My Works on Non-Abelian Zetas

Since 2009

\[\huge{\color{Brown}{\mathbb{\ Rank\ n\ Non-Abelian\ Zeta\ Functions\ }}}\] \[\] \(\bigodot\ \Large{\mathcal M_{\mathbb Q,n}\ \mathrm{denotes\ the\ moduli\ space\ of\ semi\!\!-\!\!stable\ lattices\ \Lambda\ of\ rank}\ n\ in\ \mathbb R^n}\) \[\] \(\bigodot\ \Large{h^0(\mathbb Q,\Lambda), resp. \mathrm{deg}(\Lambda),\ \mathrm{denotes\ the\ 0\!\!-\!\!th\ arithmetic\ cohomology,\ resp.\ the\ arithmetic\ degree,\ of\ the\ lattice}\ \Lambda}\) \[\] \(\huge{\heartsuit\ \color{Red}{\mathrm{Rank\ n\ Zeta\ Function\ for\ \mathbb Q:}}}\) \[\boxed{\begin{align*}\huge{\widehat\zeta_{\mathbb Q,n}(s):=}&\huge\int_{\mathcal M_{\mathbb Q,n}}\big(e^{h^0(\mathbb Q,\Lambda)}-1\big)\big(e^{-s}\big)^{\mathrm{deg}(\Lambda)}d\mu(\Lambda)\ \ \Re(s)>1\end{align*}}\] \[\] \(\huge{\spadesuit\ \color{Blue}{\mathrm{Relation\ with\ Abelian\ Zeta\ Function:}}}\) \[\boxed{\huge{ \widehat\zeta_{\mathbb Q,1}(s)=\widehat\zeta(s).}}\] \(\huge{\spadesuit\ \color{Blue}{\mathrm{Functional\ Equation:}}}\) \[\] \[\boxed{\huge{ \widehat\zeta_{\mathbb Q,n}(1-s)=\widehat\zeta_{\mathbb Q,n}(s)}}\] \[\] \(\huge{\spadesuit\ \color{Blue}{\mathrm{Geometric Interpretation of the Residue:}}}\) \(\huge{\widehat\zeta_{\mathbb Q,n}(s)\ \mathrm{admits\ only\ two\ singularities,\ namely,\ two\ simple\ poles\ at\ s=0,1.\ Moreover,}}\) \[\boxed{\huge{ \mathrm{Res}_{s=1}\Big(\widehat\zeta_{\mathbb Q,n}(s)\Big)=\mathrm{Vol}\Big(\mathcal M_{\mathbb Q,n}[1]\Big)}}\] \[\] \(\huge{\spadesuit\ \color{Blue}{\mathrm{Riemann\ Hypothesis:}}}\) \[\boxed{\huge{\widehat\zeta_{\mathbb Q,n}(s)=0\qquad\Rightarrow\qquad\mathrm{Re}(s)=\frac{1}{2} }}\] (A weak version of this is proved as Theorem 15.4 of my book on Zeta Functions of Reductive Groups and Their Zeros published by the World Scientific, 2018) \[\] \[\] \[\huge{\color{Brown}{\mathbb{\ Zeta\ Functions\ For\ (G,P)/Q}}}\] \[\] \(\bigodot\ \Large{G:\ \mathrm{a\ reductive\ group\ with\ P\ its\ maximal\ parabolic\ subgroup,\ all\ defined\ over}\ \mathbb Q }\) \[\] \(\bigodot\ \Large{\Phi^\pm:\ \mathrm{the\ associated\ system\ of\ positive,\ resp. negative, roots} }\) \[\] \(\bigodot\ \Large{\Delta:\ \mathrm{the\ set\ of\ simple\ roots} }\) \[\] \(\bigodot\ \Large{W:\ \mathrm{the\ Weyl\ group} }\) \[\] \(\bigodot\ \Large{\rho\,:=\,\frac{1}{2}\sum_{\alpha\in\Phi^+}\alpha\ \mathrm{the\ Weyl\ vector} }\) \[\] \(\bigodot\ \Large{\{\beta_1,\beta_2,\dots,\beta_{|G|-1}\}:=\Delta-\{\alpha_P\} \ \mathrm{with}\ \alpha_P\ \mathrm{the\ simple\ root\ corresponding\ to}\ P }\) \[\] \(\huge{\heartsuit\ \color{Red}{\mathrm{Weng\ Zeta\ Functions\ for\ (G,P)}/\mathbb Q:}}\) \[\boxed{\begin{align*}&\Large{\widehat\zeta^{G/P}_{\mathbb Q}(s):=}\\ &:=\Large{\mathrm{Norm}}\,\Bigg({\Large\mathrm{Res}}_{\Large{\langle \lambda-\rho,\beta_1^\vee\rangle=0,\,\langle \lambda-\rho,\beta_2^\vee\rangle=0,\, \dots,\, \langle \lambda-\rho,\beta_{|G|-1}^\vee\rangle=0}}\,\Big( {\Large\sum}_{w\in W}\frac{\Large{e^{-\langle \lambda,T\rangle}}}{{\huge\prod}_{\alpha\in\Delta}\langle \lambda-\rho,\alpha^\vee\rangle}\cdot{\Large\prod}_{\alpha\in \Phi^+, w\alpha\in\Phi^-} \frac{\widehat\zeta\big(\langle\lambda,\alpha^\vee\rangle\big)}{\widehat\zeta\big(\langle\lambda,\alpha^\vee\rangle+1\big)}\Big)\Bigg) \end{align*}}\] \[\] \[\]\(\huge{\spadesuit\ \color{Blue}{\mathrm{Relation\ with\ Non\!\!-\!\!Abelian\ Zeta\ Function:}}}\) \[\] \(\huge{\mathrm{Up\ to\ a\ constant\ factor\ and\ an\ affine\ change\ of\ variables,}}\) \[\boxed{\huge{\widehat\zeta_{\mathbb Q}^{{\mathrm SL}_n/P_{n-1,1}}(s)=\widehat\zeta_{\mathbb Q,n}(s).}}\] \[\] \[\] \(\huge{\spadesuit\ \color{Blue}{\mathrm{Functional\ Equation:}}\qquad \exists\qquad {constant}\quad c_{G/P},\quad \mathrm{s.t.}}\) \[\boxed{\huge{ \widehat\zeta^{G/P}_{\mathbb Q}(c_{P/Q}-s)=\widehat\zeta^{G/P}_{\mathbb Q}(s)}}\] \[\] \[\] \(\huge{\spadesuit\ \color{Blue}{\mathrm{Riemann\ Hypothesis:}}}\) \[\boxed{\huge{\widehat\zeta^{G/P}_{\mathbb Q}(s)=0\qquad\Rightarrow\qquad\mathrm{Re}(s)=\frac{c_{G/P}}{2} }}\] (Existence of a Functional equation was conjectured by L.Weng and proved by Y. Komori. A weak version of the Riemann Hypothesis is proved as Theorem 17.2 of my book on Zeta Functions of Reductive Groups and Their Zeros published by the World Scientific, 2018) \[\]\[\]\[\]