# Minhyong Kim, Arithmetic Field Theories and Invariants

:::{.remark}
Problem: classify principal bundles over a point $\spec k \in \spec(\Field)$.
Classified by 
\[
\Prin\Bung \slice {\spec F} \iso H^1(\pi_1 \spec F; G)
.\]
Here $G$ could be a \(p\dash \)adic Lie group, e.g. a Tate module $\cocolim_{n} A[n]$ for $A\in \Ab\Var$.
If $G$ acts trivially, this becomes a representation variety $\Hom(\pi_1 \spec F, G)/\Inn(G)$, quotienting by conjugation.
A complete description is essentially the Langlands reciprocity conjecture!
:::

:::{.remark}
Idea: replace $F$ be $\OO_F$ its ring of integers, so $X\da \spec \OO_F$ with the étale topology.
Behaves like a compact closed 3-manifold.
For $v\in\mspec \OO_F$, $\kv = \OO_F/v$ is a finite field, so $\spec \kv\injects X$ is like an embedding of a knot.
Completion at $v$, $\OO_{F, v} \da (\OO_F)\complete{v}$ is like a formal tubular neighborhood.
Completing the original ring at $v$, so $F\complete{v}$ is like a tubular neighborhood with the knot deleted $X_B$.
For $B$ a finite set of primes, set $\OO_{F, B}$ to be the set of $B\dash$integers, i.e. almost algebraic but allowing denominators from $B$.
Then $\spec \OO_{F, B}$ is like a 3-manifold with boundary, so 
\[
\bd X = \Disjoint_{v\in B} \spec F_v \to X_B \injects X
.\]

:::

:::{.remark}
What are these $\pi_1$?
Simple structure in nice cases, $\pi_1 \spec \kv = \ZZhat$, the profinite completion of $\ZZ$.
A finite field extension $K\slice F$ is **unramified** over $\mfp \in \spec \OO_F$ if the prime decomposition $\mfp \OO_K = \prod \mfq_i$ has no primes with multiplicity.
There is a maximal unramified extension, just take the compositum of all unramified extensions.
Can restrict this to just be unramified over primes not in $B$
Note that $\Ab(\pi_1 X) = \Pic \OO_F = \cl(F)$ is the ideal class group of $F$.
Old results: $\pi_1 \spec \ZZ, \pi_1 \spec \OO_F = 0$ for $F$ imaginary quadratic with $\cl(F) = 1$.
Assume GRH to get $\pi_1 \spec \OO_{K} = A_5$ for $K = \QQ\adjoin{\sqrt{653}}$, or $\PSL_2(\FF_8)\times C_{15}$ for $K = \QQ \adjoin{\sqrt{-1567}}$.
:::

:::{.question}
Central problem: classify arithmetic 3-folds, or more generally understand $H^1(\pi_1 X_B; G)$, isomorphism classes of principal $G\dash$bundles over $X_B$.
Write as $\mcm(X_B, R)$ for $R$ the group, breaks into $\prod_{b\in B} \mcm(X_b, R)$.
:::

:::{.remark}
Assume $F$ is complex, so $F\cong \QQ[x]/\gens{f}$ where $f$ has no real roots.
For gauge theory interpretations, need to write an action
\[
S: \mcm(X_B, R) = H^1(\pi_1 X_B, R)\to K
\]
and path integral
\[
\int_{\rho\in \mcm(X_B, R)} \exp(\CS(\rho)) \drho
.\]
Actions take the form of $L\dash$functions: 
\[
L: \mcm(X_B, \GL(V))\to \CC \text{ or } \CCpadic
.\]
To a representation $\pi_1(X_B) \to \GL(V)$ assign
\[
L(\rho) = \prod_{v\in \spec \OO_F} {1\over \det\qty{I-\Frob_v }V^{I_v} }
.\]
Importantly, this product may not always make sense!
It's a big problem to regularize this to get convergence.
Hasse-Weil conjecture says one can renormalise to a function of $s$ in $\CC$, get convergence for $\Re(s) \gg 0$, and analytically continue to $s=0$ or $s=1$ to recover original product.
Write $r\da \ord_{s=0} L(\rho(s))$ for the order of vanishing.
For the trivial representation $\phi$, $r = \rank \OO_{F}\units$.
For $\rho = T_p E$ for an elliptic curve, $r = \rank E(\QQ)$ (the Mordell-Weil group) assuming BSD.
Néron-Tate height pairing is a metric, its determinant appears in $L$ function for $E$.
:::

