Obstruction-free regimes for $q$th-degree power-residue reciprocity laws from Brauer-group exact sequences

T. J. Ford's 2024 preprint derives elementary $q$th power-residue reciprocity laws (of Legendre-symbol type) for function fields in one and two variables from the Grothendieck and Artin–Mumford exact sequences for the Brauer group, but only in restricted regimes: tame ramification, residue curves with vanishing $\mathrm{H}^1(C,\mathbb{Q}/\mathbb{Z})$, henselizations or completions at closed points or along simply connected curves, and nonsingular divisors. The open problem is to characterize the schemes and localizations for which these cohomological obstructions vanish and to extend the resulting reciprocity laws to wild ramification, thereby obtaining broadly applicable $q$th-degree reciprocity laws from Brauer-group exact sequences.

Background

For a field $K$, the Brauer group $\mathrm{B}(K)$ is the torsion subgroup of the étale (equivalently, Galois) cohomology group $\mathrm{H}^2(K,\mathbb{G}_m)$; its elements classify finite-dimensional central simple algebras over $K$ up to Morita equivalence (two algebras being identified when they differ only by taking matrix rings). A basic source of classes is the symbol algebra $(\alpha,\beta)_q$, the $q^2$-dimensional algebra generated by elements $u,v$ with $u^q=\alpha$, $v^q=\beta$, and $vu=\zeta uv$, where $q$ is a prime invertible in $K$ and $\zeta$ is a primitive $q$th root of unity. For a discrete valuation $\nu$ on $K$ with residue field $\kappa$ and $\alpha,\beta\in K^{\times}$, the tame symbol $\partial(\alpha,\beta)=(-1)^{\nu(\alpha)\nu(\beta)}\bar{\alpha}^{\nu(\beta)}/\bar{\beta}^{\nu(\alpha)}\bmod \kappa^{\times q}$ records the ramification of $(\alpha,\beta)_q$; it is the degree-one case of Kato's residue maps. This symbol is 'tame' because it sees nothing at primes whose residue characteristic divides $q$ — the wild case, where ramification theory is far subtler.

For a scheme $X$ with function field $K$, ramification of Brauer classes along codimension-one points $x$ is encoded by the ramification map $a\colon\mathrm{B}(K)\to\bigoplus_{x}\mathrm{H}^{1}(k(x),\mathbb{Q}/\mathbb{Z})$, where $\mathrm{H}^{1}(k(x),\mathbb{Q}/\mathbb{Z})$ is the group of continuous characters of the absolute Galois group of the residue field, i.e. of cyclic field extensions of $k(x)$. For a global field (a number field or function field of a curve over a finite field), the Hasse exact sequence makes $a$ injective with a concretely computable cokernel, and the composite $r\circ a=0$ for a summation map $r$ is precisely the classical quadratic reciprocity law: applied to $(p,q)_2$ over $\mathbb{Q}$ it yields Gauss's formula $(p/q)(q/p)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}$, where the wild prime 2 must be handled separately since the tame symbol does not apply there. Grothendieck's one-dimensional sequence (Theorem 2.1 of [1]) generalizes this to any regular integral curve with perfect residue fields, and Ford recovers reciprocity laws such as $(\alpha/\beta)(\beta/\alpha)=(-1)^{\deg\alpha\,\deg\beta}$ for real polynomials, along with $q$th-power non-residue criteria used in control theory over rings (the coefficient assignment problem).

In two variables the governing tool is the Artin–Mumford sequence (Artin and Mumford, 1972, introduced to construct threefolds — three-dimensional varieties — that are unirational, i.e. dominated by a rational map from projective space, yet not rational, i.e. not birational to projective space, using torsion in cohomology): for a nonsingular integral surface $X$ over an algebraically closed field with function field $K=K(X)$, the sequence $0\to\mathrm{B}(X)\to\mathrm{B}(K)\xrightarrow{a}\bigoplus_{C}\mathrm{H}^{1}(K(C),\mathbb{Q}/\mathbb{Z})\xrightarrow{r}\bigoplus_{p}\mu(-1)\to\mathrm{H}^{4}(X,\mu)\to 0$ (where $C$ runs over curves on $X$, $p$ over closed points, $\mu$ is the sheaf of roots of unity, and $\mu(-1)$ its Tate twist) is a complex that fails to be exact in general, the failure at the middle stage being measured by $\mathrm{im}(a)\neq\ker(r)$; it is exact when $\mathrm{H}^{3}(X,\mu)=0$, e.g. for affine or complete simply connected surfaces. Exactness at the middle stage would give a $q$th-degree reciprocity law ($r\circ a=0$ interpreted as 'if $f$ is a $q$th power mod $g$ then $g$ is essentially a $q$th power mod $f$'). Ford shows the obstruction is real: for the elliptic curve $C$ given by $g=y^{2}-x(x^{2}-1)=0$ over a field of characteristic $\neq 2$, the group $\mathrm{H}^{1}(C,\mathbb{Q}/\mathbb{Z})$ — equivalently the abelianized fundamental group of $C$ — is nonzero, and the reciprocity law fails for $(x,g)$ in $k[x,y]$. He restores it in limited regimes by passing to a henselization (the 'algebraic approximation' of a local ring in which Hensel's lemma holds) or a completion at a closed point, or along an ideal whose components are simply connected (trivial étale fundamental group, i.e. admitting no nontrivial finite étale covers), obtaining laws in $k[[x,y]]$ and $k[x][[y]]$. Two further obstructions are exhibited: a cusp singularity ($y^{2}=x^{3}$) prevents a $q$th root that exists in the function field of the divisor from lying in the divisor's coordinate ring (failure of normality, i.e. of the ring being integrally closed in its fraction field), and the whole analysis is confined to tame ramification. Wild ramification — primes whose residue characteristic divides $q$ — is handled by completely different tools in the classical theory (higher-dimensional local class field theory in the sense of Kato, which describes abelian extensions of complete higher-dimensional local fields via Milnor $K$-theory (the ring generated by symbols on field elements subject to Steinberg relations) but does not produce elementary power-residue laws of Legendre-symbol type for polynomial and power-series rings over algebraically closed fields).

