My Works on Geometric Arithmetic
Since 1999
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\[\huge{\color{Brown}{\mathbb{\ A\ Global\ Cohomology\ Theory}}}\]
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\(\bigodot\ \Large{F:\ \mathrm{number\ field\ with}\ \mathcal O_F\ \mathrm{integer\ ring}}\)
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\(\bigodot\ \Large{S:=S_{\mathrm{fin}}\cup S_\infty:\ \mathrm{set\ of\ normalized\ non\,archimedean\ and\
archimedean\ places\ of}\ F}\)
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\(\bigodot\ \Large{F_v:\ v\mathrm{-completion\ of}\ F, \mathrm{with}\ \mathcal O_v\ \mathrm{its\ integer\ ring}}\)
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\(\bigodot\ \Large{\mathbb A:\ \mathrm{adelic\ ring\ of}\ F,\ GL_n(\mathbb A)\ \mathrm{associated\ general\
linear\ group}}\)
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\(\bigodot\ \Large{\kappa_F:\ \mathrm{dualizing\ adelic\ elememnt\ of}\ F}\)
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\(\bigodot\ \Large{g=(g_{\frak p};g_v)\in GL_n(\mathbb A),\ {\frak p}\in S_{\mathrm{fin}},\ v\in S_\infty}\)
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\(\huge{\heartsuit\ \color{red}{\mathrm{Auxiliary\ Topological\ Space}\ \mathbb A^n(g):}}\)
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\(\Large{\mathrm{(i)\ Set\ theoretically:}}\)
\[\LARGE{\boxed{\mathbb A^n(g):=\Bigg\{{\bf x}\in\mathbb A^n\,\Big|\,\begin{matrix}\exists {\bf a}\in F^n,
\ \mathrm{s.t.}\ {\bf x_v}={\bf a}, &\forall v\in S_\infty\\
g_\frak p\cdot{\bf x}_\frak p\in\mathcal O_{\frak p}^n,\,g_\frak p\cdot {\bf a}\in\mathcal O_{\frak p}^n,
&\forall \frak p\in S_{\mathrm{fin}}\end{matrix}\Bigg\}}}\]
\(\Large{\mathrm{(ii)\ Topologically:}}\)
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\(\qquad\Large{\mathrm{introduce\ a\ new\ topological\ structure\ on}\ \mathbb A^n\ \mathrm{by\ keeping\
the\ finite\ part\ but\ altering\ its\ metric\ at}\ v\in S_\infty}\)
\(\qquad\Large{\mathrm{using\ the\ positive\ definite\ matrix}\
g_\sigma^t\cdot g_\sigma\ (\mathrm{resp.}\ \bar g^{t}_\tau\cdot g_\tau)\ \mathrm{when}\ v=\sigma\
\mathrm{is\ real}\ (\mathrm{resp.}\ v=\tau\ \mathrm{is complex});}\)
\(\qquad\Large{\mathrm{and\ equip}\ \mathbb A^n(g)\ \mathrm{with\ the\ induced\ topological\ structure\ from\
the\ embedding}\ \mathbb A^n(g)\subset\mathbb A^n}\)
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\(\huge{\heartsuit\ \color{red}{\mathrm{Arithmetical\ Cohomology\ Groups}\ H^i(F,g):}}\)
\[\LARGE\boxed{\displaystyle{H^0(F,g):=\mathbb A^n(g)\cap K^n\qquad\qquad\&\qquad\qquad
H^1(F,g):=\mathbb A^n\big/\big(\mathbb A^n(g)+ K^n\big)}}\]
\[\LARGE{\mathrm{with\ induced\
topology\ from\ the\ altered}\ \mathbb A^n}\]
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\(\huge{\clubsuit\ \color{Blue}{\mathrm{Topological\ Structures:}}}\)
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\(\qquad\LARGE{{\mathrm{Strong\ Approximation\ Theorem}}\qquad\Rightarrow}\)
\[\LARGE{
\boxed{\ H^0(F,g):\ \ discrete\ \qquad\&\ \qquad H^1(F,g):\ \ compact\ }}\]
\[\]
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\(\huge{\clubsuit\ \color{Blue}{\mathrm{Arithmetic\ Duality}}}\)
\[\LARGE{\boxed{\widehat{H^1(F,g)}\,\simeq\, H^0(F,\kappa_F\cdot g^{-1})}}\]
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\(\huge{\clubsuit\ \color{Blue}{\mathrm{Arithmetic\ Riemann-Roch\ Theorem}}}\)
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\(\qquad\LARGE{\mathrm{Fourier\ Aanlysis\ on\ Locally\ Compact\ Groups}\qquad\Rightarrow}\)
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\(\qquad\LARGE{\mathrm{(i)\ Numerical\ Duality}}\)
\[\LARGE\boxed{\displaystyle{h^1(F,g)\,=\,h^0(F,\kappa_F\cdot g^{-1})}}\]
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\(\qquad\LARGE{\mathrm{(ii)\ Arithmetic\ Riemann-Roch}}\)
\[\LARGE{\boxed{\displaystyle{h^0(F,g)-h^1(F,g)\,=\,\mathrm{deg}(g)-\frac{n}{2}\cdot\log \Delta_F}}}\]
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\(\huge{\clubsuit\ \color{Blue}{\mathrm{Positivity,\ Ampleness\ \&\ Vanishing\ Theorem}}}\)
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\(\qquad\LARGE{{The\ following\ conditions\ are\ equivalent:}}\)
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\[\LARGE{\mathrm{(i)}\qquad {\bf a}\in GL_1(\mathbb A)\ is\ positive;}\]
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\[\LARGE{\mathrm{
(ii)}\qquad {\bf a}\in GL_1(\mathbb A)\ is\ ample;}\]
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\[\LARGE{\mathrm{
(iii)}\qquad \boxed{\displaystyle{\lim_{m\to \infty}h^1(F, {\bf a}^m\cdot {\bf g})=0
\qquad \forall {\bf g}\in GL_n(\mathbb F)}}}\]
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\(\huge{\clubsuit\ \color{Blue}{\mathrm{Effective\ Vanishing\ Theorem:}}}\ \Large{(\mathrm{van\,der\ Geer\,-\,Schoof,
\ Groenewegen\ N=1;\ \ Weng\ N>\ 1})}\)
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\[\LARGE{\overline E:=\ semi-stable\ \mathcal O_F-lattice\ of\ rank\ n\ satisfying\qquad
\displaystyle{\mathrm{deg}(\overline E)\leq -[F:\mathbb Q]\cdot\frac{n\log n}{2}}}\]
\[\LARGE{\Downarrow}\]
\[\LARGE{\boxed{h^0(\overline E)\leq \frac{3^{\mathrm{rank}_{\mathbb Z}(\overline E)}}{1-\frac{\log 3}{\pi}}\cdot e^{-\pi\cdot [F:\mathbb Q]
\cdot e^{-\frac{2\mathrm{deg}(\overline E)}{\mathrm{rank}_{\mathbb Z}(\overline E)}}}}}\]
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