Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the Poincaré series for diagonal forms


Author: Jun Wang
Journal: Proc. Amer. Math. Soc. 116 (1992), 607-611
MSC: Primary 11E95; Secondary 11D88, 11E76, 11L03
DOI: https://doi.org/10.1090/S0002-9939-1992-1116274-3
MathSciNet review: 1116274
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, New York, 1966. MR 0195803 (33:4001)
  • [2] J. Igusa, Complex powers and asymptotic expansions. II, J. Reine Angew. Math. 278/279 (1979), 307-321.
  • [3] -, Some observations on higher degree character, Amer. J. Math. 99 (1977), 393-471. MR 0441933 (56:324)
  • [4] J. R. Goldman, Numbers of solutions of congruences: Poincaré series for strongly nondegenerate forms, Proc Amer. Math. Soc. 87 (1983), 586-590. MR 687622 (85d:11105)
  • [5] -, Numbers of solutions of congruences: Poincaré series for algebraic curves, Adv. in Math. 62 (1986), 68-83. MR 859254 (88b:11035)
  • [6] E. Stevenson, The rationality of the Poincaré series of a diagonal form, Thesis, Princeton Univ., 1978.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11E95, 11D88, 11E76, 11L03

Retrieve articles in all journals with MSC: 11E95, 11D88, 11E76, 11L03


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1116274-3
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society