Habiro lectures 10 and 12

V5A2-Habiro-Rings/Lecture 10.tex

@@ -1,17 +1,34 @@
 \section{Lecture 10 -- 17th January 2025}\label{sec: lecture 10}
-We continue our discussion of relative motivic cohomology. Just as classical motivic cohomology has realization maps that recover de Rham, Betti, \'{e}tale, and prismatic cohomology, so too does relative motivic cohomology. 
+Observe that we can consider the ring of integers Nahm number field in the relative setting over $\ZZ[t_{1},\dots,t_{N}]$ and develop a notion of the relative Habiro ring as follows. Let
+\begin{equation}\label{eqn: pre-relative Habiro}
+    R=\frac{\ZZ\left[t_{1},\dots,t_{N},z_{1},\dots,z_{N}, \frac{1}{\Delta},\frac{1}{z_{1}(1-z_{1})},\dots,\frac{1}{z_{N}(1-z_{N})}\right]}{\left(1-z_{i}=(-1)^{A_{ii}}t_{i}z_{1}^{A_{i1}}\dots z_{N}^{A_{iN}}\right)}
+\end{equation}
+of \Cref{eqn: number field of Nahm equation} as a ring over $\ZZ[t_{1},\dots,t_{N}]$. We seek a relative Habiro ring $\Hcal_{R/\ZZ[t_{1},\dots,t_{N}]}$ and an invertible module over this ring such that our Nahm sums are sections of this line bundle. 
 
-Observe that we can consider the ring of integers Nahm number field 
-$$R=\ZZ[t_{1},\dots,t_{N},z_{1},\dots,z_{N}]/\left(1-z_{i}=(-1)^{A_{ii}}t_{i}z_{1}^{A_{i1}}\dots z_{N}^{A_{iN}}\right)$$
-of \Cref{eqn: number field of Nahm equation} as a ring over $\ZZ[t_{1},\dots,t_{N}]$. Recall that for $R=\ZZ[t,\frac{1}{1-t}]$ as discussed in \Cref{sec: lecture 9}, we can define relative motivic cohomology $H^{i}(\ZZ(n)(R/\ZZ[t^{\pm}]))$ where there is a map 
-$$H^{1}(\ZZ(2)(R/\ZZ[t^{\pm}]))\longrightarrow H^{i}(\ZZ(1)(R/\ZZ[t^{\pm}]))=R[1/t]^{\times}/t^{\ZZ}$$
-induced by the logarithmic derivative taking $-\Li^{\mathrm{univ}}_{2}(t)$ to $[\frac{1}{1-t}]$. Under ``de Rham realization,'' the diloagarithm class goes to $-\log(1-t)$. This theory of realizations of motivic cohomology also explains the factor $\prod_{i=0}^{m-1}(1-\zeta_{m}t)^{i/m}$ at the expansion of Nahm sums at $m$th roots of unity \Cref{thm: Nahm sum asymptotics at roots of unity}.
+Recall that for $R=\ZZ[t,\frac{1}{1-t}]$ as discussed in \Cref{sec: lecture 9}, we can define relative motivic cohomology $H^{i}(\ZZ(n)(R/\ZZ[t^{\pm}]))$ where there is a map 
+$$\nabla^{\log}_{t}:H^{1}(\ZZ(2)(R/\ZZ[t^{\pm}]))\longrightarrow H^{1}(\ZZ(1)(R/\ZZ[t^{\pm}]))=R[1/t]^{\times}/t^{\ZZ}$$
+taking $-\Li^{\mathrm{univ}}_{2}(t)$ to $[\frac{1}{1-t}]$. Under ``de Rham realization,'' the diloagarithm class goes to $-\log(1-t)$, but this is precisely the (logarithmic) differential equation that defines the dilogarithm as a power series. 
+\begin{remark}
+    More generally, for $R$ as in (\ref{eqn: pre-relative Habiro}), there is a canonical class $V^{\univ}\in H^{1}\left(\ZZ(2)(R/\ZZ[t_{1},\dots,t_{N}])\right)$ and whose logarithmic derivative under any $\nabla^{\log}_{t_{i}}$ is the class $z_{i}$ for all $1\leq i\leq N$. 
+\end{remark}
+
+More generally, motivic cohomology has multiple realizations such as de Rham, Betti, \'{e}tale, and even prismatic cohomology. And while the de Rham realization recovers the classical dilogarithm, we can show that the \'{e}tale realization also gives rise to information appearing in asymptotic expansions of Nahm sums. In particular, it will explain the factor $\prod_{i=0}^{m-1}(1-\zeta_{m}t)^{i/m}$ at the expansion of Nahm sums at $m$th roots of unity \Cref{thm: Nahm sum asymptotics at roots of unity}.
 
