COUNTING INTEGER AND RATIONAL-POINTS ON VARIETIES

Let V/K be an algebraic variety defined over a number field, and let H be a height function on V. We study the asymptotic behavior of the counting function N(V(R),B) = #{P is an element of V(R): H(P) less than or equal to B} as B --> infinity. We describe some known and conjectural formulas for the behaviour of N(Sr(R), B) which are given in terms of elementary geometric invariants of the variety V and elementary arithmetic invariants of the ring R.