"Zeta integral"의 두 판 사이의 차이

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-==introduction==
-* many zeta integrals in the theory of automorphic forms can be produced or explained by appropriate choices of a Schwartz space of test functions on a spherical homogeneous space, which are in turn dictated by the geometry of affine spherical embeddings
-==local zeta integral==
-* quasicharacter on $F_v^{\times}$ are of the form $\omega_s(x)=\omega(x)|x|^s$ where $\omega$ is unitary
-* $\omega$ : unitary, $s\in \mathbb{C}$
-* the following converges for $\Re(s)>0$
-$$
-\zeta(f,\omega,s)=\int_{F_v^{\times}}f(x)\omega(x)|x|^s\, d^{\times}x
-$$
-* analytic continuation of $Z(f,\omega,s)$
-* functional equation
-==global zeta integral==
-===Riemann zeta function===
-* $f\in \mathcal{S}(\mathbb{A})$
-* define
-$$
-\zeta(f,s)=\int_{\mathbb{A}^{\times}}f(x)|x|^s\, d^{\times}x
-$$
-;thm
-The integral converges locally uniformly for $\Re(s)>1$ and so it defines a holomorphic function in that range, which extends to an meromorphic function on $\mathbb{C}$.
-This function is holomorphic away from the points $s=0,1$, where it has at most simple poles of residue $-f(0)$ and $\hat{f}(0)$, respectively. The zeta integral satisfies the functional equation
-One has
-$$
-\zeta(f,s)=\zeta(\widehat{f},1-s)
-$$
-===Dirichlet L-functions===
-* $f\in \mathcal{S}(\mathbb{A})$
-* $\chi$ : character of $\mathbb{A}^{\times}/\mathbb{Q}^{\times}$ with finite image
-* define
-$$
-\zeta(f,\chi,s)=\int_{\mathbb{A}^{\times}}f(x)\chi(x)|x|^s\, d^{\times}x
-$$
-;thm
-Let $\chi\neq 1$. The integral converges locally uniformly for $\Re(s)>1$ and so it defines a holomorphic function in that range, which extends to an entire function on $\mathbb{C}$. One has
-$$
-\zeta(f,\chi,s)=\zeta(\widehat{f},\overline{\chi},1-s)
-$$
-==articles==
-* http://arxiv.org/abs/1509.04835
-* Li, Wen-Wei. “Zeta Integrals, Schwartz Spaces and Local Functional Equations.” arXiv:1508.05594 [math], August 23, 2015. http://arxiv.org/abs/1508.05594.
-[[분류:L-functions and L-values]]

"Zeta integral"의 두 판 사이의 차이

2020년 11월 13일 (금) 09:55 판

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