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On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras. (English) Zbl 1173.11050
Let \(\mathcal D\) be a non-split simple algebra of dimension \(m = 4\) or \(9\) over a number field \(k\), and define a prehomogeneous vector space \((G,V) = (\mathcal D^\times \times (\mathcal D^{op})^\times \times \mathrm{GL}(2), \mathcal D \otimes k^2)\) with action \((g_1, g_2, g)\cdot (a\otimes v) = g_1ag_2 \otimes gv\), which is an inner \(k\)-form of \((\mathrm{GL}(n) \times \mathrm{GL}(n) \times \mathrm{GL}(2), k^n \otimes k^n \otimes k^2)\). Then, one may define the global zeta function (Shintani zeta function) \(Z(\Phi, s, \omega)\) associated with the above \((G,V)\), where \(\Phi\) is a Schwartz-Bruhat function on \(V_\mathbb A\), \(s \in \mathbb C\) and \(\omega\) is an adelic character of \(G\). The main result of this paper is the meromorphic continuation of \(Z(\Phi, s, \omega)\) to the region \(\text{Re}(s) > 2m -2\) with an only possible pole at \(s = 2m\).
For the case \(m = 4\), the author obtained a certain density theorem as an application in a later paper [ibid. 58, No. 2, 625–670 (2008; Zbl 1231.11111)].
There are preceding works for the split case and another non-split case by A. C. Kable, D. J. Wright and A. Yukie.

11M41 Other Dirichlet series and zeta functions
11S90 Prehomogeneous vector spaces
Full Text: DOI Numdam EuDML
[1] Bourbaki, N., Algèbre. Éléments de mathématique, (1958), Hermann
[2] Datskovsky, B.; Wright, D. J., The adelic zeta function associated with the space of binary cubic forms II: local theory., J. Reine Angew. Math., 367, 27-75, (1986) · Zbl 0575.10016
[3] Datskovsky, B.; Wright., D. J., Density of discriminants of cubic extensions, J. Reine Angew. Math., 386, 116-138, (1988) · Zbl 0632.12007
[4] Davenport, H.; Heilbronn., H., On the density of discriminants of cubic fields. II, Proc. Royal Soc., A322, 405-420, (1971) · Zbl 0212.08101
[5] Godement, R.; Jacquet, H.; Springer-Verlag, Zeta functions of simple algebras, Lecture Notes in Mathematics, 260, (1972), Berlin, Heidelberg, New York · Zbl 0244.12011
[6] Kable, A. C.; Wright, D. J., Uniform distribution of the Steinitz invariants of quadratic and cubic extensions, Compos. Math., 142, 84-100, (2006) · Zbl 1113.11065
[7] Kable, A. C.; Yukie, A., The mean value of the product of class numbers of paired quadratic fields, I, Tohoku Math. J., 54, 513-565, (2002) · Zbl 1020.11079
[8] Mumford, D.; Press, Princeton University, Lectures on curves on an algebraic surface, Annales of Mathematical Studies, 59, (1966), Princeton, New Jersey · Zbl 0187.42701
[9] Saito, H., On a classification of prehomogeneous vector spaces over local and global fields, Journal of Algebra, 187, 510-536, (1997) · Zbl 0874.14046
[10] Saito, H., Convergence of the zeta functions of prehomogeneous vector spaces, Nagoya. Math. J., 170, 1-31, (2003) · Zbl 1045.11083
[11] Sato, M.; Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65, 1-155, (1977) · Zbl 0321.14030
[12] Sato, M.; Shintani, T., On zeta functions associated with prehomogeneous vector spaces., Ann. of Math., 100, 131-170, (1974) · Zbl 0309.10014
[13] Shintani, T., On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms, J. Math. Soc. Japan, 24, 132-188, (1972) · Zbl 0227.10031
[14] Taniguchi, T., Distributions of discriminants of cubic algebras
[15] Taniguchi, T., A mean value theorem for the square of class number times regulator of quadratic extensions · Zbl 1231.11111
[16] Weil, A., Basic number theory, (1974), Springer-Verlag, Berlin, Heidelberg, New York · Zbl 0823.11001
[17] Wright, D. J., The adelic zeta function associated to the space of binary cubic forms part I: global theory, Math. Ann., 270, 503-534, (1985) · Zbl 0533.10020
[18] Wright, D. J.; Yukie, A., Prehomogeneous vector spaces and field extensions, Invent. Math., 110, 283-314, (1992) · Zbl 0803.12004
[19] Yukie, A., Lecture Note Series, 183, Shintani zeta functions, (1993), London Math. Soc. · Zbl 0801.11021
[20] Yukie, A., On the shintani zeta function for the space of pairs of binary Hermitian forms, J. Number Theory, 92, 205-256, (2002) · Zbl 1020.11078
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