The second reciprocity law for Pablos Romo's 3-cocycle symbols on a curve over a discretely valued field

For a complete, irreducible, non-singular curve $C$ defined over a perfect field $\mathscr{K}$ carrying a discrete valuation $v_{\mathscr{K}}$, Pablos Romo constructed a normalized 3-cocycle symbol $(f,g,h)_{v_x,v_{\mathscr{K}}}\in k^*$ for $f,g,h$ in the function field $\Sigma_C^*$, built from two-level tame central extensions attached to the point valuations $v_x$ of $C$ and the base valuation $v_{\mathscr{K}}$, and conjectured (recorded as Conjecture 3.8 of the source paper) the second reciprocity law $\prod_{x\in C}(f,g,h)^{\varphi}_{v_x,v_{\mathscr{K}}}=1$ for every morphism of commutative groups $\varphi:k^*\to G$. This conjecture exchanges the roles of the two valuations relative to the reciprocity laws the same author proved, generalizing the second Parshin reciprocity law on surfaces.

Background

A classical reciprocity law in arithmetic geometry is Weil reciprocity: if $C$ is a complete (i.e. proper) irreducible non-singular algebraic curve over a field and $f,g$ are non-zero rational functions on $C$, then the tame symbols $(f,g)_x=(-1)^{v_x(f)v_x(g)}\big(f^{v_x(g)}/g^{v_x(f)}\big)(x)$, taken over all closed points $x$ of $C$, multiply to $1$; here $v_x$ is the discrete valuation of the function field $\Sigma_C$ measuring the order of vanishing at $x$. Milnor generalized the tame symbol to any discrete valuation $v$ on a field $F$, and Tate gave a definition of residues on curves through traces of linear operators, deducing the residue theorem (the additive counterpart of Weil reciprocity) from the finite-dimensionality of the cohomology groups $H^0(C,\mathcal{O}_C)$ and $H^1(C,\mathcal{O}_C)$ (the space of global regular functions on $C$ and its first sheaf cohomology group); Arbarello, de Concini and Kac later re-derived the tame symbol and Weil reciprocity from the commutator of a central extension of groups (a short exact sequence $1\to k^*\to\widetilde{F}^*\to F^*\to 1$ with $k^*$ in the center).

For surfaces, Parshin introduced a three-function symbol $\langle f,g,h\rangle_{(x,C)}$ attached to a flag consisting of a curve $C$ on a surface and a closed point $x\in C$ (the sign refinement is due to Fesenko), and stated two reciprocity laws for it: a product over the points of a fixed curve, and a product obtained by exchanging the roles of the point and the curve; Brylinski and McLaughlin gave independent constructions and proofs of such multidimensional reciprocity laws in geometric and analytic settings. Pablos Romo reorganized this circle of ideas in cohomological terms. In the source paper [1] he works with a tower $F, v_F, K, v_K, k$ in which $F$ is a field with a discrete valuation $v_F$ whose residue field $K(v_F)$ is finite separable over a field $K$ that itself carries a discrete valuation $v_K$ with residue field $k$. A tame central extension attached to $(F,v_F,K)$ is a central extension $1\to K^*\to\widetilde{F}^*\to F^*\to 1$ whose commutator computes the norm to $K$ of the classical tame symbol at $v_F$. From the two-level tame central extensions attached to $(F,v_F,K)$ and $(K,v_K,k)$ one obtains, via iterated commutators, a normalized group 3-cocycle $\{f,g,h\}_{v_F,v_K}\in k^*$ (a 3-cocycle being a map satisfying Eilenberg–Mac Lane's cocycle identities for group cohomology in degree 3); inside its cohomology class there is a distinguished representative $(f,g,h)^{\varphi}_{v_F,v_K}\in G$, a 2-dimensional Steinberg symbol (multilinear in its arguments; the source proves the Steinberg relation $(f,1-f,g)=1$ under the additional hypothesis $K(v_F)=K$), whose explicit formula generalizes the Parshin–Fesenko symbol on a surface, pushed forward along any morphism of commutative groups $\varphi:k^*\to G$.

For $F=\Sigma_C$ the function field of a curve, Pablos Romo proved reciprocity laws in the orientation where the curve's function field sits at the lower level of the valuation tower, deducing them, as Tate did for the residue theorem, from the finiteness of $H^0(C,\mathcal{O}_C)$ and $H^1(C,\mathcal{O}_C)$; this is the analogue of Parshin's first reciprocity law. He then explicitly posed the exchanged orientation — the analogue of Parshin's second law — as an open conjecture: now the base field $\mathscr{K}$ is itself discretely valued, the functions live in $\Sigma_C$ at the top of the tower, and each symbol pairs the point valuation $v_x$ with the base valuation $v_{\mathscr{K}}$.

