Functional equations of Weng's zeta functions for (G,P)/Q
Functional equations of Weng's zeta functions for (G,P)/Q
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(G,P)/Q 的翁氏 zeta 函数的函数方程
DOI：
10.1353/ajm.2013.0032
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发表时间：
2013
影响因子：
1.7
通讯作者：
Y. Komori
中科院分区：
文献类型：
--
作者：
Y. Komori
It is shown that Weng's zeta functions associated with arbitrary semisimple algebraic groups defined over the rational number field and their maximal parabolic subgroups satisfy the functional equations. 1. Introduction. Recently, Lin Weng introduced a new class of abelian zeta functions associated to a pair of reductive algebraic group G and its maximal parabolic subgroup P , which are related with constant terms of Eisenstein series. In this paper, we simply refer to these zeta functions as Weng's zeta functions. These are motivated by and closely related to non-abelian zeta functions called "high rank zeta functions" associated with algebraic number fields, which were also introduced by Weng himself from a viewpoint of Arakelov geometry based on Iwasawa's interpretation and Tate's Fourier analysis on adeles. High rank zeta functions are generalizations of the Dedekind zeta functions and in fact, rank one zeta functions coincide with the Dedekind zeta functions up to constant multiples. Hence the study of Weng's zeta functions is not only interesting itself but also sug- gestive for the study of the Dedekind zeta functions. The profound background, the path to the discovery, and the development of Weng's zeta functions are detailed in his elaborated papers (13, 14, 15, 16). One of the most significant properties for Weng's zeta functions is the behav- ior of their zeros. Weng conjectured that for any pair (G, P ), Weng's zeta functions satisfy certain functional equations and the Riemann hypothesis, as is expected or shown for various kinds of zeta functions. As for the functional equations, initially, Weng proved them in the cases SL(n) (n = 2, 3, 4, 5), Sp(4) ,S O(8) and G2 ,a nd successively H. Kim-Weng, in the cases (SL(n) ,P n−1,1) for arbitrary n ≥ 2 (un- published). As for the Riemann hypothesis, its validity was shown in the cases SL(n) (n = 2, 3, 4, 5), Sp(4) and G2 with their arbitrary maximal parabolic sub- groups (see (6, 8, 10, 11, 12) for the details). In this paper, we establish the functional equations in arbitrary semisimple cases in a unified way. We will see that the functional equations are governed by the involutions on the Weyl groups (see the last paragraph of Section 3 and Lemma 5.3). Furthermore we give the explicit forms of Weng's zeta functions and
It is shown that Weng's zeta functions associated with arbitrary semisimple algebraic groups defined over the rational number field and their maximal parabolic subgroups satisfy the functional equations. 1. Introduction. Recently, Lin Weng introduced a new class of abelian zeta functions associated to a pair of reductive algebraic group G and its maximal parabolic subgroup P , which are related with constant terms of Eisenstein series. In this paper, we simply refer to these zeta functions as Weng's zeta functions. These are motivated by and closely related to non-abelian zeta functions called "high rank zeta functions" associated with algebraic number fields, which were also introduced by Weng himself from a viewpoint of Arakelov geometry based on Iwasawa's interpretation and Tate's Fourier analysis on adeles. High rank zeta functions are generalizations of the Dedekind zeta functions and in fact, rank one zeta functions coincide with the Dedekind zeta functions up to constant multiples. Hence the study of Weng's zeta functions is not only interesting itself but also sug- gestive for the study of the Dedekind zeta functions. The profound background, the path to the discovery, and the development of Weng's zeta functions are detailed in his elaborated papers (13, 14, 15, 16). One of the most significant properties for Weng's zeta functions is the behav- ior of their zeros. Weng conjectured that for any pair (G, P ), Weng's zeta functions satisfy certain functional equations and the Riemann hypothesis, as is expected or shown for various kinds of zeta functions. As for the functional equations, initially, Weng proved them in the cases SL(n) (n = 2, 3, 4, 5), Sp(4) ,S O(8) and G2 ,a nd successively H. Kim-Weng, in the cases (SL(n) ,P n−1,1) for arbitrary n ≥ 2 (un- published). As for the Riemann hypothesis, its validity was shown in the cases SL(n) (n = 2, 3, 4, 5), Sp(4) and G2 with their arbitrary maximal parabolic sub- groups (see (6, 8, 10, 11, 12) for the details). In this paper, we establish the functional equations in arbitrary semisimple cases in a unified way. We will see that the functional equations are governed by the involutions on the Weyl groups (see the last paragraph of Section 3 and Lemma 5.3). Furthermore we give the explicit forms of Weng's zeta functions and
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DOI：
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期刊：
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Arithmetic Geometry and Number Theory(World Scientific) (to appear)
影响因子：
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期刊：
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DOI：
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发表时间：
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期刊：
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影响因子：
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