On the Poincaré series for diagonal forms

Wang, Jun

doi:10.1090/S0002-9939-1992-1116274-3

On the Poincaré series for diagonal forms
HTML articles powered by AMS MathViewer

by Jun Wang PDF

Proc. Amer. Math. Soc. 116 (1992), 607-611 Request permission

Abstract:

Let $p$ be a fixed prime, $f({x_1}, \ldots ,{x_s})$ a polynomial over ${{\mathbf {Z}}_p}$, the $p$-adic integers, ${c_n}$ the number of solutions of $f = 0$ over ${\mathbf {Z}}/{p^n}{\mathbf {Z}}$, and ${P_f}(t) = \sum \nolimits _{n = 0}^\infty {{c_n}{t^n}}$ the Poincaré series. Explicit formulas for ${P_f}(t)$ are derived for diagonal forms.

References

A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803

Complex powers and asymptotic expansions

278/279

Jun-Ichi Igusa, Some observations on higher degree characters, Amer. J. Math. 99 (1977), no. 2, 393–417. MR 441933, DOI 10.2307/2373827
Jay R. Goldman, Numbers of solutions of congruences: Poincaré series for strongly nondegenerate forms, Proc. Amer. Math. Soc. 87 (1983), no. 4, 586–590. MR 687622, DOI 10.1090/S0002-9939-1983-0687622-8
Jay R. Goldman, Numbers of solutions of congruences: Poincaré series for algebraic curves, Adv. in Math. 62 (1986), no. 1, 68–83. MR 859254, DOI 10.1016/0001-8708(86)90088-5

The rationality of the Poincaré series of a diagonal form