Problem Statement

Determine for which function fields $K$ in one and two variables, and which localizations of the underlying regular schemes $X$ (curves and nonsingular surfaces over, for instance, an algebraically closed ground field), the Brauer-group exact sequences — the Hasse/Grothendieck sequence in dimension one and the Artin–Mumford sequence in dimension two — yield explicit, practically computable $q$th-degree power-residue reciprocity laws of the form 'if $f$ is a $q$th power modulo $g$ then $g$ is essentially a $q$th power modulo $f$' (with the correction term given by a concrete symbol evaluation). Concretely, following [1], '$f$ is a $q$th power modulo $g$' means that $f$ is a $q$th power in the quotient $\mathcal{O}/(g)$ of the ambient coordinate ring $\mathcal{O}$, '$g$ is essentially a $q$th power modulo $f$' means that the residue class of $g$ is a $q$th power in the normalization of the coordinate ring of the divisor $\{f=0\}$ (equivalently, $g-s^{q}=f\,t$ for some $s,t$ in the function field), and the 'concrete symbol evaluation' is the explicitly computable product $\prod_{x}\partial_{x}(f,g)$ of tame residue symbols, whose vanishing is $r\circ a=0$. This single objective comprises five jointly supporting components, each currently open beyond the regimes treated in Ford (2024): (i) give a usable, concretely computable description of $\mathrm{coker}\,a$ of the ramification map $a\colon\mathrm{B}(K)\to\bigoplus_{x}\mathrm{H}^{1}(k(x),\mathbb{Q}/\mathbb{Z})$ for classes of function fields beyond global fields and their localizations; (ii) classify the geometric conditions on residue curves $C$ under which the residue (Gysin/boundary) stage of the sequence is injective — in particular under which the obstruction group $\mathrm{H}^{1}(C,\mathbb{Q}/\mathbb{Z})$ vanishes — going beyond henselizations/completions at closed points and along simply connected components; (iii) extend the reciprocity analysis to wild ramification, where the residue characteristic divides $q$ and the tame symbol carries no information; (iv) give a general criterion relating the singularity type of a divisor $C$ to whether a $q$th root existing in $K(C)$ is integral over the coordinate ring of $C$ (equivalently, when normalization is harmless), replacing the ad hoc nonsingular and linear-lowest-degree-form hypotheses; and (v) determine geometric conditions ensuring exactness of the Artin–Mumford sequence (or explicitly computing the defect $\ker(r)/\mathrm{im}(a)$) beyond the vanishing hypothesis $\mathrm{H}^{3}(X,\mu)=0$. These five components are independent open sub-problems that jointly support the objective.

References

1. Timothy J. Ford, Division Algebras and Quadratic Reciprocity, preprint (2024), arXiv:2407.03447, DOI 10.48550/arxiv.2407.03447, https://arxiv.org/abs/2407.03447
2. M. Artin and D. Mumford, Some Elementary Examples of Unirational Varieties Which are Not Rational, Proc. London Math. Soc. (3) 25 (1972) 75–95, DOI 10.1112/plms/s3-25.1.75, https://doi.org/10.1112/plms/s3-25.1.75
3. Kazuya Kato, A generalization of local class field theory by using K-groups, I, Proc. Japan Acad. Ser. A Math. Sci. 53 (1977), DOI 10.3792/pjaa.53.140, https://doi.org/10.3792/pjaa.53.140
4. Kazuya Kato, A generalization of local class field theory by using K-groups, II, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), DOI 10.3792/pjaa.54.250, https://doi.org/10.3792/pjaa.54.250
5. J. Brewer, T. Ford, L. Klingler, W. Schmale, When does the ring K[y] have the coefficient assignment property?, J. Pure Appl. Algebra 112 (1996) 239–246, DOI 10.1016/0022-4049(95)00142-5, https://doi.org/10.1016/0022-4049(95)00142-5
6. T. J. Ford, On the Brauer group of a Laurent polynomial ring, J. Pure Appl. Algebra 51 (1988) 111–117, DOI 10.1016/0022-4049(88)90081-3, https://doi.org/10.1016/0022-4049(88)90081-3