 For $R$ a ring, motivic cohomology admits Betti and \'{e}tale realizations
-$$H^{i}(R,\ZZ(n))\to H^{i}_{\mathsf{sing}}(\spec(R)(\CC),\ZZ), H^{i}(R,\ZZ(n))\to H^{i}_{\mathsf{\acute{e}t}}(\spec(R[1/m]),\ZZ/m\ZZ(n)).$$
-We can describe this explicitly in motivic weight $n=1$. This produces maps $R^{\times}$ to $\ZZ$-torsors to $\spec(R)(\CC)$ taking $f$ to a $\ZZ$-torsor of choices of $\log(f)$ in the Betti setting and $R^{\times}$ to $\mu_{m}$-torsors over $\spec(R)$. Analogous constructions can be made for relative motivic cohomology. In relative motivic cohomology, the same constructions yield maps
-$$H^{i}(\ZZ(n)(R/\ZZ[t^{\pm}]))\to H^{i}_{\mathsf{sing}}\left(\spec(R)(\CC)\times_{(\CC^{\times})^{N}}\CC^{N}\right)$$
-$$ H^{i}(\ZZ(n)(R/\ZZ[t^{\pm}]))\to H^{i}_{\mathsf{\acute{e}t}}\left(\spec(R[1/m, t_{1}^{1/m},\dots,t_{N}^{1/m}]),\ZZ/m\ZZ(n)\right)$$
-where in the first case we the fibered product is taken over the map $\exp:\CC^{N}\to(\CC^{\times})^{N}$. Here, we observe that relative motivic cohomology still give \'{e}tale cohomology classes, but these only start to appear after extracting some roots of the coordinates. 
+\begin{align*}
+    H^{i}(R,\ZZ(n))&\longrightarrow H^{i}_{\mathsf{sing}}(\spec(R)(\CC),\ZZ) \\
+    H^{i}(R,\ZZ(n))&\longrightarrow H^{i}_{\mathsf{\acute{e}t}}(\spec(R[1/m]),\ZZ/m\ZZ(n))
+\end{align*}
+We can describe this explicitly for first cohomology $i=1$ in motivic weight $n=1$. 
+\begin{itemize}
+    \item (de Rham Realization) Takes $f\in R^{\times}$ to the $\ZZ$-torsor of choices of the logarithm $\log(f)$. 
+    \item (\'{E}tale Realization) Recalling that $\ZZ/m\ZZ(1)\cong\mu_{m}$, the realization takes $f$ to the torsor of choices of $m$th roots of $f$, which is well-defined up to an $m$th root of unity. 
+\end{itemize}
+Analogous constructions can be made for relative motivic cohomology. In relative motivic cohomology, the same constructions yield maps
+\begin{align*}
+    H^{i}(\ZZ(n)(R/\ZZ[t^{\pm}]))&\longrightarrow H^{i}_{\mathsf{sing}}\left(\spec(R)(\CC)\times_{(\CC^{\times})}\CC,\ZZ\right) \\
+    H^{i}(\ZZ(n)(R/\ZZ[t^{\pm}]))&\longrightarrow H^{i}_{\mathsf{\acute{e}t}}\left(\spec(R[1/m, t^{1/m}]),\ZZ/m\ZZ(n)\right)
+\end{align*}
+but the objects live only over appropriate covers of the spaces $\spec(R)(\CC),\spec(R[\frac{1}{m}])$ obtained after extracting the logarithm of $t$, respectively. In the first case, the fibered product is taken over the map above and the $\exp:\CC\to(\CC^{\times})^{N}$. This reinforces the intuition that relative motivic cohomology should one where the contribution of the motivic cohomology of $\ZZ[t^{\pm}]$ is ignored, and this is done in singular cohomology by extracting logarithms. Similarly in the \'{e}tale setting, we consider the space $\spec(R[1/m, t^{1/m}])$ which plays the role of the unviersal cover in the \'{e}tale algebraic setting. 
 
