Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The genus zeta function of hereditary orders in central simple algebras over global fields

Author: M. Denert
Journal: Math. Comp. 54 (1990), 449-465
MSC: Primary 11R54; Secondary 16A18
MathSciNet review: 993928
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Louis Solomon introduced the notion of a zeta function $ {\zeta _\Theta }(s)$ of an order $ \Theta $ in a finite-dimensional central simple K-algebra A, with K a number field or its completion $ {K_P}$ (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate $ {\zeta _\Theta }(s)$, it is sufficient to consider the zeta function of local orders $ {\Theta _P}$. However, since these local orders $ {\Theta _P}$ are in general not principal ideal domains, their zeta function is a finite sum of so-called 'partial zeta functions'. The most complicated term is the 'genus zeta function', $ {Z_{{\Theta _P}}}(s)$, which is related to the free $ {\Theta _P}$-ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., $ [A:K] = 4$). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function $ {Z_{{\Theta _P}}}(s)$ of any local hereditary order $ {\Theta _P}$ in a central simple algebra over a local field. We obtain $ {Z_{{\Theta _P}}}(s)$ as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.

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

  • [1] C. J. Bushnell and I. Reiner, Zeta functions of arithmetic orders and Solomon's conjectures, Math. Z. 173 (1980), 135-161. MR 583382 (81k:12019)
  • [2] -, L-functions of arithmetic orders and asymptotic distribution of ideals, J. Reine Angew. Math. 327 (1981), 156-183. MR 631314 (83a:12024a)
  • [3] -, Functional equations for L-series of arithmetic orders, J. Reine Angew. Math. 329 (1981), 88-124. MR 636447 (83a:12024b)
  • [4] -, Zeta functions of hereditary orders and integral group rings, Texas Tech. Univ. Math. Ser. 14 (1980), 71-94. MR 640732 (83j:12007)
  • [5] -, A survey of analytic methods in noncommutative number theory, in Orders and Their Applications (I. Reiner and K. W. Roggenkamp, eds.), Proc. 1984 Lecture Notes in Math., vol. 1142, Springer-Verlag, Berlin and New York, 1985, pp. 50-86. MR 812490 (87a:11119)
  • [6] L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954), 332-350. MR 0060538 (15:686a)
  • [7] M. Denert and J. Van Geel, Cancellation property for orders in non Eichler division algebras over global function fields, J. Reine Angew. Math. 368 (1986), 165-171. MR 850620 (87i:11169)
  • [8] -, The class number of hereditary orders in non-Eichler (R)-algebras over global function fields, Math. Ann. 282 (1988). MR 967020 (90d:11126)
  • [9] M. Denert, Affine and projective orders in central simple algebras over global function fields (An analytic approach to the ideal theory), Thesis, State University of Ghent, 1987.
  • [10] -, Solomon's second conjecture: a proof for local hereditary orders in central simple algebras, preprint.
  • [11] I. Reiner, Maximal orders, Academic Press, New York, 1975. MR 0393100 (52:13910)
  • [12] L. Solomon, Zeta functions and integral representation theory, Adv. in Math. 26 (1977), 306-326. MR 0460292 (57:286)
  • [13] A. Weil, Adeles and algebraic groups, Institute for Advanced Studies, Princeton, N.J., 1961.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11R54, 16A18

Retrieve articles in all journals with MSC: 11R54, 16A18

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society