:::{.remark}
Need arithmetic orientations, problematic since dualizing sheaves are often $\mu_n$.
Fact: 
\[
H^3(X; \mu_n) = H^3(\spec \OO_F; \mu_n) = {{1\over n}\ZZ \over \ZZ}
.\]
More well-known that 
\[
H^3(X; \mu_n) = H^3(X; \GG_m)[n] \cong (\QQ/\ZZ)[n]
.\]
Local CFT yields $H^2(F_v; \GG_m) \cong \QQ/\ZZ$ by classification of gerbes.
Follows from SES in global CFT:

\begin{tikzcd}
	0 && {H^2(F; \GG_m)} && {\bigoplus_v H^2(F_v; \GG_m)} && {\QQ/\ZZ} && 0
	\arrow[from=1-1, to=1-3]
	\arrow["{\text{loc}}", from=1-3, to=1-5]
	\arrow["{\text{sum}}", from=1-5, to=1-7]
	\arrow[from=1-7, to=1-9]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCIwIl0sWzIsMCwiSF4yKEY7IFxcR0dfbSkiXSxbNCwwLCJcXGJpZ29wbHVzIEheMihGX3Y7IFxcR0dfbSkiXSxbNiwwLCJcXFFRL1xcWloiXSxbOCwwLCIwIl0sWzAsMV0sWzEsMiwiXFx0ZXh0e2xvY30iXSxbMiwzLCJcXHRleHR7c3VtfSJdLFszLDRdXQ==)

Assume $\mu_n \subseteq F$, get a map
\[
\mathrm{inv}: H^2(\pi_1 X; C_n) \to H^3(X;\mu_n) \cong{ {1\over n} \ZZ \over \ZZ}
.\]
Use this to build a Chern-Simons function $\CS: \mcm(X; R)\to { {1\over n} \ZZ \over \ZZ}$, essentially by pulling back cocycles along the representation.
Use Bockstein to define a "path integral" defined as a finite sum over representations.
Closed form solution involves Legendre symbol, very nice!
Also a determinant of a quadratic form.
:::

:::{.remark}
Bockstein is common in arithmetic geometry, $d:H^1(X; C_n)\to H^2(X; C_n)$, used to construct de Rham-Witt complex for crystalline cohomology.
On general, having a SES like $0\to V\to E\to V\to 0$ allows defining such arithmetic functionals.
Can define a bilinear pairing 
\[
\mathrm{BF}: H^1(X; C_n) \times H^1(X; \mu_n) &\to 1/n \ZZ/\ZZ \\
a\times b &\mapsto \mathrm{inv}(da \cupprod b)
.\]

Nice closed form solutions for the "path integral":
\[
\sum_{(a, b)\in H^1(X; C_n) \times H^1(X; \mu_n) } \exp(2\pi i \mathrm{BF}(a, b)) = \size \Pic(X)[n] \cdot \size ( \OO_X\units/(\OO_X\units)^n)
.\]
Compare to orders of $L$ functions.
:::

:::{.remark}
This setup occurs for Néron models of elliptic curves: for $n\gg 0$, there is a SES 
\[
0\to \mce[n] \to \mce[n^2] \to \mce[n] \to 0
.\]
Path integral evaluates to $\size \Sha(A)[n] \cdot (\size E(F)/n)^2$.
:::

:::{.remark}
For boundaries: $X_B = \spec \OO_F \invert{B}$ for $B$ a finite set of primes.
Start with $\mcm(X_B, R)$, get a local version $\mcm(\bd X_B, R)$.
Here $H^2(\pi_v; C_n) \cong 1/n\ZZ/\ZZ$ and vanishes for $i\geq 3$, get global CFT SES.
Pulling back cocycles lands in $Z^3$, but $H^3$ is trivial, so look at space trivialization (torsors for $H^2$).
Get a bunch of local torsors, then sum to get a single $1/n\ZZ/\ZZ$ torsor.
Do some local/global and push/pull gymnastics to get a $\U_1$ bundle over $\mcm(\bd X_B, R)$.
Space of sections breaks up as a tensor product of local spaces of sections, can cook up a way to land in a Hilbert space, and this is the "state" you assign to the 3-manifold.
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:::{.remark}
Recent work: entanglement of primes.
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