-In this framework, it can be seen that the Betti realization of the diloagarithm is a well-defined function on $\CC\setminus(2\pi i)\ZZ$ with at most simple poles at $(2\pi i)\ZZ$, and residues $\pm(2\pi i)n$ at $2\pi i n$. In particular, the diloagarithm realizes to a $\ZZ$-local system on $\CC\setminus(2\pi i)\ZZ$ whose mononodromy around $2\pi i n$ is $n$. Similarly, the \'{e}tale realization produces precisely the product $\prod_{i=0}^{m-1}(1-\zeta_{m}t)^{i/m}$ alluded to above. This recovers a construction of Calegari-Garoufalidis-Zagier as well as the cyclic quantum dilogarithm. 
+We consider the realization of the universal dilogarithm $\Li_{2}^{\univ}$ in this setting. Recall from \Cref{lem: dilogarithm as integral on split plane} and \Cref{thm: analytic continuation for dilogarithm} that the function $\Li_{2}(t)+\log(t)\log(1-t)$ is a well-defined function $\CC\setminus(2\pi i)\ZZ\to\CC/(2\pi i)^{2}\ZZ$. Note that there is an isomorphism $\CC^{\times}\setminus\{1\}\times_{\CC^{\times}}\CC$ with $\CC\setminus(2\pi i)\ZZ$ by taking the logarithm. The $\ZZ$-torsor obtained by the map on relative motivic cohomology is the given by the $\ZZ$-torsor of choices of liftings of $\CC/(2\pi i)^{2}\ZZ$ to $\CC$, and moreover is related to the Beilinson-Deligne cohomology. In this framework, it can be seen that the Betti realization of the diloagarithm is a well-defined function on $\CC\setminus(2\pi i)\ZZ$ with at most simple poles at $(2\pi i)\ZZ$, and residues $\pm(2\pi i)n$ at $2\pi i n$. In particular, the diloagarithm realizes to a $\ZZ$-local system on $\CC\setminus(2\pi i)\ZZ$ whose mononodromy around $2\pi i n$ is $n$. Similarly, the \'{e}tale realization produces a $\ZZ/m\ZZ$ local system on $\mathbb{A}^{1}_{\ZZ[\frac{1}{m},\zeta_{m}]}\setminus\mu_{m}$. But this local system must be compatible with the Betti realization, so the monodromy at each $\zeta_{m}^{i}$ is congruent to $i\pmod{m}$ and trivial at zero. But this is sufficient data to determine to determine the torsor, forcing the \'{e}tale realization to be precisely the product $\prod_{i=0}^{m-1}(1-\zeta_{m}t)^{i/m}$ alluded to above, and which is the cyclic quantum dilogarithm. This also recovers a construction of Calegari-Garoufalidis-Zagier. 

V5A2-Habiro-Rings/Lecture 12.tex

@@ -1,8 +1,10 @@
 \section{Lecture 12 -- 31st January 2025}\label{sec: lecture 12}
 Recall from \Cref{thm: Nahm sum asymptotics at roots of unity} and the discussion of the first part of \Cref{sec: lecture 11} that we have 
-$$f_{a}(t,q)\sim\exp\left(-\frac{V(t)}{m^{2}\varepsilon}g_{a,m}(t,q)\right)$$
-as $q\to\zeta_{m}$ and $q=\zeta_{m}\exp(-\varepsilon)$ and $g_{a,m}(t,q)\in\QQ(\zeta_{m})[t][[\varepsilon]]$. Setting 
-$$R_{m}=R[\zeta_{m},t^{1/m}], S_{m}=R_{m}\left[\frac{1}{2},\sqrt{\delta}\right]$$
+$$f_{a}(t,q)\sim\exp\left(-\frac{V(t)}{m^{2}\varepsilon}\right)g_{a,m}(t,q)$$
+a convergent series on the open unit disc. We can consider asymptotics as $q\to\zeta_{m}$ setting $q=\zeta_{m}\exp(-\varepsilon)$ and $g_{a,m}(t,q)\in\QQ(\zeta_{m})[t][[\varepsilon]]$. We consider the \'{e}tale $\ZZ[t]$-algebra 
+$$R=\ZZ\left[t,z,\frac{1}{\delta}\right]/(1-z=(-1)^{a}tz^{a})$$
+with $\delta=z+a(1-z)$ and set 
+$$R_{m}=R[\zeta_{m},t^{1/m}]\text{ and }S_{m}=R_{m}\left[\frac{1}{2},\sqrt{\delta}\right]$$
 we can show the following theorem. 
 \begin{theorem}\label{thm: power series of Nahm expansion at roots of unity}
     Let $f_{a}(t,q)$ be a Nahm sum with power series term $g_{a,m}(t,q)$ as $q\to\zeta_{m}$. Then 
@@ -15,13 +17,13 @@ \section{Lecture 12 -- 31st January 2025}\label{sec: lecture 12}
 