Problem Statement

Let $\mathscr{K}$ be a perfect field equipped with a discrete valuation $v_{\mathscr{K}}$ with residue field $k$, assume that a tame central extension of groups $1\to k^*\to\widetilde{\mathscr{K}}^*\to\mathscr{K}^*\to 1$ attached to the triple $(\mathscr{K},v_{\mathscr{K}},k)$ exists (i.e. a central extension whose commutator is given by the norm to the residue field $k$ of the classical tame symbol at $v_{\mathscr{K}}$, and therefore takes values in $k^*$), and let $C$ be a complete, irreducible, non-singular curve over $\mathscr{K}$ with function field $\Sigma_C$. For each closed point $x\in C$, the tame central extension attached to the triple $(\Sigma_C,v_x,\mathscr{K})$ — which exists because the residue field $k(x)$ of $v_x$ is a finite extension of the perfect field $\mathscr{K}$, hence separable over it — combined with the tame central extension of $\mathscr{K}^*$ above determines a normalized 3-cocycle symbol $(f,g,h)_{v_x,v_{\mathscr{K}}}\in k^*$ for $f,g,h\in\Sigma_C^*$ — a 2-dimensional Steinberg symbol generalizing the Parshin–Fesenko symbol — and every morphism of commutative groups $\varphi:k^*\to G$ induces $(f,g,h)^{\varphi}_{v_x,v_{\mathscr{K}}}=\varphi\big((f,g,h)_{v_x,v_{\mathscr{K}}}\big)\in G$. The conjecture (recorded as Conjecture 3.8 of Pablos Romo's 2006 paper, the second reciprocity law exchanging the roles of the two valuations, in analogy with Parshin's second reciprocity law on surfaces) asserts that for every such $\varphi$ and all $f,g,h\in\Sigma_C^*$ one has

$$\prod_{x\in C} (f,g,h)^{\varphi}_{v_x,v_{\mathscr{K}}} = 1 \in G,$$

the product being finite since all but finitely many factors are trivial. Prove, or refute by an explicit counterexample, this reciprocity law in the stated generality.

References

1. Fernando Pablos Romo, 3-Cocycles, symbols and reciprocity laws on curves, Journal of Pure and Applied Algebra 205 (2006), no. 1, 94–116. DOI: 10.1016/j.jpaa.2005.06.004, https://doi.org/10.1016/j.jpaa.2005.06.004
2. Fernando Pablos Romo, On n-Dimensional Steinberg Symbols, Canadian Journal of Mathematics 60 (2008), no. 6, 1387–1405. DOI: 10.4153/CJM-2008-058-3, https://doi.org/10.4153/CJM-2008-058-3
3. J.-L. Brylinski and D. A. McLaughlin, Multidimensional reciprocity laws, Journal für die reine und angewandte Mathematik 481 (1996), 125–147. DOI: 10.1515/crll.1996.481.125, https://doi.org/10.1515/crll.1996.481.125
4. J.-L. Brylinski and D. A. McLaughlin, The geometry of two-dimensional symbols, K-Theory 10 (1996), no. 3, 215–237. DOI: 10.1007/BF00538183, https://doi.org/10.1007/BF00538183
5. E. Arbarello, C. de Concini, V. G. Kac, The infinite wedge representation and the reciprocity law for algebraic curves, in: Proceedings of Symposia in Pure Mathematics 49, Part I, American Mathematical Society, 1989, pp. 171–190. DOI: 10.1090/pspum/049.1/1013132, https://doi.org/10.1090/pspum/049.1/1013132
6. J. Tate, Residues of differentials on curves, Annales scientifiques de l'École Normale Supérieure (4) 1 (1968), 149–159. DOI: 10.24033/asens.1162, https://doi.org/10.24033/asens.1162
7. Fernando Pablos Romo, A General Reciprocity Law for Symbols on Arbitrary Vector Spaces, arXiv:1305.7066 (2013), https://arxiv.org/abs/1305.7066
8. Daniel Hernández Serrano and Fernando Pablos Romo, Determinants of finite potent endomorphisms, symbols and reciprocity laws, arXiv:1011.2044 (2010), https://arxiv.org/abs/1011.2044
9. José María Muñoz Porras, Luis Manuel Navas Vicente, Fernando Pablos Romo, and Francisco J. Plaza Martín, An idelic quotient related to Weil reciprocity and the Picard group, arXiv:1905.03954 (2019), https://arxiv.org/abs/1905.03954
10. Fernando Pablos Romo, On the Steinberg property of the Contou-Carrère symbol, arXiv:math/0602374 (2006), https://arxiv.org/abs/math/0602374