us to go beyond the BombieriPila bound
for certain kinds of equations.
