class: center, middle, inverse, title-slide # The Toric regulator ### Amnon Besser (Joint with Wayne Raskind) ### Automorphic p-adic L-functions and regulators, Lille ### 14/10/2019 --- `\(\newcommand{\Z}{\mathbb{Z}}\)` `\(\newcommand{\coker}{\operatorname{coker}}\)` `\(\newcommand{\rg}{\mathbb{R}\Gamma}\)` `\(\newcommand{\rhom}{\mathbb{R}\operatorname{Hom}}\)` `\(\newcommand{\Q}{\mathbb{Q}} \newcommand{\XX}{\mathcal{X}}\)` `\(\newcommand{\Qpb}{\bar{\Q}_p} \newcommand{\Kbar}{\bar{K}}\)` `\(\newcommand{\hh}{\mathcal{H}}\)` `\(\newcommand{\isom}{\cong}\)` `\(\newcommand{\et}{\operatorname{et}}\)` `\(\newcommand{\Ymm}[1]{\bar{Y}^{(#1)}}\)` `\(\newcommand{\Ym}{\Ymm{m}}\)` `\(\newcommand{\cijk}{C_j^{i,k}}\)` `\(\newcommand{\cij}{C_j^{i}}\)` `\(\newcommand{\Cijk}{CH^{i+j-k}(\Ymm{2k-i+1})}\)` `\(\newcommand{\ct}[3]{C_{#1}^{#2,#3}}\)` `\(\newcommand{\ctt}[2]{C_{#1}^{#2}}\)` `\(\newcommand{\Ct}[3]{CH^{#1+#2-#3}(\Ymm{2#3-#1+1})}\)` `\(\newcommand{\Cy}[2]{CH^{#1}(\Ymm{#2})}\)` `\(\newcommand{\nt}{\tilde{N}}\)` `\(\newcommand{\ntl}{\nt_l}\)` `\(\newcommand{\Zl}{\Z_l}\)` `\(\def\hT(#1,#2,#3){H_{\mathcal{T}}^{#2}(#1,\Z(#3))}\)` `\(\def\hmot(#1,#2,#3){H_{\mathcal{M}}^{#2}(#1,\Z(#3))}\)` `\(\newcommand{\reg}{\operatorname{reg}}\)` `\(\newcommand{\regt}{\reg_t}\)` `\(\newcommand{\Qp}{\Q_p}\)` `\(\newcommand{\Ql}{\Q_l}\)` `\(\newcommand{\hst}{H_{\operatorname{st}}}\)` `\(\newcommand{\DR}{\operatorname{DR}}\)` `\(\newcommand\aA[1]{\hh(\Gamma_{#1},\Z)}\)` `\(\newcommand\BB[1]{H^1(\Gamma_{#1},\Z)}\)` `\(\newcommand{\id}{\operatorname{id}}\)` `\(\newcommand{\het}{H_{\operatorname{et}}}\)` --- # Regaulators `\(X/K\)` proper variety Regulator map `\(H_{\mathcal{M}}^\ast(X,\Z(\ast)) \to H_?^\ast(X,\ast)\)` `\(H_?\)` "cohomology theory", often `\(\rhom^\ast(\mathbb{1}, \rg_?(X)(*))\)` Examples: - Deligne-Beilinson theory - `\(K=\mathbb{C}\)`, `\(H_{\mathcal{D}}\)` Deligne cohomology. `\(\rg_{MHS}\)` in the category of Mixed Hodge Structures. -- - l-adic etale theory - `\(\het^\ast(X, \Z_l(\ast))\)` (absolute) etale cohomology. `\(\rg_{\operatorname{et}}\)` in Galois respresentations of `\(\operatorname{Gal}(\Kbar/K)\)`. -- From now onward: `\([K:\Qp]<\infty\)`, `\(R=\mathcal{O}_K\)`, `\(R/\pi=k\)` -- - Syntomic theory - `\(H_{syn}^\ast(X,\ast)\)` (Nekovar-Niziol version), `\(\rg_{pst}\)` in Potentially Semi-Stable representations of `\(\operatorname{Gal}(\Kbar/K)\)`. - Sphoradic examples (+ Sreekantan regulator) --- # Relation with L-functions Regulators are used to make numbers, also called regulators, that are supposed to be related with special values of L-functions -- - Beilinson regulators related to classical L-functions -- - Syntomic regulators related to `\(p\)`-adic L-functions --- # The toric regulator This is a new type of regulator - It applied to `\(X/K\)` with "totally degenerate" reduction. -- - Its target is a a " `\(p\)`-adic torus" `\(\coker (T^0 \to T^{-1} \otimes K^\times)\)`, `\(T^\ast\)` finitely generated `\(\Z\)`-modules. -- - Constructed out of `\(l\)`-adic regulators for all `\(l\)`. -- - It "exponentiates" the syntomic regulator -- - We expect "refined" conjectures relating it with L-values, a-la Gross, Mazur-Tate-Teitelbaum, Bertolini-Darmon. --- # Some examples - Identity map `\(H_{\mathcal{M}}^1(K,1) = K^\times \to K^\times\)` -- - For `\(E/K\)` a Tate elliptic curve `\(\mathbb{G}_m /q^{\Z}\)` the identity map `$$H_{\mathcal{M}}^2(E,1)_0 \isom E(K) \to K^\times / q^{\Z}$$` -- - More generally, for a Mumford curve `\(X\)` with Jacobian `\(J\)`, the Abel map `\(X(K) \to J(K)\)`, -- J(K) given via its `\(p\)`-adic uniformization. -- - The Pal regulator: For `\(X\)` a Mumford curve with dual graph `\(\Gamma\)` $$ K_2(X)\to \hh(\Gamma,K^\times) $$ into `\(K^\times\)` valued harmonic cochains on `\(\Gamma\)` --- # Totally degenerate reduction and examples A variety `\(X/K\)` is said to have totally degenerate reduction if it is the generic fiber of a proper `\(\mathcal{O}_K\)`-scheme `\(\mathcal{X}\)` with strictly semi-stable reduction whose special fiber `\(Y\)` decomposes as a union `\(Y= \cup_{i=1}^n Y_i\)`. s.t., with `\(Y_I = \bigcap_{i\in I} Y_i\)`, `\(I \subset\{1,... ,n\}\)` -- - For all primes `\(l\ne p\)` all `\(l\)`-adic cohomology of `\(Y_I\)` comes from cycles. -- - Similarly at `\(p\)` for cristalline cohomology -- - For `\(p\)` also some ordinarity conditions. - Chow groups are finitely generated and satisfy Hodge index theorem -- Typical example: `\(p\)`-adically uniformized varieties. --- # Etale cohomology of varieties with totally degenerate reduction `\(k,r \ge 0\)` `\(M_l=H_{\et}^{k}(X\otimes_K \bar{K}, \Z_l)\)` -- Raskind and Xarles. On the etale cohomology of algebraic varieties with totally degenerate reduction over `\(p\)`-adic fields. J. Math. Sci. Univ. Tokyo 14(2):261--291, 2007. -- Simplifying 1: Ignore finite torsion and cotorsion. -- .inverse[Theorem: There are finitely generated `\(\Z\)`-moduels `\(T^i\)`, `\(i\in \Z\)` such that for each `\(l\)` the Galois module `\(M_l(r)\)` has an increasing filtration `\(U\)` with `\(\operatorname{gr}_U M_l(r) = \oplus_i T^i \otimes \Z_l(-i)\)`. ] -- Proof: Rapoport-Zink weight spectral sequence for `\(l\ne p\)` Mokrane + Hyodo + Tsuji when `\(l=p\)`. -- Simplifying 2: Assume trivial action of `\(\operatorname{Gal}(\Kbar/K)\)` on the `\(T\)`'s. --- # Construction of `\(T\)`'s `\(\bar{Y}_I:= Y_I\otimes \bar{F}\)` `\(\Ym = \bigcup_{|I| =m} \bar{Y}_I\)` `\(\cijk= \Cijk\)`, `\(k\ge \max(0,i)\)` `\(\cij= \bigoplus_k \cijk\)` `\(I_r = I-\{i_r\}\)` Inclusion `\(\rho_r: Y_I \to Y_{I_r}\)` `\(\theta_{i,m} = \sum_{r=1}^{m+1} (-1)^{r-1} \rho_r^\ast: CH^i(\Ym) \to CH^i(\Ymm{m+1})\)` `\(\delta_{i,m} = \sum_{r=1}^{m+1} (-1)^{r} \rho_{r\ast}: CH^i(\Ymm{m+1}) \to CH^{i+1}(\Ym)\)` --- # Construction continued `\(d' = \bigoplus_{k\ge \max(0,i)} \theta_{i+j-k,2k-i+1}\)` `\(d'' = \bigoplus_{k\ge \max(0,i)} \theta_{i+j-k,2k-i}\)` `\(d_j^i = d'+d'': \cij \to C_j^{i+1}\)` .inverse[Definition: `\(T_j^i := H^i(C_j^\bullet)\)`] Renumbering: `\(T^i = T_{i+r}^{k-2r-2i}\)` Monodromy map: `\(N: T^i \to T^{i-1}\)` given by "the identity map on identical groups". --- # The toric intermediate Jacobian `\(M'_l = U^0 M_l(r) / U^{-2} M_l(r)\)` -- `\(0\to T^{-1} \otimes \Zl(1) \to M_l^\prime \to T^0 \otimes \Zl \to 0\)` -- Use Bloch-Kato `\(H_g\)`. Boundary map `\(\ntl: T^0 \otimes \Zl \to H_g^1(K,T^{-1}\otimes \Zl(1))\isom T^{-1} \otimes K^{\times(l)}\)` -- .inverse[Lemma: `\(T^0 \otimes \Zl \xrightarrow{\ntl} T^{-1} \otimes