 Fix a prime $p>3$ and consider the embedding $\QQ\to\QQ_{p}$, where we have $\widehat{R}=R_{p}^{\wedge},\widehat{S}=S_{p}^{\wedge}$ are the $p$-adic completions of $R,S$, respectively. We obtain $\widehat{R_{m}}=\widehat{R}\widehat{\otimes}_{\ZZ_{p}\langle t\rangle}\ZZ_{p}\langle\zeta_{p^{m}},t^{1/p^{m}}\rangle,\widehat{S_{m}}=\widehat{S}\widehat{\otimes}_{\ZZ_{p}\langle t\rangle}\ZZ_{p}\langle\zeta_{p^{m}},t^{1/m}\rangle$ analogously. Passing to the limit, we have 
 $$\widehat{R_{\infty}}=\lim_{m}\widehat{R_{m}}$$
-which is an integral perfectoid ring over $R_{\infty}^{0}=\ZZ_{p}\langle\zeta_{p^{\infty}},t^{1/p^{\infty}}\rangle$. In this setting, we can consider the tilt $R_{\infty}^{0,\flat}=\lim_{x\mapsto x^{p}}(\ZZ_{p}\langle\zeta_{p},t^{1/m}\rangle/p), \widehat{R_{\infty}}^{\flat}=\lim_{x\mapsto x^{p}}(\widehat{R_{\infty}}/p)$. 
+which is an integral perfectoid ring over $R_{\infty}^{0}=\ZZ_{p}\langle\zeta_{p^{\infty}},t^{1/p^{\infty}}\rangle$. In this setting, we can consider the tilt $R_{\infty}^{0,\flat}=\lim_{x\mapsto x^{p}}(\ZZ_{p}\langle\zeta_{p},t^{1/p^{m}}\rangle/p), \widehat{R_{\infty}}^{\flat}=\lim_{x\mapsto x^{p}}(\widehat{R_{\infty}}/p)$. 
 \begin{example}
     Consider $\ZZ_{p}\langle\zeta_{1/p^{\infty}}\rangle$. The tilt $\ZZ\langle\zeta_{p^{\infty}}\rangle^{\flat}$ contains the element $(\overline{1},\overline{\zeta_{p}},\overline{\zeta_{p^{2}}},\dots)=\varepsilon$ and we can produce an isomorphism $\ZZ\langle\zeta_{p^{\infty}}\rangle^{\flat}\cong\FF_{p}[[(\varepsilon-1)^{1/p^{\infty}}]]$.
 \end{example}
 Per the previous example, there is a map $\ZZ_{p}\langle \zeta_{p^{\infty}}\rangle^{\flat}\mapsto R_{\infty}^{0,\flat}$ taking the element $\varepsilon$ to $(t,t^{1/p},t^{1/p^{2}},\dots)$ which we denote $t^{\flat}$. 
 
-In $p$-adic Hodge theory, we also have the $A_{\inf}(-)$ construction taking a ring to the ring of its $p$-typical Witt vectors. We have $A_{\inf}(\ZZ_{p}\langle\zeta_{p^{\infty}}\rangle)\cong\ZZ_{p}\langle q^{1/p^{\infty}}\rangle^{\wedge}_{(q-1)}, A_{\inf}(R^{0}_{\infty})\cong\ZZ_{p}\langle q^{1/p^{\infty}}[t^{\flat}]^{1/p^{\infty}}\rangle^{\wedge}_{(q-1)}$. These rings are still defined by the Nahm equation, replacing $t$ by the Teichm\"{u}ller lift $[t^{\flat}]$ of $t$. This gives the pushout diagram 
+In $p$-adic Hodge theory, we also have the $A_{\inf}(-)$ construction taking a ring to the ring of its $p$-typical Witt vectors. We have $A_{\inf}(\ZZ_{p}\langle\zeta_{p^{\infty}}\rangle)\cong\ZZ_{p}\langle q^{1/p^{\infty}}\rangle^{\wedge}_{(q-1)}, A_{\inf}(R^{0}_{\infty})\cong\ZZ_{p}\langle q^{1/p^{\infty}},[t^{\flat}]^{1/p^{\infty}}\rangle^{\wedge}_{(q-1)}$. These rings are still defined by the Nahm equation, replacing $t$ by the Teichm\"{u}ller lift $[t^{\flat}]$ of $t$. This gives the pushout diagram 
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 \begin{tikzcd}
 	{\ZZ_{p}\langle q,\widetilde{t}\rangle^{\wedge}_{(q-1)}} && {A_{\inf}(R^{0}_{\infty})} \\