# My Works on Geometric Arithmetic

## Since 1999

 $\huge{\color{Brown}{\mathbb{\ A\ Global\ Cohomology\ Theory}}}$  $\bigodot\ \Large{F:\ \mathrm{number\ field\ with}\ \mathcal O_F\ \mathrm{integer\ ring}}$  $\bigodot\ \Large{S:=S_{\mathrm{fin}}\cup S_\infty:\ \mathrm{set\ of\ normalized\ non\,archimedean\ and\ archimedean\ places\ of}\ F}$  $\bigodot\ \Large{F_v:\ v\mathrm{-completion\ of}\ F, \mathrm{with}\ \mathcal O_v\ \mathrm{its\ integer\ ring}}$  $\bigodot\ \Large{\mathbb A:\ \mathrm{adelic\ ring\ of}\ F,\ GL_n(\mathbb A)\ \mathrm{associated\ general\ linear\ group}}$  $\bigodot\ \Large{\kappa_F:\ \mathrm{dualizing\ adelic\ elememnt\ of}\ F}$  $\bigodot\ \Large{g=(g_{\frak p};g_v)\in GL_n(\mathbb A),\ {\frak p}\in S_{\mathrm{fin}},\ v\in S_\infty}$  $\huge{\heartsuit\ \color{red}{\mathrm{Auxiliary\ Topological\ Space}\ \mathbb A^n(g):}}$  $\Large{\mathrm{(i)\ Set\ theoretically:}}$ $\LARGE{\boxed{\mathbb A^n(g):=\Bigg\{{\bf x}\in\mathbb A^n\,\Big|\,\begin{matrix}\exists {\bf a}\in F^n, \ \mathrm{s.t.}\ {\bf x_v}={\bf a}, &\forall v\in S_\infty\\ g_\frak p\cdot{\bf x}_\frak p\in\mathcal O_{\frak p}^n,\,g_\frak p\cdot {\bf a}\in\mathcal O_{\frak p}^n, &\forall \frak p\in S_{\mathrm{fin}}\end{matrix}\Bigg\}}}$ $\Large{\mathrm{(ii)\ Topologically:}}$  $\qquad\Large{\mathrm{introduce\ a\ new\ topological\ structure\ on}\ \mathbb A^n\ \mathrm{by\ keeping\ the\ finite\ part\ but\ altering\ its\ metric\ at}\ v\in S_\infty}$ $\qquad\Large{\mathrm{using\ the\ positive\ definite\ matrix}\ g_\sigma^t\cdot g_\sigma\ (\mathrm{resp.}\ \bar g^{t}_\tau\cdot g_\tau)\ \mathrm{when}\ v=\sigma\ \mathrm{is\ real}\ (\mathrm{resp.}\ v=\tau\ \mathrm{is complex});}$ $\qquad\Large{\mathrm{and\ equip}\ \mathbb A^n(g)\ \mathrm{with\ the\ induced\ topological\ structure\ from\ the\ embedding}\ \mathbb A^n(g)\subset\mathbb A^n}$  $\huge{\heartsuit\ \color{red}{\mathrm{Arithmetical\ Cohomology\ Groups}\ H^i(F,g):}}$ $\LARGE\boxed{\displaystyle{H^0(F,g):=\mathbb A^n(g)\cap K^n\qquad\qquad\&\qquad\qquad H^1(F,g):=\mathbb A^n\big/\big(\mathbb A^n(g)+ K^n\big)}}$ $\LARGE{\mathrm{with\ induced\ topology\ from\ the\ altered}\ \mathbb A^n}$  $\huge{\clubsuit\ \color{Blue}{\mathrm{Topological\ Structures:}}}$  $\qquad\LARGE{{\mathrm{Strong\ Approximation\ Theorem}}\qquad\Rightarrow}$ $\LARGE{ \boxed{\ H^0(F,g):\ \ discrete\ \qquad\&\ \qquad H^1(F,g):\ \ compact\ }}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Arithmetic\ Duality}}}$ $\LARGE{\boxed{\widehat{H^1(F,g)}\,\simeq\, H^0(F,\kappa_F\cdot g^{-1})}}$  $\huge{\clubsuit\ \color{Blue}{\mathrm{Arithmetic\ Riemann-Roch\ Theorem}}}$  $\qquad\LARGE{\mathrm{Fourier\ Aanlysis\ on\ Locally\ Compact\ Groups}\qquad\Rightarrow}$  $\qquad\LARGE{\mathrm{(i)\ Numerical\ Duality}}$ $\LARGE\boxed{\displaystyle{h^1(F,g)\,=\,h^0(F,\kappa_F\cdot g^{-1})}}$  $\qquad\LARGE{\mathrm{(ii)\ Arithmetic\ Riemann-Roch}}$ $\LARGE{\boxed{\displaystyle{h^0(F,g)-h^1(F,g)\,=\,\mathrm{deg}(g)-\frac{n}{2}\cdot\log \Delta_F}}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Positivity,\ Ampleness\ \&\ Vanishing\ Theorem}}}$  $\qquad\LARGE{{The\ following\ conditions\ are\ equivalent:}}$  $\LARGE{\mathrm{(i)}\qquad {\bf a}\in GL_1(\mathbb A)\ is\ positive;}$  $\LARGE{\mathrm{ (ii)}\qquad {\bf a}\in GL_1(\mathbb A)\ is\ ample;}$  $\LARGE{\mathrm{ (iii)}\qquad \boxed{\displaystyle{\lim_{m\to \infty}h^1(F, {\bf a}^m\cdot {\bf g})=0 \qquad \forall {\bf g}\in GL_n(\mathbb F)}}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Effective\ Vanishing\ Theorem:}}}\ \Large{(\mathrm{van\,der\ Geer\,-\,Schoof, \ Groenewegen\ N=1;\ \ Weng\ N>\ 1})}$  $\LARGE{\overline E:=\ semi-stable\ \mathcal O_F-lattice\ of\ rank\ n\ satisfying\qquad \displaystyle{\mathrm{deg}(\overline E)\leq -[F:\mathbb Q]\cdot\frac{n\log n}{2}}}$ $\LARGE{\Downarrow}$ $\LARGE{\boxed{h^0(\overline E)\leq \frac{3^{\mathrm{rank}_{\mathbb Z}(\overline E)}}{1-\frac{\log 3}{\pi}}\cdot e^{-\pi\cdot [F:\mathbb Q] \cdot e^{-\frac{2\mathrm{deg}(\overline E)}{\mathrm{rank}_{\mathbb Z}(\overline E)}}}}}$