K^{\times(l)}\xrightarrow{\operatorname{val}} T^{-1} \otimes \Zl\)` is `\(N\otimes \Zl\)`. ] -- .inverse[Corollary: Exists `\(T^0 \xrightarrow{\nt} T^{-1}\otimes K^\times\)` s.t. `\(\ntl = \nt \otimes \Zl\)` for each `\(l\)`] -- We define the toric intermediate Jacobian `$$\hT(X,k+1,r) := \coker \nt$$` --- # Toric regulator completed at `\(l\)` The etale regulator map `\(\operatorname{reg}_l: \hmot(X,k+1,r)_0 \to H_g^1(K,M_l(r))\)` factors via `\(H_g^1(K,U^0 M_l(r))\)` because `\(H^0(K,\Ql(j))= H_g^1(K,\Ql(j))=0\)` for `\(j<0\)`. -- Applying `\(U^0 \to M'_l\)` we get `\(\operatorname{reg}_l^{\prime\prime} : \hmot(X,k+1,r)_0 \to H_g^1(K,M_l^\prime)\)` -- Recall short exact sequence `\(0\to T^{-1} \otimes \Zl(1) \to M_l^\prime \to T^0 \otimes \Zl \to 0\)` -- .inverse[Lemma: `\(H_g^1(K,T^{0} \otimes \Zl) \to H^2(K,T^{-1} \otimes \Zl(1))\)` is injective ] -- .inverse[Corollary: `\(H_g^1(K,M_l^\prime)\cong \coker \left( T^0\otimes \Zl \xrightarrow{\ntl} T^{-1}\otimes K^{\times(l)} \right)\)` ] -- Completed toric regulator: `\(\hmot(X,k+1,r)_0\to \coker \left( T^0\otimes \Zl \xrightarrow{\ntl} T^{-1}\otimes K^{\times(l)} \right)\)` --- # The Sreekantan regulator Sreekantan. Non-Archimedean regulator maps and special values of L-functions. In Cycles, motives and Shimura varieties, Tata Inst. Fund. Res. Stud. Math., pages 469--492. Tata Inst. Fund. Res., Mumbai, 2010. Idea: To have Beilinson conjectures over function fields (but we stay mixed...) Drop for now the totally degenerate assumption (keep ss reduction) -- `\(\newcommand{\Cons}{\mathcal{C}}\)` .inverse[Definition: The Consani complex is `\(\Cons(r) = \operatorname{Cone}(N: C_{r}^{*-2r} \to C_{r-1}^{*-2r+2} )\otimes \Q\)` The Deligne cohomology is `\(H_{\mathcal{D}}^{k+1}(X,\Q(r)) := H^{k+1-2r}(\Cons(r))\)` ] -- By definition we have, assuming again totally degenerate `\(T_{\Q}^0 \to T_{\Q}^1 \to H_{\mathcal{D}}^{k+1}(X,\Q(r)) \to T_{r}^{k+1-2r} \otimes \Q \to T_{r-1}^{k+3-2r}\otimes \Q\)` and right most map is injective in "motivic" range (not for cycles) --- # The Sreekantan regulator Assuming standard conjectures (that hold for all known cases of totally degenerate varieties) Sreekantan's Deligne cohomology groups are isomorphic to Higher Chow groups `$$H_{\mathcal{D}}^{k+1}(X,\Q(r)) \isom CH^{r-1}(Y,2r-k-2)\otimes \Q$$` The Sreekantan regulator `\(r_{\mathcal{D}}: \hmot(X,k+1,r) \to H_{\mathcal{D}}^{k+1}(X,\Q(r))\)` is just the boundary map in K-theory --- # The toric regulator .inverse[Conjecture: For each prime `\(l\)` the valuation of the toric regulator at `\(l\)` is the Sreekantan regulator tensored with `\(\Q_l\)` ] -- Probably not difficult. -- .inverse[Theorem: Assuming Conjecture, Exists `\(\hmot(X,k+1,r) \xrightarrow{\regt} \hT(X,k+1,r)\)` giving the toric regulator at `\(l\)` for each `\(l\)` by completion] Example: Chow groups. Raskind and Xarles. On `\(p\)`-adic intermediate Jacobians. Trans. Amer. Math. Soc., 359(12):6057--6077, 2007. Unconditional construction Tate and Mumford curve cases essentially tautological. --- # The Pal regulator Pal. A rigid analytical regulator for the `\(K_2\)` of Mumford curves. Publ. Res. Inst. Math. Sci., 46(2):219--253, 2010. -- `\(X\)` - totally degenerate curve with dual graph `\(\Gamma\)`. -- Can check: `\(\hT(X,2,2)\isom \hh(\Gamma,K^\times)\)` `\(=K^\times\)` valued harmonic cochains on `\(\Gamma\)`. -- The Pal regulator: `\(K_2(X) \to \hh(\Gamma,K^\times)\)` Construction: Define a "tame symbol" on an oriented annuli `\(e=\{r\le |z|\le s\}\)` (by the inner disc `\(|z|<r\)` ) first on rational functions: `\(t_e(f,g) = \prod_{|x|<r} t_x(f,g)\)` -- Then for `\(f,g\in \mathcal{O}(e)\)`, `\(t_e(f,g) = \lim_{n\to \infty} t_e(f_n,g_n)\)` `\(f_n \to f\)`, `\(g_n \to g\)` --- # The Pal regulator For `\(\Gamma=(V,E)\)` `\(v \in V\to\)` overconvergent domain `\(U_v\)` `\(e\in E\)` - annulus Pal regulator `\(\{f,g\} \mapsto\)` harmonic cochain `\(e\mapsto t_e(f,g)\)`. -- .inverse[Theorem: Pal regulator = Toric regulator in this case] checked only for `\(l\ne p\)`. --- # Relations with L-functions - Tate parametrization of rational points on elliptic curves via square root p-adic L-functions (Bertolini-Darmon) - Gross's refined p-adic stark conjectures involve the identity map on `\(K^\times\)` - Work of Pal relating his regulator with `\(\infty\)`-adic L-functions for Tate elliptic curves in the function field case --- # Relation with the log-syntomic regulator Moto: The log-syntomic regulator is the logarithm of the toric regulator. -- Natural since: "The syntomic Abel-Jacobi map is the log composed with Albanese" -- `\(\reg_{\operatorname{syn}}: \hmot(X,k+1,r)_0 \to \hst^1(K,V)\)`, `\(V=H_{\et}^{k}(X\otimes_K \bar{K}, \Qp(r))\)` `\(\hst^1\)` is semi-stable cohomology - can be computed explicitly in terms of the Fontaine functors. -- Totally degenerate case is explicit. Get a map "fairly close to isomorphism" `\(\hst^1(K,V) \to \DR(V)/(F^0+ T^0\otimes \Qp)\)`. -- Precise statement of relation: Syntomic regulator composed with `\(\DR(V)/(F^0+ T^0\otimes \Qp)\to T^{-1}\otimes K / T^0\otimes \Qp\)` is the `\(\operatorname{log}: T^{-1}\otimes K^\times / T^0 \to T^{-1}\otimes K / T^0\otimes \Qp\)` of the toric regulator. -- We check compatibility with the result on the Pal regulator. --- # Conjectural regulator for `\(K_1\)` Guess by exponentiating a conjectural result for the syntomic regulator. -- Mumford curves `\(X_i\)`, `\(i=1,2\)` dual graphs `\(\Gamma_i\)` `\(X=X_1\times X_2\)` -- `\(\regt: \hmot(X,3,2) \to \hT(X,3,2)\)` -- Use Kunneth to isolate in `\(\het^2(X,\Zl(2))\)` a piece `\(M''\)` with $$ 0 \to \left(\aA{1} \otimes \BB{2}\right) \otimes \Zl(1) \to M'' \to \aA{1} \otimes \aA{2} \to 0$$ -- Toric intermediate Jacobian = coker of `$$\id_{\aA{1}} \otimes \nt_2: \aA{1} \otimes \aA{2} \to \aA{1} \otimes \BB{2}\otimes K^\times$$` --- # Conjectural regulator for `\(K_1\)` Want: The (piece of) toric regulator of `\(\Theta = \sum (C_j, g_j)\)` - `\(C_j\subset X\)` curves - `\(g_j\)` a rational functions on `\(C_j\)` -- As a bilinear form `\(\BB{1}\times \aA{2}\to K^\times\)` -- construction relies on: To an `\(\alpha\in \aA{2}\)` correspond functions `\(f_\alpha^v \in \mathcal{O}(U_v)\)` such that -- - `\(d\log f_\alpha^v\)` patch to a holomorphic form `\(\omega\)` on `\(X_2\)` with residue `\(\alpha(e)\)` on `\(e\)` -- - the cocycle `\(e\mapsto f_\alpha^{e+}/f_\alpha^{e-}\in K^\times\)` is harmonic -- The regulator on `\((C,g)\)` on `\((\beta,\alpha)\)` should be `$$\sum_e t_e(g,f_\alpha^{\pi_2(e+)})^{\beta(\pi_1(e))}$$`