--- date: 2023-02-05 00:04 title: Arithmetic Geometry aliases: - Arithmetic Geometry flashcard: created: 2023-03-30T12:54 updated: 2024-01-29T17:51 --- Progress: `= "" + " " + round((length(filter(this.file.tasks.completed, (t) => t = true)) / length(this.file.tasks.text)) * 100) + "%"` # Arithmetic Geometry ## Basics - [ ] What is a local field? - [ ] What is a global field? - [x] Define a DVR ✅ 2023-02-12 - [ ] A PID with a unique nonzero prime, and thus a uniformizer: any $\pi$ with $v(\pi) = 1$. - [x] Give an example of a local ring which is not a DVR. ✅ 2023-02-12 - [ ] $R = k\fps{x^2, x^3}$ is local with $\mfm_R = \gens{x^2, x^3}$, so not a PID. - [x] Discuss valuation rings for $K$ a DVR. ✅ 2023-02-12 - [ ] $R = \ts{x\in K \st v(x) \geq 0}$ - [ ] Has unique maximal ideal $\mfm_R = \ts{\in K \st v(x) > 0} = \gens{\pi}$ for $\pi$ a uniformizer with $v(\pi) = 1$. - [ ] Always a PID and $\spec R = \ts{\mfm_R}$. - [x] Define a Dedekind domain. ✅ 2023-02-12 - [ ] Noetherian, integral, integrally closed, with $\spec R = \mspec R$. - [ ] Implies every localization is a DVR. - [x] Why are complete DVRs $K$ with valuation ring $A$ great? ✅ 2023-02-12 - [ ] Every $x\in \ff(A)$ is of the form $\sum_{n\geq n_0} s_n \pi^n$ with $s_n\in S\subset A$ a set of representatives of $A/\mfm$. - [ ] Compare to $x = \sum a_n p^n$ with $0\leq a_n\leq p-1$ for $\ZZpadic$. - [ ] If $[L: \ff(A)] = n$ then $B \da \intcl_L(A)$ is a DVR and $B \in \mods{A}^{\free, \rank =n}$ and complete wrt the topology induced by $B$. ## ANTII - [x] Define $\abs{a\over b}_p$. ✅ 2023-02-12 - [ ] $\qty{1\over p}^{v_p(a/b)}$. - [x] What is the ultrametric triangle inequality? ✅ 2023-02-12 - [ ] $\abs{x+y}_p \leq \max\ts{\abs{x}_p, \abs{y}_p}\leq \abs{x}_p + \abs{y}_p$. - [ ] $v_p(x+y) \geq \min\ts{v_p(x), v_p(y)}$. - [x] Describe $\ZZpadic$ in terms of valuations. ✅ 2023-02-12 - [ ] $\ts{x\in \QQ \st v_p(x) \geq 0}$ or $\ts{x\in \QQ \st \abs{x}_p \leq 1}$. - [ ] Explicitly $\ZZpadic \da \ts{{a\over b} \in \QQ \st p\nmid b} = \ZZ\adjoin{{1\over \ell} \st \ell \neq p}$. - [x] Discuss ring-theoretic properties of $\ZZpadic$. ✅ 2023-02-12 - [ ] Has maximal ideal $\gens{p} = \ts{{a\over b} \st p\nmid a}$ - [ ] Every element in $\ZZpadic\sm\gens{p}$ is a unit, so this is a local ring with residue field $\FF_p$. - [ ] Take completion to get $\ZZpadic$. - [x] Explicitly describe elements in $\ZZpadic$. ✅ 2023-02-12 - [ ] $\ZZpadic = \ts{\sum a_i p^i \st 0 \leq a_i \leq p-1}$ - [ ] Equivalently $\ZZpadic = \cocolim_n \ZZ/\gens{p^n}$ which contains sequences $\vector a = \tv{a_0, a_0 + a_1 p, a_0 + a_1 p + a_2 p^2, \cdots}$ of partial sums so that $\vector a_n \mod p^n \cong \vector a_{n-1}$. - [x] For a DVR $A$, setting $\abs{\wait}_v \da a^{v_p(\wait)}$, why should one choose $a \da 1/q$ for $q \da \size A/p$? ✅ 2023-02-12 - [ ] For the Haar measure, satisfies $\mu(xT) = \abs{x} \mu(T)$. - [x] Interpret local fields geometrically. ✅ 2023-02-12 - [ ] For $X$ a variety with coordinate ring $R$, points $p\in X$ correspond to $p\in \spec R$ and localizing $L_p R$ yields the ring of functions with no pole at $p$. - [ ] E.g. $L_3 \ZZ$ allows fractions with no 3 in the denominator. - [x] Discuss the AKLB setup when $L/K$ is finite and $B = \intcl_L(A)$ is a DVR. ✅ 2023-02-12 - [ ] The extension satisfies $\mfm_A B = \mfm_B^e$ with $[\kappa_B : \kappa_A] = f$ and $n \da [L: K] = ef$. Two extreme cases: - [ ] If $e=1, f=n$ (so no ramification) and $\kappa_B = \kappa_A(\bar\alpha)$ is simple (e.g. if the residue field extension is separable) then $B$ is monogenic: $B\cong A[x]/\gens{f}$ where $f$ is the minimal polynomial over $K$ of a lift of $\bar\alpha$ to $B$. - [ ] If $e=n, f=1$ then $B = A[x]/\gens{f}$ where $f$ is the minimal polynomial of $\pi_B$ - [ ] Useful: rings of integers of extensions become much easier to express! Ramification and splitting is simpler because of the single prime. - [x] What is the decomposition group? ✅ 2023-02-12 - [ ] For $L/K$ Galois and $p\in \spec A$, write $pB = \prod_{i\leq g} q_i^e$ where $q_i\in \spec B$, all $e$ the same since Galois, and $f \da [B/b_i : A/p]$. Then $D_{q_i}\da \Stab_{\Gal(L/K)}(q_i)$, elements of Galois fixing $q_i$. - [ ] All $D_{q_i}$ are conjugate in $\Gal(L/K)$ and thus the corresponding fields $K_{D_{q_i}}$ are isomorphic. - [ ] Not necessarily normal. - [x] Why is the decomposition group useful? ✅ 2023-02-12 - [ ] Have $n=efg$ and $[\Gal(L/K): D_{q_i}] = g \implies [K_{D_{q_i}} : K] = g$. - [ ] All conjugate, so suffices to consider a single $P\mid p$ and the tower: ![](attachments/2023-02-05-towers.png) - [ ] Precisely the condition to induce a map on residue fields: $\sigma \in D\implies \bar \sigma: B/P\to B/P$ where $b+P\mapsto \sigma(b) + P$, well-defined since $\sigma$ fixed $P$. - [ ] Also fixes $A$, so descends to the local Galois group, yielding a group morphism $D\surjects \Aut(\kappa_B / \kappa_A)$. - [x] What is a **normal** extension $L/K$? ✅ 2023-02-12 - [ ] If $\alpha\in L$ then all Galois conjugates are again in $L$, so $L$ contains all roots of $\minpoly_K(\alpha)$. - [x] What is the inertia group? ✅ 2023-02-12 - [ ] $T_P \da \ker\qty{D_P\surjects \Aut(\kappa_B/\kappa_A)}$. - [ ] Idea: elements in Galois that make sense on residue fields and satisfy $\sigma(b) = b \mod P$ for all $b\in B$. - [x] Discuss the inertia/decomposition SES. ✅ 2023-02-12 - [ ] $T_P \injects D_P \surjects \Aut(\kappa_B/\kappa_A)$. - [x] Why care about inertia/decomposition groups? ✅ 2023-02-12 - [ ] Factors $L/K$ into a tower of extensions where all splitting happens first and all ramification happens last: ![](attachments/2023-02-05-inert-decomposition.png) - [x] Describe the filtration on $D/T \cong \Gal(\hat L / \hat K)$. ✅ 2023-02-12 - [ ] $\sigma\in G_i \iff \sigma$ is the identity on $\hat B/P^{i+1}$. - [ ] What this is doing: write $B\ni b = \sum b_i \pi^i$, then - [ ] $\sigma \in T = G_0 \iff \sigma(b) = b_0 + \bigo(\pi)$ - [ ] $\sigma\in G_1\iff \sigma(b) = b_0 + b_1\pi + \bigo(\pi^2)$ - [ ] Allowed to change coefficients $k\geq i+1$. - [ ] Terminates, so $1 = G_i \normal \cdots \normal G_1 \normal G_0 = D$. - [x] Describe tame and wld ramification ✅ 2023-02-12 - [ ] Locally: ![](attachments/2023-02-05-tame-wild.png) - [ ] Globally: ![](attachments/2023-02-05-globally.png) - [ ] What is the Tate-Shaferevich group? - [ ] What is the Selmer group? ## Etale - [x] What is the profinite completion? ✅ 2023-02-12 - [ ] Regard objects $X$ in $\pro \cat{C}$ as sequences $X = \ts{X_\alpha}_{\alpha\in I}$ where $X_\alpha\in \cat{C}$. - [ ] Let $\Spaces_{\Fin\Grp}$ be the full subcategory of spaces with $\pi_i \in \Fin\Grp$. - [ ] The profinite completion of $X\in\Spaces$ is the left-adjoint to the inclusion $\pro \Spaces_{\Fin\Grp} \to \pro\Spaces$. ## Perfectoid - [x] What are Witt vectors? ✅ 2023-02-12 - [ ] For a $K\in \algs{\FF_p}^\perf$, the unique lift $W(K)\in \algs{\ZZpadic}$ which is $p\dash$adically complete and $p\dash$torsionfree. - [ ] $W(\FF_p) = \ZZpadic$. - [x] What is the ghost map for Witt vectors? ✅ 2023-02-12 - [ ] $w: W_r(A)\to A\cartpower{r}$ where $\vector x\mapsto \tv{w_0(x_0), w_1(x_0, x_1) \cdots, w_{r-1}(x_0,\cdots, x_{r-1})}$ where $w_i(x_0,\cdots, x_n) \da x_0^{p^n} + p x_1^{p^{n-1}} + \cdots + p^n x_n$. - [x] What is tilting? ✅ 2023-02-12 - [ ] For $A\in\zalg$ $\pi\dash$adically complete for some $\pi\mid p$, the tilt is $A^\flat \da \cocolim_{\Frob} A/pA$ where $\Frob: A/pA \to A/pA$. - [ ] Always have $A^\flat \in \algs{\FF_p}^\perf$. - [ ] Think of $\vector x\in A^\flat \cong \cocolim_{x\mapsto x^p} A$ as $x_{i+1}^p =x_i$. - [x] What is $\AA_\inf(A)$? ✅ 2023-02-12 - [ ] $\AA_\inf(A) \da W(A^\flat)$. - [ ] Interpolates between characteristic zero geometry of $A$ and characteristic $p$ geometry of $A^\flat$. - [ ] $A$ may not have a Frobenius, but $\AA_\inf(A)$ does. - [x] What is a perfectoid ring $S$? ✅ 2023-02-12 - [ ] $S$ is $\pi\dash$adically complete for some $\pi\in S$ with $\pi^p \mid p$ - [ ] $\Frob: S/pS \to S/pS$ is surjective. - [ ] $\ker \qty{\AA_\inf(S) \mapsvia\theta S} = \gens{\xi}$ is principal. - [ ] Precisely the rings so that $\theta$ is a pro-infinitesimal 1-parameter degeneration, deforming $S$ in a preferred $\xi$ direction. - [ ] If $S$ is characteristic $p$, then $S$ is perfect iff perfectoid. - [x] What is $A_{\crys}$? ✅ 2023-02-12 - [ ] Setup: $K = \CCpadic, S\da \OO_K, A_\inf \da \AA_\inf(\OO_K)$. - [ ] $A_\crys$ is the $p\dash$adic completion of the subalgebra $\gens{\xi^m/m!