The genus zeta function of hereditary orders in central simple algebras over global fields
Author:
M. Denert
Journal:
Math. Comp. 54 (1990), 449465
MSC:
Primary 11R54; Secondary 16A18
MathSciNet review:
993928
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Louis Solomon introduced the notion of a zeta function of an order in a finitedimensional central simple Kalgebra A, with K a number field or its completion (P a nonArchimedean 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 , it is sufficient to consider the zeta function of local orders . However, since these local orders are in general not principal ideal domains, their zeta function is a finite sum of socalled 'partial zeta functions'. The most complicated term is the 'genus zeta function', , which is related to the free ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., ). 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 of any local hereditary order in a central simple algebra over a local field. We obtain 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 qEulerian polynomials. We conclude with some remarks on a conjectured identity.
 [1]
Colin
J. Bushnell and Irving
Reiner, Zeta functions of arithmetic orders and Solomon’s
conjectures, Math. Z. 173 (1980), no. 2,
135–161. MR
583382 (81k:12019), http://dx.doi.org/10.1007/BF01159955
 [2]
Colin
J. Bushnell and Irving
Reiner, 𝐿functions of arithmetic orders and asymptotic
distribution of ideals, J. Reine Angew. Math. 327
(1981), 156–183. MR 631314
(83a:12024a), http://dx.doi.org/10.1515/crll.1981.327.156
 [3]
Colin
J. Bushnell and Irving
Reiner, Functional equations for 𝐿functions of arithmetic
orders, J. Reine Angew. Math. 329 (1981),
88–124. MR
636447 (83a:12024b), http://dx.doi.org/10.1515/crll.1981.329.88
 [4]
Colin
J. Bushnell and Irving
Reiner, Zeta functions of hereditary orders and integral group
rings, Visiting scholars’ lectures—1980 (Lubbock, Tex.,
1980) Math. Ser., vol. 14, Texas Tech Univ., Lubbock, Tex., 1981,
pp. 71–94. MR 640732
(83j:12007)
 [5]
Colin
J. Bushnell and Irving
Reiner, A survey of analytic methods in noncommutative number
theory, Orders and their applications (Oberwolfach, 1984) Lecture
Notes in Math., vol. 1142, Springer, Berlin, 1985,
pp. 50–87. MR 812490
(87a:11119), http://dx.doi.org/10.1007/BFb0074792
 [6]
L.
Carlitz, 𝑞Bernoulli and Eulerian
numbers, Trans. Amer. Math. Soc. 76 (1954), 332–350. MR 0060538
(15,686a), http://dx.doi.org/10.1090/S00029947195400605382
 [7]
Marleen
Denert and Jan
Van Geel, Cancellation property for orders in nonEichler division
algebras over global functionfields, J. Reine Angew. Math.
368 (1986), 165–171. MR 850620
(87i:11169)
 [8]
Marleen
Denert and Jan
Van Geel, The class number of hereditary orders in nonEichler
algebras over global function fields, Math. Ann. 282
(1988), no. 3, 379–393. MR 967020
(90d:11126), http://dx.doi.org/10.1007/BF01460041
 [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 [A subsidiary of
Harcourt Brace Jovanovich, Publishers], LondonNew York, 1975. London
Mathematical Society Monographs, No. 5. MR 0393100
(52 #13910)
 [12]
Louis
Solomon, Zeta functions and integral representation theory,
Advances in Math. 26 (1977), no. 3, 306–326. MR 0460292
(57 #286)
 [13]
A. Weil, Adeles and algebraic groups, Institute for Advanced Studies, Princeton, N.J., 1961.
 [1]
 C. J. Bushnell and I. Reiner, Zeta functions of arithmetic orders and Solomon's conjectures, Math. Z. 173 (1980), 135161. MR 583382 (81k:12019)
 [2]
 , Lfunctions of arithmetic orders and asymptotic distribution of ideals, J. Reine Angew. Math. 327 (1981), 156183. MR 631314 (83a:12024a)
 [3]
 , Functional equations for Lseries of arithmetic orders, J. Reine Angew. Math. 329 (1981), 88124. MR 636447 (83a:12024b)
 [4]
 , Zeta functions of hereditary orders and integral group rings, Texas Tech. Univ. Math. Ser. 14 (1980), 7194. 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, SpringerVerlag, Berlin and New York, 1985, pp. 5086. MR 812490 (87a:11119)
 [6]
 L. Carlitz, qBernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954), 332350. 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), 165171. MR 850620 (87i:11169)
 [8]
 , The class number of hereditary orders in nonEichler (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), 306326. 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
DOI:
http://dx.doi.org/10.1090/S00255718199009939287
PII:
S 00255718(1990)09939287
Article copyright:
© Copyright 1990
American Mathematical Society
