Poincaré series

Abstract

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series$\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } }$. called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ϕ(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ϕ_n(x, y) + ϕ_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.