On the p-parts of quadratic Weyl group multiple Dirichlet series
On the p-parts of quadratic Weyl group multiple Dirichlet series
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关于二次Weyl群多重Dirichlet级数的p部分
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Paul E. Gunnells
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作者：
G. Chinta;S. Friedberg;Paul E. Gunnells
摘要 设 Φ 为秩为 r 的简化根系。 Φ 的 Weyl 群多重狄利克雷级数是 r 复数变量 s 1,…, sr 中的狄利克雷级数，最初收敛于足够大的 ℜ(si )，它对 ℂ r 具有亚纯延拓，并且在对应于 Φ 的 Weyl 群的 ℂ r 变换下满足函数方程。该系列有两种结构可供选择，一种是[B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, Weyl 将多重狄利克雷级数 I 分组，见：多重狄利克雷级数、自同构形式和解析数论，S. Friedberg, D. Bump, D. Goldfeld, and J. Hoffstein, eds., Proc.症状。纯数学。 75（2006），91-114。] [B。 Brubaker、D. Bump 和 S. Friedberg、Twisted Weyl 分组多重狄利克雷级数：稳定情况，见：Eisenstein 系列和应用，Gan、Kudla 和 Tschinkel，eds.，Progr。数学。 258（2008），1-26。] [B。 Brubaker、D. Bump、S. Friedberg 和 J. Hoffstein、Weyl 分组多重狄利克雷级数 III：爱森斯坦级数和扭曲不稳定 Ar，Ann。数学。 166 (2007), 293–316.] [B. Brubaker、D. Bump 和 S. Friedberg、Weyl 分组多重狄利克雷级数 II，稳定情况，发明。数学。 165（2006），没有。 2, 325–355.] 基于 n 阶高斯和的乘积求和，第二个 [G. Chinta 和 P. E. Gunnells、Weyl 将由二次特征构造的多个狄利克雷级数组合在一起，Invent。数学。 167（2007），没有。 2, 327–353.] 基于对 Weyl 群的某个群动作的平均。在每种情况下，基本功都发生在通用素数 p 处；然后将满足局部函数方程的局部因素拼凑成全局对象。在本文中，我们研究这些结构以及它们之间的关系。首先，我们扩展平均结构以获得扭曲Weyl群多重狄利克雷级数，其p部分是通过评估r变量中的某些有理函数给出的。然后我们开发这样一个有理函数的性质，给出它的精确分母，表明其分子的非零系数由包含在某个凸多胞形中的点索引，确定对应于顶点的系数，并表明在未扭曲的情况下，有理函数是根据其极性行为和局部函数方程唯一确定的。我们还证明，在 Φ = Ar 的情况下，这里获得的 p 部分与 [B. Brubaker、D. Bump、S. Friedberg 和 J. Hoffstein、Weyl 分组多重狄利克雷级数 III：爱森斯坦级数和扭曲不稳定 Ar，Ann。数学。 166 (2007), 293–316.] 当 n = 2 时。
Abstract Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s 1,…, sr , initially converging for ℜ(si ) sufficiently large, which has meromorphic continuation to ℂ r and satisfies functional equations under the transformations of ℂ r corresponding to the Weyl group of Φ. Two constructions of such series are available, one [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series I, in: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, S. Friedberg, D. Bump, D. Goldfeld, and J. Hoffstein, eds., Proc. Symp. Pure Math. 75 (2006), 91–114.] [B. Brubaker, D. Bump, and S. Friedberg, Twisted Weyl group multiple Dirichlet series: the stable case, in: Eisenstein Series and Applications, Gan, Kudla, and Tschinkel, eds., Progr. Math. 258 (2008), 1–26.] [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293–316.] [B. Brubaker, D. Bump, and S. Friedberg, Weyl group multiple Dirichlet series II, The stable case, Invent. Math. 165 (2006), no. 2, 325–355.] based on summing products of n-th order Gauss sums, the second [G. Chinta and P. E. Gunnells, Weyl group multiple Dirichlet series constructed from quadratic characters, Invent. Math. 167 (2007), no. 2, 327–353.] based on averaging a certain group action over the Weyl group. In each case, the essential work occurs at a generic prime p; the local factors, satisfying local functional equations, are then pieced into a global object. In this paper we study these constructions and the relationship between them. First we extend the averaging construction to obtain twisted Weyl group multiple Dirichlet series, whose p-parts are given by evaluating certain rational functions in r variables. Then we develop properties of such a rational function, giving its precise denominator, showing that the nonzero coefficients of its numerator are indexed by points that are contained in a certain convex polytope, determining the coefficients corresponding to the vertices, and showing that in the untwisted case the rational function is uniquely determined from its polar behavior and the local functional equations. We also give evidence that in the case Φ = Ar , the p-part obtained here exactly matches the p-part of the twisted multiple Dirichlet series introduced in [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar , Ann. Math. 166 (2007), 293–316.] when n = 2.