The two-dimensional Contou-Carrère symbol: adelic two-dimensional class field theory

Osipov and Zhu (J. Algebraic Geom. 25, 2016) defined the two-dimensional Contou-Carrère symbol, a tri-multiplicative deformation of the two-dimensional tame symbol on the double Laurent series algebra over a ring, and proved reciprocity laws for it on algebraic surfaces. The problem is the integration of the symbol into Parshin's two-dimensional class field theory, namely constructing the restricted adelic product of the groups $K_2(K_{x,C})$ over flags on a surface over a finite field and a global reciprocity map $\theta=\sum_{x\in C}\theta_{x,C}$ which, according to Parshin's program, should exist, and proving that it annihilates the diagonally embedded groups $K_2^M(K_x)$ and $K_2(K_C)$ beyond the maximal Kummer and maximal abelian $p$-quotients treated by Osipov–Zhu.

Background

For a commutative ring $R$, the Laurent series algebra $R((t))$ consists of formal series with finitely many negative powers of $t$. The one-dimensional tame symbol is the map $R((t))^{\times}\times R((t))^{\times}\to R^{\times}$, $(f,g)\mapsto (-1)^{v(f)v(g)}f^{v(g)}/g^{v(f)}$, where $v$ is the $t$-adic valuation. The Contou-Carrère symbol is a functorial deformation of the tame symbol, constructed by Contou-Carrère from the autoduality of the generalized Jacobian of a family of curves, and reinterpreted by Anderson–Pablos Romo via determinant groupoids and by Deligne in 'Le symbole modéré'; it interpolates the tame symbol as the base ring degenerates and encodes residues and the classical reciprocity laws on curves. Osipov and Zhu (2016) introduced the two-dimensional analogue: for a ring $R$, a tri-multiplicative, antisymmetric, functorial map $CC_2:R((u))((t))^{\times}\times R((u))((t))^{\times}\times R((u))((t))^{\times}\to R^{\times}$ on the double iterated Laurent algebra, defined through central extensions of categorical groupoids (equivalently, 2-gerbes — higher-categorical analogues of groupoid central extensions), and recoverable from boundary maps in algebraic $K$-theory. When $R$ contains $\mathbb{Q}$ (or the nil-radical of $R$ is nilpotent) it is given by an explicit formula $CC_2(f,g,h)=\exp\,\mathrm{Res}\big(\log f\wedge dg/g\wedge dh/h\big)$, where $\mathrm{Res}$ is the two-dimensional residue. On a smooth algebraic surface $X$ over a field, Parshin flags $x\in C$ (a closed point $x$ lying on an irreducible curve $C$) give two-dimensional local fields $K_{x,C}$ — iterated completions of the function field, first along $C$ and then at $x$ — and the symbol satisfies two reciprocity laws: along a curve (product over the points of a fixed curve) and around a point (product over the formal branches of curves through a fixed point). The arithmetic motivation is two-dimensional class field theory, developed by Parshin, Kato, and Kato–Saito: whereas one-dimensional class field theory of a function field describes its abelian extensions through the idèle group (the restricted product of the multiplicative groups of the completions at all places) and the tame/Contou-Carrère symbol, the two-dimensional theory of Kato–Saito (1983) describes the abelianized fundamental group of a surface over a finite field by étale-cohomological means (étale cohomology being the standard cohomology theory for varieties over finite fields). Parshin envisioned a more explicit adelic formulation: for $X$ smooth projective over a finite field, a restricted adelic product $K_{2,A_X}=\prod'_{x\in C}K_2(K_{x,C})$ over all flags (here $K_2$ is the second Quillen $K$-group of the two-dimensional local field) and a global reciprocity map $\theta:K_{2,A_X}\to\mathrm{Gal}(\mathbb{F}_q(X)^{ab}/\mathbb{F}_q(X))$ should exist, factoring as the product of the local reciprocity maps of two-dimensional local class field theory, with the kernel governed by reciprocity laws of symbols; the two-dimensional Contou-Carrère symbol is the expected local ingredient. Osipov–Zhu proved that their reciprocity laws imply that $\theta$ annihilates (after passage to the maximal Kummer quotient, corresponding to abelian extensions generated by adjoining $n$-th roots with $n$ prime to the characteristic, or to the maximal abelian pro-$p$ quotient) the diagonally embedded groups $K_2^M(K_x)$ and $K_2(K_C)$ (Milnor $K_2$ of the local field $K_x$ at the closed point $x$, and Quillen $K_2$ of the one-dimensional local field $K_C$ along the irreducible curve $C$; here $k(x)$ and $k(C)$ are the residue fields of $K_x$ and $K_C$ respectively), as evidence for this picture, without constructing $K_{2,A_X}$ or $\theta$; the paper's introduction describes these reciprocity laws as an indication about some subgroup in the kernel of the conjectural global reciprocity map. Separately, Beilinson–Bloch–Esnault showed that the one-dimensional Contou-Carrère symbol arises as the epsilon-factor of the Gauss–Manin determinant of a family of curves — a determinantal avatar of the de Rham epsilon-factor of a rank-one connection.

Problem Statement

The problem is the global arithmetic meaning of the two-dimensional Contou-Carrère symbol. Adelic two-dimensional class field theory: for a smooth projective surface $X$ over a finite field $\mathbb{F}_q$, construct the adelic $K_2$-object $K_{2,A_X}$ as the appropriate restricted product of the groups $K_2(K_{x,C})$ over all flags $x\in C$ (a closed point $x$ lying on an irreducible curve $C$) and a global reciprocity map $\theta=\sum_{x\in C}\theta_{x,C}:K_{2,A_X}\to\mathrm{Gal}(\mathbb{F}_q(X)^{ab}/\mathbb{F}_q(X))$ that factorizes as the product of the local reciprocity maps of two-dimensional local class field theory — objects which, according to Parshin's program, should exist — and prove that $\theta$ annihilates the diagonal images of $K_2^M(K_x)$ for every closed point $x$ and of $K_2(K_C)$ for every irreducible curve $C$, extending Osipov–Zhu's annihilation results $\gamma\circ\theta(K_2^M(K_x))=0$ and $\gamma\circ\theta(K_2(K_C))=0$ beyond the maximal Kummer and maximal abelian $p$-quotients (in the source these are obtained only after composing with the natural quotient $\gamma:\mathrm{Gal}(\mathbb{F}_q(X)^{ab}/\mathbb{F}_q(X))\to\mathrm{Gal}(N/\mathbb{F}_q(X))$ to the maximal Kummer extension or the maximal abelian $p$-extension $N$).

References

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