} \leq A_\inf\invert{p}$ . - [ ] Universal $p\dash$adically complete divided power thickening of $\OO_K$ over $\ZZpadic$. - [x] What is $B_\crys$ and $B_\crys^+$? $B_\dR$? ✅ 2023-02-12 - [ ] $B^+_\crys \da A_\crys\invert{p}$ and $B_\crys = B_\crys^+\invert{\mu}$ - [ ] $B_\dR^+$ is the $\xi\dash$adic completion of $B^+_\crys$ and $B_\dR = \ff B^+_\dR$. - [ ] $B^+_\dR$ is a DVR with residue field $\CCpadic$. ## Homotopy and stacks - [ ] What is the nerve of a category? - [x] What is the etale homotopy type of a scheme $X$? ✅ 2023-02-12 - [ ] For $X$ quasiprojective, $\ho\Et(X) \da \ts{\pi_0 \realize{\nerve{\mcu}}}_{\mcu\in X_\et} \in \pro \Spaces$ - [ ] For general $X$, replace $\nerve{\mcu}$ by hypercovers, and $X_\et$ only provides a left-filtering category up to homotopy, so generally in $\ho\pro\Spaces$. - [ ] Since $\ts{\realize{\nerve{\mcu}}}_{\mcu \in X_\et}$ is a cofibrant replacement, $\ho\Et(\wait) = \lderive\pi_0(\wait)$ is a derived functor. - [x] What is the sheaf-theoretic definition of a DM stack? ✅ 2023-02-12 - [ ] An etale sheaf $F\in \Sh_\et$ which admits a surjective etale morphism $\disjoint \spec R_i\to F$. - [x] What is an etale sheaf? ✅ 2023-02-12 - [ ] A full subcategory $\Sh_\et \da \Sh_\et(\calg\slice \ZZ) \leq \Fun(\calg\slice \ZZ, \Spaces)$ where if $R\to S$ is etale anred faithfully flat, there is a weak equivalence $F(R) \to \Tot F(S\tensorpower{R}{\bullet})$. ## Major Results - [ ] What are the Weil bounds? - [x] What is the section conjecture? ✅ 2023-02-12 - [ ] For $X$ smooth hyperbolic and proper over $k$, there is a set bijection $$X(k) \iso \pi_0 \pro\Spaces\slice {\spec k}(\Et(\spec k) \to \Et(X)) \iso \Et(X_{\bar k})^{h G_k},$$regarding the RHS as homotopy classes of morphisms. - [ ] Thus conjecturally $X(k)\iso \Et(X_{\bar k})^{h G_k}$ can be recovered from homotopy fixed points. - [ ] This is a section since morphisms are triangles over $\Et(\spec k)$, so such maps are sections to the structure morphism $\Et(X)\to \Et(\spec k)$. - [x] What inspires the anabelian conjectures? ✅ 2023-02-12 - [ ] For $K\da \QQ(\sqrt a), L\da \QQ(\sqrt b)$, $$\Gal(K) \cong \Gal(L) \iff K\cong L \iff a=b\in \QQ\units/\squares{\QQ\units}$$ so $\Sch(\spec L \iso \spec K) \cong \ho\pro\Spaces(\Et(L)\iso \Et(K))$. - [ ] For $K, L\in \NF$, $\NF(K \iso L) \cong \Top\Grp(G_K\iso_{\Out} G_L)$. - [x] What are the anabelian conjectures? ✅ 2023-02-12 - [ ] Existence of $\mathsf{An}_k \leq \Sch\slice k$ a full subcategory including smooth hyperbolic curves, closed under taking fibrations with base and fiber in $\mathsf{An}_k$, and $$\Sch(X_1\to X_2 \st \text{open image}) \cong \Top\Grp\slice{G_k}(\pi_1 X_1 \to \pi_1 X_2 \st \text{open morphisms})$$ # Conceptual - [ ] Describe geometric Langlands. - [ ] $\rho \in \mcx(\pi_1\Sigma_g, G)\leadsto$ a flat connection $\nabla_\rho$ on a principal $G\dash$bundle over $C$ - [ ] Irreducible such connections $\mapstofrom$ $R\in \dmod(\Bun_{{}^L G})$.